https://en.xen.wiki/api.php?action=feedcontributions&feedformat=atom&user=ArrowHead294Xenharmonic Wiki - User contributions [en]2026-07-21T00:06:28ZUser contributionsMediaWiki 1.43.6https://en.xen.wiki/index.php?title=MOS_scale&diff=234331MOS scale2026-07-18T16:30:39Z<p>ArrowHead294: /* Basic properties */</p>
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<div>{{interwiki<br />
| en = MOS scale<br />
| de = MOS-Skala<br />
| es =<br />
| ja = MOSスケール<br />
| ro = G2S<br />
}}{{Beginner|Mathematics of MOS}}<br />
A '''moment of symmetry''' ('''MOS''' or '''mos'''<ref group="note">The acronym "MOS" is generally pronounced ''em-oh-ess'', while the {{w|anacronym}} "mos", more common in informal and experimental settings, is generally pronounced ''moss''. Sometimes "MOSS" or "moss", standing for "moment of symmetry scale", are used instead, although there is no significant difference in meaning.</ref>) '''scale''' is a [[periodic scale]] where every 2nd (that is, every interval formed by ascending a step) is either small or large with no in-between, and the same goes for 3rds, 4ths, etc. Multiples of the period (which is usually the octave or a fraction thereof), however, come in only one size.<br />
<br />
MOS scales are often referred to as MOSes, thus MOS can be used as either an adjective or a noun.<br />
<br />
== Examples ==<br />
The most widely used MOS scale is the [[5L 2s|diatonic scale]]. It has 7 steps: 5 large ones (major 2nds) and 2 small ones (minor 2nds), and thus is named 5L&nbsp;2s. The major mode is LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode).<br />
<br />
{| class="wikitable"<br />
|+ style="font-size: 105%;" | Interval classes in the 5L&nbsp;2s MOS scale<br />
|-<br />
! rowspan="2" | Interval class<br />
! colspan="2" | Small version<br />
! colspan="2" | Large version<br />
|-<br />
! Quality<br />
! Size<br />
! Quality<br />
! Size<br />
|-<br />
! 2nds (1 step)<br />
| minor<br />
| s<br />
| major<br />
| L<br />
|-<br />
! 3rds (2 steps)<br />
| minor<br />
| {{nowrap|1L + 1s}}<br />
| major<br />
| 2L<br />
|-<br />
! 4ths (3 steps)<br />
| perfect<br />
| {{nowrap|2L + 1s}}<br />
| augmented<br />
| 3L<br />
|-<br />
! 5ths (4 steps)<br />
| diminished<br />
| {{nowrap|2L + 2s}}<br />
| perfect<br />
| {{nowrap|3L + 1s}}<br />
|-<br />
! 6ths (5 steps)<br />
| minor<br />
| {{nowrap|3L + 2s}}<br />
| major<br />
| {{nowrap|4L + 1s}}<br />
|-<br />
! 7ths (6 steps)<br />
| minor<br />
| {{nowrap|4L + 2s}}<br />
| major<br />
| {{nowrap|5L + 1s}}<br />
|-<br />
! 8ves (7 steps)<br />
| perfect<br />
| {{nowrap|5L + 2s}}<br />
| colspan="2" | (only one version)<br />
|}<br />
<br />
Note that the melodic minor scale (LsLLLLs) has only two step sizes, but it is not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.<br />
<br />
Other MOS scales include [[2L&nbsp;3s]], where among its 5 modes are the major pentatonic scale (ssLsL) and the minor pentatonic scale (LssLs), and less commonly [[4L&nbsp;4s]], also known as the octatonic scale in 12edo, which alternates between large and small steps and comes in two modes (LsLsLsLs and sLsLsLsL).<br />
<br />
See the [[catalog of MOS]] for other MOS scales.<br />
<br />
== Periods and generators ==<br />
Every MOS scale can be ''generated'' by stacking a certain interval called the [[generator]] and octave-reducing (or more generally, [[period]]-reducing). For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, 2L&nbsp;3s is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as {{nowrap| C D E F G A C }} does not produces a MOS, because there are more than 2 sizes of each interval class. <br />
<br />
The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53, …. However for a quarter-comma meantone fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50, …. <br />
<br />
== Step ratio spectrum ==<br />
The [[step ratio]] is the ratio of the larger step size to the smaller step size. MOSes with smaller step ratios sound smooth and soft. MOSes with larger step ratios sound jagged and hard. Different step ratios produce different corresponding potential temperament interpretations. The [[TAMNAMS #Step ratio spectrum|TAMNAMS]] system has names for specific ratios and also ranges of ratios.<br />
<br />
When the step ratio is a rational number, the MOS is tuned to an edo. A counter-example is 5L&nbsp;2s tuned to quarter-comma meantone, which has a step ratio of about 1.649.<br />
<br />
{| class="wikitable"<br />
|+ style="font-size: 105%;" | 5L&nbsp;2s step ratios in various edos<br />
|-<br />
! Example edo<br />
! Step ratio<br />
! TAMNAMS name<br />
! Likely temperament<br />interpretations<br />
|-<br />
! 12<br />
| 2:1<br />
| basic<br />
| [[Meantone]] or [[Schismatic]]<br />
|-<br />
! 19<br />
| 3:2<br />
| soft<br />
| [[Meantone]]<br />
|-<br />
! 22<br />
| 4:1<br />
| superhard<br />
| [[Archy]] or [[Superpyth]]<br />
|}<br />
<br />
== Naming ==<br />
Every MOS can be uniquely specified by giving its [[signature]], i.e. the number of large and small steps, which is typically notated e.g. "5L&nbsp;2s,". Every possible signature corresponds to a valid MOS scale. Sometimes, if one simply wants to talk about step sizes without specifying which is large and small, the notation "5a&nbsp;2b" is used (which could refer to either diatonic or {{nowrap| [[2L 5s|anti-diatonic]] {{=}} 2L 5s }}).<br />
<br />
By default, the [[equave]] of a MOS is assumed to be [[2/1]]. To specify a non-octave equave, "{{angbr|equave}}" is placed after the signature, e.g. {{mos scalesig|4L 5s<3/1>|link=1}}. Using angle brackets (<code>&#x26;#x27E8;</code> and <code>&#x26;#x27E9;</code>) is recommended; using greater-than and less-than signs ("&#x3C;equave&#x3E;") can also be done, but this can conflict with HTML and other uses of these symbols.<br />
<br />
Several naming systems have been proposed for MOSes, which can be seen at [[MOS naming]].<br />
<br />
== History and terminology ==<br />
The term ''MOS'', and the method of scale construction it entails, were invented by [[Erv Wilson]] in 1975. His original paper is archived on Anaphoria.com here: [https://anaphoria.com/mos.pdf ''Moments of Symmetry'']. There is also an introduction by [[Kraig Grady]] here: [https://anaphoria.com/wilsonintroMOS.html ''Introduction to Erv Wilson's Moments of Symmetry''].<br />
<br />
Sometimes, scales are defined with respect to a period and an additional [[equivalence interval]], the interval at which pitch classes repeat. MOSes in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called '''Multi-MOSes'''. For example, a MOS with a half-octave period is called a '''2mos''', with a 1/3-octave period a '''3mos''', and so on. MOSes in which the equivalence interval is equal to the period are sometimes called '''Strict MOSes'''. MOSes in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.<br />
<br />
With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional evenness|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt<ref>Norman Carey and David Clampitt. "Aspects of Well-Formed Scales", ''Music Theory Spectrum'', Vol. 11, No. 2 (Autumn, 1989), pp. 187-206.</ref>. A great deal of work has been done in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilson's concept. They are in a sense the MOS of MOS patterns. This is used to explain the [[pentatonic]]s used in traditional [[Japanese music]] (e.g. {{nowrap| A B C E F A }}), where the 5-tone cycles are derived from a 7-tone MOS, which are not found in the concept of DE.<br />
<br />
== Equivalent definitions and generalizations ==<br />
A scale is a MOS if and only if it satisfies one of the following equivalent criteria:<br />
# [[Maximum variety]] 2: Ascending by a certain number of steps is equivalent to ascending by one of at most two intervals, and the maximum of two is achieved (i. e. it is not true that ascending by a certain number of steps is always equivalent to ascending by one interval.) <br />
# [[Binary]] and has a [[generator]]: The scale step comes in exactly two sizes, and the scale is formable from stacking some interval called a generator and octave-reducing.<br />
# Mode of a Christoffel word: The scale can be formed by creating a 2D lattice where the period is on the lattice, then taking pitches by travelling vertically and horizontally from the origin, maintaining as close to the line from the origin to the octave as possible without going above it.<br />
<br />
Each definition generalizes to scales with three or more step sizes, but these generalizations are not equivalent. The concepts of [[balanced word|balance]] and [[distributional evenness]] provide still different generalizations, although defining MOS through these terms is less helpful. For more information, see [[Mathematics of MOS]].<br />
<br />
== Properties ==<br />
=== Basic properties ===<br />
* For every MOS scale with an [[octave]] period (which is usually the [[octave]]), if ''x''-[[edo]] is the [[collapsed]] tuning (where the small step vanishes) and ''y''-[[edo]] is the [[equalized]] tuning (where the large (''L'') step and small (''s'') step are the same size), then by definition it is an {{nowrap| ''x''L (''y'' − ''x'')s }} MOS scale, and the [[basic]] tuning where {{nowrap| ''L'' {{=}} 2''s'' }} is thus {{nowrap|(''x'' + ''y'')}}-[[edo]]. This is also true if the period is 1\''p'', that is, 1 step of ''p''-[[edo]], which implies that ''x'' and ''y'' are divisible by ''p'', though note that in that case (if {{nowrap| ''p'' > 1 }}) you are considering a "multiperiod" MOS scale.<br />
* More generally, whenever ''px''-[[edo]] and ''py''-[[edo]] are used to define two [[val]]s (usually but not necessarily through taking the [[Patent val|patent vals]]) while simultaneously also being used to define the {{nowrap|''px''L (''py'' − ''px'')s}} MOS scale (where ''p'' is the number of periods per octave), then the ''px'' & ''py'' temperament corresponds to that MOS scale, and adding ''x'' and/or ''y'' corresponds to tuning closer to ''x''-[[edo]] and/or ''y''-[[edo]] respectively. (Optionally, see the below more precise statement for the mathematically-inclined.)<br />
* For the mathematically-inclined, we can say that whenever we consider a MOS with ''X''/''p'' notes per period in the [[collapsed]] tuning and ''Y''/''p'' notes per period in the [[equalized]] tuning and ''p'' periods per [[Octave stretching|tempered octave]] (or more generally tempered [[equave]]), and whenever we want to associate that MOS with the {{nowrap| ''X'' & ''Y'' }} rank 2 temperament'''*''', we can say that any {{w|natural number|natural}}-coefficient {{w|linear combination}} of vals {{val| ''X'' … }} and {{val| ''Y'' … }} (where {{nowrap| ''X'' < ''Y'' }}) corresponds uniquely to a tuning of the {{nowrap| ''X'' & ''Y'' }} rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive) iff {{nowrap| gcd(''a'', ''b'') {{=}} 1 }}, because if {{nowrap| ''k'' {{=}} gcd(''a'', ''b'') > 1 }} then the val {{nowrap| ''a''{{val| ''X'' … }} + ''b''{{val| ''Y'' … }} }} has a common factor ''k'' in all of its terms, meaning it is guaranteed to be [[contorted]]. The tuning corresponding to the {{w|Rational number|rational}} ''a''/''b'' is technically only unique up to (discarding of) [[octave stretching]] (or more generally [[equave]]-tempering).<br />
<br />
: The period of this temperament is {{nowrap|1\gcd(''X'', ''Y'')}}, and the rational ''a''/''b'' is very closely related to the [[step ratio]] of the corresponding MOS scale, because {{nowrap| 1{{val| ''X'' … }} + 0{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 0}} tuning while {{nowrap| 0{{val| ''X'' … }} + 1{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 1}} tuning and {{nowrap| 1{{val| ''X'' … ;}} + 1{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 2|''s'' {{=}} 1}} tuning, so that {{nowrap|''L'' {{=}} ''a'' + ''b''}} and {{nowrap|''s'' {{=}} ''b''}} and therefore:<br />
<br />
: {{nowrap|1/([[step ratio]]) {{=}} ''s''/''L''}} {{nowrap|{{=}} ''b''/(''a'' + ''b'')}} implying [[step ratio]] {{nowrap| ''r'' {{=}} (''a'' + ''b'')/''b'' ≥ 1 }} for {{w|Natural number|natural}} ''a'' and ''b'', where if {{nowrap| ''b'' {{=}} 0 }} then the step ratio is infinite, corresponding to the [[collapsed]] tuning.<ref group="note">It is ''important to note'' that the correspondence to the {{nowrap| ''X'' & ''Y'' }} rank-2 temperament only works in all cases if we allow the temperament to be [[contorted]] on its [[subgroup]]; alternatively, it works if we exclude cases where {{nowrap| ''X'' & ''Y'' }} describe a contorted temperament on the subgroup given. An example is the {{nowrap| 5 & 19 }} temperament is contorted in the [[5-limit]] (having a generator of a semifourth, corresponding to [[5L&nbsp;14s]]), so we either need to consider the temperament itself to be contorted (generated by something lacking an interpretation in the subgroup given, two of which yielding a meantone-tempered [[~]][[4/3]]) or we exclude it because of its contortion.</ref><br />
<br />
* Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale ''a''L&nbsp;''b''s are {{nowrap| (''a'' + ''b'')L ''a''s }} (generated by generators of soft-of-basic ''a''L ''b''s) and {{nowrap| ''a''L (''a'' + ''b'')s }} (generated by generators of hard-of-basic ''a''L''&nbsp;b''s).<br />
* Every MOS scale (with a specified [[equave]] ''Ɛ''), excluding {{nowrap|''a''L ''a''s{{angbr|''Ɛ''|-)}} }}, has a ''parent MOS''. If {{nowrap| ''a'' > ''b'' }}, the parent of ''a''L&nbsp;''b''s is {{nowrap| ''b''L (''a'' − ''b'')s }}; if {{nowrap| ''a'' < ''b'' }}, the parent of ''a''L&nbsp;''b''s is {{nowrap| ''a''L (''b'' − ''a'')s }}.<br />
<br />
=== Advanced discussion ===<br />
See:<br />
* [[Mathematics of MOS]], a more formal definition and a discussion of the mathematical properties.<br />
** [[Recursive structure of MOS scales]], a description of how MOS scales are recursive and how one scale can be converted into a related scale.<br />
** [[MOS scale family tree]], a tree initially described by Erv Wilson that organizes scales by parent-and-child relationship, which also helps illustrate mos recursion.<br />
* [[Generator ranges of MOS]], organized by number of scale steps and quantity of L/s steps.<br />
* [[MOS diagrams]], visualizations of the MOS process.<br />
* [http://x31eq.com/temper/method.html How to Find Linear Temperaments], by [[Graham Breed]]<br />
<br />
== Individual pages for MOS scales ==<br />
=== L ≤ 12, s ≤ 12 ===<br />
{| class="wikitable center-all"<br />
|+ style="font-size: 105%; white-space: nowrap;" | Pages for MOS scales ({{nowrap|L ≤ 12|s ≤ 12}})<br />
|-<br />
| [[1L&nbsp;1s]]<br />
| [[1L&nbsp;2s]]<br />
| [[1L&nbsp;3s]]<br />
| [[1L&nbsp;4s]]<br />
| [[1L&nbsp;5s]]<br />
| [[1L&nbsp;6s]]<br />
| [[1L&nbsp;7s]]<br />
| [[1L&nbsp;8s]]<br />
| [[1L&nbsp;9s]]<br />
| [[1L&nbsp;10s]]<br />
| [[1L&nbsp;11s]]<br />
| [[1L&nbsp;12s]]<br />
|-<br />
| [[2L&nbsp;1s]]<br />
| [[2L&nbsp;2s]]<br />
| [[2L&nbsp;3s]]<br />
| [[2L&nbsp;4s]]<br />
| [[2L&nbsp;5s]]<br />
| [[2L&nbsp;6s]]<br />
| [[2L&nbsp;7s]]<br />
| [[2L&nbsp;8s]]<br />
| [[2L&nbsp;9s]]<br />
| [[2L&nbsp;10s]]<br />
| [[2L&nbsp;11s]]<br />
| [[2L&nbsp;12s]]<br />
|-<br />
| [[3L&nbsp;1s]]<br />
| [[3L&nbsp;2s]]<br />
| [[3L&nbsp;3s]]<br />
| [[3L&nbsp;4s]]<br />
| [[3L&nbsp;5s]]<br />
| [[3L&nbsp;6s]]<br />
| [[3L&nbsp;7s]]<br />
| [[3L&nbsp;8s]]<br />
| [[3L&nbsp;9s]]<br />
| [[3L&nbsp;10s]]<br />
| [[3L&nbsp;11s]]<br />
| [[3L&nbsp;12s]]<br />
|-<br />
| [[4L&nbsp;1s]]<br />
| [[4L&nbsp;2s]]<br />
| [[4L&nbsp;3s]]<br />
| [[4L&nbsp;4s]]<br />
| [[4L&nbsp;5s]]<br />
| [[4L&nbsp;6s]]<br />
| [[4L&nbsp;7s]]<br />
| [[4L&nbsp;8s]]<br />
| [[4L&nbsp;9s]]<br />
| [[4L&nbsp;10s]]<br />
| [[4L&nbsp;11s]]<br />
| [[4L&nbsp;12s]]<br />
|-<br />
| [[5L&nbsp;1s]]<br />
| [[5L&nbsp;2s]]<br />
| [[5L&nbsp;3s]]<br />
| [[5L&nbsp;4s]]<br />
| [[5L&nbsp;5s]]<br />
| [[5L&nbsp;6s]]<br />
| [[5L&nbsp;7s]]<br />
| [[5L&nbsp;8s]]<br />
| [[5L&nbsp;9s]]<br />
| [[5L&nbsp;10s]]<br />
| [[5L&nbsp;11s]]<br />
| [[5L&nbsp;12s]]<br />
|-<br />
| [[6L&nbsp;1s]]<br />
| [[6L&nbsp;2s]]<br />
| [[6L&nbsp;3s]]<br />
| [[6L&nbsp;4s]]<br />
| [[6L&nbsp;5s]]<br />
| [[6L&nbsp;6s]]<br />
| [[6L&nbsp;7s]]<br />
| [[6L&nbsp;8s]]<br />
| [[6L&nbsp;9s]]<br />
| [[6L&nbsp;10s]]<br />
| [[6L&nbsp;11s]]<br />
| [[6L&nbsp;12s]]<br />
|-<br />
| [[7L&nbsp;1s]]<br />
| [[7L&nbsp;2s]]<br />
| [[7L&nbsp;3s]]<br />
| [[7L&nbsp;4s]]<br />
| [[7L&nbsp;5s]]<br />
| [[7L&nbsp;6s]]<br />
| [[7L&nbsp;7s]]<br />
| [[7L&nbsp;8s]]<br />
| [[7L&nbsp;9s]]<br />
| [[7L&nbsp;10s]]<br />
| [[7L&nbsp;11s]]<br />
| [[7L&nbsp;12s]]<br />
|-<br />
| [[8L&nbsp;1s]]<br />
| [[8L&nbsp;2s]]<br />
| [[8L&nbsp;3s]]<br />
| [[8L&nbsp;4s]]<br />
| [[8L&nbsp;5s]]<br />
| [[8L&nbsp;6s]]<br />
| [[8L&nbsp;7s]]<br />
| [[8L&nbsp;8s]]<br />
| [[8L&nbsp;9s]]<br />
| [[8L&nbsp;10s]]<br />
| [[8L&nbsp;11s]]<br />
| [[8L&nbsp;12s]]<br />
|-<br />
| [[9L&nbsp;1s]]<br />
| [[9L&nbsp;2s]]<br />
| [[9L&nbsp;3s]]<br />
| [[9L&nbsp;4s]]<br />
| [[9L&nbsp;5s]]<br />
| [[9L&nbsp;6s]]<br />
| [[9L&nbsp;7s]]<br />
| [[9L&nbsp;8s]]<br />
| [[9L&nbsp;9s]]<br />
| [[9L&nbsp;10s]]<br />
| [[9L&nbsp;11s]]<br />
| [[9L&nbsp;12s]]<br />
|-<br />
| [[10L&nbsp;1s]]<br />
| [[10L&nbsp;2s]]<br />
| [[10L&nbsp;3s]]<br />
| [[10L&nbsp;4s]]<br />
| [[10L&nbsp;5s]]<br />
| [[10L&nbsp;6s]]<br />
| [[10L&nbsp;7s]]<br />
| [[10L&nbsp;8s]]<br />
| [[10L&nbsp;9s]]<br />
| [[10L&nbsp;10s]]<br />
| [[10L&nbsp;11s]]<br />
| [[10L&nbsp;12s]]<br />
|-<br />
| [[11L&nbsp;1s]]<br />
| [[11L&nbsp;2s]]<br />
| [[11L&nbsp;3s]]<br />
| [[11L&nbsp;4s]]<br />
| [[11L&nbsp;5s]]<br />
| [[11L&nbsp;6s]]<br />
| [[11L&nbsp;7s]]<br />
| [[11L&nbsp;8s]]<br />
| [[11L&nbsp;9s]]<br />
| [[11L&nbsp;10s]]<br />
| [[11L&nbsp;11s]]<br />
| [[11L&nbsp;12s]]<br />
|-<br />
| [[12L&nbsp;1s]]<br />
| [[12L&nbsp;2s]]<br />
| [[12L&nbsp;3s]]<br />
| [[12L&nbsp;4s]]<br />
| [[12L&nbsp;5s]]<br />
| [[12L&nbsp;6s]]<br />
| [[12L&nbsp;7s]]<br />
| [[12L&nbsp;8s]]<br />
| [[12L&nbsp;9s]]<br />
| [[12L&nbsp;10s]]<br />
| [[12L&nbsp;11s]]<br />
| [[12L&nbsp;12s]]<br />
|}<br />
<br />
=== L ≤ 12, 13 ≤ s ≤ 24 ===<br />
{| class="wikitable mw-collapsible mw-collapsed center-all"<br />
|+ style="font-size: 105%; white-space: nowrap;" | Pages for MOS scales ({{nowrap|L ≤ 12|13 ≤ s ≤ 24}})<br />
|-<br />
| [[1L&nbsp;13s]]<br />
| [[1L&nbsp;14s]]<br />
| [[1L&nbsp;15s]]<br />
| [[1L&nbsp;16s]]<br />
| [[1L&nbsp;17s]]<br />
| [[1L&nbsp;18s]]<br />
| [[1L&nbsp;19s]]<br />
| [[1L&nbsp;20s]]<br />
| [[1L&nbsp;21s]]<br />
| [[1L&nbsp;22s]]<br />
| [[1L&nbsp;23s]]<br />
| [[1L&nbsp;24s]]<br />
|-<br />
| [[2L&nbsp;13s]]<br />
| [[2L&nbsp;14s]]<br />
| [[2L&nbsp;15s]]<br />
| [[2L&nbsp;16s]]<br />
| [[2L&nbsp;17s]]<br />
| [[2L&nbsp;18s]]<br />
| [[2L&nbsp;19s]]<br />
| [[2L&nbsp;20s]]<br />
| [[2L&nbsp;21s]]<br />
| [[2L&nbsp;22s]]<br />
| [[2L&nbsp;23s]]<br />
| [[2L&nbsp;24s]]<br />
|-<br />
| [[3L&nbsp;13s]]<br />
| [[3L&nbsp;14s]]<br />
| [[3L&nbsp;15s]]<br />
| [[3L&nbsp;16s]]<br />
| [[3L&nbsp;17s]]<br />
| [[3L&nbsp;18s]]<br />
| [[3L&nbsp;19s]]<br />
| [[3L&nbsp;20s]]<br />
| [[3L&nbsp;21s]]<br />
| [[3L&nbsp;22s]]<br />
| [[3L&nbsp;23s]]<br />
| [[3L&nbsp;24s]]<br />
|-<br />
| [[4L&nbsp;13s]]<br />
| [[4L&nbsp;14s]]<br />
| [[4L&nbsp;15s]]<br />
| [[4L&nbsp;16s]]<br />
| [[4L&nbsp;17s]]<br />
| [[4L&nbsp;18s]]<br />
| [[4L&nbsp;19s]]<br />
| [[4L&nbsp;20s]]<br />
| [[4L&nbsp;21s]]<br />
| [[4L&nbsp;22s]]<br />
| [[4L&nbsp;23s]]<br />
| [[4L&nbsp;24s]]<br />
|-<br />
| [[5L&nbsp;13s]]<br />
| [[5L&nbsp;14s]]<br />
| [[5L&nbsp;15s]]<br />
| [[5L&nbsp;16s]]<br />
| [[5L&nbsp;17s]]<br />
| [[5L&nbsp;18s]]<br />
| [[5L&nbsp;19s]]<br />
| [[5L&nbsp;20s]]<br />
| [[5L&nbsp;21s]]<br />
| [[5L&nbsp;22s]]<br />
| [[5L&nbsp;23s]]<br />
| [[5L&nbsp;24s]]<br />
|-<br />
| [[6L&nbsp;13s]]<br />
| [[6L&nbsp;14s]]<br />
| [[6L&nbsp;15s]]<br />
| [[6L&nbsp;16s]]<br />
| [[6L&nbsp;17s]]<br />
| [[6L&nbsp;18s]]<br />
| [[6L&nbsp;19s]]<br />
| [[6L&nbsp;20s]]<br />
| [[6L&nbsp;21s]]<br />
| [[6L&nbsp;22s]]<br />
| [[6L&nbsp;23s]]<br />
| [[6L&nbsp;24s]]<br />
|-<br />
| [[7L&nbsp;13s]]<br />
| [[7L&nbsp;14s]]<br />
| [[7L&nbsp;15s]]<br />
| [[7L&nbsp;16s]]<br />
| [[7L&nbsp;17s]]<br />
| [[7L&nbsp;18s]]<br />
| [[7L&nbsp;19s]]<br />
| [[7L&nbsp;20s]]<br />
| [[7L&nbsp;21s]]<br />
| [[7L&nbsp;22s]]<br />
| [[7L&nbsp;23s]]<br />
| [[7L&nbsp;24s]]<br />
|-<br />
| [[8L&nbsp;13s]]<br />
| [[8L&nbsp;14s]]<br />
| [[8L&nbsp;15s]]<br />
| [[8L&nbsp;16s]]<br />
| [[8L&nbsp;17s]]<br />
| [[8L&nbsp;18s]]<br />
| [[8L&nbsp;19s]]<br />
| [[8L&nbsp;20s]]<br />
| [[8L&nbsp;21s]]<br />
| [[8L&nbsp;22s]]<br />
| [[8L&nbsp;23s]]<br />
| [[8L&nbsp;24s]]<br />
|-<br />
| [[9L&nbsp;13s]]<br />
| [[9L&nbsp;14s]]<br />
| [[9L&nbsp;15s]]<br />
| [[9L&nbsp;16s]]<br />
| [[9L&nbsp;17s]]<br />
| [[9L&nbsp;18s]]<br />
| [[9L&nbsp;19s]]<br />
| [[9L&nbsp;20s]]<br />
| [[9L&nbsp;21s]]<br />
| [[9L&nbsp;22s]]<br />
| [[9L&nbsp;23s]]<br />
| [[9L&nbsp;24s]]<br />
|-<br />
| [[10L&nbsp;13s]]<br />
| [[10L&nbsp;14s]]<br />
| [[10L&nbsp;15s]]<br />
| [[10L&nbsp;16s]]<br />
| [[10L&nbsp;17s]]<br />
| [[10L&nbsp;18s]]<br />
| [[10L&nbsp;19s]]<br />
| [[10L&nbsp;20s]]<br />
| [[10L&nbsp;21s]]<br />
| [[10L&nbsp;22s]]<br />
| [[10L&nbsp;23s]]<br />
| [[10L&nbsp;24s]]<br />
|-<br />
| [[11L&nbsp;13s]]<br />
| [[11L&nbsp;14s]]<br />
| [[11L&nbsp;15s]]<br />
| [[11L&nbsp;16s]]<br />
| [[11L&nbsp;17s]]<br />
| [[11L&nbsp;18s]]<br />
| [[11L&nbsp;19s]]<br />
| [[11L&nbsp;20s]]<br />
| [[11L&nbsp;21s]]<br />
| [[11L&nbsp;22s]]<br />
| [[11L&nbsp;23s]]<br />
| [[11L&nbsp;24s]]<br />
|-<br />
| [[12L&nbsp;13s]]<br />
| [[12L&nbsp;14s]]<br />
| [[12L&nbsp;15s]]<br />
| [[12L&nbsp;16s]]<br />
| [[12L&nbsp;17s]]<br />
| [[12L&nbsp;18s]]<br />
| [[12L&nbsp;19s]]<br />
| [[12L&nbsp;20s]]<br />
| [[12L&nbsp;21s]]<br />
| [[12L&nbsp;22s]]<br />
| [[12L&nbsp;23s]]<br />
| [[12L&nbsp;24s]]<br />
|}<br />
<br />
=== 13 ≤ L ≤ 24, s ≤ 12 ===<br />
{| class="wikitable mw-collapsible mw-collapsed center-all"<br />
|+ style="font-size: 105%; white-space: nowrap;" | Pages for MOS scales ({{nowrap|13 ≤ L ≤ 24|s ≤ 12}})<br />
|-<br />
| [[13L&nbsp;1s]]<br />
| [[13L&nbsp;2s]]<br />
| [[13L&nbsp;3s]]<br />
| [[13L&nbsp;4s]]<br />
| [[13L&nbsp;5s]]<br />
| [[13L&nbsp;6s]]<br />
| [[13L&nbsp;7s]]<br />
| [[13L&nbsp;8s]]<br />
| [[13L&nbsp;9s]]<br />
| [[13L&nbsp;10s]]<br />
| [[13L&nbsp;11s]]<br />
| [[13L&nbsp;12s]]<br />
|-<br />
| [[14L&nbsp;1s]]<br />
| [[14L&nbsp;2s]]<br />
| [[14L&nbsp;3s]]<br />
| [[14L&nbsp;4s]]<br />
| [[14L&nbsp;5s]]<br />
| [[14L&nbsp;6s]]<br />
| [[14L&nbsp;7s]]<br />
| [[14L&nbsp;8s]]<br />
| [[14L&nbsp;9s]]<br />
| [[14L&nbsp;10s]]<br />
| [[14L&nbsp;11s]]<br />
| [[14L&nbsp;12s]]<br />
|-<br />
| [[15L&nbsp;1s]]<br />
| [[15L&nbsp;2s]]<br />
| [[15L&nbsp;3s]]<br />
| [[15L&nbsp;4s]]<br />
| [[15L&nbsp;5s]]<br />
| [[15L&nbsp;6s]]<br />
| [[15L&nbsp;7s]]<br />
| [[15L&nbsp;8s]]<br />
| [[15L&nbsp;9s]]<br />
| [[15L&nbsp;10s]]<br />
| [[15L&nbsp;11s]]<br />
| [[15L&nbsp;12s]]<br />
|-<br />
| [[16L&nbsp;1s]]<br />
| [[16L&nbsp;2s]]<br />
| [[16L&nbsp;3s]]<br />
| [[16L&nbsp;4s]]<br />
| [[16L&nbsp;5s]]<br />
| [[16L&nbsp;6s]]<br />
| [[16L&nbsp;7s]]<br />
| [[16L&nbsp;8s]]<br />
| [[16L&nbsp;9s]]<br />
| [[16L&nbsp;10s]]<br />
| [[16L&nbsp;11s]]<br />
| [[16L&nbsp;12s]]<br />
|-<br />
| [[17L&nbsp;1s]]<br />
| [[17L&nbsp;2s]]<br />
| [[17L&nbsp;3s]]<br />
| [[17L&nbsp;4s]]<br />
| [[17L&nbsp;5s]]<br />
| [[17L&nbsp;6s]]<br />
| [[17L&nbsp;7s]]<br />
| [[17L&nbsp;8s]]<br />
| [[17L&nbsp;9s]]<br />
| [[17L&nbsp;10s]]<br />
| [[17L&nbsp;11s]]<br />
| [[17L&nbsp;12s]]<br />
|-<br />
| [[18L&nbsp;1s]]<br />
| [[18L&nbsp;2s]]<br />
| [[18L&nbsp;3s]]<br />
| [[18L&nbsp;4s]]<br />
| [[18L&nbsp;5s]]<br />
| [[18L&nbsp;6s]]<br />
| [[18L&nbsp;7s]]<br />
| [[18L&nbsp;8s]]<br />
| [[18L&nbsp;9s]]<br />
| [[18L&nbsp;10s]]<br />
| [[18L&nbsp;11s]]<br />
| [[18L&nbsp;12s]]<br />
|-<br />
| [[19L&nbsp;1s]]<br />
| [[19L&nbsp;2s]]<br />
| [[19L&nbsp;3s]]<br />
| [[19L&nbsp;4s]]<br />
| [[19L&nbsp;5s]]<br />
| [[19L&nbsp;6s]]<br />
| [[19L&nbsp;7s]]<br />
| [[19L&nbsp;8s]]<br />
| [[19L&nbsp;9s]]<br />
| [[19L&nbsp;10s]]<br />
| [[19L&nbsp;11s]]<br />
| [[19L&nbsp;12s]]<br />
|-<br />
| [[20L&nbsp;1s]]<br />
| [[20L&nbsp;2s]]<br />
| [[20L&nbsp;3s]]<br />
| [[20L&nbsp;4s]]<br />
| [[20L&nbsp;5s]]<br />
| [[20L&nbsp;6s]]<br />
| [[20L&nbsp;7s]]<br />
| [[20L&nbsp;8s]]<br />
| [[20L&nbsp;9s]]<br />
| [[20L&nbsp;10s]]<br />
| [[20L&nbsp;11s]]<br />
| [[20L&nbsp;12s]]<br />
|-<br />
| [[21L&nbsp;1s]]<br />
| [[21L&nbsp;2s]]<br />
| [[21L&nbsp;3s]]<br />
| [[21L&nbsp;4s]]<br />
| [[21L&nbsp;5s]]<br />
| [[21L&nbsp;6s]]<br />
| [[21L&nbsp;7s]]<br />
| [[21L&nbsp;8s]]<br />
| [[21L&nbsp;9s]]<br />
| [[21L&nbsp;10s]]<br />
| [[21L&nbsp;11s]]<br />
| [[21L&nbsp;12s]]<br />
|-<br />
| [[22L&nbsp;1s]]<br />
| [[22L&nbsp;2s]]<br />
| [[22L&nbsp;3s]]<br />
| [[22L&nbsp;4s]]<br />
| [[22L&nbsp;5s]]<br />
| [[22L&nbsp;6s]]<br />
| [[22L&nbsp;7s]]<br />
| [[22L&nbsp;8s]]<br />
| [[22L&nbsp;9s]]<br />
| [[22L&nbsp;10s]]<br />
| [[22L&nbsp;11s]]<br />
| [[22L&nbsp;12s]]<br />
|-<br />
| [[23L&nbsp;1s]]<br />
| [[23L&nbsp;2s]]<br />
| [[23L&nbsp;3s]]<br />
| [[23L&nbsp;4s]]<br />
| [[23L&nbsp;5s]]<br />
| [[23L&nbsp;6s]]<br />
| [[23L&nbsp;7s]]<br />
| [[23L&nbsp;8s]]<br />
| [[23L&nbsp;9s]]<br />
| [[23L&nbsp;10s]]<br />
| [[23L&nbsp;11s]]<br />
| [[23L&nbsp;12s]]<br />
|-<br />
| [[24L&nbsp;1s]]<br />
| [[24L&nbsp;2s]]<br />
| [[24L&nbsp;3s]]<br />
| [[24L&nbsp;4s]]<br />
| [[24L&nbsp;5s]]<br />
| [[24L&nbsp;6s]]<br />
| [[24L&nbsp;7s]]<br />
| [[24L&nbsp;8s]]<br />
| [[24L&nbsp;9s]]<br />
| [[24L&nbsp;10s]]<br />
| [[24L&nbsp;11s]]<br />
| [[24L&nbsp;12s]]<br />
|}<br />
<br />
=== Larger MOS scales ===<br />
[[7L&nbsp;34s]], [[9L&nbsp;29s]], [[12L&nbsp;29s]], [[12L&nbsp;41s]], [[13L&nbsp;14s]], [[14L&nbsp;13s]], [[17L&nbsp;14s]], [[25L&nbsp;6s]], [[41L&nbsp;12s]]<br />
<br />
== Variations ==<br />
* [[MODMOS scales]] are derived from chromatic alterations of one or more tones of an MOS scale, typically by the interval of {{nowrap| L − s }}, the "chroma".<br />
* [[Muddle]]s are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.<br />
* [[MOS cradle]] is a technique of embedding MOS-like structures inside MOS scales and may or may not produce subsets of MOS scales.<br />
* [[Operations on MOSes]]<br />
<br />
== Listen ==<br />
This is an algorithmically generated recording of every MOS scale that has 14 or fewer notes for a total of 91 scales being showcased here. Each MOS scale played has its simplest step ratio (large step is 2 small step is 1) and therefore is inside the smallest EDO that can support it. Each MOS scale is also in its brightest mode. And rhythmically, each scale is being played with its respective MOS rhythm. Note that changing the mode or step ratio of any of these MOSes may dramatically alter the sound and therefore this recording is not thoroughly representative of each MOS but rather a small taste.<br />
<br />
[[File:Every-MOS-Scale-With-14-Or-Fewer-Notes.mp3|left|800x800px]] {{clear}}<br />
<br />
== See also ==<br />
* Pailiaq's [https://lkorr.github.io/mos-explorer/ MOS explorer], an interactive tool for visualizing MOSses and the MOS spectrum.<br />
* [[Diamond-mos notation]], a microtonal [[notation]] system focused on MOS scales<br />
* [[Metallic MOS]], an article focusing on MOS scales based on metallic means, such as [[phi]]<br />
* [[MOS rhythm]]<br />
* [[:Category:MOS scales|Category:MOS scales]], the category including all MOS-related articles on this wiki<br />
* [[Gallery of MOS patterns]]<br />
<br />
== Notes ==<br />
<references group="note" /><br />
<br />
== References ==<br />
<references /><br />
<br />
[[Category:Math]]<br />
[[Category:MOS scale| ]] <!-- Sort order in category: this page shows above A --><br />
[[Category:Scale]]<br />
[[Category:Erv Wilson]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Template:Angbr&diff=234330Template:Angbr2026-07-18T16:29:59Z<p>ArrowHead294: Try adding space since some characters may collide when italicised</p>
<hr />
<div><includeonly>{{#if: {{{1|}}}|<span class{{=}}"Unicode">&#x27E8;</span>{{{1}}}<span class{{=}}"Unicode" {{#ifeq: {{{2|}}}|-)|style{{=}}"padding-left: 0.1em; white-space: nowrap;"|}}>&#x27E9;</span>|}}</includeonly><noinclude><br />
{{documentation}}<br />
<br />
[[Category:Character templates]]<br />
[[Category:Formatting templates]]<br />
</noinclude></div>ArrowHead294https://en.xen.wiki/index.php?title=Xenharmonic_Wiki:I%27m_a_new_editor,_what_is_everything_I_can_do_to_help%3F&diff=233317Xenharmonic Wiki:I'm a new editor, what is everything I can do to help?2026-07-06T15:24:13Z<p>ArrowHead294: </p>
<hr />
<div>There are many possible things you can do. This is a sampling of some of them. Pick and choose whichever sound the most interesting to you.<br />
<br />
== Small tasks ==<br />
=== Add examples ===<br />
Add examples to one of the pages in [[:Category:Todo:add examples]].<br />
<br />
=== Write a missing introduction ===<br />
Write an introduction for one of the pages in [[:Category:Todo:intro]].<br />
<br />
=== Finish a list or table ===<br />
Complete one of the incomplete lists or tables from [[:Category:Todo:complete list]] or [[:Category:Todo:complete table]] respectively.<br />
<br />
=== Resurrect a dead link ===<br />
Go to one of the pages in [[:Category:Pages containing dead links]]. <br />
<br />
Try to find a live version of the link's target online. Perhaps [https://web.archive.org/ Wayback Machine] might have one.<br />
<br />
If you can find a live version, then update the link to point to that. <br />
<br />
If you can't find one, '''please do not delete''' the link. Instead, post on the talk page and list every place you've looked so far, so that future searchers don't have to tread the same ground.<br />
<br />
== Intermediate tasks ==<br />
=== Rate a piece of music ===<br />
Add your ratings of xenharmonic pieces of music to [[List of xenharmonic music by community ratings]]. Carefully follow the instructions on that page to add them.<br />
<br />
=== Answer a question ===<br />
If you know the answers to any questions on [[FAQ]] or any other page in [[:Category:Todo:answer questions]], then edit the page to add your answers.<br />
<br />
=== Add an audio example ===<br />
Almost every page on the wiki is in need of audio examples, so that readers can hear what the concepts actually sound like.<br />
<br />
If you can add such audio examples, please do so.<br />
<br />
=== Add an illustration ===<br />
Add an illustration to one of the pages in [[:Category:Todo:add illustration]].<br />
<br />
=== Create a choose-an-EDO flow chart ===<br />
Create a flow chart to help readers choose an [[EDO]] to make music with. Link your flow chart in [[Edo recommendation hub page]].<br />
<br />
=== Add a rank-2 temperaments table ===<br />
Add a rank-2 temperaments table to one of the pages in [[:Category:Todo:add rank 2 temperaments table]].<br />
<br />
Check out some edo pages (e.g. [[26edo]], [[27edo]], [[28edo]]) to get an idea of what rank-2 temperaments tables look like.<br />
<br />
=== Finish a section ===<br />
Complete one of the incomplete page subsections from [[:Category:Todo:complete section]].<br />
<br />
== Large tasks ==<br />
=== Recommend the EDOs you like ===<br />
Create a page where you recommend a list of [[edo]]s to beginners. <br />
<br />
Follow the standardised structure shown on the page [[Edo recommendation hub page]] to make it easy for readers to compare with the other lists.<br />
<br />
When your page is done, link it on [[Edo recommendation hub page]].<br />
<br />
=== Become a resident expert ===<br />
Choose an article about a tuning or scale from [[:Category:Stubs]] or [[:Category:Todo:expand]]. <br />
<br />
Make some music with that tuning, and document your process every step along your way:<br />
* What subset scales you tried that didn't work. <br />
** What ones did.<br />
* What instruments or synth settings sounded bad.<br />
** What ones sounded good.<br />
* Document as much as you can, even the smallest details.<br />
<br />
By the time you're done, you will probably know more about that tuning than anybody else in the world ever has. You will now be more qualified to write its page than anyone else.<br />
<br />
Go ahead and expand the stub or todo:expand page with those notes you gathered as a guide, and add a link to your musical example(s) too.<br />
<br />
Now you have taken the page from almost empty, to one of the best written, most complete on the wiki!<br />
<br />
== For special skillsets ==<br />
=== Mathematics ===<br />
==== Correct the mathematics on a page ====<br />
If you have some experience with mathematics, go to one of the pages in [[:Category:Todo:correct maths]] and make sure all the mathematics written on the page is correct. If it's not, correct it.<br />
<br />
==== Make a page more accessible ====<br />
Write a simplified version of one of the pages listed in either [[:Category:Inaccessible pages]] or [[:Category:Todo:reduce mathslang]]. <br />
<br />
Make it fully understandable to non-mathematicians.<br />
<br />
Post it as a new, separate page on the wiki.<br />
<br />
=== Journalism, modern history or social science ===<br />
==== Fact check ====<br />
Do any of the following, whichever ones interest you:<br />
* Check facts for pages in [[:Category:Todo:confirm]] and [[:Category:Todo:research]]<br />
* Update time-sensitive facts (e.g. whether someone is or was active, whethee they are studying or did study at x, etc.) for pages in [[:Category:Todo:update]]<br />
<br />
==== Find sources ====<br />
Do any of the following, whichever ones interest you:<br />
* Find and add etymology for pages in [[:Category:Todo:add etymology]]<br />
* Find and add sources for pages in [[:Category:Todo:add source]] and [[:Category:Pages with unsourced statements]]<br />
* Find how a page in [[:Category:Todo:explain its xenharmonic value]] relates to microtonality or musical tuning, and edit the page to explain how<br />
<br />
=== Lua ===<br />
==== Write documentation for a template or module ====<br />
If you have some experience with Lua, we need your help to write documentation for the templates or modules in [[:Category:Todo:add documentation]]<br />
<br />
=== Ethnomusicology ===<br />
==== Improve a page ====<br />
The following pages are in need of drastic improvement by experts in the field, please improve them:<br />
* [[African music]]<br />
* [[Arabic, Turkish, Persian music]]<br />
* [[Georgian]] music<br />
* [[Indian music]]<br />
* [[Indonesian]] music<br />
** [[Gamelan]]<br />
** [[Pelog]]<br />
** [[Slendro]]<br />
* [[Pre-Columbian South American music]]<br />
<br />
==== Create a page ====<br />
The following pages are needed, but have not yet been created. If you are an expert in the field, please create them. <br />
<br />
Note that given the nature of this wiki, the pages should focus on musical ''tuning'' first and foremost:<br />
* Separate pages for some different musical traditions within Africa<br />
* Separate pages for Arabic music, Turkish music and Iranian music<br />
* A page for Thai music<br />
* A page for Chinese music (ancient & modern)<br />
* Pages for non-Western composers and musicians who use microtuning<br />
<br />
=== Music history ===<br />
==== Improve a page ====<br />
The following articles are in need of drastic improvement by experts in the field, please improve them:<br />
* [[Historical temperaments]]<br />
** And all the examples listed therein<br />
* [[Ancient Greek music]]<br />
** [[Teleic scales]]<br />
** [[Tetrachord]]<br />
<br />
==== Create a page ====<br />
The following pages are needed, but have not yet been created. If you are an expert in the field, please create them. <br />
<br />
Note that given the nature of this wiki, the pages should focus on musical ''tuning'' first and foremost:<br />
* A page for Byzantine music<br />
* A page for Medieval European music<br />
* All the red links on the page [[Historical temperaments]]<br />
* Pages for historical composers who used microtuning<br />
<br />
=== Any non-English language ===<br />
==== Contribute ====<br />
If the language you know already has a Xen Wiki interwik—contribute to it! Write pages for it.<br />
<br />
==== Initiate ====<br />
If it does not yet have an interwiki, start writing pages in your language in the main English Xen Wiki and eventually they will be moved into their own site.<br />
<br />
=== Large EDOs ===<br />
==== Curate a table ====<br />
Go to one of the pages in [[:Category:Todo:Replace auto-generated table of intervals with manually curated table]].<br />
<br />
Create a manual, annotated version of the table using a wikitable. <br />
<br />
Google how to do make wikitables if you're not sure how, or use the visual editor. [https://excel2wiki.toolforge.org/index.php Excel2Wiki] may also prove helpful.<br />
<br />
=== MOS scales ===<br />
==== Write an introductory guide ====<br />
Write a thorough introduction to MOS scales from the point of view of a musician who knows nothing about them and just wants to know how to make music with them and how different ones sound and feel.<br />
<br />
=== Comma pumps ===<br />
==== Add a comma pump to a comma page ====<br />
Go to any page in [[:Category:Commas]] and its subcategories. Edit the page and describe a comma pump that uses the comma. Follow the style of the [[Frameshift comma]] page.<br />
<br />
== Want more to do? ==<br />
Check out [[Xenharmonic Wiki:Things to do]] and [[Wikifuture]].<br />
<br />
[[Category:Xenharmonic Wiki]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Template:MOS_intro&diff=232543Template:MOS intro2026-06-21T23:51:20Z<p>ArrowHead294: </p>
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<div><includeonly>{{#invoke: MOS_intro | mos_intro_frame<br />
| Scale Signature={{{Scale Signature|{{PAGENAME}}}}}<br />
| Other Names={{{Other Names|}}}<br />
| debug={{lc: {{{debug|}}}}}<br />
}}</includeonly><noinclude><br />
{{documentation}}<br />
<br />
[[Category:Intro templates]]<br />
[[Category:MOS scale templates]]<br />
</noinclude></div>ArrowHead294https://en.xen.wiki/index.php?title=Xenharmonic_Wiki:Manual_of_Style/MOS_pages&diff=232542Xenharmonic Wiki:Manual of Style/MOS pages2026-06-21T23:49:19Z<p>ArrowHead294: Formatting and dashes</p>
<hr />
<div>{{mbox|text=This page is maintained by [[Xenharmonic Wiki:WikiProject Mospage|Project Mospage]].}}<br />
<br />
{{Infobox MOS|Tuning=5L 2s|debug=1}}<br />
This is a '''style guide for mos pages'''.<br />
<br />
The minimum templates for a mos page, for a page that can be considered a stub or otherwise has very little information, is the following:<br />
<br />
* [[Template:Infobox MOS|Infobox MOS]]; required, as it categorizes the page<br />
* [[Template:MOS intro|MOS intro]]; recommended<br />
* [[Template:MOS intervals|MOS intervals]]; optional, but recommended for small mosses (no more than around 10 notes)<br />
* [[Template:MOS modes|MOS modes]]; optional, but recommended for small mosses (no more than around 10 notes)<br />
* [[Template:MOS tuning spectrum|MOS tuning spectrum]]; recommended<br />
<br />
For guidance on how to expand or improve existing pages, this page's sections provide on what to include, and how to name each section. Sections should be added ''as needed''. Sections in ''italics'' do not denote actual sections, but rather items to add in addition to existing sections or outside of any section. This page uses 5L&nbsp;2s as its running example.<br />
<br />
Scale name is included in the lead section rather than its own section, since the main focus of a mos page is ''what it is'', not ''what it's called''. For TAMNAMS-named scales, this is accomplished using the [[Template:TAMNAMS name|TAMNAMS name]] template; in most cases, this is enough information. A name section ''may'' be added if there are contentious names (EG, misnomers), or an in-depth explanation is needed, but for TAMNAMS-named scales, the template already includes either further reasoning or a link to said reasoning.<br />
<br />
'''Disclaimer''': as this guide cannot account for every possible use case, editors are advised to use their best judgment on what to add or not. Additionally, nothing about this guide is set in stone; this guide can (and should!) be updated to fit current developments regarding mos pages.<br />
<br />
== ''Lead section'' (precedes the first section) ==<br />
<pre><br />
{{Infobox MOS}}<br />
{{MOS intro}}<br />
</pre><br />
The lead section should, at the minimum, consist of [[Template:Infobox MOS]] and the [[Template:MOS intro]]. To ensure proper order of templates when viewed on mobile devices, the infobox should be placed before the intro. Hatnotes should be placed before the infobox. The mos intro is as follows:<br />
<br />
{{MOS intro|Scale Signature=5L 2s}}<br />
Other important info can include the following:<br />
<br />
* Whether the mos can be thought of a warping of another, more familiar mos or edo. Examples:<br />
** 4L&nbsp;3s can be seen as a warped diatonic scale (5L&nbsp;2s), where one large step is replaced with a small step.<br />
** 5L&nbsp;1s can be seen as the equal-tempered whole-tone scale (6edo) but with one step that is larger than the others.<br />
* Usage, discovery, and noteworthy temperaments the mos corresponds to.<br />
* Under what conditions is the mos proper. This is only recommended for mosses with notable near-mos forms, as this information is included in the intro already.<br />
Sections that would be otherwise too short, such as temperament-related information, can be added in the lead section instead.<br />
<br />
What the mos is referred to may also be included here. Names that see common use should be added here; proposed names are assumed to be idiosyncratic until proven otherwise.<br />
<br />
Subsections for this follow the format for TAMNAMS, in which the following are described in their own subsections:<br />
<br />
== Scale properties ==<br />
=== Intervals ===<br />
The use of TAMNAMS is advised for describing the names of its intervals and scale degrees. The mos intervals template gives ranges for each of the intervals present in the mos.<br />
<pre><br />
{{MOS intervals}}<br />
</pre>{{MOS intervals|Scale Signature=5L 2s}}<br />
<br />
=== Modes ===<br />
Listings of the scale's modes, or rotations of its step patterns, are added here. This can include mode names in use by the general community. Names that have not received use by the broader community use should be treated as idiosyncratic, and if applicable, the person who had proposed the names should be stated. Example: ''person-name'' has proposed/advocated for the following names...<br />
<br />
Listing mode names ''specific'' to a tuning or temperament is not advised. Linking to such mode names is allowed.<br />
<pre><br />
{{MOS modes}}<br />
</pre>{{MOS modes|Scale Signature=5L 2s}}<br />
<br />
Scale degree qualities (major, minor, etc) can be show using the template shown.<pre><br />
{{MOS mode degrees}}<br />
</pre>{{MOS mode degrees|Scale Signature=5L 2s}}<br />
<br />
== Theory ==<br />
Subsections include:<br />
<br />
* Low harmonic entropy scales<br />
* Temperament interpretations<br />
Other subsections include:<br />
<br />
* Any theory described by musicians/theorists<br />
* Tetrachordal analysis or similar<br />
<br />
== Tuning ranges ==<br />
Discussions about specific tuning ranges, and what JI ratios are approximated, can be given. Hatnotes to appropriate temperament pages are also advised.<br />
<br />
Tuning ranges include:<br />
<br />
* '''Simple tunings''' – Step ratios 2:1, 3:1, and 3:2<br />
* '''Soft-of-basic''' '''tunings''' – Step ratios softer than 2:1<br />
* '''Hard-of-basic''' '''tunings''' – Step ratios harder than 2:1<br />
<br />
Sections for smaller ranges can be used instead. Such ranges include:<br />
* '''Ultrasoft tunings''' – Step ratio tunings between 1:1 to 4:3<br />
* '''Parasoft tunings''' – Step ratio tunings between 4:3 to 3:2<br />
* '''Hyposoft tunings''' – Step ratio tunings between 3:2 to 2:1, which can be split even further if needed:<br />
** '''Quasisoft tunings''' – Step ratio tunings between 3:2 to 5:3<br />
** '''Minisoft tunings''' – Step ratio tunings between 5:3 to 2:1<br />
* '''Hypohard tunings''' – Step ratio tunings between 2:1 to 3:1, which can be split even further if needed:<br />
** '''Minihard tunings''' – Step ratio tunings between 2:1 to 5:2<br />
** '''Quasihard tunings''' – Step ratio tunings between 5:2 to 3:1<br />
* '''Parahard tunings''' – Step ratio tunings between 3:1 to 4:1<br />
* '''Ultrahard tunings''' – Step ratio tunings between 4:1 to 1:0<br />
<br />
The mos tunings template shows cent values for varying step ratios.<br />
<pre><br />
{{MOS tunings|Step Ratio=2/1; 3/1; 3/2}}<br />
</pre><br />
{{MOS tunings|Step Ratio=2/1; 3/1; 3/2|Scale Signature=5L 2s}}<br />
<br />
== Scales ==<br />
Links to scala file pages can be added here, as well as links to pages regarding step variations (typically modmos scales) of the mos.<br />
<br />
== Scale tree ==<br />
The tuning spectrum template (or scale tree) can be provided using the corresponding template. A handmade table can also be used if the template is found too limiting, such as by substituting the template.<br />
<pre><br />
{{MOS tuning spectrum}}<br />
</pre>{{MOS tuning spectrum|Scale Signature=5L 2s}}<br />
<br />
== Music ==<br />
Links to music can be added here.<br />
<br />
== See also ==<br />
Links to other pages can be added here.<br />
<br />
== External links ==<br />
Links to outside-wiki resources regarding the mos and/or its tunings can be added here.<br />
<br />
== References ==<br />
If any works are cited, add them here.<br />
<br />
== ''Categories'' (succeeds the last section) ==<br />
The mos infobox automatically categorizes the mos by note count and under the category of abstract mos patterns. Any other categories can be added here.<br />
<br />
== ''Other sections not discussed here'' ==<br />
* Gallery – recommended location is before Music<br />
* Notation – recommended location is after Intervals. Notation schemes shouldn't be added unless it has notable use by multiple composers or its use is ubiquitous, as with 5L&nbsp;2s. If there are multiple notation schemes, this section may be made into a subpage, though this should be a last resort.<br />
* Approaches<br />
* Genchain<br />
* Trivia<br />
<br />
[[Category:Xenharmonic Wiki guidelines]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Template:Szg/doc&diff=230209Template:Szg/doc2026-05-13T15:23:30Z<p>ArrowHead294: Fix link</p>
<hr />
<div>{{dochead}}<br />
The '''szg''' template converts a pictorial ASCII code into a [[Stein–Zimmermann–Gould notation|Stein–Zimmermann–Gould]] symbol in the BravuraText font. In some cases, the symbol is made up of multiple symbols offset and overlaid.<br />
<br />
=== Usage notes ===<br />
This template accepts one unnamed argument and two named arguments:<br />
# Text representing an SZG or related symbol.<br />
# (optional) '''size =''' a CSS font-size value; defaults to "250%".<br />
# (optional) '''height =''' a CSS line-height value, e.g. "40px"; defaults to "1.5em".<br />
<br />
=== Examples ===<br />
{| class="wikitable" style="text-align: center;"<br />
|-<br />
! You type !! You get<br />
|-<br />
| <code><nowiki>{{szg| ^t# }}</nowiki></code> || {{szg| ^t# }}<br />
|-<br />
| <code><nowiki>{{szg| vvvdb }}</nowiki></code> || {{szg| vvvdb }}<br />
|}<br />
<br />
=== See also ===<br />
* [[Template:Sagittal]]<br />
* [[Template:Bravura]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Template:Szg/doc&diff=230208Template:Szg/doc2026-05-13T15:22:26Z<p>ArrowHead294: </p>
<hr />
<div>{{dochead}}<br />
The '''szg''' template converts a pictorial ASCII code into a [[Stein–Zimmermann–Gould]] symbol in the BravuraText font. In some cases, the symbol is made up of multiple symbols offset and overlaid.<br />
<br />
=== Usage notes ===<br />
This template accepts one unnamed argument and two named arguments:<br />
# Text representing an SZG or related symbol.<br />
# (optional) '''size =''' a CSS font-size value; defaults to "250%".<br />
# (optional) '''height =''' a CSS line-height value, e.g. "40px"; defaults to "1.5em".<br />
<br />
=== Examples ===<br />
{| class="wikitable" style="text-align: center;"<br />
|-<br />
! You type !! You get<br />
|-<br />
| <code><nowiki>{{szg| ^t# }}</nowiki></code> || {{szg| ^t# }}<br />
|-<br />
| <code><nowiki>{{szg| vvvdb }}</nowiki></code> || {{szg| vvvdb }}<br />
|}<br />
<br />
=== See also ===<br />
* [[Template:Sagittal]]<br />
* [[Template:Bravura]]</div>ArrowHead294https://en.xen.wiki/index.php?title=S-expression&diff=230207S-expression2026-05-13T15:19:13Z<p>ArrowHead294: </p>
<hr />
<div>An '''S-expression''' is any product, or ratio of products, of the '''square superparticulars''' S''k'', which are defined as the fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. Commas defined by S-expressions turn out to represent intuitive and wide-reaching families of tempered equivalences, and therefore present a very useful framework to learn for a good understanding of the [[commas]] that appear frequently in xen.<br />
<br />
== Quick rules of S-expressions ==<br />
As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply when they are [[tempering out|tempered out]]. The linked sections provide deeper information into each comma family.<br />
<br />
* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. <br>Tempering out S''k'' equates {{sfrac|''k'' + 1|''k''}} with {{sfrac|''k''|''k'' − 1}} and splits {{sfrac|''k'' + 1|''k'' − 1}} in two.<br />
* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: {{nowrap|'''S''k'' * S(''k'' + 1)'''}}, superparticular fractions of the form {{sfrac|''k''(''k'' + 1)/2|(''k'' − 1)(''k'' + 2)/2}}. <br>Tempering out {{nowrap|S''k'' * S(''k'' + 1)}} equates {{sfrac|''k'' + 2|''k'' + 1}} with {{sfrac|''k''|''k'' − 1}}, and {{sfrac|''k'' + 2|''k''}} with {{sfrac|''k'' + 1|''k'' − 1}}.<br />
* [[#Sk2 * S(k + 1) and S(k − 1) * Sk2 (lopsided commas)|Lopsided commas]]: {{nowrap|'''(S''k'')<sup>2</sup> * S(''k'' + 1)'''}} and {{nowrap|'''(S''k'')<sup>2</sup> * S(''k'' − 1)'''}}. <br>Tempering out the former equates {{sfrac|''k'' + 2|''k''}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>2</sup> and {{sfrac|''k'' + 2|''k'' − 1}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>3</sup>, and tempering out the latter equates {{sfrac|''k''|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>2</sup> and {{sfrac|''k'' + 1|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.<br />
* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out splits {{sfrac|''k'' + 2|''k'' − 1}} into {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.<br />
* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out splits {{sfrac|''k'' + 3|''k'' − 1}} into {{pars|{{sfrac|''k'' + 2|''k''}}}}<sup>2</sup>.<br />
<br />
== S''k'' (square-particulars) ==<br />
A '''square superparticular''', or ''square-particular'' for short, is a [[superparticular]] [[interval]] whose numerator is a square number, which is to say, a superparticular of the form<br />
<br />
<math>\displaystyle \frac {k^2}{k^2 - 1} = \frac {k/(k - 1)}{(k + 1)/k} </math><br />
<br />
which is square-(super)particular ''k'' for a given integer {{nowrap|''k'' &gt; 1}}. A suggested shorthand for this interval is '''S''k''''' for the ''k''-th square superparticular, where the ''S'' stands for "(Shorthand for) Second-order/Square Superparticular". This will be used later in this article as the notation will prove powerful in understanding the commas and implied tempered structures of [[regular temperament]]s. Note that this means {{nowrap|S2 {{=}} [[4/3]]}} is the first musically meaningful square-particular, as {{nowrap|S1 {{=}} 1/0}}.<br />
<br />
Also note that we use the notation S''k''<sup>''p''</sup> to mean (S''k'')<sup>''p''</sup> rather than S(''k''<sup>''p''</sup>) for convenience in the practical analysis of regular temperaments using [[S-expression]]s.<br />
<br />
=== Significance/motivation ===<br />
Square-superparticulars are the intervals between consecutive [[superparticular]] [[interval]]s, such that tempering Sk out makes the harmonic segment centered around k have equal steps; eg. tempering out {{nowrap|S9 {{=}} 81/80}} equalizes 8:9:10, as in [[meantone]]. Understanding the mappings of S''k'' in a given temperament is equivalent to understanding the spacing of consecutive superparticulars and the way it represents (or tries to represent) the harmonic series.<br />
<br />
=== Table of square-particulars ===<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 31-limit square-particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2<br />
| ([[2/1]])/([[3/2]])<br />
| [[4/3]]<br />
| 3<br />
|-<br />
| S3<br />
| ([[3/2]])/([[4/3]])<br />
| [[9/8]]<br />
| 3<br />
|-<br />
| S4<br />
| ([[4/3]])/([[5/4]])<br />
| [[16/15]]<br />
| 5<br />
|-<br />
| S5<br />
| ([[5/4]])/([[6/5]])<br />
| [[25/24]]<br />
| 5<br />
|-<br />
| S6 = S8*S9<br />
| ([[6/5]])/([[7/6]])<br />
| [[36/35]]<br />
| 7<br />
|-<br />
| S7<br />
| ([[7/6]])/([[8/7]])<br />
| [[49/48]]<br />
| 7<br />
|-<br />
| S8<br />
| ([[8/7]])/([[9/8]])<br />
| [[64/63]]<br />
| 7<br />
|-<br />
| S9 = S6/S8<br />
| ([[9/8]])/([[10/9]])<br />
| [[81/80]]<br />
| 5<br />
|-<br />
| S10<br />
| ([[10/9]])/([[11/10]])<br />
| [[100/99]]<br />
| 11<br />
|-<br />
| S11<br />
| ([[11/10]])/([[12/11]])<br />
| [[121/120]]<br />
| 11<br />
|-<br />
| S12<br />
| ([[12/11]])/([[13/12]])<br />
| [[144/143]]<br />
| 13<br />
|-<br />
| S13<br />
| ([[13/12]])/([[14/13]])<br />
| [[169/168]]<br />
| 13<br />
|-<br />
| S14<br />
| ([[14/13]])/([[15/14]])<br />
| [[196/195]]<br />
| 13<br />
|-<br />
| S15<br />
| ([[15/14]])/([[16/15]])<br />
| [[225/224]]<br />
| 7<br />
|-<br />
| S16<br />
| ([[16/15]])/([[17/16]])<br />
| [[256/255]]<br />
| 17<br />
|-<br />
| S17<br />
| ([[17/16]])/([[18/17]])<br />
| [[289/288]]<br />
| 17<br />
|-<br />
| S18<br />
| ([[18/17]])/([[19/18]])<br />
| [[324/323]]<br />
| 19<br />
|-<br />
| S19<br />
| ([[19/18]])/([[20/19]])<br />
| [[361/360]]<br />
| 19<br />
|-<br />
| S20<br />
| ([[20/19]])/([[21/20]])<br />
| [[400/399]]<br />
| 19<br />
|-<br />
| S21<br />
| ([[21/20]])/([[22/21]])<br />
| [[441/440]]<br />
| 11<br />
|-<br />
| S22<br />
| ([[22/21]])/([[23/22]])<br />
| [[484/483]]<br />
| 23<br />
|-<br />
| S23<br />
| ([[23/22]])/([[24/23]])<br />
| [[529/528]]<br />
| 23<br />
|-<br />
| S24<br />
| ([[24/23]])/([[25/24]])<br />
| [[576/575]]<br />
| 23<br />
|-<br />
| S25<br />
| ([[25/24]])/([[26/25]])<br />
| [[625/624]]<br />
| 13<br />
|-<br />
| S26 = S13/S15<br />
| ([[26/25]])/([[27/26]])<br />
| [[676/675]]<br />
| 13<br />
|-<br />
| S27<br />
| ([[27/26]])/([[28/27]])<br />
| [[729/728]]<br />
| 13<br />
|-<br />
| S28<br />
| ([[28/27]])/([[29/28]])<br />
| [[784/783]]<br />
| 29<br />
|-<br />
| S29<br />
| ([[29/28]])/([[30/29]])<br />
| [[841/840]]<br />
| 29<br />
|-<br />
| S30<br />
| ([[30/29]])/([[31/30]])<br />
| [[900/899]]<br />
| 31<br />
|-<br />
| S31<br />
| ([[31/30]])/([[32/31]])<br />
| [[961/960]]<br />
| 31<br />
|-<br />
| S32<br />
| ([[32/31]])/([[33/32]])<br />
| [[1024/1023]]<br />
| 31<br />
|-<br />
| S33<br />
| ([[33/32]])/([[34/33]])<br />
| [[1089/1088]]<br />
| 17<br />
|-<br />
| S34<br />
| ([[34/33]])/([[35/34]])<br />
| [[1156/1155]]<br />
| 17<br />
|-<br />
| S35 = S49*S50<br />
| ([[35/34]])/([[36/35]])<br />
| [[1225/1224]]<br />
| 17<br />
|-<br />
| S39<br />
| ([[39/38]])/([[40/39]])<br />
| [[1521/1520]]<br />
| 19<br />
|-<br />
| S45<br />
| ([[45/44]])/([[46/45]])<br />
| [[2025/2024]]<br />
| 23<br />
|-<br />
| S49<br />
| ([[49/48]])/([[50/49]])<br />
| [[2401/2400]]<br />
| 7<br />
|-<br />
| S50<br />
| ([[50/49]])/([[51/50]])<br />
| [[2500/2499]]<br />
| 17<br />
|-<br />
| S51<br />
| ([[51/50]])/([[52/51]])<br />
| [[2601/2600]]<br />
| 17<br />
|-<br />
| S55 = S22/S24<br />
| ([[55/54]])/([[56/55]])<br />
| [[3025/3024]]<br />
| 11<br />
|-<br />
| S56<br />
| ([[56/55]])/([[57/56]])<br />
| [[3136/3135]]<br />
| 19<br />
|-<br />
| S57<br />
| ([[57/56]])/([[58/57]])<br />
| [[3249/3248]]<br />
| 29<br />
|-<br />
| S63<br />
| ([[63/62]])/([[64/63]])<br />
| [[3969/3968]]<br />
| 31<br />
|-<br />
| S64<br />
| ([[64/63]])/([[65/64]])<br />
| [[4096/4095]]<br />
| 13<br />
|-<br />
| S65<br />
| ([[65/64]])/([[66/65]])<br />
| [[4225/4224]]<br />
| 13<br />
|-<br />
| S69<br />
| ([[69/68]])/([[70/69]])<br />
| [[4761/4760]]<br />
| 23<br />
|-<br />
| S76<br />
| ([[76/75]])/([[77/76]])<br />
| [[5776/5775]]<br />
| 19<br />
|-<br />
| S77<br />
| ([[77/76]])/([[78/77]])<br />
| [[5929/5928]]<br />
| 19<br />
|-<br />
| S91<br />
| ([[91/90]])/([[92/91]])<br />
| [[8281/8280]]<br />
| 23<br />
|-<br />
| S92<br />
| ([[92/91]])/([[93/92]])<br />
| [[8464/8463]]<br />
| 31<br />
|-<br />
| S99 = S33/S35<br />
| ([[99/98]])/([[100/99]])<br />
| [[9801/9800]]<br />
| 11<br />
|-<br />
| S115<br />
| ([[115/114]])/([[116/115]])<br />
| [[13225/13224]]<br />
| 29<br />
|-<br />
| S116<br />
| ([[116/115]])/([[117/116]])<br />
| [[13456/13455]]<br />
| 29<br />
|-<br />
| S120<br />
| ([[120/119]])/([[121/120]])<br />
| [[14400/14399]]<br />
| 17<br />
|-<br />
| S125<br />
| ([[125/124]])/([[126/125]])<br />
| [[15625/15624]]<br />
| 31<br />
|-<br />
| S144<br />
| ([[144/143]])/([[145/144]])<br />
| [[20736/20735]]<br />
| 29<br />
|-<br />
| S153<br />
| ([[153/152]])/([[154/153]])<br />
| [[23409/23408]]<br />
| 19<br />
|-<br />
| S154<br />
| ([[154/153]])/([[155/154]])<br />
| [[23716/23715]]<br />
| 31<br />
|-<br />
| S155<br />
| ([[155/154]])/([[156/155]])<br />
| [[24025/24024]]<br />
| 31<br />
|-<br />
| S161 = S46/S48<br />
| ([[161/160]])/([[162/161]])<br />
| [[25921/25920]]<br />
| 23<br />
|-<br />
| S169<br />
| ([[169/168]])/([[170/169]])<br />
| [[28561/28560]]<br />
| 17<br />
|-<br />
| S170<br />
| ([[170/169]])/([[171/170]])<br />
| [[28900/28899]]<br />
| 19<br />
|-<br />
| S175<br />
| ([[175/174]])/([[176/175]])<br />
| [[30625/30624]]<br />
| 29<br />
|-<br />
| S208<br />
| ([[208/207]])/([[209/208]])<br />
| [[43264/43263]]<br />
| 23<br />
|-<br />
| S209<br />
| ([[209/208]])/([[210/209]])<br />
| [[43681/43680]]<br />
| 19<br />
|-<br />
| S231<br />
| ([[231/230]])/([[232/231]])<br />
| [[53361/53360]]<br />
| 29<br />
|-<br />
| S289<br />
| ([[289/288]])/([[290/289]])<br />
| [[83521/83520]]<br />
| 29<br />
|-<br />
| S323<br />
| ([[323/322]])/([[324/323]])<br />
| [[104329/104328]]<br />
| 23<br />
|-<br />
| S324<br />
| ([[324/323]])/([[325/324]])<br />
| [[104976/104975]]<br />
| 19<br />
|-<br />
| S341<br />
| ([[341/340]])/([[342/341]])<br />
| [[116281/116280]]<br />
| 31<br />
|-<br />
| S342<br />
| ([[342/341]])/([[343/342]])<br />
| [[116964/116963]]<br />
| 31<br />
|-<br />
| S351 = S78/S80<br />
| ([[351/350]])/([[352/351]])<br />
| [[123201/123200]]<br />
| 13<br />
|-<br />
| S391<br />
| ([[391/390]])/([[392/391]])<br />
| [[152881/152880]]<br />
| 23<br />
|-<br />
| S441<br />
| ([[441/440]])/([[442/441]])<br />
| [[194481/194480]]<br />
| 17<br />
|-<br />
| S494<br />
| ([[494/493]])/([[495/494]])<br />
| [[244036/244035]]<br />
| 29<br />
|-<br />
| S495<br />
| ([[495/494]])/([[496/495]])<br />
| [[245025/245024]]<br />
| 31<br />
|-<br />
| S528<br />
| ([[528/527]])/([[529/528]])<br />
| [[278784/278783]]<br />
| 31<br />
|-<br />
| S551<br />
| ([[551/550]])/([[552/551]])<br />
| [[303601/303600]]<br />
| 29<br />
|-<br />
| S714<br />
| ([[714/713]])/([[715/714]])<br />
| [[509796/509795]]<br />
| 31<br />
|-<br />
| S783<br />
| ([[783/782]])/([[784/783]])<br />
| [[613089/613088]]<br />
| 29<br />
|-<br />
| S1275<br />
| ([[1275/1274]])/([[1276/1275]])<br />
| [[1625625/1625624]]<br />
| 29<br />
|-<br />
| S1519<br />
| ([[1519/1518]])/([[1520/1519]])<br />
| [[2307361/2307360]]<br />
| 31<br />
|-<br />
| S1520<br />
| ([[1520/1519]])/([[1521/1520]])<br />
| [[2310400/2310399]]<br />
| 31<br />
|-<br />
| S2001<br />
| ([[2001/2000]])/([[2002/2001]])<br />
| [[4004001/4004000]]<br />
| 29<br />
|-<br />
| S2024<br />
| ([[2024/2023]])/([[2025/2024]])<br />
| [[4096576/4096575]]<br />
| 23<br />
|-<br />
| S2431<br />
| ([[2431/2430]])/([[2432/2431]])<br />
| [[5909761/5909760]]<br />
| 19<br />
|-<br />
| S3249<br />
| ([[3249/3248]])/([[3250/3249]])<br />
| <font style="font-size:0.88em">[[10556001/10556000]]</font><br />
| 29<br />
|-<br />
| S9801<br />
| ([[9801/9800]])/([[9802/9801]])<br />
| <font style="font-size:0.88em">[[96059601/96059600]]</font><br />
| 29<br />
|-<br />
| S13311<br />
| <font style="font-size:0.86em">([[13311/13310]])/([[13312/13311]])</font><br />
| <font style="font-size:0.79em">[[177182721/177182720]]</font><br />
| 29<br />
|-<br />
| S13455<br />
| <font style="font-size:0.86em">([[13455/13454]])/([[13456/13455]])</font><br />
| <font style="font-size:0.79em">[[181037025/181037024]]</font><br />
| 31<br />
|}<br />
<br />
=== Alternatives to tempering square-particulars ===<br />
It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas {{nowrap|U''k'' {{=}} {{sfrac|S''k''|S(''k'' + 1)}}}}, we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]<ref group="note">In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as U''k'' = S(''k'' - 1)/S''k'' and U''k'' = S(''k'' + 1)/S(''k'' + 2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.</ref>. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because {{sfrac|''k'' + 2|''k'' + 1}} and {{sfrac|''k''|''k'' − 1}} are equidistant from {{sfrac|''k'' + 1|''k''}} (because of tempering {{sfrac|S''k''|S(''k'' + 1)}}), this means that another expression for {{sfrac|S''k''|S(''k'' + 1)}} is the following:<br />
<br />
<math>\displaystyle {\rm S}k / {\rm S} (k + 1) = \frac{(k + 2) / (k - 1)}{((k + 1)/k)^3} </math><br />
<br />
This means you can read the ''k'' and {{nowrap|''k'' + 1}} from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in [[#Sk/S(k + 1) (ultraparticulars)|the table of ultraparticulars]]).<br />
<br />
Furthermore, defining another sequence of commas with [[semiparticular|formula {{sfrac|S''k''|S(''k'' + 2)}} leads to semiparticulars]] which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering consecutive ultraparticulars.<br />
<br />
== {{nowrap|S''k''*S(''k'' + 1)}} (triangle-particulars) ==<br />
=== Significance ===<br />
1. Every triangle-particular is superparticular, so these are efficient commas. (See also the [[#Short proof of the superparticularity of triangle-particulars]].)<br />
<br />
2. Often each individual triangle-particular, taken as a comma, implies other useful equivalences not necessarily corresponding to the general form, speaking of which …<br />
<br />
3. Every triangle-particular is the difference between two nearly-adjacent superparticular intervals {{nowrap|''k'' + 2|''k'' + 1}} and {{nowrap|''k''|''k'' − 1}}.<br />
<br />
4. Tempering any two consecutive square-particulars S''k'' and {{nowrap|S(''k'' + 1)}} implies tempering a triangle-particular, so these are common commas. (See also: [[lopsided comma]]s.)<br />
<br />
5. If we temper {{nowrap|S''k'' * S(''k'' + 1)}} but not S''k'' or {{nowrap|S(''k'' + 1)}}, then one or more intervals of {{sfrac|''k''|''k'' − 1}}, {{sfrac|''k'' + 1|''k''}}, and {{sfrac|''k'' + 2|''k'' + 1}} ''must'' be mapped inconsistently, because:<br />
: If {{nowrap|''k'' + 1|''k''}} is mapped above {{nowrap|{{sfrac|''k'' + 2|''k'' + 1}} ~ {{sfrac|''k''|''k'' − 1}}}} we have {{nowrap|{{sfrac|''k'' + 1|''k''}} &gt; {{sfrac|''k''|''k'' − 1}}}} and if it is mapped below we have {{nowrap|{{sfrac|''k'' + 1|''k''}} &lt; {{sfrac|''k'' + 2|''k'' + 1}}}}.<br />
: (Generalisations of this and their implications for consistency are discussed in [[#Sk*S(k + 1)*…*S(k + n − 1) (1/n-square-particulars)|the section covering 1/''n''-square-particulars]].)<br />
<br />
=== Meaning ===<br />
Notice that if we equate {{sfrac|''k'' + 2|''k'' + 1}} with {{sfrac|''k''|''k'' − 1}} (by [[tempering out]] their difference), then multiply both sides by {{sfrac|''k'' + 1|''k''}}, we have:<br />
<br />
<math>\displaystyle \left(\frac{k + 2}{k + 1}\right)\left(\frac{k + 1}{k}\right) = \left(\frac{k + 1}{k}\right)\left(\frac{k}{k - 1}\right) </math><br />
<br />
which simplifies to:<br />
<br />
<math>\displaystyle \frac{k + 2}{k} = \frac{k + 1}{k - 1} </math><br />
<br />
This means that if we temper: <math>{\rm S}k \cdot {\rm S}(k+1) = \frac{k/(k-1)}{(k+1)/k} \cdot \frac{(k+1)/k}{(k+2)/(k+1)} = \frac{k/(k-1)}{(k+2)/(k+1)}</math><br />
<br />
… then this equivalence is achieved. Note that there is little to no reason to not also temper S''k'' and {{nowrap|S(''k'' + 1)}} individually unless other considerations seem to force your hand.<br />
<br />
=== Short proof of the superparticularity of triangle-particulars ===<br />
<math>\displaystyle S(k)*S(k + 1) = \frac{\frac{k}{k - 1}}{\frac{k + 2}{k + 1}} = \frac{k(k + 1)}{(k - 1)(k + 2)} = \frac{k^2 + k}{k^2 + k - 2} </math><br />
<br />
Then notice that {{nowrap|''k''<sup>2</sup> + ''k''}} is always a multiple of 2, therefore the above always simplifies to a superparticular. Half of this superparticular is halfway between the corresponding square-particulars, and because of its composition it could therefore be reasoned that it'd likely be half as accurate as tempering either of the square-particulars individually, so these are "1/2-square-particulars" in a sense, and half of a square is a triangle, which is not a coincidence here because the numerators of all of these superparticular intervals/commas are [[triangular number]]s! (Hence the alternative name "[[triangle-particular]]".)<br />
<br />
=== Table of triangle-particulars ===<br />
For completeness, all the intervals of this form are included, because of their structural importance for JI, and for the possibility of (in)consistency of mappings when tempered for the above reason.<br />
<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 31-limit triangle-particulars<ref group="note">After 75, 76, 77, 78, streaks of four consecutive harmonics in the 23-limit become very sparse. The last few streaks are deeply related to the consistency and structure of [[311edo]], as [[311edo]] can be described as the unique 23-limit temperament that tempers all triangle-particulars from [[595/594]] up to [[21736/21735]]. It also tempers all the square-particulars composing those triangle-particulars with the exception of S169 and S170. It also maps the corresponding intervals of the 77-odd-limit consistently. 170/169 is the only place where the logic seems to "break" as it is mapped to 2 steps instead of 3 meaning the mapping of that superparticular is inconsistent.</ref><br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3<br />
| ([[3/1]])/([[2/1]])<br />
| [[3/2]]<br />
| 3<br />
|-<br />
| S3*S4<br />
| ([[3/2]])/([[5/4]])<br />
| [[6/5]]<br />
| 5<br />
|-<br />
| S4*S5<br />
| ([[4/3]])/([[6/5]])<br />
| [[10/9]]<br />
| 5<br />
|-<br />
| S5*S6<br />
| ([[5/4]])/([[7/6]])<br />
| [[15/14]]<br />
| 7<br />
|-<br />
| S6*S7<br />
| ([[6/5]])/([[8/7]])<br />
| [[21/20]]<br />
| 7<br />
|-<br />
| S7*S8 = S4/S6<br />
| ([[7/6]])([[9/8]])<br />
| [[28/27]]<br />
| 7<br />
|-<br />
| S8*S9 = S6<br />
| ([[8/7]])/([[10/9]])<br />
| [[36/35]]<br />
| 7<br />
|-<br />
| S9*S10<br />
| ([[9/8]])/([[11/10]])<br />
| [[45/44]]<br />
| 11<br />
|-<br />
| S10*S11<br />
| ([[10/9]])/([[12/11]])<br />
| [[55/54]]<br />
| 11<br />
|-<br />
| S11*S12<br />
| ([[11/10]])/([[13/12]])<br />
| [[66/65]]<br />
| 13<br />
|-<br />
| S12*S13<br />
| ([[12/11]])/([[14/13]])<br />
| [[78/77]]<br />
| 13<br />
|-<br />
| S13*S14<br />
| ([[13/12]])/([[15/14]])<br />
| [[91/90]]<br />
| 13<br />
|-<br />
| S14*S15<br />
| ([[14/13]])/([[16/15]])<br />
| [[105/104]]<br />
| 13<br />
|-<br />
| S15*S16<br />
| ([[15/14]])/([[17/16]])<br />
| [[120/119]]<br />
| 17<br />
|-<br />
| S16*S17<br />
| ([[16/15]])/([[18/17]])<br />
| [[136/135]]<br />
| 17<br />
|-<br />
| S17*S18<br />
| ([[17/16]])/([[19/18]])<br />
| [[153/152]]<br />
| 19<br />
|-<br />
| S18*S19<br />
| ([[18/17]])/([[20/19]])<br />
| [[171/170]]<br />
| 19<br />
|-<br />
| S19*S20<br />
| ([[19/18]])/([[21/20]])<br />
| [[190/189]]<br />
| 19<br />
|-<br />
| S20*S21<br />
| ([[20/19]])/([[22/21]])<br />
| [[210/209]]<br />
| 19<br />
|-<br />
| S21*S22<br />
| ([[21/20]])/([[23/22]])<br />
| [[231/230]]<br />
| 23<br />
|-<br />
| S22*S23<br />
| ([[22/21]])/([[24/23]])<br />
| [[253/252]]<br />
| 23<br />
|-<br />
| S23*S24<br />
| ([[23/22]])/([[25/24]])<br />
| [[276/275]]<br />
| 23<br />
|-<br />
| S24*S25<br />
| ([[24/23]])/([[26/25]])<br />
| [[300/299]]<br />
| 23<br />
|-<br />
| S25*S26 = S10/S12<br />
| ([[25/24]])/([[27/26]])<br />
| [[325/324]]<br />
| 13<br />
|-<br />
| S26*S27<br />
| ([[26/25]])/([[28/27]])<br />
| [[351/350]]<br />
| 13<br />
|-<br />
| S27*S28<br />
| ([[27/26]])/([[29/28]])<br />
| [[378/377]]<br />
| 29<br />
|-<br />
| S28*S29<br />
| ([[28/27]])/([[30/29]])<br />
| [[406/405]]<br />
| 29<br />
|-<br />
| S29*S30<br />
| ([[29/28]])/([[31/30]])<br />
| [[435/434]]<br />
| 31<br />
|-<br />
| S30*S31<br />
| ([[30/29]])/([[32/31]])<br />
| [[465/464]]<br />
| 31<br />
|-<br />
| S31*S32<br />
| ([[31/30]])/([[33/32]])<br />
| [[496/495]]<br />
| 31<br />
|-<br />
| S32*S33<br />
| ([[32/31]])/([[34/33]])<br />
| [[528/527]]<br />
| 31<br />
|-<br />
| S33*S34<br />
| ([[33/32]])/([[35/34]])<br />
| [[561/560]]<br />
| 17<br />
|-<br />
| S34*S35<br />
| ([[34/33]])/([[36/35]])<br />
| [[595/594]]<br />
| 17<br />
|-<br />
| S49*S50 = S35<br />
| ([[49/48]])/([[51/50]])<br />
| [[1225/1224]]<br />
| 17<br />
|-<br />
| S50*S51<br />
| ([[50/49]])/([[52/51]])<br />
| [[1275/1274]]<br />
| 17<br />
|-<br />
| S55*S56<br />
| ([[55/54]])/([[57/56]])<br />
| [[1540/1539]]<br />
| 19<br />
|-<br />
| S63*S64<br />
| ([[63/62]])/([[65/64]])<br />
| [[2016/2015]]<br />
| 31<br />
|-<br />
| S64*S65<br />
| ([[64/63]])/([[66/65]])<br />
| [[2080/2079]]<br />
| 13<br />
|-<br />
| S76*S77<br />
| ([[76/75]])/([[78/77]])<br />
| [[2926/2925]]<br />
| 19<br />
|-<br />
| S91*S92<br />
| ([[91/90]])/([[93/92]])<br />
| [[4186/4185]]<br />
| 31<br />
|-<br />
| S115*S116<br />
| ([[115/114]])/([[117/116]])<br />
| [[6670/6669]]<br />
| 29<br />
|-<br />
| S153*S154<br />
| ([[153/152]])/([[155/154]])<br />
| [[11781/11780]]<br />
| 31<br />
|-<br />
| S154*S155<br />
| ([[154/153]])/([[156/155]])<br />
| [[11935/11934]]<br />
| 31<br />
|-<br />
| S169*S170<br />
| ([[169/168]])/([[171/170]])<br />
| [[14365/14364]]<br />
| 19<br />
|-<br />
| S208*S209<br />
| ([[208/207]])/([[210/209]])<br />
| [[21736/21735]]<br />
| 19<br />
|-<br />
| S323*S324<br />
| ([[323/322]])/([[325/324]])<br />
| [[52326/52325]]<br />
| 23<br />
|-<br />
| S341*S342<br />
| ([[341/340]])/([[343/342]])<br />
| [[58311/58310]]<br />
| 31<br />
|-<br />
| S494*S495<br />
| ([[494/493]])/([[496/495]])<br />
| [[122265/122264]]<br />
| 31<br />
|-<br />
| S1519*S1520<br />
| ([[1519/1518]])/([[1521/1520]])<br />
| [[1154440/1154439]]<br />
| 31<br />
|}<br />
<br />
== {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} (1/''n''-square-particulars) ==<br />
=== Motivation ===<br />
1/''n''-square-particulars are a generalization of square- and 1/2-square-particulars to a comma/interval whose S-expression is can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}.<br />
<br />
In other words, each and every S-expression of a comma as a 1/''n''-square-particular corresponds exactly to expressing it as the ratio between two [[superparticular]] intervals, with ''n'' distance between them, where, for example, 10/9 and 11/10 are considered as having 1 distance between them, corresponding to (1/1-)square-particulars (in this case [[100/99|S10]]).<br />
<br />
These commas are important in a few ways:<br />
1. As a generalization of important special cases {{nowrap|''n'' {{=}} 0|''n'' {{=}} 1}}, and {{nowrap|''n'' {{=}} 2}}, (which are almost all superparticular; the only case where they aren't is that {{nowrap|''n'' {{=}} 3}} (1/3-square-particulars) are throdd-particular one third of the time, so this suggests these are efficient commas. A cursory look will show that many 1/n-square-particulars for small n are superparticular, and many more are the next best things (odd-particular, throdd-particular, quodd-particular, etc.) so this confirms them being a family of efficient commas.<br />
<br />
2. Because of being the ratio of two superparticular intervals, in higher-complexity cases they often correspond to small commas between large commas which we don't want to temper, for example {{nowrap|{{sfrac|[[81/80]]|[[91/90]]}} {{=}} S81 * S82 * … * S90}} {{nowrap|{{=}} [[729/728]]}} {{nowrap|{{=}} S27}}. They also often simplify in cases like these; note that a suggested shorthand is S81..90 for {{nowrap|S81 * S82 * … * S90}} and thus more generally S''a''..''b'' for {{nowrap|S''a'' * S(''a'' + 1) * … * S''b''}}.<br />
<br />
3. They often correspond to "nontrivial" equivalences that need to be dug up which are not obvious from their expression as a ratio of two superparticular intervals, for example, [[385/384|S33*S34*S35]], suggesting they are a goldmine for valuable tempering opportunities. <br />
<br />
4. Their expressions naturally make them implied by tempering consecutive square-particulars, so if you notice them present and that the individual square-particulars aren't tempered, if you want to extend your temperament and/or reduce its rank (tempering it down) and/or hope to make your temperament more efficient, you can try tempering the untempered square-particulars that a tempered 1/''n''-square-particular is composed of (although this is not always possible). There is also good theoretical motivation for wanting to do this, as the next section will discuss.<br />
<br />
5. They're relevant to understanding how much damage is present in a temperament's harmonic series representation, because they show how many superparticular intervals are either not distinguished or worse mapped inconsistently, bringing us finally to …<br />
<br />
6. They're relevant to understanding limitations of consistency (or more precisely, monotonicity) of any given temperament, as the next section will discuss.<br />
<br />
=== Significance/implications for consistency ===<br />
1/n-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} have implications for the [[consistency]] of the ({{nowrap|''k'' + ''n''}})-[[odd-limit]] when tempered. Specifically:<br />
<br />
If a temperament tempers a 1/''n''-square-particular of the form {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, it must temper all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', {{nowrap|S(''k'' + 1)}}, …, {{nowrap|S(''k'' + ''n'' − 1)}}. If it does not, it is ''necessarily'' inconsistent (more formally and weakly, not monotonic) in the ({{nowrap|''k'' + ''n''}})-odd-limit.<ref group="note">Note that this statement is a slight inaccuracy, because technically the tuning of the higher rank temperament corresponding to the lower rank temperament that tempers all of these commas is the unique ''and only'' (continuum of) tuning(s) for which this statement is false, but it's reasonable to simplify this technicality as this (continuum of) tuning(s) corresponds exactly and uniquely to tempering all the square-particulars we said were not tempered.</ref> A proof is as follows:<br />
<br />
Consider the following sequence of superparticular intervals, all of which in the ({{nowrap|''k'' + ''n''}})-odd-limit:<br />
<br />
<math>\displaystyle\frac{k + n}{k + n - 1}, \frac{k + n - 1}{k + n - 2}, …, \frac{k + 1}{k}, \frac{k}{k - 1}</math><br />
<br />
Because of tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, we require that {{nowrap|{{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} {{=}} {{sfrac|''k''|''k'' − 1}}}} consistently. Therefore, if any superparticular {{sfrac|''x''|''x'' − 1}} imbetween (meaning {{nowrap|''k'' + ''n'' &gt; ''x'' &gt; ''k''}}) is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true:<br />
<br />
<math>\displaystyle\begin{align}<br />
\operatorname{mapping}\left(\frac{k + n}{k + n - 1}\right) &> \operatorname{mapping}\left(\frac{x}{x - 1}\right) \\<br />
\operatorname{mapping}\left(\frac{k}{k - 1}\right) &< \operatorname{mapping}\left(\frac{x}{x - 1}\right)<br />
\end{align}</math><br />
<br />
Therefore any superparticular interval {{sfrac|''x''|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap|''k'' − 1}})-odd-limit.<br />
<br />
<br />
=== {{nowrap|S(''k'' − 1)*S''k''*S(''k'' + 1)}} (1/3-square-particulars) ===<br />
This section concerns commas of the form {{nowrap|S(''k'' − 1) * S''k'' * S(''k'' + 1) {{=}} {{sfrac|&nbsp;{{sfrac|''k'' − 1|''k'' − 2}}&nbsp;|&nbsp;{{sfrac|''k'' + 2|''k'' + 1}}&nbsp;}}}} which therefore do not (directly) involve the ''k''th harmonic. These are a special case of 1/''n''-square-particulars.<br />
<br />
==== Significance ====<br />
1. Two-thirds of all {{frac|1|3}}-square-particulars are superparticular and the other third are [[#Glossary|throdd-particular]], so these are efficient commas. (See also the [[#Proof of simplification of 1/3-square-particulars]].)<br />
<br />
2. They are often implied in a variety of ways by combinations of other commas discussed on this page.<br />
<br />
3. Their omission of direct relation to the ''k''th harmonic make them theoretically interesting and potentially useful. (The other type of comma on this page that does this is [[semiparticular]]s.)<br />
<br />
4. Square-particulars, {{frac|1|2}}-square-particulars (a.k.a. [[triangle-particular]]s), and {{frac|1|3}}-square-particulars are part of a more general sequence with interesting properties: [[1/n-square-particular|1/''n''-square-particular]]s.<br />
<br />
==== Proof of simplification of 1/3-square-particulars ====<br />
We can check the general algebraic expression of any 1/3-square-particular for any potential simplifications:<br />
<br />
<math>\displaystyle\begin{align}<br />
S(k-1) * S(k) * S(k+1) &= \left(\frac{\frac{k-1}{k-2}}{\frac{k}{k-1}}\right)\left(\frac{\frac{k}{k-1}}{\frac{k+1}{k}}\right)\left(\frac{\frac{k+1}{k}}{\frac{k+2}{k+1}}\right) \\<br />
&= \frac{\frac{k-1}{k-2}}{\frac{k+2}{k+1}} \\<br />
&= \frac{(k-1)(k+1)}{(k-2)(k+2)} \\<br />
&= \frac{k^2 - 1}{k^2 - 4}<br />
\end{align}</math><br />
<br />
If {{nowrap|''k'' {{=}} 3''n'' + 1}} then:<br />
<br />
<math>\displaystyle S(k-1) * Sk * S(k+1) = \frac{9n^2 + 6n}{9n^2 + 6n - 3} = \frac{3n^2 + 2n}{3n^2 + 2n - 1}</math><br />
<br />
if {{nowrap|''k'' {{=}} 3''n'' + 2}} then:<br />
<br />
<math>\displaystyle S(k-1) * Sk * S(k+1) = \frac{9n^2 + 12n + 3}{9n^2 + 12n} = \frac{3n^2 + 4n + 1}{3n^2 + 4n}</math><br />
<br />
if {{nowrap|''k'' {{=}} 3''n''}} then:<br />
<br />
<math>\displaystyle S(k-1) * Sk * S(k+1) = \frac{9n^2 - 1}{9n^2 - 4}</math><br />
<br />
In other words, what this shows is all {{frac|1|3}}-square-particulars of the form S(''k'' − 1) * S''k'' * S(''k'' + 1) are superparticular iff ''k'' is throdd (not a multiple of 3), and all {{frac|1|3}}-square-particulars of the form {{nowrap|S(3''k'' − 1) * S(3''k'') * S(3''k'' + 1)}} are throdd-particular with the numerator and denominator always being one less than a multiple of 3 (which is to say, commas of this form are throdd-particular iff ''k'' is threven and superparticular iff ''k'' is throdd).<br />
<br />
=== Tables of 1/n-square-particulars ===<br />
<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 41-limit {{frac|1|3}}-square-particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3*S4<br />
| ([[2/1]])/([[5/4]])<br />
| [[8/5]]<br />
| 5<br />
|-<br />
| S3*S4*S5<br />
| ([[3/2]])/([[6/5]])<br />
| [[5/4]]<br />
| 5<br />
|-<br />
| S4*S5*S6<br />
| ([[4/3]])/([[7/6]])<br />
| [[8/7]]<br />
| 7<br />
|-<br />
| S5*S6*S7<br />
| ([[5/4]])/([[8/7]])<br />
| [[35/32]]<br />
| 7<br />
|-<br />
| S6*S7*S8<br />
| ([[6/5]])/([[9/8]])<br />
| [[16/15]]<br />
| 5<br />
|-<br />
| S7*S8*S9<br />
| ([[7/6]])/([[10/9]])<br />
| [[21/20]]<br />
| 7<br />
|-<br />
| S8*S9*S10<br />
| ([[8/7]])/([[11/10]])<br />
| [[80/77]]<br />
| 11<br />
|-<br />
| S9*S10*S11<br />
| ([[9/8]])/([[12/11]])<br />
| [[33/32]]<br />
| 11<br />
|-<br />
| S10*S11*S12<br />
| ([[10/9]])/([[13/12]])<br />
| [[40/39]]<br />
| 13<br />
|-<br />
| S11*S12*S13<br />
| ([[11/10]])/([[14/13]])<br />
| [[143/140]]<br />
| 13<br />
|-<br />
| S12*S13*S14<br />
| ([[12/11]])/([[15/14]])<br />
| [[56/55]]<br />
| 11<br />
|-<br />
| S13*S14*S15<br />
| ([[13/12]])/([[16/15]])<br />
| [[65/64]]<br />
| 13<br />
|-<br />
| S14*S15*S16<br />
| ([[14/13]])/([[17/16]])<br />
| [[224/221]]<br />
| 17<br />
|-<br />
| S15*S16*S17<br />
| ([[15/14]])/([[18/17]])<br />
| [[85/84]]<br />
| 17<br />
|-<br />
| S16*S17*S18<br />
| ([[16/15]])/([[19/18]])<br />
| [[96/95]]<br />
| 19<br />
|-<br />
| S17*S18*S19<br />
| ([[17/16]])/([[20/19]])<br />
| [[323/320]]<br />
| 19<br />
|-<br />
| S18*S19*S20<br />
| ([[18/17]])/([[21/20]])<br />
| [[120/119]]<br />
| 17<br />
|-<br />
| S19*S20*S21<br />
| ([[19/18]])/([[22/21]])<br />
| [[133/132]]<br />
| 19<br />
|-<br />
| S20*S21*S22<br />
| ([[20/19]])/([[23/22]])<br />
| [[440/437]]<br />
| 23<br />
|-<br />
| S21*S22*S23<br />
| ([[21/20]])/([[24/23]])<br />
| [[161/160]]<br />
| 23<br />
|-<br />
| S22*S23*S24<br />
| ([[22/21]])/([[25/24]])<br />
| [[176/175]]<br />
| 11<br />
|-<br />
| S23*S24*S25<br />
| ([[23/22]])/([[26/25]])<br />
| [[575/572]]<br />
| 23<br />
|-<br />
| S24*S25*S26<br />
| ([[24/23]])/([[27/26]])<br />
| [[208/207]]<br />
| 23<br />
|-<br />
| S25*S26*S27<br />
| ([[25/24]])/([[28/27]])<br />
| [[225/224]]<br />
| 7<br />
|-<br />
| S26*S27*S28<br />
| ([[26/25]])/([[29/28]])<br />
| [[728/725]]<br />
| 29<br />
|-<br />
| S27*S28*S29<br />
| ([[27/26]])/([[30/29]])<br />
| [[261/260]]<br />
| 29<br />
|-<br />
| S28*S29*S30<br />
| ([[28/27]])/([[31/30]])<br />
| [[280/279]]<br />
| 31<br />
|-<br />
| S29*S30*S31<br />
| ([[29/28]])/([[32/31]])<br />
| [[899/896]]<br />
| 31<br />
|-<br />
| S30*S31*S32<br />
| ([[30/29]])/([[33/32]])<br />
| [[320/319]]<br />
| 29<br />
|-<br />
| S31*S32*S33<br />
| ([[31/30]])/([[34/33]])<br />
| [[341/340]]<br />
| 31<br />
|-<br />
| S32*S33*S34<br />
| ([[32/31]])/([[35/34]])<br />
| [[1088/1085]]<br />
| 31<br />
|-<br />
| S33*S34*S35<br />
| ([[33/32]])/([[36/35]])<br />
| [[385/384]]<br />
| 11<br />
|-<br />
| S34*S35*S36<br />
| ([[34/33]])/([[37/36]])<br />
| [[408/407]]<br />
| 37<br />
|-<br />
| S35*S36*S37<br />
| ([[35/34]])/([[38/37]])<br />
| [[1295/1292]]<br />
| 37<br />
|-<br />
| S36*S37*S38<br />
| ([[36/35]])/([[39/38]])<br />
| [[456/455]]<br />
| 19<br />
|-<br />
| S37*S38*S39<br />
| ([[37/36]])/([[40/39]])<br />
| [[481/480]]<br />
| 37<br />
|-<br />
| S38*S39*S40<br />
| ([[38/37]])/([[41/40]])<br />
| [[1520/1517]]<br />
| 41<br />
|-<br />
| S39*S40*S41<br />
| ([[39/38]])/([[42/41]])<br />
| [[533/532]]<br />
| 41<br />
|-<br />
| S42*S43*S44<br />
| ([[42/41]])/([[45/44]])<br />
| [[616/615]]<br />
| 41<br />
|-<br />
| S46*S47*S48<br />
| ([[46/45]])/([[49/48]])<br />
| [[736/735]]<br />
| 23<br />
|-<br />
| S49*S50*S51<br />
| ([[49/48]])/([[52/51]])<br />
| [[833/832]]<br />
| 17<br />
|-<br />
| S52*S53*S54<br />
| ([[52/51]])/([[55/54]])<br />
| [[936/935]]<br />
| 17<br />
|-<br />
| S55*S56*S57<br />
| ([[55/54]])/([[58/57]])<br />
| [[1045/1044]]<br />
| 29<br />
|-<br />
| S63*S64*S65<br />
| ([[63/62]])/([[66/65]])<br />
| [[1365/1364]]<br />
| 31<br />
|-<br />
| S66*S67*S68<br />
| ([[66/65]])/([[69/68]])<br />
| [[1496/1495]]<br />
| 23<br />
|-<br />
| S75*S76*S77<br />
| ([[75/74]])/([[78/77]])<br />
| [[1925/1924]]<br />
| 37<br />
|-<br />
| S78*S79*S80<br />
| ([[78/77]])/([[81/80]])<br />
| [[2080/2079]]<br />
| 13<br />
|-<br />
| S82*S83*S84<br />
| ([[82/81]])/([[85/84]])<br />
| [[2296/2295]]<br />
| 41<br />
|-<br />
| S85*S86*S87<br />
| ([[85/84]])/([[88/87]])<br />
| [[2465/2464]]<br />
| 29<br />
|-<br />
| S88*S89*S90<br />
| ([[88/87]])/([[91/90]])<br />
| [[2640/2639]]<br />
| 29<br />
|-<br />
| S93*S94*S95<br />
| ([[93/92]])/([[96/95]])<br />
| [[2945/2944]]<br />
| 31<br />
|-<br />
| S96*S97*S98<br />
| ([[96/95]])/([[99/98]])<br />
| [[3136/3135]]<br />
| 19<br />
|-<br />
| S112*S113*S114<br />
| ([[112/111]])/([[115/114]])<br />
| [[4256/4255]]<br />
| 37<br />
|-<br />
| S117*S118*S119<br />
| ([[117/116]])/([[120/119]])<br />
| [[4641/4640]]<br />
| 29<br />
|-<br />
| S121*S122*S123<br />
| ([[121/120]])/([[124/123]])<br />
| [[4961/4960]]<br />
| 41<br />
|-<br />
| S133*S134*S135<br />
| ([[133/132]])/([[136/135]])<br />
| [[5985/5984]]<br />
| 19<br />
|-<br />
| S145*S146*S147<br />
| ([[145/144]])/([[148/147]])<br />
| [[7105/7104]]<br />
| 37<br />
|-<br />
| S153*S154*S155<br />
| ([[153/152]])/([[156/155]])<br />
| [[7905/7904]]<br />
| 31<br />
|-<br />
| S162*S163*S164<br />
| ([[162/161]])/([[165/164]])<br />
| [[8856/8855]]<br />
| 41<br />
|-<br />
| S187*S188*S189<br />
| ([[187/186]])/([[190/189]])<br />
| [[11781/11780]]<br />
| 31<br />
|-<br />
| S205*S206*S207<br />
| ([[205/204]])/([[208/207]])<br />
| [[14145/14144]]<br />
| 41<br />
|-<br />
| S222*S223*S224<br />
| ([[222/221]])/([[225/224]])<br />
| [[16576/16575]]<br />
| 37<br />
|-<br />
| S243*S244*S245<br />
| ([[243/242]])/([[246/245]])<br />
| [[19845/19844]]<br />
| 41<br />
|-<br />
| S253*S254*S255<br />
| ([[253/252]])/([[256/255]])<br />
| [[21505/21504]]<br />
| 23<br />
|-<br />
| S273*S274*S275<br />
| ([[273/272]])/([[276/275]])<br />
| [[25025/25024]]<br />
| 23<br />
|-<br />
| S286*S287*S288<br />
| ([[286/285]])/([[289/288]])<br />
| [[27456/27455]]<br />
| 19<br />
|-<br />
| S287*S288*S289<br />
| ([[287/286]])/([[290/289]])<br />
| [[82943/82940]]<br />
| 41<br />
|-<br />
| S297*S298*S299<br />
| ([[297/296]])/([[300/299]])<br />
| [[29601/29600]]<br />
| 37<br />
|-<br />
| S320*S321*S322<br />
| ([[320/319]])/([[323/322]])<br />
| [[103040/103037]]<br />
| 29<br />
|-<br />
| S361*S362*S363<br />
| ([[361/360]])/([[364/363]])<br />
| [[43681/43680]]<br />
| 19<br />
|-<br />
| S375*S376*S377<br />
| ([[375/374]])/([[378/377]])<br />
| [[47125/47124]]<br />
| 29<br />
|}<br />
<br />
{| class="wikitable center-all\<br />
|+ style="font-size: 105%;" | 23-limit {{frac|1|4}}-square particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3*S4*S5<br />
| ([[2/1]])/([[6/5]])<br />
| [[5/3]]<br />
| 5<br />
|-<br />
| S3*S4*S5*S6<br />
| ([[3/2]])/([[7/6]])<br />
| [[9/7]]<br />
| 7<br />
|-<br />
| S4*S5*S6*S7<br />
| ([[4/3]])/([[8/7]])<br />
| [[7/6]]<br />
| 7<br />
|-<br />
| S5*S6*S7*S8<br />
| ([[5/4]])/([[9/8]])<br />
| [[10/9]]<br />
| 5<br />
|-<br />
| S6*S7*S8*S9<br />
| ([[6/5]])/([[10/9]])<br />
| [[27/25]]<br />
| 5<br />
|-<br />
| S7*S8*S9*S10<br />
| ([[7/6]])/([[11/10]])<br />
| [[35/33]]<br />
| 11<br />
|-<br />
| S8*S9*S10*S11<br />
| ([[8/7]])/([[12/11]])<br />
| [[22/21]]<br />
| 11<br />
|-<br />
| S9*S10*S11*S12<br />
| ([[9/8]])/([[13/12]])<br />
| [[27/26]]<br />
| 13<br />
|-<br />
| S10*S11*S12*S13<br />
| ([[10/9]])/([[14/13]])<br />
| [[65/63]]<br />
| 13<br />
|-<br />
| S11*S12*S13*S14<br />
| ([[11/10]])/([[15/14]])<br />
| [[77/75]]<br />
| 11<br />
|-<br />
| S12*S13*S14*S15<br />
| ([[12/11]])/([[16/15]])<br />
| [[45/44]]<br />
| 11<br />
|-<br />
| S13*S14*S15*S16<br />
| ([[13/12]])/([[17/16]])<br />
| [[52/51]]<br />
| 17<br />
|-<br />
| S14*S15*S16*S17<br />
| ([[14/13]])/([[18/17]])<br />
| [[119/117]]<br />
| 17<br />
|-<br />
| S15*S16*S17*S18<br />
| ([[15/14]])/([[19/18]])<br />
| [[135/133]]<br />
| 19<br />
|-<br />
| S16*S17*S18*S19<br />
| ([[16/15]])/([[20/19]])<br />
| [[76/75]]<br />
| 19<br />
|-<br />
| S17*S18*S19*S20<br />
| ([[17/16]])/([[21/20]])<br />
| [[85/84]]<br />
| 17<br />
|-<br />
| S18*S19*S20*S21<br />
| ([[18/17]])/([[22/21]])<br />
| [[189/187]]<br />
| 17<br />
|-<br />
| S19*S20*S21*S22<br />
| ([[19/18]])/([[23/22]])<br />
| [[209/207]]<br />
| 23<br />
|-<br />
| S20*S21*S22*S23<br />
| ([[20/19]])/([[24/23]])<br />
| [[115/114]]<br />
| 23<br />
|-<br />
| S21*S22*S23*S24<br />
| ([[21/20]])/([[25/24]])<br />
| [[126/125]]<br />
| 7<br />
|-<br />
| S22*S23*S24*S25<br />
| ([[22/21]])/([[26/25]])<br />
| [[275/273]]<br />
| 13<br />
|-<br />
| S23*S24*S25*S26<br />
| ([[23/22]])/([[27/26]])<br />
| [[299/297]]<br />
| 23<br />
|-<br />
| S24*S25*S26*S27<br />
| ([[24/23]])/([[28/27]])<br />
| [[162/161]]<br />
| 23<br />
|-<br />
| S35*S36*S37*S38<br />
| ([[35/34]])/([[39/38]])<br />
| [[665/663]]<br />
| 19<br />
|-<br />
| S36*S37*S38*S39<br />
| ([[36/35]])/([[40/39]])<br />
| [[351/350]]<br />
| 13<br />
|-<br />
| S45*S46*S47*S48<br />
| ([[45/44]])/([[49/48]])<br />
| [[540/539]]<br />
| 11<br />
|-<br />
| S46*S47*S48*S49<br />
| ([[46/45]])/([[50/49]])<br />
| [[1127/1125]]<br />
| 23<br />
|-<br />
| S51*S52*S53*S54<br />
| ([[51/50]])/([[55/54]])<br />
| [[1377/1375]]<br />
| 17<br />
|-<br />
| S52*S53*S54*S55<br />
| ([[52/51]])/([[56/55]])<br />
| [[715/714]]<br />
| 17<br />
|-<br />
| S65*S66*S67*S68<br />
| ([[65/64]])/([[69/68]])<br />
| [[1105/1104]]<br />
| 23<br />
|-<br />
| S66*S67*S68*S69<br />
| ([[66/65]])/([[70/69]])<br />
| [[2277/2275]]<br />
| 23<br />
|-<br />
| S77*S78*S79*S80<br />
| ([[77/76]])/([[81/80]])<br />
| [[1540/1539]]<br />
| 19<br />
|-<br />
| S81*S82*S83*S84<br />
| ([[81/80]])/([[85/84]])<br />
| [[1701/1700]]<br />
| 17<br />
|-<br />
| S92*S93*S94*S95<br />
| ([[92/91]])/([[96/95]])<br />
| [[2185/2184]]<br />
| 23<br />
|-<br />
| S96*S97*S98*S99<br />
| <span style="font-size:0.94em">([[96/95]])/([[100/99]])</span><br />
| [[2376/2375]]<br />
| 19<br />
|-<br />
| <span style="font-size: 0.79em;">S221*S222*S223*S224</span><br />
| <span style="font-size: 0.79em;">([[221/220]])/([[225/224]])</span><br />
| [[12376/12375]]<br />
| 17<br />
|}<br />
<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 23-limit {{frac|1|5}}-square particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3*S4*S5*S6<br />
| ([[2/1]])/([[7/6]])<br />
| [[12/7]]<br />
| 7<br />
|-<br />
| S3*S4*S5*S6*S7<br />
| ([[3/2]])/([[8/7]])<br />
| [[21/16]]<br />
| 7<br />
|-<br />
| S4*S5*S6*S7*S8<br />
| ([[4/3]])/([[9/8]])<br />
| [[32/27]]<br />
| 3<br />
|-<br />
| S5*S6*S7*S8*S9<br />
| ([[5/4]])/([[10/9]])<br />
| [[9/8]]<br />
| 3<br />
|-<br />
| S6*S7*S8*S9*S10<br />
| ([[6/5]])/([[11/10]])<br />
| [[12/11]]<br />
| 11<br />
|-<br />
| S7*S8*S9*S10*S11<br />
| ([[7/6]])/([[12/11]])<br />
| [[77/72]]<br />
| 11<br />
|-<br />
| S8*S9*S10*S11*S12<br />
| ([[8/7]])/([[13/12]])<br />
| [[96/91]]<br />
| 13<br />
|-<br />
| S9*S10*S11*S12*S13<br />
| ([[9/8]])/([[14/13]])<br />
| [[117/112]]<br />
| 13<br />
|-<br />
| S10*S11*S12*S13*S14<br />
| ([[10/9]])/([[15/14]])<br />
| [[28/27]]<br />
| 7<br />
|-<br />
| S11*S12*S13*S14*S15<br />
| ([[11/10]])/([[16/15]])<br />
| [[33/32]]<br />
| 11<br />
|-<br />
| S12*S13*S14*S15*S16<br />
| ([[12/11]])/([[17/16]])<br />
| [[192/187]]<br />
| 17<br />
|-<br />
| S13*S14*S15*S16*S17<br />
| ([[13/12]])/([[18/17]])<br />
| [[221/216]]<br />
| 17<br />
|-<br />
| S14*S15*S16*S17*S18<br />
| ([[14/13]])/([[19/18]])<br />
| [[252/247]]<br />
| 19<br />
|-<br />
| S15*S16*S17*S18*S19<br />
| ([[15/14]])/([[20/19]])<br />
| [[57/56]]<br />
| 19<br />
|-<br />
| S16*S17*S18*S19*S20<br />
| ([[16/15]])/([[21/20]])<br />
| [[64/63]]<br />
| 7<br />
|-<br />
| S17*S18*S19*S20*S21<br />
| ([[17/16]])/([[22/21]])<br />
| [[357/352]]<br />
| 17<br />
|-<br />
| S18*S19*S20*S21*S22<br />
| ([[18/17]])/([[23/22]])<br />
| [[396/391]]<br />
| 23<br />
|-<br />
| S19*S20*S21*S22*S23<br />
| ([[19/18]])/([[24/23]])<br />
| [[437/432]]<br />
| 23<br />
|-<br />
| S20*S21*S22*S23*S24<br />
| ([[20/19]])/([[25/24]])<br />
| [[96/95]]<br />
| 19<br />
|-<br />
| S21*S22*S23*S24*S25<br />
| ([[21/20]])/([[26/25]])<br />
| [[105/104]]<br />
| 13<br />
|-<br />
| S22*S23*S24*S25*S26<br />
| ([[22/21]])/([[27/26]])<br />
| [[572/567]]<br />
| 13<br />
|-<br />
| S23*S24*S25*S26*S27<br />
| ([[23/22]])/([[28/27]])<br />
| [[621/616]]<br />
| 23<br />
|-<br />
| S28*S29*S30*S31*S32<br />
| ([[28/27]])/([[33/32]])<br />
| [[896/891]]<br />
| 11<br />
|-<br />
| S34*S35*S36*S37*S38<br />
| ([[34/33]])/([[39/38]])<br />
| [[1292/1287]]<br />
| 19<br />
|-<br />
| S35*S36*S37*S38*S39<br />
| ([[35/34]])/([[40/39]])<br />
| [[273/272]]<br />
| 17<br />
|-<br />
| S40*S41*S42*S43*S44<br />
| ([[40/39]])/([[45/44]])<br />
| [[352/351]]<br />
| 13<br />
|-<br />
| S45*S46*S47*S48*S49<br />
| ([[45/44]])/([[50/49]])<br />
| [[441/440]]<br />
| 11<br />
|-<br />
| S46*S47*S48*S49*S50<br />
| ([[46/45]])/([[51/50]])<br />
| [[460/459]]<br />
| 23<br />
|-<br />
| S50*S51*S52*S53*S54<br />
| ([[50/49]])/([[55/54]])<br />
| [[540/539]]<br />
| 11<br />
|-<br />
| S51*S52*S53*S54*S55<br />
| ([[51/50]])/([[56/55]])<br />
| [[561/560]]<br />
| 17<br />
|-<br />
| S52*S53*S54*S55*S56<br />
| ([[52/51]])/([[57/56]])<br />
| [[2912/2907]]<br />
| 19<br />
|-<br />
| S64*S65*S66*S67*S68<br />
| ([[64/63]])/([[69/68]])<br />
| [[4352/4347]]<br />
| 23<br />
|-<br />
| S65*S66*S67*S68*S69<br />
| ([[65/64]])/([[70/69]])<br />
| [[897/896]]<br />
| 23<br />
|-<br />
| S76*S77*S78*S79*S80<br />
| ([[76/75]])/([[81/80]])<br />
| [[1216/1215]]<br />
| 19<br />
|-<br />
| S91*S92*S93*S94*S95<br />
| ([[91/90]])/([[96/95]])<br />
| [[1729/1728]]<br />
| 19<br />
|-<br />
| <span style="font-size: 0.79em;">S100*S101*S102*S103*S104</span><br />
| <span style="font-size: 0.83em;">([[100/99]])/([[105/104]])</span><br />
| [[2080/2079]]<br />
| 13<br />
|-<br />
| <span style="font-size: 0.79em;">S115*S116*S117*S118*S119</span><br />
| <span style="font-size: 0.79em;">([[115/114]])/([[120/119]])</span><br />
| [[2737/2736]]<br />
| 23<br />
|-<br />
| <span style="font-size: 0.79em;">S121*S122*S123*S124*S125</span><br />
| <span style="font-size: 0.79em;">([[121/120]])/([[126/125]])</span><br />
| [[3025/3024]]<br />
| 11<br />
|-<br />
| <span style="font-size: 0.79em;">S171*S172*S173*S174*S175</span><br />
| <span style="font-size: 0.79em;">([[171/170]])/([[176/175]])</span><br />
| [[5985/5984]]<br />
| 19<br />
|}<br />
<br />
== {{nowrap|S''k''/S(''k'' + 1)}} (ultraparticulars) ==<br />
=== Motivational example ===<br />
Often it is desirable to make consecutive [[superparticular]] intervals equidistant. This has a number of nice consequences, many of which not explained here—see the motivation section for each infinite family of commas defined on this page.<br />
<br />
For example, if you want 6/5 equidistant from 5/4 and 7/6, you must equate {{nowrap|{{sfrac|[[5/4]]|[[6/5]]}} {{=}} [[25/24]]}} {{nowrap|{{=}} S5}} with {{nowrap|{{sfrac|[[6/5]]|[[7/6]]}} {{=}} [[36/35]]}} {{nowrap|{{=}} S6}}, hence tempering {{nowrap|{{sfrac|S5|S6}} {{=}} {{sfrac|25/24|36/35}}}} {{nowrap|{{=}} [[875/864]]}}, but it's actually often not necessary to know the specific numbers, often familiarizing yourself with and understanding the "S''k''" notation will give you a lot of insight, as we'll see.<br />
<br />
Back to our example: we know that {{nowrap|S5 ~ S6}} (because we're tempering S5/S6); from this we can deduce that the intervals must be arranged like this: {{nowrap|7/6 &larr; S5~S6 &rarr; 6/5 &larr; S5~S6 &rarr; 5/4}}.<br />
<br />
From this you can deduce that {{nowrap|([[6/5]])<sup>3</sup> &rarr; [[7/4]]}}, because you can lower one of the 6/5's to [[7/6]] (lowering it by S6) and raise another of the 6/5's to [[5/4]] (raising it by S5). Then because we've tempered S5 and S6 together, we've lowered and raised by the same amount, so the result of {{nowrap|7/6 * 6/5 * 5/4 {{=}} 7/4}} must be the same as the result of {{nowrap|6/5 * 6/5 * 6/5}} in this temperament.<br />
<br />
Familiarize yourself with the structure of this argument, as [[S-expression/Advanced results#Mathematical derivations|it generalizes to arbitrary S''k'']]; the algebraic proof is tedious, but the intuition is the same:<br />
<br />
<math>\displaystyle \frac{k+2}{k+1} \leftarrow S(k+1)~Sk \rightarrow \frac{k+1}{k} \leftarrow S(k+1)~Sk \rightarrow \frac{k}{k-1}</math><br />
<br />
… implies that three {{sfrac|''k'' + 1|''k''}} give {{sfrac|''k'' + 2|''k'' − 1}} iff we temper {{sfrac|S''k''|S(''k'' + 1)}}.&nbsp;{{qed}}<br />
<br />
=== Significance ===<br />
1. Tempering any two consecutive square-particulars S''k'' and S({{nowrap|''k'' + 1}}) will naturally imply tempering the ultraparticular between them, {{sfrac|S''k''|S(''k'' + 1)}}, meaning they are very common implicit commas.<br />
<br />
2. Tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness.<br />
<br />
3. Tempering the ultraparticular S''k''/S({{nowrap|''k'' + 1}}) along with either the corresponding 1/2-square-particular {{nowrap|S''k'' * S(''k'' + 1)}} or one of the two corresponding lopsided commas {{nowrap|S''k''<sup>2</sup> * S(''k'' + 1)}} or {{nowrap|S''k'' * S(''k'' + 1)<sup>2</sup>}} implies tempering both of S''k'' and S({{nowrap|''k'' + 1}}) individually, and vice versa, so that there is a total of ''five'' equivalences—corresponding to ''five'' infinite families of commas—for every such S''k'' and S({{nowrap|''k''+1}}). This only gets better if you temper a third consecutive square-particular. This is an abundance of "at a glance" essential tempering information that is fully general so only needs to be learned once, and is the motivation of the use of '''S-expressions'''. (For example, {{nowrap|{S16, S17} &rarr; {{(}}S16 * S17, S16/S17, S16<sup>2</sup> * S17, S16 * S17<sup>2</sup>{{)}} }}, and any of the two commas in the latter set imply all the other commas too.)<br />
<br />
=== Table of ultraparticulars ===<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Cube Relation<br />
! Comma<br />
! Cents<br />
|-<br />
| S2/S3 = ([[4/3]])/([[9/8]])<br />
| ([[4/1]])/([[3/2]])<sup>3</sup><br />
| [[32/27]]<br />
| 294.135<br />
|-<br />
| S3/S4 = ([[9/8]])/([[16/15]])<br />
| ([[5/2]])/([[4/3]])<sup>3</sup><br />
| [[135/128]]<br />
| 92.179<br />
|-<br />
| S4/S5 = ([[16/15]])/([[25/24]])<br />
| ([[2/1]])/([[5/4]])<sup>3</sup><br />
| [[128/125]]<br />
| 41.059<br />
|-<br />
| S5/S6 = ([[25/24]])/([[36/35]])<br />
| ([[7/4]])/([[6/5]])<sup>3</sup><br />
| [[875/864]]<br />
| 21.902<br />
|-<br />
| S6/S7 = ([[36/35]])/([[49/48]])<br />
| ([[8/5]])/([[7/6]])<sup>3</sup><br />
| [[1728/1715]]<br />
| 13.074<br />
|-<br />
| S7/S8 = ([[49/48]])/([[64/63]])<br />
| ([[3/2]])/([[8/7]])<sup>3</sup><br />
| [[1029/1024]]<br />
| 8.433<br />
|-<br />
| S8/S9 = ([[64/63]])/([[81/80]])<br />
| ([[10/7]])/([[9/8]])<sup>3</sup><br />
| [[5120/5103]]<br />
| 5.758<br />
|-<br />
| S9/S10 = ([[81/80]])/([[100/99]])<br />
| ([[11/8]])/([[10/9]])<sup>3</sup><br />
| [[8019/8000]]<br />
| 4.107<br />
|-<br />
| S10/S11 = ([[100/99]])/([[121/120]])<br />
| ([[4/3]])/([[11/10]])<sup>3</sup><br />
| [[4000/3993]]<br />
| 3.032<br />
|-<br />
| S11/S12 = ([[121/120]])/([[144/143]])<br />
| ([[13/10]])/([[12/11]])<sup>3</sup><br />
| [[17303/17280]]<br />
| 2.303<br />
|-<br />
| S12/S13 = ([[144/143]])/([[169/168]])<br />
| ([[14/11]])/([[13/12]])<sup>3</sup><br />
| [[24192/24167]]<br />
| 1.79<br />
|-<br />
| S13/S14 = ([[169/168]])/([[196/195]])<br />
| ([[5/4]])/([[14/13]])<sup>3</sup><br />
| [[10985/10976]]<br />
| 1.419<br />
|-<br />
| S14/S15 = ([[196/195]])/([[225/224]])<br />
| ([[16/13]])/([[15/14]])<sup>3</sup><br />
| [[43904/43875]]<br />
| 1.144<br />
|-<br />
| S15/S16 = ([[225/224]])/([[256/255]])<br />
| ([[17/14]])/([[16/15]])<sup>3</sup><br />
| [[57375/57344]]<br />
| 0.936<br />
|-<br />
| S16/S17 = ([[256/255]])/([[289/288]])<br />
| ([[6/5]])/([[17/16]])<sup>3</sup><br />
| [[24576/24565]]<br />
| 0.775<br />
|-<br />
| S17/S18 = ([[289/288]])/([[324/323]])<br />
| ([[19/16]])/([[18/17]])<sup>3</sup><br />
| [[93347/93312]]<br />
| 0.649<br />
|-<br />
| S18/S19 = ([[324/323]])/([[361/360]])<br />
| ([[20/17]])/([[19/18]])<sup>3</sup><br />
| [[116640/116603]]<br />
| 0.549<br />
|-<br />
| S19/S20 = ([[361/360]])/([[400/399]])<br />
| ([[7/6]])/([[20/19]])<sup>3</sup><br />
| [[48013/48000]]<br />
| 0.469<br />
|-<br />
| S20/S21 = ([[400/399]])/([[441/440]])<br />
| ([[22/19]])/([[21/20]])<sup>3</sup><br />
| [[176000/175959]]<br />
| 0.403<br />
|-<br />
| S21/S22 = ([[441/440]])/([[484/483]])<br />
| ([[23/20]])/([[22/21]])<sup>3</sup><br />
| [[213003/212960]]<br />
| 0.35<br />
|-<br />
| S22/S23 = ([[484/483]])/([[529/528]])<br />
| ([[8/7]])/([[23/22]])<sup>3</sup><br />
| [[85184/85169]]<br />
| 0.305<br />
|-<br />
| S23/S24 = ([[529/528]])/([[576/575]])<br />
| ([[25/22]])/([[24/23]])<sup>3</sup><br />
| [[304175/304128]]<br />
| 0.268<br />
|-<br />
| S24/S25 = ([[576/575]])/([[625/624]])<br />
| ([[26/23]])/([[25/24]])<sup>3</sup><br />
| [[359424/359375]]<br />
| 0.236<br />
|-<br />
| S25/S26 = ([[625/624]])/([[676/675]])<br />
| ([[9/8]])/([[26/25]])<sup>3</sup><br />
| [[140625/140608]]<br />
| 0.209<br />
|-<br />
| S26/S27 = ([[676/675]])/([[729/728]])<br />
| ([[28/25]])/([[27/26]])<sup>3</sup><br />
| [[492128/492075]]<br />
| 0.186<br />
|-<br />
| S27/S28 = ([[729/728]])/([[784/783]])<br />
| ([[29/26]])/([[28/27]])<sup>3</sup><br />
| [[570807/570752]]<br />
| 0.167<br />
|-<br />
| S28/S29 = ([[784/783]])/([[841/840]])<br />
| ([[10/9]])/([[29/28]])<sup>3</sup><br />
| [[219520/219501]]<br />
| 0.15<br />
|-<br />
| S31/S32 = ([[961/960]])/([[1024/1023]])<br />
| ([[11/10]])/([[32/31]])<sup>3</sup><br />
| [[327701/327680]]<br />
| 0.111<br />
|-<br />
| S33/S34 = ([[1089/1088]])/([[1156/1155]])<br />
| ([[35/32]])/([[34/33]])<sup>3</sup><br />
| [[1257795/1257728]]<br />
| 0.092<br />
|-<br />
| S34/S35 = ([[1156/1155]])/([[1225/1224]])<br />
| ([[12/11]])/([[35/34]])<sup>3</sup><br />
| [[471648/471625]]<br />
| 0.084<br />
|-<br />
| S37/S38 = ([[1369/1368]])/([[1444/1443]])<br />
| ([[13/12]])/([[38/37]])<sup>3</sup><br />
| [[658489/658464]]<br />
| 0.066<br />
|-<br />
| S40/S41 = ([[1600/1599]])/([[1681/1680]])<br />
| ([[14/13]])/([[41/40]])<sup>3</sup><br />
| [[896000/895973]]<br />
| 0.052<br />
|-<br />
| S43/S44 = ([[1849/1848]])/([[1936/1935]])<br />
| ([[15/14]])/([[44/43]])<sup>3</sup><br />
| [[1192605/1192576]]<br />
| 0.042<br />
|-<br />
| S46/S47 = ([[2116/2115]])/([[2209/2208]])<br />
| ([[16/15]])/([[47/46]])<sup>3</sup><br />
| [[1557376/1557345]]<br />
| 0.034<br />
|-<br />
| S49/S50 = ([[2401/2400]])/([[2500/2499]])<br />
| ([[17/16]])/([[50/49]])<sup>3</sup><br />
| [[2000033/2000000]]<br />
| 0.029<br />
|-<br />
| S50/S51 = ([[2500/2499]])/([[2601/2600]])<br />
| ([[52/49]])/([[51/50]])<sup>3</sup><br />
| [[6500000/6499899]]<br />
| 0.027<br />
|-<br />
| S55/S56 = ([[3025/3024]])/([[3136/3135]])<br />
| ([[19/18]])/([[56/55]])<sup>3</sup><br />
| [[3161125/3161088]]<br />
| 0.02<br />
|-<br />
| S64/S65 = ([[4096/4095]])/([[4225/4224]])<br />
| ([[22/21]])/([[65/64]])<sup>3</sup><br />
| [[5767168/5767125]]<br />
| 0.013<br />
|}<br />
<br />
The above table is a list of all [[23-limit]] ultraparticulars corresponding to S''k'' with ''k'' < 77, plus ultraparticulars corresponding to dividing a [[superparticular interval]] into three equal parts up to [[17/16]] (or up to [[19/18]] but excluding 18/17 because of it requiring a large prime, 53), plus S27/S28 so that we have all ultraparticulars up to S28/S29 listed rather than up to S26/S27.<br />
<br />
This table has been expanded following every ultraparticular from S2/S3 to S16/S17 having its own page. Note that ultraparticulars are, in general, extremely precise commas so that usually one wouldn't consider tempering them directly rather than through tempering the square-particulars S''k'' which they are composed of. As an example of this, notice that [[4000/3993|S10/S11]] is the largest ultraparticular categorised as an [[unnoticeable comma]], which means not unnoticeable in the absolute sense but rather in the sense of being smaller than the melodic just-noticeable difference, despite only dividing a superparticular as simple and unremarkable as [[4/3]]. For this reason, a [[cent]]s column has been included to aid an appreciation of their precision. The cent value of a [[semiparticular]] is roughly double that of any of the two ultraparticulars it is composed of; this becomes more true the higher you go.<br />
<br />
Note also from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably [[consistent]]) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3''k'' + 1)/S(3''k''+ 2) for a positive integer ''k'', because then the superparticular can be expressed as:<br />
<br />
<math>\frac{(3k + 3)/3k}{((3k + 2)(3k + 1))^3} = \frac{(k + 1)/k}{((3k + 2)(3k + 1))^3}</math><br />
<br />
Also note that if you temper multiple adjacent ultraparticulars, you sometimes are not required to use those ultraparticulars in the comma list as description of (the bulk of) the tempering may be possible through [[#Sk/S(k + 2) (semiparticulars)|semiparticulars]], discussed next.<br />
<br />
== {{nowrap|S''k''/S(''k'' + 2)}} (semiparticulars) ==<br />
=== Motivational examples ===<br />
If we want to halve one JI interval into two of another JI interval, there is a powerful and elegant pattern for doing so:<br />
* [[4/3]] is approximately half of [[9/5]]<br />
* [[9/7]] is approximately half of [[5/3]] (=&nbsp;10/6)<br />
* [[5/4]] is approximately half of [[11/7]]<br />
* [[11/9]] is approximately half of [[3/2]] (=&nbsp;12/8)<br />
* [[6/5]] is approximately half of [[13/9]]<br />
* [[13/11]] is approximately half of [[7/5]] (=&nbsp;14/10)<br />
* [[7/6]] is approximately half of [[15/11]]<br />
* [[15/13]] is approximately half of [[4/3]] (=&nbsp;16/12)<br />
* [[8/7]] is approximately half of [[17/13]]<br />
* [[17/15]] is approximately half of [[9/7]] (=&nbsp;18/14)<br />
* [[9/8]] is approximately half of [[19/15]]<br />
* [[19/17]] is approximately half of [[5/4]] (=&nbsp;20/16)<br />
<br />
These properties show a pattern: take some arbitrary [[#Glossary|quodd-particular]] (''k'' + 4)/''k''; observe that we can split it into (''k'' + 4)/(''k'' + 2) * (''k'' + 2)/''k''.<br />
<br />
Now observe that (''k'' + 2)/''k'' > (''k'' + 3)/(''k'' + 1) > (''k'' + 4)/(''k'' + 2); in fact, it can be shown fairly easily that (''k'' + 3)/(''k'' + 1) is the [[mediant]] of (''k'' + 4)/(''k'' + 2) and (''k'' + 2)/''k''.<br />
<br />
It turns out that making this mediant — (''k'' + 3)/(''k'' + 1) — equal to half of (''k'' + 4)/''k'' is equivalent to tempering S(''k'' + 1)/S(''k'' + 3).<br />
<br />
=== Significance ===<br />
1. For differences between square-particulars of the form S''k''/S(''k'' + 2), the resulting comma is either [[superparticular]] or [[#Glossary|odd-particular]], so these are efficient commas. (This terminology also suggests [[#Glossary|throdd-particular]] and [[#Glossary|quodd-particular]] as generalizations.)<br />
<br />
2. Tempering any two consecutive [[ultraparticular]]s implies tempering a semiparticular, so from two adjacent "thirding" equivalences you get a "halving" equivalence for free!<br />
<br />
3. Tempering any two nearly-consecutive square-particulars (S''k'' and S(''k'' + 2)) implies tempering a semiparticular; this is generally much more ideal than tempering two consecutive S''k'' because it is a lot lower damage (see [[lopsided comma]]s for (relatively) large commas implied by this higher-damage strategy).<br />
<br />
4. On top of the halving equivalence, there is a number of subtler structural implications, [[discussed below, that may be desirable to the temperament designer.<br />
<br />
=== Meaning ===<br />
: '''Reader notes:''' In the below, we use S(''k'' - 1)/S(''k'' + 1) for symmetry around ''k'' to make the math visually simpler, but keep in mind it's equivalent to using an offset ''k''.<br />
: '''Also:''' keep in mind that ''k'' - ''a'' (for positive ''a'') is smaller than ''k'', so that ''k''/(''k'' - ''a'') > (''k'' + ''a'')/''k'' (because the former appears earlier in the harmonic series & is thus larger); this is an important and useful intuition to learn.<br />
<br />
Tempering S(''k'' - 1)/S(''k'' + 1) implies that (''k'' + 2)/(''k'' - 2) is divisible exactly into two halves of (''k'' + 1)/(''k'' - 1). It also implies that the intervals (''k'' + 2)/''k'' (=s) and ''k''/(''k'' - 2) (=L) are equidistant from (''k'' + 1)/(''k'' - 1) (=M) because to make them equidistant we need to temper:<br />
<br />
<math>\displaystyle \frac{L/M}{M/s} = \frac{ \left(\frac{k}{k-2}\right)/\left(\frac{k+1}{k-1}\right) }{ \left(\frac{k+1}{k-1}\right)/\left(\frac{k+2}{k}\right) } = \frac{Ls}{M^2} = \frac{\frac{k+2}{k-2}}{\left(\frac{k+1}{k-1}\right)^2} </math><br />
<br />
… and notice that the latter expression is the one we've [[S-expression/Advanced results#Mathematical derivations|shown is equal to S(''k''-1)/S(''k''+1)]] (up to an offset ''k''). In other words, you could interpret that a reason that tempering S(''k''-1)/S(''k''+1) results in (''k''+1)/(''k''-1) being half of (''k''+2)/(''k''-2) is because it makes the following three intervals equidistant:<br />
(''k''+2)/''k'', (''k''+1)/(''k''-1), ''k''/(''k''-2)<br />
<br />
Also note that in the above, (''k'' + 1)/(''k'' - 1) is the [[mediant]] of the adjacent two intervals, meaning that division of an interval into two via tempering a semiparticular is in some sense 'optimal' relative to the complexity. This also means that if ''k'' is a multiple of 2, this corresponds to a natural way to split the square superparticular S(''k''/2) into two parts. For example, if ''k'' = 10 then we have (10+2)/10, (10+1)/(10-1), 10/(10-2) as equidistant, which simplified is 6/5, 11/9, 5/4, with 11/9 being the mediant of 6/5 and 5/4, and therefore the corresponding superparticular S5 = (5/4)/(6/5) is split into two parts which are tempered together: (5/4)/(11/9) = 45/44 and (11/9)/(6/5) = 55/54. The semiparticular is therefore S(10-1)/S(10+1) = S9/S11 = 243/242 = (45/44)/(55/54) = ((10+2)/(10-2))/((10+1)/(10-1))<sup>2</sup>.<br />
<br />
This form of comma has been named "semiparticular", because most of the time it is superparticular but less often it is odd-particular, and because when tempered out they all cause an interval to be divided into two equal parts where each part is a (tempered version of a) superparticular or odd-particular, and the interval being divided in half is sometimes quodd-particular, sometimes odd-particular and sometimes superparticular. Specifically:<br />
<br />
* To find out what a superparticular (''a''+1)/''a'' is approximately half of, temper the semiparticular S(2''a'')/S(2''a''+2) and you can observe that (2''a''+3)/(2''a''-1) is the interval it is approximately half of.<br />
<br />
* To find out what an odd-particular (2''a''+1)/(2''a''-1) is approximately half of, temper the semiparticular S(2''a''-1)/S(2''a''+1) and you can observe that (2''a''+2)/(2''a''-2) = (''a''+1)/(''a''-1), a superparticular or odd-particular, is the interval it is approximately half of.<br />
<br />
* To find out what splits a superparticular (''a''+1)/''a'' in half, temper the semiparticular S(4''a''+1)/S(4''a''+3) and you can observe that (4''a''+3)/(4''a''+1), an odd-particular, is the interval that is approximately half of it.<br />
<br />
* To find out what splits an odd-particular (2''a''+1)/(2''a''-1) in half, temper the semiparticular S(4''a''-2)/S(4''a''+2) and you can observe that (4''a''-1)/(4''a''+1), an odd-particular, is the interval that is approximately half of it.<br />
<br />
Also, the interval in the denominator of an expression of a semiparticular of the form (a/b)/(c/d)<sup>2</sup> is significant in that it has a special relationship: specifically, consider tempering (a/b)/(c/d)<sup>2</sup> so therefore the interval c/d is equal to the interval (a/b)/(c/d). This is significant because it allows the intuitive replacement of two consecutive superparticulars (whose product is a superparticular or odd-particular) with the two superparticulars directly adjacent to them.<br />
<br />
For example, as 9/8 = 18/17 * 17/16 we can replace 18/17 with 19/18 and 17/16 with 16/15 by tempering S16/S18 = (19/15)/(9/8)<sup>2</sup> because we can multiply 9/8 by the tempered comma (19/15)/(9/8)<sup>2</sup> to get (19/15)/(9/8) = (19/18)(16/15) (because 9/8 = 18/16), or as 13/11 = 13/12 * 12/11 we can replace 13/12 with 14/13 and 12/11 with 11/10 by tempering S11/S13 = (7/5)/(13/11)<sup>2</sup> because we can multiply 13/11 by the tempered comma (7/5)/(13/11)<sup>2</sup> to get (7/5)/(13/11) = (14/13)(11/10) (because 7/5 = 14/10). Note we have to replace ''both'' intervals ''simultaneously'' as this is lower error, and note that if we want to be able to replace them individually we must pick the higher error route of tempering S16 and S18 or S11 and S13 individually (for which tempering the semiparticular is then an implied consequence). (The broader lesson is that you can rewrite exact JI equivalences with the commas you are tempering to find new interesting consequences of those commas.)<br />
<br />
=== Table of semiparticulars ===<br />
Here follows a table of [[23-limit]] semiparticulars corresponding to square-particulars S''k'' for ''k'' < 96, plus all semiparticulars up to [[9801/9800|S33/S35 = S99]], an exceptional [[11-limit]] comma, plus all semiparticulars dividing [[superparticular interval]]s up to [[13/12]] (corresponding to the [[17-limit]] semiparticular [[31213/31212|S49/S51]]) for completeness. This table also shows all semiparticulars corresponding to splitting an [[#Glossary|odd-particular]] in two up to [[17/15]] (although a common strategy is to temper the square-particular that is the difference between the two superparticular intervals the odd-particular is composed of instead). The bound ''k'' < 96 was chosen as it corresponds to another remarkable semiparticular [[123201/123200|S78/S80 = S351]]. Perhaps many of the patterns will become clearer if you examine this table:<br />
<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Square Relation<br />
! Ratio<br />
|-<br />
| S2/S4 = ([[4/3]])/([[16/15]])<br />
| ([[5/1]])/([[2/1]])<sup>2</sup><br />
| [[5/4]]<br />
|-<br />
| S3/S5 = ([[9/8]])/([[25/24]])<br />
| ([[3/1]])/([[5/3]])<sup>2</sup><br />
| [[27/25]]<br />
|-<br />
| S4/S6 = ([[16/15]])/([[36/35]])<br />
| ([[7/3]])/([[3/2]])<sup>2</sup><br />
| [[28/27]]<br />
|-<br />
| S5/S7 = ([[25/24]])/([[49/48]])<br />
| ([[2/1]])/([[7/5]])<sup>2</sup><br />
| [[50/49]]<br />
|-<br />
| S6/S8 = ([[36/35]])/([[64/63]])<br />
| ([[9/5]])/([[4/3]])<sup>2</sup><br />
| [[81/80]]<br />
|-<br />
| S7/S9 = ([[49/48]])/([[81/80]])<br />
| ([[5/3]])/([[9/7]])<sup>2</sup><br />
| [[245/243]]<br />
|-<br />
| S8/S10 = ([[64/63]])/([[100/99]])<br />
| ([[11/7]])/([[5/4]])<sup>2</sup><br />
| [[176/175]]<br />
|-<br />
| S9/S11 = ([[81/80]])/([[121/120]])<br />
| ([[3/2]])/([[11/9]])<sup>2</sup><br />
| [[243/242]]<br />
|-<br />
| S10/S12 = ([[100/99]])/([[144/143]])<br />
| ([[13/9]])/([[6/5]])<sup>2</sup><br />
| [[325/324]]<br />
|-<br />
| S11/S13 = ([[121/120]])/([[169/168]])<br />
| ([[7/5]])/([[13/11]])<sup>2</sup><br />
| [[847/845]]<br />
|-<br />
| S12/S14 = ([[144/143]])/([[196/195]])<br />
| ([[15/11]])/([[7/6]])<sup>2</sup><br />
| [[540/539]]<br />
|-<br />
| S13/S15 = ([[169/168]])/([[225/224]])<br />
| ([[4/3]])/([[15/13]])<sup>2</sup><br />
| [[676/675]]<br />
|-<br />
| S14/S16 = ([[196/195]])/([[256/255]])<br />
| ([[17/13]])/([[8/7]])<sup>2</sup><br />
| [[833/832]]<br />
|-<br />
| S15/S17 = ([[225/224]])/([[289/288]])<br />
| ([[9/7]])/([[17/15]])<sup>2</sup><br />
| [[2025/2023]]<br />
|-<br />
| S16/S18 = ([[256/255]])/([[324/323]])<br />
| ([[19/15]])/([[9/8]])<sup>2</sup><br />
| [[1216/1215]]<br />
|-<br />
| S17/S19 = ([[289/288]])/([[361/360]])<br />
| ([[5/4]])/([[19/17]])<sup>2</sup><br />
| [[1445/1444]]<br />
|-<br />
| S18/S20 = ([[324/323]])/([[400/399]])<br />
| ([[21/17]])/([[10/9]])<sup>2</sup><br />
| [[1701/1700]]<br />
|-<br />
| S19/S21 = ([[361/360]])/([[441/440]])<br />
| ([[11/9]])/([[21/19]])<sup>2</sup><br />
| [[3971/3969]]<br />
|-<br />
| S20/S22 = ([[400/399]])/([[484/483]])<br />
| ([[23/19]])/([[11/10]])<sup>2</sup><br />
| [[2300/2299]]<br />
|-<br />
| S21/S23 = ([[441/440]])/([[529/528]])<br />
| ([[6/5]])/([[23/21]])<sup>2</sup><br />
| [[2646/2645]]<br />
|-<br />
| S22/S24 = ([[484/483]])/([[576/575]])<br />
| ([[25/21]])/([[12/11]])<sup>2</sup><br />
| [[3025/3024]]<br />
|-<br />
| S23/S25 = ([[529/528]])/([[625/624]])<br />
| ([[13/11]])/([[25/23]])<sup>2</sup><br />
| [[6877/6875]]<br />
|-<br />
| S24/S26 = ([[576/575]])/([[676/675]])<br />
| ([[27/23]])/([[13/12]])<sup>2</sup><br />
| [[3888/3887]]<br />
|-<br />
| S25/S27 = ([[625/624]])/([[729/728]])<br />
| ([[7/6]])/([[27/25]])<sup>2</sup><br />
| [[4375/4374]]<br />
|-<br />
| S26/S28 = ([[676/675]])/([[784/783]])<br />
| ([[29/25]])/([[14/13]])<sup>2</sup><br />
| [[4901/4900]]<br />
|-<br />
| S27/S29 = ([[729/728]])/([[841/840]])<br />
| ([[15/13]])/([[29/27]])<sup>2</sup><br />
| [[10935/10933]]<br />
|-<br />
| S28/S30 = ([[784/783]])/([[900/899]])<br />
| ([[31/27]])/([[15/14]])<sup>2</sup><br />
| [[6076/6075]]<br />
|-<br />
| S29/S31 = ([[841/840]])/([[961/960]])<br />
| ([[8/7]])/([[31/29]])<sup>2</sup><br />
| [[6728/6727]]<br />
|-<br />
| S30/S32 = ([[900/899]])/([[1024/1023]])<br />
| ([[33/29]])/([[16/15]])<sup>2</sup><br />
| [[7425/7424]]<br />
|-<br />
| S31/S33 = ([[961/960]])/([[1089/1088]])<br />
| ([[17/15]])/([[33/31]])<sup>2</sup><br />
| [[16337/16335]]<br />
|-<br />
| S32/S34 = ([[1024/1023]])/([[1156/1155]])<br />
| ([[35/31]])/([[17/16]])<sup>2</sup><br />
| [[8960/8959]]<br />
|-<br />
| S33/S35 = ([[1089/1088]])/([[1225/1224]])<br />
| ([[9/8]])/([[35/33]])<sup>2</sup><br />
| [[9801/9800]]<br />
|-<br />
| S36/S38 = ([[1296/1295]])/([[1444/1443]])<br />
| ([[39/35]])/([[19/18]])<sup>2</sup><br />
| [[12636/12635]]<br />
|-<br />
| S37/S39 = ([[1369/1368]])/([[1521/1520]])<br />
| ([[10/9]])/([[39/37]])<sup>2</sup><br />
| [[13690/13689]]<br />
|-<br />
| S41/S43 = ([[1681/1680]])/([[1849/1848]])<br />
| ([[11/10]])/([[43/41]])<sup>2</sup><br />
| [[18491/18490]]<br />
|-<br />
| S45/S47 = ([[2025/2024]])/([[2209/2208]])<br />
| ([[12/11]])/([[47/45]])<sup>2</sup><br />
| [[24300/24299]]<br />
|-<br />
| S46/S48 = ([[2116/2115]])/([[2304/2303]])<br />
| ([[49/45]])/([[24/23]])<sup>2</sup><br />
| [[25921/25920]]<br />
|-<br />
| S49/S51 = ([[2401/2400]])/([[2601/2600]])<br />
| ([[13/12]])/([[51/49]])<sup>2</sup><br />
| [[31213/31212]]<br />
|-<br />
| S52/S54 = ([[2704/2703]])/([[2916/2915]])<br />
| ([[55/51]])/([[27/26]])<sup>2</sup><br />
| [[37180/37179]]<br />
|-<br />
| S66/S68 = ([[4356/4355]])/([[4624/4623]])<br />
| ([[69/65]])/([[34/33]])<sup>2</sup><br />
| [[75141/75140]]<br />
|-<br />
| S78/S80 = ([[6084/6083]])/([[6400/6399]])<br />
| ([[81/77]])/([[40/39]])<sup>2</sup><br />
| [[123201/123200]]<br />
|}<br />
<br />
(Note that while a lot of these have pages, not all of them do, although that doesn't mean they shouldn't. A noticeable streak of commas currently without pages correspond to when dividing a superparticular interval implicates intervals from a higher [[prime limit]], as a surprising amount of 23-limit semiparticulars shown here already have pages.)<br />
<br />
== {{nowrap|S''k''² * S(''k'' + 1)}} and {{nowrap|S(''k'' − 1) * S''k''²}} (lopsided commas) ==<br />
=== Significance ===<br />
1. Tempering any two consecutive square-particulars, S''k'' and S(''k'' + 1), implies tempering the two associated lopsided commas as well as the associated [[triangle-particular]] and [[ultraparticular]], so the lopsided commas represent the general form of the highest-damage relations/consequences of doing so.<br />
<br />
2. If a comma (such as the diaschisma, [[2048/2025]]), admits an expression as a lopsided comma, it means that one is likely missing out on tempering opportunities by not also tempering the square-particulars composing it (such as [[256/255|S16]] and [[289/288|S17]] in the case of the diaschisma), often involving expanding the subgroup and adding a number of new equivalence relations (as previously explained) while simultaneously making the temperament more efficient and more precise.<br />
<br />
3. It is surprising that there are fairly simple general equivalence relations for these S-expressions, essentially being "free" to read off of an S-expression-based comma list, once you know the general form.<br />
<br />
=== Derivation of equivalence relation ===<br />
Using the clarity of [[S-expression/Advanced results#Using S-factorizations to understand the significance of S-expressions|S-factorizations]], we can show the interval relations implicated by these two new "lopsided" forms, which will make clear the reason for their name:<br />
<br />
S''k''<sup>2</sup> * S(''k''+1) = [''k''-1, ''k'', ''k''+1, ''k''+2]^(2[-1, 2, -1, 0] + [0, -1, 2, -1] = [-2, 4, -2, 0] + [0, -1, 2, -1] = [-2, 3, 0, -1]) implies:<br />
<br />
S''k''<sup>2</sup> * S(''k''+1) = (''k''/(''k''-1))<sup>2</sup> / ((''k''+2)/''k'') through [-2, 3, 0, -1] = [-2, 2, 0, 0] - [0, -1, 0, 1].<br />
<br />
S(''k''-1) * S''k''<sup>2</sup> = [''k''-2, ''k''-1, ''k'', ''k''+1]^([-1, 2, -1, 0] + 2[0, -1, 2, -1] = [-1, 2, -1, 0] + [0, -2, 4, -2] = [-1, 0, 3, -2]) implies:<br />
<br />
S(''k''-1) * S''k''<sup>2</sup> = (''k''/(''k''-2)) / ((''k''+1)/''k'')<sup>2</sup> through [-1, 0, 3, -2] = [-1, 0, 1, 0] - [0, 0, -2, 2].<br />
<br />
=== Tables ===<br />
Below are two tables of [[43-limit]] lopsided commas. First, the "top heavy" lopsided commas, where the squared interval is in the numerator, then the "bottom heavy" lopsided commas, where the squared interval is in the denominator. These tables are so big because these commas are quite large so the more interesting commas appear later. For this reason and for completeness, the tables show up to until a little past the largest known lopsided commas that have their own page: the [[olympia]] and the [[phaotic comma]].<br />
<br />
==== Top-heavy lopsided commas ====<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Square Relation<br />
! Ratio<br />
|-<br />
| S2<sup>2</sup>*S3 = [[3/2]] * [[4/3]]<br />
| ([[2/1]])<sup>2</sup> / ([[2/1]])<br />
| [[2/1]]<br />
|-<br />
| S3<sup>2</sup>*S4 = [[6/5]] * [[9/8]]<br />
| ([[3/2]])<sup>2</sup> / ([[5/3]])<br />
| [[27/20]]<br />
|-<br />
| S4<sup>2</sup>*S5 = [[10/9]] * [[16/15]]<br />
| ([[4/3]])<sup>2</sup> / ([[3/2]])<br />
| [[32/27]]<br />
|-<br />
| S5<sup>2</sup>*S6 = [[15/14]] * [[25/24]]<br />
| ([[5/4]])<sup>2</sup> / ([[7/5]])<br />
| [[125/112]]<br />
|-<br />
| S6<sup>2</sup>*S7 = [[21/20]] * [[36/35]]<br />
| ([[6/5]])<sup>2</sup> / ([[4/3]])<br />
| [[27/25]]<br />
|-<br />
| S7<sup>2</sup>*S8 = [[28/27]] * [[49/48]]<br />
| ([[7/6]])<sup>2</sup> / ([[9/7]])<br />
| [[343/324]]<br />
|-<br />
| S8<sup>2</sup>*S9 = [[36/35]] * [[64/63]]<br />
| ([[8/7]])<sup>2</sup> / ([[5/4]])<br />
| [[256/245]]<br />
|-<br />
| S9<sup>2</sup>*S10 = [[45/44]] * [[81/80]]<br />
| ([[9/8]])<sup>2</sup> / ([[11/9]])<br />
| [[729/704]]<br />
|-<br />
| S10<sup>2</sup>*S11 = [[55/54]] * [[100/99]]<br />
| ([[10/9]])<sup>2</sup> / ([[6/5]])<br />
| [[250/243]]<br />
|-<br />
| S11<sup>2</sup>*S12 = [[66/65]] * [[121/120]]<br />
| ([[11/10]])<sup>2</sup> / ([[13/11]])<br />
| [[1331/1300]]<br />
|-<br />
| S12<sup>2</sup>*S13 = [[78/77]] * [[144/143]]<br />
| ([[12/11]])<sup>2</sup> / ([[7/6]])<br />
| [[864/847]]<br />
|-<br />
| S13<sup>2</sup>*S14 = [[91/90]] * [[169/168]]<br />
| ([[13/12]])<sup>2</sup> / ([[15/13]])<br />
| [[2197/2160]]<br />
|-<br />
| S14<sup>2</sup>*S15 = [[105/104]] * [[196/195]]<br />
| ([[14/13]])<sup>2</sup> / ([[8/7]])<br />
| [[343/338]]<br />
|-<br />
| S15<sup>2</sup>*S16 = [[120/119]] * [[225/224]]<br />
| ([[15/14]])<sup>2</sup> / ([[17/15]])<br />
| [[3375/3332]]<br />
|-<br />
| S16<sup>2</sup>*S17 = [[136/135]] * [[256/255]]<br />
| ([[16/15]])<sup>2</sup> / ([[9/8]])<br />
| [[2048/2025]]<br />
|-<br />
| S17<sup>2</sup>*S18 = [[153/152]] * [[289/288]]<br />
| ([[17/16]])<sup>2</sup> / ([[19/17]])<br />
| [[4913/4864]]<br />
|-<br />
| S18<sup>2</sup>*S19 = [[171/170]] * [[324/323]]<br />
| ([[18/17]])<sup>2</sup> / ([[10/9]])<br />
| [[1458/1445]]<br />
|-<br />
| S19<sup>2</sup>*S20 = [[190/189]] * [[361/360]]<br />
| ([[19/18]])<sup>2</sup> / ([[21/19]])<br />
| [[6859/6804]]<br />
|-<br />
| S20<sup>2</sup>*S21 = [[210/209]] * [[400/399]]<br />
| ([[20/19]])<sup>2</sup> / ([[11/10]])<br />
| [[4000/3971]]<br />
|-<br />
| S21<sup>2</sup>*S22 = [[231/230]] * [[441/440]]<br />
| ([[21/20]])<sup>2</sup> / ([[23/21]])<br />
| [[9261/9200]]<br />
|-<br />
| S22<sup>2</sup>*S23 = [[253/252]] * [[484/483]]<br />
| ([[22/21]])<sup>2</sup> / ([[12/11]])<br />
| [[1331/1323]]<br />
|-<br />
| S23<sup>2</sup>*S24 = [[276/275]] * [[529/528]]<br />
| ([[23/22]])<sup>2</sup> / ([[25/23]])<br />
| [[12167/12100]]<br />
|-<br />
| S24<sup>2</sup>*S25 = [[300/299]] * [[576/575]]<br />
| ([[24/23]])<sup>2</sup> / ([[13/12]])<br />
| [[6912/6877]]<br />
|-<br />
| S25<sup>2</sup>*S26 = [[325/324]] * [[625/624]]<br />
| ([[25/24]])<sup>2</sup> / ([[27/25]])<br />
| [[15625/15552]]<br />
|-<br />
| S26<sup>2</sup>*S27 = [[351/350]] * [[676/675]]<br />
| ([[26/25]])<sup>2</sup> / ([[14/13]])<br />
| [[4394/4375]]<br />
|-<br />
| S27<sup>2</sup>*S28 = [[378/377]] * [[729/728]]<br />
| ([[27/26]])<sup>2</sup> / ([[29/27]])<br />
| [[19683/19604]]<br />
|-<br />
| S28<sup>2</sup>*S29 = [[406/405]] * [[784/783]]<br />
| ([[28/27]])<sup>2</sup> / ([[15/14]])<br />
| [[10976/10935]]<br />
|-<br />
| S29<sup>2</sup>*S30 = [[435/434]] * [[841/840]]<br />
| ([[29/28]])<sup>2</sup> / ([[31/29]])<br />
| [[24389/24304]]<br />
|-<br />
| S30<sup>2</sup>*S31 = [[465/464]] * [[900/899]]<br />
| ([[30/29]])<sup>2</sup> / ([[16/15]])<br />
| [[3375/3364]]<br />
|-<br />
| S31<sup>2</sup>*S32 = [[496/495]] * [[961/960]]<br />
| ([[31/30]])<sup>2</sup> / ([[33/31]])<br />
| [[29791/29700]]<br />
|-<br />
| S32<sup>2</sup>*S33 = [[528/527]] * [[1024/1023]]<br />
| ([[32/31]])<sup>2</sup> / ([[17/16]])<br />
| [[16384/16337]]<br />
|-<br />
| S33<sup>2</sup>*S34 = [[561/560]] * [[1089/1088]]<br />
| ([[33/32]])<sup>2</sup> / ([[35/33]])<br />
| [[35937/35840]]<br />
|-<br />
| S34<sup>2</sup>*S35 = [[595/594]] * [[1156/1155]]<br />
| ([[34/33]])<sup>2</sup> / ([[18/17]])<br />
| [[9826/9801]]<br />
|-<br />
| S35<sup>2</sup>*S36 = [[630/629]] * [[1225/1224]]<br />
| ([[35/34]])<sup>2</sup> / ([[37/35]])<br />
| [[42875/42772]]<br />
|-<br />
| S36<sup>2</sup>*S37 = [[666/665]] * [[1296/1295]]<br />
| ([[36/35]])<sup>2</sup> / ([[19/18]])<br />
| [[23328/23275]]<br />
|-<br />
| S37<sup>2</sup>*S38 = [[703/702]] * [[1369/1368]]<br />
| ([[37/36]])<sup>2</sup> / ([[39/37]])<br />
| [[50653/50544]]<br />
|-<br />
| S38<sup>2</sup>*S39 = [[741/740]] * [[1444/1443]]<br />
| ([[38/37]])<sup>2</sup> / ([[20/19]])<br />
| [[6859/6845]]<br />
|-<br />
| S39<sup>2</sup>*S40 = [[780/779]] * [[1521/1520]]<br />
| ([[39/38]])<sup>2</sup> / ([[41/39]])<br />
| [[59319/59204]]<br />
|-<br />
| S40<sup>2</sup>*S41 = [[820/819]] * [[1600/1599]]<br />
| ([[40/39]])<sup>2</sup> / ([[21/20]])<br />
| [[32000/31941]]<br />
|-<br />
| S41<sup>2</sup>*S42 = [[861/860]] * [[1681/1680]]<br />
| ([[41/40]])<sup>2</sup> / ([[43/41]])<br />
| [[68921/68800]]<br />
|-<br />
| S42<sup>2</sup>*S43 = [[903/902]] * [[1764/1763]]<br />
| ([[42/41]])<sup>2</sup> / ([[22/21]])<br />
| [[18522/18491]]<br />
|-<br />
| S43<sup>2</sup>*S44 = [[946/945]] * [[1849/1848]]<br />
| ([[43/42]])<sup>2</sup> / ([[45/43]])<br />
| [[79507/79380]]<br />
|-<br />
| S44<sup>2</sup>*S45 = [[990/989]] * [[1936/1935]]<br />
| ([[44/43]])<sup>2</sup> / ([[23/22]])<br />
| [[42592/42527]]<br />
|-<br />
| S46<sup>2</sup>*S47 = [[1081/1080]] * [[2116/2115]]<br />
| ([[46/45]])<sup>2</sup> / ([[24/23]])<br />
| [[12167/12150]]<br />
|-<br />
| S49<sup>2</sup>*S50 = [[1225/1224]] * [[2401/2400]]<br />
| ([[49/48]])<sup>2</sup> / ([[51/49]])<br />
| [[117649/117504]]<br />
|-<br />
| S50<sup>2</sup>*S51 = [[1275/1274]] * [[2500/2499]]<br />
| ([[50/49]])<sup>2</sup> / ([[26/25]])<br />
| [[31250/31213]]<br />
|-<br />
| S52<sup>2</sup>*S53 = [[1378/1377]] * [[2704/2703]]<br />
| ([[52/51]])<sup>2</sup> / ([[27/26]])<br />
| [[70304/70227]]<br />
|-<br />
| S55<sup>2</sup>*S56 = [[1540/1539]] * [[3025/3024]]<br />
| ([[55/54]])<sup>2</sup> / ([[57/55]])<br />
| [[166375/166212]]<br />
|-<br />
| S56<sup>2</sup>*S57 = [[1596/1595]] * [[3136/3135]]<br />
| ([[56/55]])<sup>2</sup> / ([[29/28]])<br />
| [[87808/87725]]<br />
|-<br />
| S58<sup>2</sup>*S59 = [[1711/1710]] * [[3364/3363]]<br />
| ([[58/57]])<sup>2</sup> / ([[30/29]])<br />
| [[48778/48735]]<br />
|-<br />
| S63<sup>2</sup>*S64 = [[2016/2015]] * [[3969/3968]]<br />
| ([[63/62]])<sup>2</sup> / ([[65/63]])<br />
| [[250047/249860]]<br />
|-<br />
| S64<sup>2</sup>*S65 = [[2080/2079]] * [[4096/4095]]<br />
| ([[64/63]])<sup>2</sup> / ([[33/32]])<br />
| [[131072/130977]]<br />
|-<br />
| S66<sup>2</sup>*S67 = [[2211/2210]] * [[4356/4355]]<br />
| ([[66/65]])<sup>2</sup> / ([[34/33]])<br />
| [[71874/71825]]<br />
|-<br />
| S70<sup>2</sup>*S71 = [[2485/2484]] * [[4900/4899]]<br />
| ([[70/69]])<sup>2</sup> / ([[36/35]])<br />
| [[42875/42849]]<br />
|-<br />
| S75<sup>2</sup>*S76 = [[2850/2849]] * [[5625/5624]]<br />
| ([[75/74]])<sup>2</sup> / ([[77/75]])<br />
| [[421875/421652]]<br />
|-<br />
| S76<sup>2</sup>*S77 = [[2926/2925]] * [[5776/5775]]<br />
| ([[76/75]])<sup>2</sup> / ([[39/38]])<br />
| [[219488/219375]]<br />
|-<br />
| S78<sup>2</sup>*S79 = [[3081/3080]] * [[6084/6083]]<br />
| ([[78/77]])<sup>2</sup> / ([[40/39]])<br />
| [[59319/59290]]<br />
|}<br />
<br />
==== Bottom-heavy lopsided commas ====<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Square Relation<br />
! Ratio<br />
|-<br />
| S3<sup>2</sup>*S2 = [[3/2]] * [[9/8]]<br />
| ([[3/1]]) / ([[4/3]])<sup>2</sup><br />
| [[27/16]]<br />
|-<br />
| S4<sup>2</sup>*S3 = [[6/5]] * [[16/15]]<br />
| ([[2/1]]) / ([[5/4]])<sup>2</sup><br />
| [[32/25]]<br />
|-<br />
| S5<sup>2</sup>*S4 = [[10/9]] * [[25/24]]<br />
| ([[5/3]]) / ([[6/5]])<sup>2</sup><br />
| [[125/108]]<br />
|-<br />
| S6<sup>2</sup>*S5 = [[15/14]] * [[36/35]]<br />
| ([[3/2]]) / ([[7/6]])<sup>2</sup><br />
| [[54/49]]<br />
|-<br />
| S7<sup>2</sup>*S6 = [[21/20]] * [[49/48]]<br />
| ([[7/5]]) / ([[8/7]])<sup>2</sup><br />
| [[343/320]]<br />
|-<br />
| S8<sup>2</sup>*S7 = [[28/27]] * [[64/63]]<br />
| ([[4/3]]) / ([[9/8]])<sup>2</sup><br />
| [[256/243]]<br />
|-<br />
| S9<sup>2</sup>*S8 = [[36/35]] * [[81/80]]<br />
| ([[9/7]]) / ([[10/9]])<sup>2</sup><br />
| [[729/700]]<br />
|-<br />
| S10<sup>2</sup>*S9 = [[45/44]] * [[100/99]]<br />
| ([[5/4]]) / ([[11/10]])<sup>2</sup><br />
| [[125/121]]<br />
|-<br />
| S11<sup>2</sup>*S10 = [[55/54]] * [[121/120]]<br />
| ([[11/9]]) / ([[12/11]])<sup>2</sup><br />
| [[1331/1296]]<br />
|-<br />
| S12<sup>2</sup>*S11 = [[66/65]] * [[144/143]]<br />
| ([[6/5]]) / ([[13/12]])<sup>2</sup><br />
| [[864/845]]<br />
|-<br />
| S13<sup>2</sup>*S12 = [[78/77]] * [[169/168]]<br />
| ([[13/11]]) / ([[14/13]])<sup>2</sup><br />
| [[2197/2156]]<br />
|-<br />
| S14<sup>2</sup>*S13 = [[91/90]] * [[196/195]]<br />
| ([[7/6]]) / ([[15/14]])<sup>2</sup><br />
| [[686/675]]<br />
|-<br />
| S15<sup>2</sup>*S14 = [[105/104]] * [[225/224]]<br />
| ([[15/13]]) / ([[16/15]])<sup>2</sup><br />
| [[3375/3328]]<br />
|-<br />
| S16<sup>2</sup>*S15 = [[120/119]] * [[256/255]]<br />
| ([[8/7]]) / ([[17/16]])<sup>2</sup><br />
| [[2048/2023]]<br />
|-<br />
| S17<sup>2</sup>*S16 = [[136/135]] * [[289/288]]<br />
| ([[17/15]]) / ([[18/17]])<sup>2</sup><br />
| [[4913/4860]]<br />
|-<br />
| S18<sup>2</sup>*S17 = [[153/152]] * [[324/323]]<br />
| ([[9/8]]) / ([[19/18]])<sup>2</sup><br />
| [[729/722]]<br />
|-<br />
| S19<sup>2</sup>*S18 = [[171/170]] * [[361/360]]<br />
| ([[19/17]]) / ([[20/19]])<sup>2</sup><br />
| [[6859/6800]]<br />
|-<br />
| S20<sup>2</sup>*S19 = [[190/189]] * [[400/399]]<br />
| ([[10/9]]) / ([[21/20]])<sup>2</sup><br />
| [[4000/3969]]<br />
|-<br />
| S21<sup>2</sup>*S20 = [[210/209]] * [[441/440]]<br />
| ([[21/19]]) / ([[22/21]])<sup>2</sup><br />
| [[9261/9196]]<br />
|-<br />
| S22<sup>2</sup>*S21 = [[231/230]] * [[484/483]]<br />
| ([[11/10]]) / ([[23/22]])<sup>2</sup><br />
| [[2662/2645]]<br />
|-<br />
| S23<sup>2</sup>*S22 = [[253/252]] * [[529/528]]<br />
| ([[23/21]]) / ([[24/23]])<sup>2</sup><br />
| [[12167/12096]]<br />
|-<br />
| S24<sup>2</sup>*S23 = [[276/275]] * [[576/575]]<br />
| ([[12/11]]) / ([[25/24]])<sup>2</sup><br />
| [[6912/6875]]<br />
|-<br />
| S25<sup>2</sup>*S24 = [[300/299]] * [[625/624]]<br />
| ([[25/23]]) / ([[26/25]])<sup>2</sup><br />
| [[15625/15548]]<br />
|-<br />
| S26<sup>2</sup>*S25 = [[325/324]] * [[676/675]]<br />
| ([[13/12]]) / ([[27/26]])<sup>2</sup><br />
| [[2197/2187]]<br />
|-<br />
| S27<sup>2</sup>*S26 = [[351/350]] * [[729/728]]<br />
| ([[27/25]]) / ([[28/27]])<sup>2</sup><br />
| [[19683/19600]]<br />
|-<br />
| S28<sup>2</sup>*S27 = [[378/377]] * [[784/783]]<br />
| ([[14/13]]) / ([[29/28]])<sup>2</sup><br />
| [[10976/10933]]<br />
|-<br />
| S29<sup>2</sup>*S28 = [[406/405]] * [[841/840]]<br />
| ([[29/27]]) / ([[30/29]])<sup>2</sup><br />
| [[24389/24300]]<br />
|-<br />
| S30<sup>2</sup>*S29 = [[435/434]] * [[900/899]]<br />
| ([[15/14]]) / ([[31/30]])<sup>2</sup><br />
| [[6750/6727]]<br />
|-<br />
| S31<sup>2</sup>*S30 = [[465/464]] * [[961/960]]<br />
| ([[31/29]]) / ([[32/31]])<sup>2</sup><br />
| [[29791/29696]]<br />
|-<br />
| S32<sup>2</sup>*S31 = [[496/495]] * [[1024/1023]]<br />
| ([[16/15]]) / ([[33/32]])<sup>2</sup><br />
| [[16384/16335]]<br />
|-<br />
| S33<sup>2</sup>*S32 = [[528/527]] * [[1089/1088]]<br />
| ([[33/31]]) / ([[34/33]])<sup>2</sup><br />
| [[35937/35836]]<br />
|-<br />
| S34<sup>2</sup>*S33 = [[561/560]] * [[1156/1155]]<br />
| ([[17/16]]) / ([[35/34]])<sup>2</sup><br />
| [[4913/4900]]<br />
|-<br />
| S35<sup>2</sup>*S34 = [[595/594]] * [[1225/1224]]<br />
| ([[35/33]]) / ([[36/35]])<sup>2</sup><br />
| [[42875/42768]]<br />
|-<br />
| S36<sup>2</sup>*S35 = [[630/629]] * [[1296/1295]]<br />
| ([[18/17]]) / ([[37/36]])<sup>2</sup><br />
| [[23328/23273]]<br />
|-<br />
| S37<sup>2</sup>*S36 = [[666/665]] * [[1369/1368]]<br />
| ([[37/35]]) / ([[38/37]])<sup>2</sup><br />
| [[50653/50540]]<br />
|-<br />
| S38<sup>2</sup>*S37 = [[703/702]] * [[1444/1443]]<br />
| ([[19/18]]) / ([[39/38]])<sup>2</sup><br />
| [[13718/13689]]<br />
|-<br />
| S39<sup>2</sup>*S38 = [[741/740]] * [[1521/1520]]<br />
| ([[39/37]]) / ([[40/39]])<sup>2</sup><br />
| [[59319/59200]]<br />
|-<br />
| S40<sup>2</sup>*S39 = [[780/779]] * [[1600/1599]]<br />
| ([[20/19]]) / ([[41/40]])<sup>2</sup><br />
| [[32000/31939]]<br />
|-<br />
| S41<sup>2</sup>*S40 = [[820/819]] * [[1681/1680]]<br />
| ([[41/39]]) / ([[42/41]])<sup>2</sup><br />
| [[68921/68796]]<br />
|-<br />
| S42<sup>2</sup>*S41 = [[861/860]] * [[1764/1763]]<br />
| ([[21/20]]) / ([[43/42]])<sup>2</sup><br />
| [[9261/9245]]<br />
|-<br />
| S43<sup>2</sup>*S42 = [[903/902]] * [[1849/1848]]<br />
| ([[43/41]]) / ([[44/43]])<sup>2</sup><br />
| [[79507/79376]]<br />
|-<br />
| S44<sup>2</sup>*S43 = [[946/945]] * [[1936/1935]]<br />
| ([[22/21]]) / ([[45/44]])<sup>2</sup><br />
| [[42592/42525]]<br />
|-<br />
| S45<sup>2</sup>*S44 = [[990/989]] * [[2025/2024]]<br />
| ([[45/43]]) / ([[46/45]])<sup>2</sup><br />
| [[91125/90988]]<br />
|-<br />
| S48<sup>2</sup>*S47 = [[1128/1127]] * [[2304/2303]]<br />
| ([[24/23]]) / ([[49/48]])<sup>2</sup><br />
| [[55296/55223]]<br />
|-<br />
| S50<sup>2</sup>*S49 = [[1225/1224]] * [[2500/2499]]<br />
| ([[25/24]]) / ([[51/50]])<sup>2</sup><br />
| [[15625/15606]]<br />
|-<br />
| S51<sup>2</sup>*S50 = [[1275/1274]] * [[2601/2600]]<br />
| ([[51/49]]) / ([[52/51]])<sup>2</sup><br />
| [[132651/132496]]<br />
|-<br />
| S54<sup>2</sup>*S53 = [[1431/1430]] * [[2916/2915]]<br />
| ([[27/26]]) / ([[55/54]])<sup>2</sup><br />
| [[39366/39325]]<br />
|-<br />
| S56<sup>2</sup>*S55 = [[1540/1539]] * [[3136/3135]]<br />
| ([[28/27]]) / ([[57/56]])<sup>2</sup><br />
| [[87808/87723]]<br />
|-<br />
| S57<sup>2</sup>*S56 = [[1596/1595]] * [[3249/3248]]<br />
| ([[57/55]]) / ([[58/57]])<sup>2</sup><br />
| [[185193/185020]]<br />
|-<br />
| S62<sup>2</sup>*S61 = [[1891/1890]] * [[3844/3843]]<br />
| ([[31/30]]) / ([[63/62]])<sup>2</sup><br />
| [[59582/59535]]<br />
|-<br />
| S64<sup>2</sup>*S63 = [[2016/2015]] * [[4096/4095]]<br />
| ([[32/31]]) / ([[65/64]])<sup>2</sup><br />
| [[131072/130975]]<br />
|-<br />
| S65<sup>2</sup>*S64 = [[2080/2079]] * [[4225/4224]]<br />
| ([[65/63]]) / ([[66/65]])<sup>2</sup><br />
| [[274625/274428]]<br />
|-<br />
| S68<sup>2</sup>*S67 = [[2278/2277]] * [[4624/4623]]<br />
| ([[34/33]]) / ([[69/68]])<sup>2</sup><br />
| [[157216/157113]]<br />
|-<br />
| S74<sup>2</sup>*S73 = [[2701/2700]] * [[5476/5475]]<br />
| ([[37/36]]) / ([[75/74]])<sup>2</sup><br />
| [[50653/50625]]<br />
|-<br />
| S76<sup>2</sup>*S75 = [[2850/2849]] * [[5776/5775]]<br />
| ([[38/37]]) / ([[77/76]])<sup>2</sup><br />
| [[219488/219373]]<br />
|-<br />
| S77<sup>2</sup>*S76 = [[2926/2925]] * [[5929/5928]]<br />
| ([[77/75]]) / ([[78/77]])<sup>2</sup><br />
| [[456533/456300]]<br />
|-<br />
| S80<sup>2</sup>*S79 = [[3160/3159]] * [[6400/6399]]<br />
| ([[40/39]]) / ([[81/80]])<sup>2</sup><br />
| [[256000/255879]]<br />
|}<br />
<br />
== Equivalent S-expressions ==<br />
=== Significance and meaning ===<br />
All S-expressions have other equivalent S-expressions, however when the equivalence makes one comma a member of two of the infinite families discussed on this page, or otherwise makes it equal to a product or ratio between two such commas, this often means exceptional and nontrivial ("deep") tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending how you temper further. Generally we exclude 1/n-square-particulars, only noting up to 1/3-square-particulars, because equivalent 1/n-square-particular expressions become very common as you allow higher n, but are still quite rare for small n.<br />
<br />
=== A useful general rule ===<br />
While there are likely arbitrarily many ways of rewriting S-expressions due to the redundancy in representation, the following equivalence is, due to its simplicity and elegance, arguably most likely to be useful:<br />
<br />
<math><br />
\large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)<br />
</math><br />
<br />
This is important to note because using this simple rule we can derive an infinite amount of trivially and obviously equivalent S-expressions that should be discarded from [[#Examples]]. See [[S-expression/Advanced results]] for mathematical details.<br />
<br />
=== Examples ===<br />
Here is an incomplete list of examples (feel free to expand with any equivalences you find that you think are valuable).<br />
<br />
'''Importantly:''' examples that can ''easily'' (with one or two algebraic rewriting steps) be shown to result from the aforementioned [[#A useful general rule|useful general rule]] are considered invalid/trivial.<br />
<br />
{| class="wikitable center-all<br />
|-<br />
! Comma<br />
! S-expressions<br />
|-<br />
| [[64/63]]<br />
| (S4*S5*S6)/S3 = S4/(S6*S7) = S8<br />
|-<br />
| [[81/80]]<br />
| S6/S8 = S9<br />
|-<br />
| [[176/175]]<br />
| S8/S10 = S22*S23*S24<br />
|-<br />
| [[243/242]]<br />
| S9/S11 = S15/([[3025/3024|S22/S24 = S55]])<br />
|-<br />
| [[325/324]]<br />
| S10/S12 = S25*S26<br />
|-<br />
| [[540/539]]<br />
| S12/S14 = (S9*S10)/S7 = (S6/S7)/(S8/S10)<br />
|-<br />
| [[676/675]]<br />
| S13/S15 = S26<br />
|-<br />
| [[1225/1224]]<br />
| S35 = S49*S50<br />
|-<br />
| [[3025/3024]]<br />
| S22/S24 = S55 = S25/S27 * S99<br />
|-<br />
| [[2601/2600]]<br />
| S17/(S25*S26) = S51<br />
|-<br />
| [[9801/9800]]<br />
| S99 = S33/S35<br />
|-<br />
| [[25921/25920]]<br />
| S161 = S46/S48<br />
|-<br />
| [[123201/123200]]<br />
| S351 = S78/S80<br />
|}<br />
<br />
Note: Where a comma written in the form a/b is used in an S-expression, this means to replace that comma with any equivalent S-expression. This is done in the case of [[3025/3024]] as there are many S-expressions for it so restating them each time it appears seems inconvenient.<br />
<br />
A proof that every positive rational number (and thus every JI interval) can be written as an S-expression follows.<br />
<br />
It suffices to show every superparticular number including 2/1 has an expression using square-particulars:<br />
<br />
<math>\displaystyle<br />
\begin{align}<br />
& 2/1 = S_2 \cdot S_2 \cdot S_3\ ,\\<br />
& 3/2 = S_2 \cdot S_3\ ,\\<br />
& 4/3 = S_2\ ,\\<br />
& \frac{a/(a - 1)}{(b + 1)/b} = \prod_{k=a}^b \left( S_k = \frac{k/(k - 1)}{(k + 1)/k} \right) \\<br />
& \ \ \ = \frac{a/(a - 1)}{(a + 1)/a} \cdot \frac{(a + 1)/a}{(a + 2)/(a + 1)} \cdot \frac{(a + 2)/(a + 1)}{(a + 3)/(a + 2)} \cdot\ \ldots \cdot \frac{b/(b - 1)}{(b + 1)/b} = \frac{a/(a - 1)}{(b + 1)/b} \\<br />
& \implies \frac{a/(a - 1)}{(b + 1)/b} = S_a \cdot S_{a + 1} \cdot S_{a + 2} \cdot\ \ldots \cdot S_b \\<br />
& \implies \frac{S_2 \cdot S_2 \cdot S_3}{\prod_{a = 2}^k S_a} = 2 \cdot \left( \frac{2/(2 - 1)}{(k + 1)/k} \right)^{-1} = 2 \cdot \left( \frac{(k + 1)/k}{2} \right) = (k + 1)/k<br />
\end{align}<br />
</math><br />
<br />
From here it should not be hard to see how to make any positive rational number. For 11/6, for example, we can do (11/10)(10/9)(9/8)…(2/1) = 11 and then divide that by (6/5)(5/4)(4/3)(3/2)(2/1), meaning 11/6 = (11/10)(10/9)(9/8)(8/7)(7/6) because of the cancellations, then each of those superparticulars we replace with the corresponding S-expression to get the final S-expression. This final S-expression is likely to be far from the most efficient or interesting expression; the redundancy in S-expressions is a strength and feature, as it tells us that there are more than the trivial connections between commas and intervals and that S-expressions can be wielded as a mathematical tool/language to investigate and identify them.<br />
<br />
== Glossary ==<br />
<br />
; Superparticular<br />
: The interval/comma between two consecutive harmonics. See [[superparticular]].<br />
: These are of the form {{sfrac|''k'' + 1|''k''}}.<br />
<br />
; Square-particular<br />
: A superparticular interval/comma whose numerator is a square number. A shorthand (nick)name for square superparticular.<br />
: These are of the form {{nowrap|{{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}} {{=}} S''k''}}.<br />
<br />
; Triangle-particular<br />
: A superparticular interval/comma whose numerator is a [[triangular number]]. A shorthand (nick)name for triangular superparticular. An alternative name for 1/2-square-particular.<br />
: These are of the form {{sfrac|''k''<sup>2</sup> + ''k''|''k''<sup>2</sup> + ''k'' − 2}}. (This always simplifies to a superparticular.)<br />
<br />
; 1/''n''-square-particular<br />
: A comma which is the product of ''n'' consecutive square-particulars and which can therefore be expressed as the ratio between two superparticulars.<br />
: These are of the form {{nowrap|S''a'' * S(''a'' + 1) * … * S''b'' {{=}} {{sfrac|{{sfrac|''a''|''a'' − 1}}|{{sfrac|''b'' + 1|''b''}}}}}} {{nowrap|{{=}} {{sfrac|''ab''|(''a'' − 1)(''b'' + 1)}}}}.<br />
: Replacing/substituting ''a'' with ''k'' and ''b'' with ''k'' + ''n'' - 1 gives us an equivalent expression that includes the number of square-particulars ''n'':<br />
: {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1) {{=}} {{sfrac|{{sfrac|''k''|''k'' − 1}}|{{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}}}}} {{nowrap|{{=}} {{sfrac|''k''(''k'' + ''n'' − 1)|(''k'' − 1)(''k'' + ''n''}}}}<br />
: For {{nowrap|''b'' {{=}} ''a'' + 1}} these can also be called triangle-particulars, in which case they are always superparticular.<br />
: These have implications for whether consistency in the {{nowrap|(''n'' + ''k'') {{=}} (''b'' + 1)}}-[[odd-limit]] is ''potentially'' possible in a given temperament; see the [[#Sk*S(k + 1)*…*S(k + n - 1) (1/n-square-particulars)|section on 1/''n''-square-particulars]].<br />
<br />
; Odd-particular<br />
: An interval/comma between two consecutive odd harmonics. The odd analogue of superparticular.<br />
: These are of the form {{sfrac|2''k'' + 1|2''k'' − 1}}.<br />
<br />
; Throdd-particular<br />
: An interval/comma between two harmonics 3 apart which is not superparticular.<br />
: These are of the form {{sfrac|3''k'' + 1|3''k'' − 2}} or {{sfrac|3''k'' + 2|3''k'' − 1}}.<br />
<br />
; Quodd-particular<br />
: An interval/comma between two harmonics 4 apart which is not superparticular or odd-particular.<br />
: These are of the form {{sfrac|4''k'' + 1|4''k'' − 3}} or {{sfrac|4''k'' + 3|4''k'' − 1}}.<br />
<br />
; ''n''-odd-particular<br />
: An interval/comma between two coprime harmonics ''n'' apart (also called as [[Delta-N ratio|delta-''n'' ratio]]). It is the generalization of superparticular, odd-particular, throdd-particular, and quodd-particular.<br />
: If ''n'' is a prime, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is not superparticular. For example, 5-odd-particular intervals are of the form {{sfrac|5''k'' + 1|5''k'' − 4}}, {{sfrac|5''k'' + 2|5''k'' − 3}}, {{sfrac|5''k'' + 3|5''k'' − 2}}, or {{sfrac|5''k'' + 4|5''k'' − 1}}.<br />
: If ''n'' is a composite, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is neither superparticular nor of ''m''-odd-particular intervals where ''m'' is any other divisor of ''n''. For example, 6-odd-particular intervals are of the form {{sfrac|6''k'' + 1|6''k'' − 5}} or {{sfrac|6''k'' + 5|6''k'' − 1}}.<br />
<br />
; Ultraparticular<br />
: An interval/comma which is the ratio of two consecutive square-particulars.<br />
: These are of the form {{sfrac|S''k''|S(''k'' + 1)}}.<br />
<br />
; Semiparticular<br />
: A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:<br />
: These are of the form {{sfrac|S''k''|S(''k'' + 2)}}.<br />
<br />
; S-expression<br />
: An expression using the S''k'' shorthand notation corresponding strictly to multiplying and dividing only (arbitrary) square-particulars. S-expressions include singular square superparticulars and expressions for other superparticulars in terms of square superparticulars.<br />
<br />
; S-factorization<br />
: An expression that takes a list of consecutive integer harmonics including the ''k''th harmonic and raises them to integer powers, similar to a [[smonzo]] but uniquely suited to analysing S-expressions.<br />
: For example: {{nowrap|S''k'' {{=}} [''k'' − 1, ''k'', ''k'' + 1]<sup>[−1, 2, −1]</sup>}} because {{nowrap|S''k'' {{=}} (''k'' − 1)<sup>−1</sup>''k''<sup>2</sup>(''k'' + 1)<sup>−1</sup>}}.<br />
<br />
; S-comma<br />
: Any comma within one of the infinite families of commas discussed here, excluding 1/n-square-particulars for n>5 (so square-particulars, triangle-particulars, 1/3-square-particulars, 1/4-square-particulars and 1/5-square-particulars are included, but not anything beyond; this bound is used for exclusion (rather than n>3) to allow the utility of 1/5-square-particulars in avoiding twin primes by equating superparticular intervals on either side of the twin primes).<br />
<br />
; Indirect S-comma<br />
: Any comma that is the product or ratio of two S-commas. These appear frequently as S-expressions for commas that are more challenging/nontrivial to represent from the perspective of S-expressions; for example, the [[schisma]] admits at least three such representations.<br />
<br />
== See further ==<br />
* [[S-expression/Advanced results|Advanced results]] – for the harder-to-reach algebra<br />
<br />
== Notes ==<br />
<references group="note" /><br />
<br />
[[Category:Elementary math]]<br />
[[Category:Pages with proofs]]<br />
[[Category:Ratio]]<br />
[[Category:Terms]]</div>ArrowHead294https://en.xen.wiki/index.php?title=MOS_scale&diff=229269MOS scale2026-05-01T13:22:13Z<p>ArrowHead294: </p>
<hr />
<div>{{interwiki<br />
| en = MOS scale<br />
| de = MOS-Skala<br />
| es =<br />
| ja = MOSスケール<br />
| ro = G2S<br />
}}{{Beginner|Mathematics of MOS}}<br />
A '''moment of symmetry''' ('''MOS''' or '''mos'''<ref group="note">The acronym "MOS" is generally pronounced ''em-oh-ess'', while the {{w|anacronym}} "mos", more common in informal and experimental settings, is generally pronounced ''moss''. Sometimes "MOSS" or "moss", standing for "moment of symmetry scale", are used instead, although there is no significant difference in meaning.</ref>) '''scale''' is a [[periodic scale]] where every 2nd (that is, every interval formed by ascending a step) is either small or large with no in-between, and the same goes for 3rds, 4ths, etc. Multiples of the period (which is usually the octave or a fraction thereof), however, come in only one size.<br />
<br />
MOS scales are often referred to as MOSes, thus MOS can be used as either an adjective or a noun.<br />
<br />
== Examples ==<br />
The most widely used MOS scale is the [[5L 2s|diatonic scale]]. It has 7 steps: 5 large ones (major 2nds) and 2 small ones (minor 2nds), and thus is named 5L&nbsp;2s. The major mode is LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode).<br />
<br />
{| class="wikitable"<br />
|+ style="font-size: 105%;" | Interval classes in the 5L&nbsp;2s MOS scale<br />
|-<br />
! rowspan="2" | Interval class<br />
! colspan="2" | Small version<br />
! colspan="2" | Large version<br />
|-<br />
! Quality<br />
! Size<br />
! Quality<br />
! Size<br />
|-<br />
! 2nds (1 step)<br />
| minor<br />
| s<br />
| major<br />
| L<br />
|-<br />
! 3rds (2 steps)<br />
| minor<br />
| {{nowrap|1L + 1s}}<br />
| major<br />
| 2L<br />
|-<br />
! 4ths (3 steps)<br />
| perfect<br />
| {{nowrap|2L + 1s}}<br />
| augmented<br />
| 3L<br />
|-<br />
! 5ths (4 steps)<br />
| diminished<br />
| {{nowrap|2L + 2s}}<br />
| perfect<br />
| {{nowrap|3L + 1s}}<br />
|-<br />
! 6ths (5 steps)<br />
| minor<br />
| {{nowrap|3L + 2s}}<br />
| major<br />
| {{nowrap|4L + 1s}}<br />
|-<br />
! 7ths (6 steps)<br />
| minor<br />
| {{nowrap|4L + 2s}}<br />
| major<br />
| {{nowrap|5L + 1s}}<br />
|-<br />
! 8ves (7 steps)<br />
| perfect<br />
| {{nowrap|5L + 2s}}<br />
| colspan="2" | (only one version)<br />
|}<br />
<br />
Note that the melodic minor scale (LsLLLLs) has only two step sizes, but it is not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.<br />
<br />
Other MOS scales include [[2L&nbsp;3s]], where among its 5 modes are the major pentatonic scale (ssLsL) and the minor pentatonic scale (LssLs), and less commonly [[4L&nbsp;4s]], also known as the octatonic scale in 12edo, which alternates between large and small steps and comes in two modes (LsLsLsLs and sLsLsLsL).<br />
<br />
See the [[catalog of MOS]] for other MOS scales.<br />
<br />
== Periods and generators ==<br />
Every MOS scale can be ''generated'' by stacking a certain interval called the [[generator]] and octave-reducing (or more generally, [[period]]-reducing). For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, 2L&nbsp;3s is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as {{nowrap| C D E F G A C }} does not produces a MOS, because there are more than 2 sizes of each interval class. <br />
<br />
The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53, …. However for a quarter-comma meantone fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50, …. <br />
<br />
== Step ratio spectrum ==<br />
The [[step ratio]] is the ratio of the larger step size to the smaller step size. MOSes with smaller step ratios sound smooth and soft. MOSes with larger step ratios sound jagged and hard. Different step ratios produce different corresponding potential temperament interpretations. The [[TAMNAMS #Step ratio spectrum|TAMNAMS]] system has names for specific ratios and also ranges of ratios.<br />
<br />
When the step ratio is a rational number, the MOS is tuned to an edo. A counter-example is 5L&nbsp;2s tuned to quarter-comma meantone, which has a step ratio of about 1.649.<br />
<br />
{| class="wikitable"<br />
|+ style="font-size: 105%;" | 5L&nbsp;2s step ratios in various edos<br />
|-<br />
! Example edo<br />
! Step ratio<br />
! TAMNAMS name<br />
! Likely temperament<br />interpretations<br />
|-<br />
! 12<br />
| 2:1<br />
| basic<br />
| [[Meantone]] or [[Schismatic]]<br />
|-<br />
! 19<br />
| 3:2<br />
| soft<br />
| [[Meantone]]<br />
|-<br />
! 22<br />
| 4:1<br />
| superhard<br />
| [[Archy]] or [[Superpyth]]<br />
|}<br />
<br />
== Naming ==<br />
Every MOS can be uniquely specified by giving its [[signature]], i.e. the number of large and small steps, which is typically notated e.g. "5L&nbsp;2s,". Every possible signature corresponds to a valid MOS scale. Sometimes, if one simply wants to talk about step sizes without specifying which is large and small, the notation "5a&nbsp;2b" is used (which could refer to either diatonic or {{nowrap| [[2L 5s|anti-diatonic]] {{=}} 2L 5s }}).<br />
<br />
By default, the [[equave]] of a MOS is assumed to be [[2/1]]. To specify a non-octave equave, "{{angbr|equave}}" is placed after the signature, e.g. {{mos scalesig|4L 5s<3/1>|link=1}}. Using angle brackets (<code>&#x26;#x27E8;</code> and <code>&#x26;#x27E9;</code>) is recommended; using greater-than and less-than signs ("&#x3C;equave&#x3E;") can also be done, but this can conflict with HTML and other uses of these symbols.<br />
<br />
Several naming systems have been proposed for MOSes, which can be seen at [[MOS naming]].<br />
<br />
== History and terminology ==<br />
The term ''MOS'', and the method of scale construction it entails, were invented by [[Erv Wilson]] in 1975. His original paper is archived on Anaphoria.com here: [https://anaphoria.com/mos.pdf ''Moments of Symmetry'']. There is also an introduction by [[Kraig Grady]] here: [https://anaphoria.com/wilsonintroMOS.html ''Introduction to Erv Wilson's Moments of Symmetry''].<br />
<br />
Sometimes, scales are defined with respect to a period and an additional [[equivalence interval]], the interval at which pitch classes repeat. MOSes in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called '''Multi-MOSes'''. For example, a MOS with a half-octave period is called a '''2mos''', with a 1/3-octave period a '''3mos''', and so on. MOSes in which the equivalence interval is equal to the period are sometimes called '''Strict MOSes'''. MOSes in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.<br />
<br />
With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional evenness|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt<ref>Norman Carey and David Clampitt. "Aspects of Well-Formed Scales", ''Music Theory Spectrum'', Vol. 11, No. 2 (Autumn, 1989), pp. 187-206.</ref>. A great deal of work has been done in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilson's concept. They are in a sense the MOS of MOS patterns. This is used to explain the [[pentatonic]]s used in traditional [[Japanese music]] (e.g. {{nowrap| A B C E F A }}), where the 5-tone cycles are derived from a 7-tone MOS, which are not found in the concept of DE.<br />
<br />
== Equivalent definitions and generalizations ==<br />
A scale is a MOS if and only if it satisfies one of the following equivalent criteria:<br />
# [[Maximum variety]] 2: Ascending by a certain number of steps is equivalent to ascending by one of at most two intervals, and the maximum of two is achieved (i. e. it is not true that ascending by a certain number of steps is always equivalent to ascending by one interval.) <br />
# [[Binary]] and has a [[generator]]: The scale step comes in exactly two sizes, and the scale is formable from stacking some interval called a generator and octave-reducing.<br />
# Mode of a Christoffel word: The scale can be formed by creating a 2D lattice where the period is on the lattice, then taking pitches by travelling vertically and horizontally from the origin, maintaining as close to the line from the origin to the octave as possible without going above it.<br />
<br />
Each definition generalizes to scales with three or more step sizes, but these generalizations are not equivalent. The concepts of [[balanced word|balance]] and [[distributional evenness]] provide still different generalizations, although defining MOS through these terms is less helpful. For more information, see [[Mathematics of MOS]].<br />
<br />
== Properties ==<br />
=== Basic properties ===<br />
* For every MOS scale with an [[octave]] period (which is usually the [[octave]]), if ''x''-[[edo]] is the [[collapsed]] tuning (where the small step vanishes) and ''y''-[[edo]] is the [[equalized]] tuning (where the large (''L'') step and small (''s'') step are the same size), then by definition it is an {{nowrap| ''x''L (''y'' − ''x'')s }} MOS scale, and the [[basic]] tuning where {{nowrap| ''L'' {{=}} 2''s'' }} is thus {{nowrap|(''x'' + ''y'')}}-[[edo]]. This is also true if the period is 1\''p'', that is, 1 step of ''p''-[[edo]], which implies that ''x'' and ''y'' are divisible by ''p'', though note that in that case (if {{nowrap| ''p'' > 1 }}) you are considering a "multiperiod" MOS scale.<br />
* More generally, whenever ''px''-[[edo]] and ''py''-[[edo]] are used to define two [[val]]s (usually but not necessarily through taking the [[Patent val|patent vals]]) while simultaneously also being used to define the {{nowrap|''px''L (''py'' − ''px'')s}} MOS scale (where ''p'' is the number of periods per octave), then the ''px'' & ''py'' temperament corresponds to that MOS scale, and adding ''x'' and/or ''y'' corresponds to tuning closer to ''x''-[[edo]] and/or ''y''-[[edo]] respectively. (Optionally, see the below more precise statement for the mathematically-inclined.)<br />
* For the mathematically-inclined, we can say that whenever we consider a MOS with ''X''/''p'' notes per period in the [[collapsed]] tuning and ''Y''/''p'' notes per period in the [[equalized]] tuning and ''p'' periods per [[Octave stretching|tempered octave]] (or more generally tempered [[equave]]), and whenever we want to associate that MOS with the {{nowrap| ''X'' & ''Y'' }} rank 2 temperament'''*''', we can say that any {{w|natural number|natural}}-coefficient {{w|linear combination}} of vals {{val| ''X'' … }} and {{val| ''Y'' … }} (where {{nowrap| ''X'' < ''Y'' }}) corresponds uniquely to a tuning of the {{nowrap| ''X'' & ''Y'' }} rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive) iff {{nowrap| gcd(''a'', ''b'') {{=}} 1 }}, because if {{nowrap| ''k'' {{=}} gcd(''a'', ''b'') > 1 }} then the val {{nowrap| ''a''{{val| ''X'' … }} + ''b''{{val| ''Y'' … }} }} has a common factor ''k'' in all of its terms, meaning it is guaranteed to be [[contorted]]. The tuning corresponding to the {{w|Rational number|rational}} ''a''/''b'' is technically only unique up to (discarding of) [[octave stretching]] (or more generally [[equave]]-tempering).<br />
<br />
: The period of this temperament is {{nowrap|1\gcd(''X'', ''Y'')}}, and the rational ''a''/''b'' is very closely related to the [[step ratio]] of the corresponding MOS scale, because {{nowrap| 1{{val| ''X'' … }} + 0{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 0}} tuning while {{nowrap| 0{{val| ''X'' … }} + 1{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 1}} tuning and {{nowrap| 1{{val| ''X'' … ;}} + 1{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 2|''s'' {{=}} 1}} tuning, so that {{nowrap|''L'' {{=}} ''a'' + ''b''}} and {{nowrap|''s'' {{=}} ''b''}} and therefore:<br />
<br />
: {{nowrap|1/([[step ratio]]) {{=}} ''s''/''L''}} {{nowrap|{{=}} ''b''/(''a'' + ''b'')}} implying [[step ratio]] {{nowrap| ''r'' {{=}} (''a'' + ''b'')/''b'' ≥ 1 }} for {{w|Natural number|natural}} ''a'' and ''b'', where if {{nowrap| ''b'' {{=}} 0 }} then the step ratio is infinite, corresponding to the [[collapsed]] tuning.<ref group="note">It is ''important to note'' that the correspondence to the {{nowrap| ''X'' & ''Y'' }} rank-2 temperament only works in all cases if we allow the temperament to be [[contorted]] on its [[subgroup]]; alternatively, it works if we exclude cases where {{nowrap| ''X'' & ''Y'' }} describe a contorted temperament on the subgroup given. An example is the {{nowrap| 5 & 19 }} temperament is contorted in the [[5-limit]] (having a generator of a semifourth, corresponding to [[5L&nbsp;14s]]), so we either need to consider the temperament itself to be contorted (generated by something lacking an interpretation in the subgroup given, two of which yielding a meantone-tempered [[~]][[4/3]]) or we exclude it because of its contortion.</ref><br />
<br />
* Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale ''a''L&nbsp;''b''s are {{nowrap| (''a'' + ''b'')L ''a''s }} (generated by generators of soft-of-basic ''a''L ''b''s) and {{nowrap| ''a''L (''a'' + ''b'')s }} (generated by generators of hard-of-basic ''a''L''&nbsp;b''s).<br />
* Every MOS scale (with a specified [[equave]] ''Ɛ''), excluding {{nowrap|''a''L ''a''s{{angbr|''Ɛ''}} }}, has a ''parent MOS''. If {{nowrap| ''a'' > ''b'' }}, the parent of ''a''L&nbsp;''b''s is {{nowrap| ''b''L (''a'' − ''b'')s }}; if {{nowrap| ''a'' < ''b'' }}, the parent of ''a''L&nbsp;''b''s is {{nowrap| ''a''L (''b'' − ''a'')s }}.<br />
<br />
=== Advanced discussion ===<br />
See:<br />
* [[Mathematics of MOS]], a more formal definition and a discussion of the mathematical properties.<br />
** [[Recursive structure of MOS scales]], a description of how MOS scales are recursive and how one scale can be converted into a related scale.<br />
** [[MOS scale family tree]], a tree initially described by Erv Wilson that organizes scales by parent-and-child relationship, which also helps illustrate mos recursion.<br />
* [[Generator ranges of MOS]], organized by number of scale steps and quantity of L/s steps.<br />
* [[MOS diagrams]], visualizations of the MOS process.<br />
* [http://x31eq.com/temper/method.html How to Find Linear Temperaments], by [[Graham Breed]]<br />
<br />
== Individual pages for MOS scales ==<br />
=== L ≤ 12, s ≤ 12 ===<br />
{| class="wikitable center-all"<br />
|+ style="font-size: 105%; white-space: nowrap;" | Pages for MOS scales ({{nowrap|L ≤ 12|s ≤ 12}})<br />
|-<br />
| [[1L&nbsp;1s]]<br />
| [[1L&nbsp;2s]]<br />
| [[1L&nbsp;3s]]<br />
| [[1L&nbsp;4s]]<br />
| [[1L&nbsp;5s]]<br />
| [[1L&nbsp;6s]]<br />
| [[1L&nbsp;7s]]<br />
| [[1L&nbsp;8s]]<br />
| [[1L&nbsp;9s]]<br />
| [[1L&nbsp;10s]]<br />
| [[1L&nbsp;11s]]<br />
| [[1L&nbsp;12s]]<br />
|-<br />
| [[2L&nbsp;1s]]<br />
| [[2L&nbsp;2s]]<br />
| [[2L&nbsp;3s]]<br />
| [[2L&nbsp;4s]]<br />
| [[2L&nbsp;5s]]<br />
| [[2L&nbsp;6s]]<br />
| [[2L&nbsp;7s]]<br />
| [[2L&nbsp;8s]]<br />
| [[2L&nbsp;9s]]<br />
| [[2L&nbsp;10s]]<br />
| [[2L&nbsp;11s]]<br />
| [[2L&nbsp;12s]]<br />
|-<br />
| [[3L&nbsp;1s]]<br />
| [[3L&nbsp;2s]]<br />
| [[3L&nbsp;3s]]<br />
| [[3L&nbsp;4s]]<br />
| [[3L&nbsp;5s]]<br />
| [[3L&nbsp;6s]]<br />
| [[3L&nbsp;7s]]<br />
| [[3L&nbsp;8s]]<br />
| [[3L&nbsp;9s]]<br />
| [[3L&nbsp;10s]]<br />
| [[3L&nbsp;11s]]<br />
| [[3L&nbsp;12s]]<br />
|-<br />
| [[4L&nbsp;1s]]<br />
| [[4L&nbsp;2s]]<br />
| [[4L&nbsp;3s]]<br />
| [[4L&nbsp;4s]]<br />
| [[4L&nbsp;5s]]<br />
| [[4L&nbsp;6s]]<br />
| [[4L&nbsp;7s]]<br />
| [[4L&nbsp;8s]]<br />
| [[4L&nbsp;9s]]<br />
| [[4L&nbsp;10s]]<br />
| [[4L&nbsp;11s]]<br />
| [[4L&nbsp;12s]]<br />
|-<br />
| [[5L&nbsp;1s]]<br />
| [[5L&nbsp;2s]]<br />
| [[5L&nbsp;3s]]<br />
| [[5L&nbsp;4s]]<br />
| [[5L&nbsp;5s]]<br />
| [[5L&nbsp;6s]]<br />
| [[5L&nbsp;7s]]<br />
| [[5L&nbsp;8s]]<br />
| [[5L&nbsp;9s]]<br />
| [[5L&nbsp;10s]]<br />
| [[5L&nbsp;11s]]<br />
| [[5L&nbsp;12s]]<br />
|-<br />
| [[6L&nbsp;1s]]<br />
| [[6L&nbsp;2s]]<br />
| [[6L&nbsp;3s]]<br />
| [[6L&nbsp;4s]]<br />
| [[6L&nbsp;5s]]<br />
| [[6L&nbsp;6s]]<br />
| [[6L&nbsp;7s]]<br />
| [[6L&nbsp;8s]]<br />
| [[6L&nbsp;9s]]<br />
| [[6L&nbsp;10s]]<br />
| [[6L&nbsp;11s]]<br />
| [[6L&nbsp;12s]]<br />
|-<br />
| [[7L&nbsp;1s]]<br />
| [[7L&nbsp;2s]]<br />
| [[7L&nbsp;3s]]<br />
| [[7L&nbsp;4s]]<br />
| [[7L&nbsp;5s]]<br />
| [[7L&nbsp;6s]]<br />
| [[7L&nbsp;7s]]<br />
| [[7L&nbsp;8s]]<br />
| [[7L&nbsp;9s]]<br />
| [[7L&nbsp;10s]]<br />
| [[7L&nbsp;11s]]<br />
| [[7L&nbsp;12s]]<br />
|-<br />
| [[8L&nbsp;1s]]<br />
| [[8L&nbsp;2s]]<br />
| [[8L&nbsp;3s]]<br />
| [[8L&nbsp;4s]]<br />
| [[8L&nbsp;5s]]<br />
| [[8L&nbsp;6s]]<br />
| [[8L&nbsp;7s]]<br />
| [[8L&nbsp;8s]]<br />
| [[8L&nbsp;9s]]<br />
| [[8L&nbsp;10s]]<br />
| [[8L&nbsp;11s]]<br />
| [[8L&nbsp;12s]]<br />
|-<br />
| [[9L&nbsp;1s]]<br />
| [[9L&nbsp;2s]]<br />
| [[9L&nbsp;3s]]<br />
| [[9L&nbsp;4s]]<br />
| [[9L&nbsp;5s]]<br />
| [[9L&nbsp;6s]]<br />
| [[9L&nbsp;7s]]<br />
| [[9L&nbsp;8s]]<br />
| [[9L&nbsp;9s]]<br />
| [[9L&nbsp;10s]]<br />
| [[9L&nbsp;11s]]<br />
| [[9L&nbsp;12s]]<br />
|-<br />
| [[10L&nbsp;1s]]<br />
| [[10L&nbsp;2s]]<br />
| [[10L&nbsp;3s]]<br />
| [[10L&nbsp;4s]]<br />
| [[10L&nbsp;5s]]<br />
| [[10L&nbsp;6s]]<br />
| [[10L&nbsp;7s]]<br />
| [[10L&nbsp;8s]]<br />
| [[10L&nbsp;9s]]<br />
| [[10L&nbsp;10s]]<br />
| [[10L&nbsp;11s]]<br />
| [[10L&nbsp;12s]]<br />
|-<br />
| [[11L&nbsp;1s]]<br />
| [[11L&nbsp;2s]]<br />
| [[11L&nbsp;3s]]<br />
| [[11L&nbsp;4s]]<br />
| [[11L&nbsp;5s]]<br />
| [[11L&nbsp;6s]]<br />
| [[11L&nbsp;7s]]<br />
| [[11L&nbsp;8s]]<br />
| [[11L&nbsp;9s]]<br />
| [[11L&nbsp;10s]]<br />
| [[11L&nbsp;11s]]<br />
| [[11L&nbsp;12s]]<br />
|-<br />
| [[12L&nbsp;1s]]<br />
| [[12L&nbsp;2s]]<br />
| [[12L&nbsp;3s]]<br />
| [[12L&nbsp;4s]]<br />
| [[12L&nbsp;5s]]<br />
| [[12L&nbsp;6s]]<br />
| [[12L&nbsp;7s]]<br />
| [[12L&nbsp;8s]]<br />
| [[12L&nbsp;9s]]<br />
| [[12L&nbsp;10s]]<br />
| [[12L&nbsp;11s]]<br />
| [[12L&nbsp;12s]]<br />
|}<br />
<br />
=== 13 ≤ L ≤ 24, s ≤ 12 ===<br />
{| class="wikitable mw-collapsible mw-collapsed center-all"<br />
|+ style="font-size: 105%; white-space: nowrap;" | Pages for MOS scales ({{nowrap|13 ≤ L ≤ 24|s ≤ 12}})<br />
|-<br />
| [[13L&nbsp;1s]]<br />
| [[13L&nbsp;2s]]<br />
| [[13L&nbsp;3s]]<br />
| [[13L&nbsp;4s]]<br />
| [[13L&nbsp;5s]]<br />
| [[13L&nbsp;6s]]<br />
| [[13L&nbsp;7s]]<br />
| [[13L&nbsp;8s]]<br />
| [[13L&nbsp;9s]]<br />
| [[13L&nbsp;10s]]<br />
| [[13L&nbsp;11s]]<br />
| [[13L&nbsp;12s]]<br />
|-<br />
| [[14L&nbsp;1s]]<br />
| [[14L&nbsp;2s]]<br />
| [[14L&nbsp;3s]]<br />
| [[14L&nbsp;4s]]<br />
| [[14L&nbsp;5s]]<br />
| [[14L&nbsp;6s]]<br />
| [[14L&nbsp;7s]]<br />
| [[14L&nbsp;8s]]<br />
| [[14L&nbsp;9s]]<br />
| [[14L&nbsp;10s]]<br />
| [[14L&nbsp;11s]]<br />
| [[14L&nbsp;12s]]<br />
|-<br />
| [[15L&nbsp;1s]]<br />
| [[15L&nbsp;2s]]<br />
| [[15L&nbsp;3s]]<br />
| [[15L&nbsp;4s]]<br />
| [[15L&nbsp;5s]]<br />
| [[15L&nbsp;6s]]<br />
| [[15L&nbsp;7s]]<br />
| [[15L&nbsp;8s]]<br />
| [[15L&nbsp;9s]]<br />
| [[15L&nbsp;10s]]<br />
| [[15L&nbsp;11s]]<br />
| [[15L&nbsp;12s]]<br />
|-<br />
| [[16L&nbsp;1s]]<br />
| [[16L&nbsp;2s]]<br />
| [[16L&nbsp;3s]]<br />
| [[16L&nbsp;4s]]<br />
| [[16L&nbsp;5s]]<br />
| [[16L&nbsp;6s]]<br />
| [[16L&nbsp;7s]]<br />
| [[16L&nbsp;8s]]<br />
| [[16L&nbsp;9s]]<br />
| [[16L&nbsp;10s]]<br />
| [[16L&nbsp;11s]]<br />
| [[16L&nbsp;12s]]<br />
|-<br />
| [[17L&nbsp;1s]]<br />
| [[17L&nbsp;2s]]<br />
| [[17L&nbsp;3s]]<br />
| [[17L&nbsp;4s]]<br />
| [[17L&nbsp;5s]]<br />
| [[17L&nbsp;6s]]<br />
| [[17L&nbsp;7s]]<br />
| [[17L&nbsp;8s]]<br />
| [[17L&nbsp;9s]]<br />
| [[17L&nbsp;10s]]<br />
| [[17L&nbsp;11s]]<br />
| [[17L&nbsp;12s]]<br />
|-<br />
| [[18L&nbsp;1s]]<br />
| [[18L&nbsp;2s]]<br />
| [[18L&nbsp;3s]]<br />
| [[18L&nbsp;4s]]<br />
| [[18L&nbsp;5s]]<br />
| [[18L&nbsp;6s]]<br />
| [[18L&nbsp;7s]]<br />
| [[18L&nbsp;8s]]<br />
| [[18L&nbsp;9s]]<br />
| [[18L&nbsp;10s]]<br />
| [[18L&nbsp;11s]]<br />
| [[18L&nbsp;12s]]<br />
|-<br />
| [[19L&nbsp;1s]]<br />
| [[19L&nbsp;2s]]<br />
| [[19L&nbsp;3s]]<br />
| [[19L&nbsp;4s]]<br />
| [[19L&nbsp;5s]]<br />
| [[19L&nbsp;6s]]<br />
| [[19L&nbsp;7s]]<br />
| [[19L&nbsp;8s]]<br />
| [[19L&nbsp;9s]]<br />
| [[19L&nbsp;10s]]<br />
| [[19L&nbsp;11s]]<br />
| [[19L&nbsp;12s]]<br />
|-<br />
| [[20L&nbsp;1s]]<br />
| [[20L&nbsp;2s]]<br />
| [[20L&nbsp;3s]]<br />
| [[20L&nbsp;4s]]<br />
| [[20L&nbsp;5s]]<br />
| [[20L&nbsp;6s]]<br />
| [[20L&nbsp;7s]]<br />
| [[20L&nbsp;8s]]<br />
| [[20L&nbsp;9s]]<br />
| [[20L&nbsp;10s]]<br />
| [[20L&nbsp;11s]]<br />
| [[20L&nbsp;12s]]<br />
|-<br />
| [[21L&nbsp;1s]]<br />
| [[21L&nbsp;2s]]<br />
| [[21L&nbsp;3s]]<br />
| [[21L&nbsp;4s]]<br />
| [[21L&nbsp;5s]]<br />
| [[21L&nbsp;6s]]<br />
| [[21L&nbsp;7s]]<br />
| [[21L&nbsp;8s]]<br />
| [[21L&nbsp;9s]]<br />
| [[21L&nbsp;10s]]<br />
| [[21L&nbsp;11s]]<br />
| [[21L&nbsp;12s]]<br />
|-<br />
| [[22L&nbsp;1s]]<br />
| [[22L&nbsp;2s]]<br />
| [[22L&nbsp;3s]]<br />
| [[22L&nbsp;4s]]<br />
| [[22L&nbsp;5s]]<br />
| [[22L&nbsp;6s]]<br />
| [[22L&nbsp;7s]]<br />
| [[22L&nbsp;8s]]<br />
| [[22L&nbsp;9s]]<br />
| [[22L&nbsp;10s]]<br />
| [[22L&nbsp;11s]]<br />
| [[22L&nbsp;12s]]<br />
|-<br />
| [[23L&nbsp;1s]]<br />
| [[23L&nbsp;2s]]<br />
| [[23L&nbsp;3s]]<br />
| [[23L&nbsp;4s]]<br />
| [[23L&nbsp;5s]]<br />
| [[23L&nbsp;6s]]<br />
| [[23L&nbsp;7s]]<br />
| [[23L&nbsp;8s]]<br />
| [[23L&nbsp;9s]]<br />
| [[23L&nbsp;10s]]<br />
| [[23L&nbsp;11s]]<br />
| [[23L&nbsp;12s]]<br />
|-<br />
| [[24L&nbsp;1s]]<br />
| [[24L&nbsp;2s]]<br />
| [[24L&nbsp;3s]]<br />
| [[24L&nbsp;4s]]<br />
| [[24L&nbsp;5s]]<br />
| [[24L&nbsp;6s]]<br />
| [[24L&nbsp;7s]]<br />
| [[24L&nbsp;8s]]<br />
| [[24L&nbsp;9s]]<br />
| [[24L&nbsp;10s]]<br />
| [[24L&nbsp;11s]]<br />
| [[24L&nbsp;12s]]<br />
|}<br />
<br />
=== L ≤ 12, 13 ≤ s ≤ 24 ===<br />
{| class="wikitable mw-collapsible mw-collapsed center-all"<br />
|+ style="font-size: 105%; white-space: nowrap;" | Pages for MOS scales ({{nowrap|L ≤ 12|13 ≤ s ≤ 24}})<br />
|-<br />
| [[1L&nbsp;13s]]<br />
| [[1L&nbsp;14s]]<br />
| [[1L&nbsp;15s]]<br />
| [[1L&nbsp;16s]]<br />
| [[1L&nbsp;17s]]<br />
| [[1L&nbsp;18s]]<br />
| [[1L&nbsp;19s]]<br />
| [[1L&nbsp;20s]]<br />
| [[1L&nbsp;21s]]<br />
| [[1L&nbsp;22s]]<br />
| [[1L&nbsp;23s]]<br />
| [[1L&nbsp;24s]]<br />
|-<br />
| [[2L&nbsp;13s]]<br />
| [[2L&nbsp;14s]]<br />
| [[2L&nbsp;15s]]<br />
| [[2L&nbsp;16s]]<br />
| [[2L&nbsp;17s]]<br />
| [[2L&nbsp;18s]]<br />
| [[2L&nbsp;19s]]<br />
| [[2L&nbsp;20s]]<br />
| [[2L&nbsp;21s]]<br />
| [[2L&nbsp;22s]]<br />
| [[2L&nbsp;23s]]<br />
| [[2L&nbsp;24s]]<br />
|-<br />
| [[3L&nbsp;13s]]<br />
| [[3L&nbsp;14s]]<br />
| [[3L&nbsp;15s]]<br />
| [[3L&nbsp;16s]]<br />
| [[3L&nbsp;17s]]<br />
| [[3L&nbsp;18s]]<br />
| [[3L&nbsp;19s]]<br />
| [[3L&nbsp;20s]]<br />
| [[3L&nbsp;21s]]<br />
| [[3L&nbsp;22s]]<br />
| [[3L&nbsp;23s]]<br />
| [[3L&nbsp;24s]]<br />
|-<br />
| [[4L&nbsp;13s]]<br />
| [[4L&nbsp;14s]]<br />
| [[4L&nbsp;15s]]<br />
| [[4L&nbsp;16s]]<br />
| [[4L&nbsp;17s]]<br />
| [[4L&nbsp;18s]]<br />
| [[4L&nbsp;19s]]<br />
| [[4L&nbsp;20s]]<br />
| [[4L&nbsp;21s]]<br />
| [[4L&nbsp;22s]]<br />
| [[4L&nbsp;23s]]<br />
| [[4L&nbsp;24s]]<br />
|-<br />
| [[5L&nbsp;13s]]<br />
| [[5L&nbsp;14s]]<br />
| [[5L&nbsp;15s]]<br />
| [[5L&nbsp;16s]]<br />
| [[5L&nbsp;17s]]<br />
| [[5L&nbsp;18s]]<br />
| [[5L&nbsp;19s]]<br />
| [[5L&nbsp;20s]]<br />
| [[5L&nbsp;21s]]<br />
| [[5L&nbsp;22s]]<br />
| [[5L&nbsp;23s]]<br />
| [[5L&nbsp;24s]]<br />
|-<br />
| [[6L&nbsp;13s]]<br />
| [[6L&nbsp;14s]]<br />
| [[6L&nbsp;15s]]<br />
| [[6L&nbsp;16s]]<br />
| [[6L&nbsp;17s]]<br />
| [[6L&nbsp;18s]]<br />
| [[6L&nbsp;19s]]<br />
| [[6L&nbsp;20s]]<br />
| [[6L&nbsp;21s]]<br />
| [[6L&nbsp;22s]]<br />
| [[6L&nbsp;23s]]<br />
| [[6L&nbsp;24s]]<br />
|-<br />
| [[7L&nbsp;13s]]<br />
| [[7L&nbsp;14s]]<br />
| [[7L&nbsp;15s]]<br />
| [[7L&nbsp;16s]]<br />
| [[7L&nbsp;17s]]<br />
| [[7L&nbsp;18s]]<br />
| [[7L&nbsp;19s]]<br />
| [[7L&nbsp;20s]]<br />
| [[7L&nbsp;21s]]<br />
| [[7L&nbsp;22s]]<br />
| [[7L&nbsp;23s]]<br />
| [[7L&nbsp;24s]]<br />
|-<br />
| [[8L&nbsp;13s]]<br />
| [[8L&nbsp;14s]]<br />
| [[8L&nbsp;15s]]<br />
| [[8L&nbsp;16s]]<br />
| [[8L&nbsp;17s]]<br />
| [[8L&nbsp;18s]]<br />
| [[8L&nbsp;19s]]<br />
| [[8L&nbsp;20s]]<br />
| [[8L&nbsp;21s]]<br />
| [[8L&nbsp;22s]]<br />
| [[8L&nbsp;23s]]<br />
| [[8L&nbsp;24s]]<br />
|-<br />
| [[9L&nbsp;13s]]<br />
| [[9L&nbsp;14s]]<br />
| [[9L&nbsp;15s]]<br />
| [[9L&nbsp;16s]]<br />
| [[9L&nbsp;17s]]<br />
| [[9L&nbsp;18s]]<br />
| [[9L&nbsp;19s]]<br />
| [[9L&nbsp;20s]]<br />
| [[9L&nbsp;21s]]<br />
| [[9L&nbsp;22s]]<br />
| [[9L&nbsp;23s]]<br />
| [[9L&nbsp;24s]]<br />
|-<br />
| [[10L&nbsp;13s]]<br />
| [[10L&nbsp;14s]]<br />
| [[10L&nbsp;15s]]<br />
| [[10L&nbsp;16s]]<br />
| [[10L&nbsp;17s]]<br />
| [[10L&nbsp;18s]]<br />
| [[10L&nbsp;19s]]<br />
| [[10L&nbsp;20s]]<br />
| [[10L&nbsp;21s]]<br />
| [[10L&nbsp;22s]]<br />
| [[10L&nbsp;23s]]<br />
| [[10L&nbsp;24s]]<br />
|-<br />
| [[11L&nbsp;13s]]<br />
| [[11L&nbsp;14s]]<br />
| [[11L&nbsp;15s]]<br />
| [[11L&nbsp;16s]]<br />
| [[11L&nbsp;17s]]<br />
| [[11L&nbsp;18s]]<br />
| [[11L&nbsp;19s]]<br />
| [[11L&nbsp;20s]]<br />
| [[11L&nbsp;21s]]<br />
| [[11L&nbsp;22s]]<br />
| [[11L&nbsp;23s]]<br />
| [[11L&nbsp;24s]]<br />
|-<br />
| [[12L&nbsp;13s]]<br />
| [[12L&nbsp;14s]]<br />
| [[12L&nbsp;15s]]<br />
| [[12L&nbsp;16s]]<br />
| [[12L&nbsp;17s]]<br />
| [[12L&nbsp;18s]]<br />
| [[12L&nbsp;19s]]<br />
| [[12L&nbsp;20s]]<br />
| [[12L&nbsp;21s]]<br />
| [[12L&nbsp;22s]]<br />
| [[12L&nbsp;23s]]<br />
| [[12L&nbsp;24s]]<br />
|}<br />
<br />
=== Larger MOS scales ===<br />
[[12L&nbsp;29s]], [[12L&nbsp;41s]], [[13L&nbsp;14s]], [[14L&nbsp;13s]], [[17L&nbsp;14s]], [[25L&nbsp;6s]]<br />
<br />
== Variations ==<br />
* [[MODMOS scales]] are derived from chromatic alterations of one or more tones of an MOS scale, typically by the interval of {{nowrap| L − s }}, the "chroma".<br />
* [[Muddle]]s are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.<br />
* [[MOS cradle]] is a technique of embedding MOS-like structures inside MOS scales and may or may not produce subsets of MOS scales.<br />
* [[Operations on MOSes]]<br />
<br />
== Listen ==<br />
This is an algorithmically generated recording of every MOS scale that has 14 or fewer notes for a total of 91 scales being showcased here. Each MOS scale played has its simplest step ratio (large step is 2 small step is 1) and therefore is inside the smallest EDO that can support it. Each MOS scale is also in its brightest mode. And rhythmically, each scale is being played with its respective MOS rhythm. Note that changing the mode or step ratio of any of these MOSes may dramatically alter the sound and therefore this recording is not thoroughly representative of each MOS but rather a small taste.<br />
<br />
[[File:Every-MOS-Scale-With-14-Or-Fewer-Notes.mp3|left|800x800px]] {{clear}}<br />
<br />
== See also ==<br />
* [[Diamond-mos notation]], a microtonal [[notation]] system focused on MOS scales<br />
* [[Metallic MOS]], an article focusing on MOS scales based on metallic means, such as [[phi]]<br />
* [[MOS rhythm]]<br />
* [[:Category:MOS scales|Category:MOS scales]], the category including all MOS-related articles on this wiki<br />
* [[Gallery of MOS patterns]]<br />
<br />
== Notes ==<br />
<references group="note" /><br />
<br />
== References ==<br />
<references /><br />
<br />
[[Category:Math]]<br />
[[Category:MOS scale| ]] <!-- Sort order in category: this page shows above A --><br />
[[Category:Scale]]<br />
[[Category:Erv Wilson]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Modus&diff=227916Modus2026-04-16T21:00:20Z<p>ArrowHead294: </p>
<hr />
<div>{{Infobox regtemp<br />
| Title = Modus<br />
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13<br />
| Comma basis = [[64/63]], [[4375/4374]] (7-limit);<br>[[64/63]], [[100/99]], [[243/242]] (11-limit)<br>[[64/63]], [[78/77]], [[100/99]], [[144/143]]<br>(13-limit)<br />
| Edo join 1 = 27e | Edo join 2 = 34d<br />
| Mapping = 1; 4 9 -8 10 -2<br />
| Generators = 10/9<br />
| Generators tuning = 176.8<br />
| Optimization method = CWE<br />
| MOS scales = [[6L&nbsp;1s]], [[7L&nbsp;6s]], [[7L&nbsp;13s]], [[7L&nbsp;20s]]<br />
| Odd limit 1 = 9 | Mistuning 1 = 13.6 | Complexity 1 = 20<br />
| Odd limit 2 = 13 | Mistuning 2 = 16.7 | Complexity 2 = 20<br />
}}<br />
The '''modus''' [[regular temperament|temperament]] is one of the [[7-limit]] [[extension]]s of [[tetracot]], the [[5-limit]] temperament [[tempering out]] the [[tetracot comma]] (20000/19683), and is naturally a full [[13-limit]] temperament. <br />
<br />
In addition to the tetracot comma, modus tempers out [[64/63]], making it a member of the [[archytas clan]]. As such, septimal intervals are tempered together with Pythagorean intervals; in particular, a stack of two perfect fifths [[octave reduction|octave reduced]] represents {{nowrap|[[8/7]][[~]][[9/8]]}} at 8 generator steps. Modus also tempers out [[4375/4374]], making it a [[ragismic microtemperaments|ragismic temperament]]. In the 11- and 13-limit it can be viewed as a [[weak extension]] of [[suhajira]] as well. <br />
<br />
Additionally, the generator can be taken to represent [[21/19]], which gives us an extension for prime 19 at −5 generator steps. <br />
<br />
Modus was named by [[Mike Battaglia]] in 2012 for its fantastic [[modmos]] structures<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_102416.html#102467 Yahoo! Tuning Group | ''Guaranteed meantone successor'']</ref>.<br />
<br />
See [[Tetracot family #Modus]] for technical data. <br />
<br />
== Interval chain ==<br />
In the following tables, odd harmonics 1–13 and their inverses are in '''bold'''. <br />
<br />
{| class="wikitable center-1 right-2"<br />
|-<br />
! #<br />
! Cents*<br />
! Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| 176.9<br />
| 10/9, 11/10<br />
|-<br />
| 2<br />
| 353.7<br />
| 11/9, '''16/13'''<br />
|-<br />
| 3<br />
| 530.6<br />
| 15/11<br />
|-<br />
| 4<br />
| 707.5<br />
| '''3/2'''<br />
|-<br />
| 5<br />
| 884.4<br />
| 5/3<br />
|-<br />
| 6<br />
| 1061.2<br />
| 11/6, 13/7, 24/13<br />
|-<br />
| 7<br />
| 38.1<br />
| 36/35, 40/39, 45/44, 55/54<br />
|-<br />
| 8<br />
| 215.0<br />
| '''8/7''', '''9/8'''<br />
|-<br />
| 9<br />
| 391.9<br />
| '''5/4'''<br />
|-<br />
| 10<br />
| 568.7<br />
| '''11/8''', 18/13<br />
|-<br />
| 11<br />
| 745.6<br />
| 20/13<br />
|-<br />
| 12<br />
| 922.5<br />
| 12/7, 22/13<br />
|-<br />
| 13<br />
| 1099.4<br />
| 15/8, 40/21<br />
|-<br />
| 14<br />
| 76.2<br />
| 22/21, 25/24, 27/26<br />
|-<br />
| 15<br />
| 253.1<br />
| 15/13<br />
|-<br />
| 16<br />
| 430.0<br />
| 9/7<br />
|-<br />
| 17<br />
| 606.8<br />
| 10/7<br />
|-<br />
| 18<br />
| 783.7<br />
| 11/7<br />
|-<br />
| 19<br />
| 960.6<br />
| 45/26<br />
|-<br />
| 20<br />
| 1137.5<br />
| 27/14<br />
|}<br />
<nowiki/>* in 13-limit CWE tuning<br />
<br />
== Tunings ==<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings<br />
|-<br />
! rowspan="2" | <br />
! colspan="3" | Euclidean<br />
|-<br />
! Constrained<br />
! Constrained & skewed<br />
! Destretched<br />
|-<br />
! Tenney<br />
| CTE: ~10/9 = 176.8176{{c}}<br />
| CWE: ~10/9 = 177.1188{{c}}<br />
| POTE: ~10/9 = 177.2035{{c}}<br />
|}<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings<br />
|-<br />
! rowspan="2" | <br />
! colspan="3" | Euclidean<br />
|-<br />
! Constrained<br />
! Constrained & skewed<br />
! Destretched<br />
|-<br />
! Tenney<br />
| CTE: ~10/9 = 176.4456{{c}}<br />
| CWE: ~10/9 = 176.9286{{c}}<br />
| POTE: ~10/9 = 177.0530{{c}}<br />
|}<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings<br />
|-<br />
! rowspan="2" | <br />
! colspan="3" | Euclidean<br />
|-<br />
! Constrained<br />
! Constrained & skewed<br />
! Destretched<br />
|-<br />
! Tenney<br />
| CTE: ~10/9 = 176.4708{{c}}<br />
| CWE: ~10/9 = 176.8735{{c}}<br />
| POTE: ~10/9 = 176.9532{{c}}<br />
|}<br />
<br />
=== Tuning spectrum ===<br />
{| class="wikitable center-all left-4"<br />
|-<br />
! Edo<br>generator<br />
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*<br />
! Generator (¢)<br />
! Comments<br />
|-<br />
| <br />
| 11/10<br />
| 165.004<br />
| <br />
|-<br />
| 1\7<br />
| <br />
| 171.429<br />
| <br />
|-<br />
| <br />
| 11/9<br />
| 173.704<br />
| <br />
|-<br />
| <br />
| 11/6<br />
| 174.894<br />
| <br />
|-<br />
| <br />
| 11/8<br />
| 175.132<br />
| <br />
|-<br />
| <br />
| 3/2<br />
| 175.489<br />
| <br />
|-<br />
| <br />
| 13/11<br />
| 175.899<br />
| <br />
|-<br />
| <br />
| 15/8<br />
| 176.021<br />
| <br />
|-<br />
| <br />
| 5/4<br />
| 176.257<br />
| 5-odd-limit minimax<br />
|-<br />
| <br />
| 13/9<br />
| 176.338<br />
| <br />
|-<br />
| 5\34<br />
| <br />
| 176.471<br />
| 34d val, lower bound of 7- to 15-odd-limit diamond monotone<br />
|-<br />
| <br />
| 15/13<br />
| 176.516<br />
| <br />
|-<br />
| <br />
| 11/7<br />
| 176.805<br />
| 11-, 13- and 15-odd-limit minimax<br />
|-<br />
| <br />
| 5/3<br />
| 176.872<br />
| <br />
|-<br />
| <br />
| 13/10<br />
| 176.890<br />
| <br />
|-<br />
| <br />
| 13/12<br />
| 176.905<br />
| <br />
|-<br />
| 9\61<br />
| <br />
| 177.049<br />
| 61de val<br />
|-<br />
| <br />
| 15/14<br />
| 177.116<br />
| <br />
|-<br />
| <br />
| 9/7<br />
| 177.193<br />
| 9-odd-limit minimax<br />
|-<br />
| <br />
| 7/5<br />
| 177.499<br />
| 7-odd-limit minimax<br />
|-<br />
| <br />
| 7/6<br />
| 177.761<br />
| <br />
|-<br />
| 4\27<br />
| <br />
| 177.778<br />
| 27e val, upper bound of 11- to 15-odd-limit diamond monotone<br />
|-<br />
| <br />
| 13/7<br />
| 178.617<br />
| <br />
|-<br />
| <br />
| 7/4<br />
| 178.897<br />
| <br />
|-<br />
| <br />
| 15/11<br />
| 178.984<br />
| <br />
|-<br />
| <br />
| 13/8<br />
| 179.736<br />
| <br />
|-<br />
| 3\20<br />
| <br />
| 180.000<br />
| 20ce val, upper bound of 7- and 9-odd-limit diamond monotone<br />
|-<br />
| <br />
| 9/5<br />
| 182.404<br />
| <br />
|}<br />
<nowiki/>* Besides the octave<br />
<br />
== Music ==<br />
See [[Tetracot #Music]]. <br />
<br />
== References ==<br />
<br />
[[Category:Modus| ]] <!-- main article --><br />
[[Category:Rank-2 temperaments]]<br />
[[Category:Tetracot family]]<br />
[[Category:Archytas clan]]<br />
[[Category:Ragismic microtemperaments]]</div>ArrowHead294https://en.xen.wiki/index.php?title=User:ArrowHead294&diff=227181User:ArrowHead2942026-03-31T22:02:49Z<p>ArrowHead294: /* Miscellaneous */</p>
<hr />
<div>{{subpage|sandbox|l|text=Hi!}} I'm primarily a music theorist and singer nowadays, though most of my musical training is in classical piano and cello.<br />
__NOTOC__<br />
== My musical experience ==<br />
Piano and keyboard: 2004–2017 (K–12), 2022–present<br />
<br />
Cello: 2009–2017 (Grades 5–12)<br />
<br />
Voice: 2020–present<br />
: Voice type: Baritone / bass-baritone<br />
:: Chest voice: E♭2 to E4<br />
::: Can go down to D2 if needed<br />
::: Can belt up to ~G4, A♭4 on occasion<br />
:: Mixed voice: C4–C5<br />
:: Falsetto and head voice: D4 to E♭5<br />
::: Can reach E5 and F5<br />
<br />
I've also been told I have "perfect pitch" since I acquired the ability to recognise notes (in {{nowrap|A {{=}} 440 Hz}} and 12edo) from a young age, though I have grown increasingly disdainful towards the terms "perfect pitch" and "absolute pitch" since late 2022 (the start of my last term of college) since it locked me into 12edo with {{nowrap|A {{=}} 440 Hz}} and anything else sounded "wrong" to me. This is even more true nowadays since I now prefer other tunings over 12edo in a lot of cases.<br />
<br />
== Favourite tunings within Western music ==<br />
{{Main|User:ArrowHead294/EDO impressions|l1=EDO impressions}}<br />
Most Western musicians only know 12edo, and my musical background is mainly classical, so my main interests have been in Pythagorean and meantone. I'm quite active in church despite being non-religious, and most of the alternative tunings I introduce to others are flatter-than-12 meantones as they're relatively easy to get into.<br />
<br />
I've known about quarter tones ([[24edo]]) since I was very young. I've even dabbled around with it before 2022, and used its absolute frequencies with {{nowrap|A {{=}} 440 Hz}} as a way to compare just intonation and equal temperament, though for most of my life that was the only microtonal tuning I knew about. As a result, my knowledge of just intonation was very much limited to the 2.3.5.11.37 subgroup.<br />
<br />
Since late 2022, I've also gotten into sixth tones ([[36edo]]) for exploring septimal harmonies.<br />
<br />
[[31edo]] is the first alternative tuning I learned about which helped me break out of 12-TET's walled garden, and [[Sevish]]'s song ''Better Left Unanswered'' is the first microtonal work I've ever heard that wasn't in standard quarter tones. I learned about it mainly after exploring quarter-comma meantone in classical music, and I think it's the most practical alternate tuning for most Western musicians to get into. It has excellent 7-limit and 11-limit harmonies as well (even better than 36edo), so it could also be useful for blues and jazz.<br />
<br />
This was followed by [[19edo]], after hearing ''Sunsrise'' by Supahstar Saga. 19 has its unique quirks that make it a good tuning for a lot of Western music, though I think its sound is best suited to songs with largely pentatonic melodies since the diatonic scale sounds quite loose to me. Augmented and diminished chords sound very weird, though, even more jarring than 31.<br />
<br />
=== Beyond traditional Western music ===<br />
For xenharmony and music beyond meantone, I've been mainly interested in [[22edo]] for [[superpyth]]agorean and [[Porcupine]] temperament, [[Orwell]] in 31edo, [[34edo]] for [[Tetracot]] (including [[68edo]] for [[Octacot]] by extension), and [[46edo]] for [[Sensi]] and 5-limit harmony beyond meantone.<br />
<br />
== Miscellaneous ==<br />
{| class="mw-collapsible mw-collapsed"<br />
|-<br />
! <br />
|-<br />
| The contemporary version of the {{w|Lord's Prayer}} goes:<br />
<br />
<pre><br />
Our Father in heaven,<br />
hallowed be your name,<br />
your kingdom come,<br />
your will be done,<br />
on earth as in heaven.<br />
Give us today our daily bread.<br />
Forgive us our sins<br />
as we forgive those who sin against us.<br />
Save us from the time of trial<br />
and deliver us from evil.<br />
For the kingdom, the power, and the glory are yours<br />
now and for ever.<br />
Amen.<br />
</pre><br />
|}<br />
<br />
When encoded in {{w|hexadecimal}}, this becomes:<br />
<br />
<pre><br />
4f 75 72 20 46 61 74 68 65 72 20 69 6e 20 68 65 61 76 65 6e 2c 0d 0a 68 61 6c 6c 6f 77 65 64 20 62 65 20 79 6f 75 72 20 6e 61 6d 65 2c 0d 0a 79 6f 75 72 20 6b 69 6e 67 64 6f 6d 20 63 6f 6d 65 2c 0d 0a 79 6f 75 72 20 77 69 6c 6c 20 62 65 20 64 6f 6e 65 2c 0d 0a 6f 6e 20 65 61 72 74 68 20 61 73 20 69 6e 20 68 65 61 76 65 6e 2e 0d 0a 47 69 76 65 20 75 73 20 74 6f 64 61 79 20 6f 75 72 20 64 61 69 6c 79 20 62 72 65 61 64 2e 0d 0a 46 6f 72 67 69 76 65 20 75 73 20 6f 75 72 20 73 69 6e 73 0d 0a 61 73 20 77 65 20 66 6f 72 67 69 76 65 20 74 68 6f 73 65 20 77 68 6f 20 73 69 6e 20 61 67 61 69 6e 73 74 20 75 73 2e 0d 0a 53 61 76 65 20 75 73 20 66 72 6f 6d 20 74 68 65 20 74 69 6d 65 20 6f 66 20 74 72 69 61 6c 0d 0a 61 6e 64 20 64 65 6c 69 76 65 72 20 75 73 20 66 72 6f 6d 20 65 76 69 6c 2e 0d 0a 46 6f 72 20 74 68 65 20 6b 69 6e 67 64 6f 6d 2c 20 74 68 65 20 70 6f 77 65 72 2c 20 61 6e 64 20 74 68 65 20 67 6c 6f 72 79 20 61 72 65 20 79 6f 75 72 73 0d 0a 6e 6f 77 20 61 6e 64 20 66 6f 72 20 65 76 65 72 2e 0d 0a 41 6d 65 6e 2e<br />
</pre><br />
<br />
When encoded in {{w|Base64}} with the RFC&nbsp;4648 and numeral-first alphabets, this becomes (respectively):<br />
<br />
<pre style="word-break: break-all;"><br />
T3VyIEZhdGhlciBpbiBoZWF2ZW4sDQpoYWxsb3dlZCBiZSB5b3VyIG5hbWUsDQp5b3VyIGtpbmdkb20gY29tZSwNCnlvdXIgd2lsbCBiZSBkb25lLA0Kb24gZWFydGggYXMgaW4gaGVhdmVuLg0KR2l2ZSB1cyB0b2RheSBvdXIgZGFpbHkgYnJlYWQuDQpGb3JnaXZlIHVzIG91ciBzaW5zDQphcyB3ZSBmb3JnaXZlIHRob3NlIHdobyBzaW4gYWdhaW5zdCB1cy4NClNhdmUgdXMgZnJvbSB0aGUgdGltZSBvZiB0cmlhbA0KYW5kIGRlbGl2ZXIgdXMgZnJvbSBldmlsLg0KRm9yIHRoZSBraW5nZG9tLCB0aGUgcG93ZXIsIGFuZCB0aGUgZ2xvcnkgYXJlIHlvdXJzDQpub3cgYW5kIGZvciBldmVyLg0KQW1lbi4=<br />
</pre><br />
<br />
<pre style="word-break: break-all;"><br />
JtLo84PXT6XbSY1fRY1ePM5sPMui3GfeOMniRtTbP21YPI1vRtLo86vXRMKi3GfvRtLo86jfRcTaRsqWOszjPImD2dblTN8WTsbiR21YPI1aRsvbB0qARsuWPM5oT6WWONCWQMuWQ6LXTcLkBWqAHsbsPI1rSo1qRsHXUI1lTN8WP65fR7aWOd9bOMGk3Gf6Rt9dQNPb87Lp86zrSY1pQMvp3GfXSo1tPI1cRt9dQNPb87HeRtDb87TeRo1pQMuWOMTXQMvpT21rSouD2bDXTcKWTNCWPd9lRI1qQ6KWT6bjPI1lPY1qScbXR0qAOMva86HbR6bsPN8WTNCWPd9lRI1bTcbiBWqAHczo87HePI1hQMvdP6zjB21qQ6KWS6ztPN8i865kP21qQ6KWPsnlSdaWON9b87blTN9p3GfkRtSWOMva86PlSY1bTcLoBWqAGMrbRYu=<br />
</pre><br />
<br />
== See also ==<br />
* {{subpage|Purely consistent EDOs by odd limit}}<br />
* {{subpage|Regex snippets}}<br />
* {{subpage|The Star-Spangled Banner}}<br />
* {{subpage|UTF-8 extensions}}<br />
<br />
[[Category:User en-N]]<br />
[[Category:User zh-3]]<br />
[[Category:User de-1]]<br />
[[Category:User on Discord]]<br />
[[Category:User in EST/EDT]]</div>ArrowHead294https://en.xen.wiki/index.php?title=MOS_scale&diff=227096MOS scale2026-03-30T02:56:31Z<p>ArrowHead294: </p>
<hr />
<div>{{interwiki<br />
| en = MOS scale<br />
| de = MOS-Skala<br />
| es =<br />
| ja = MOSスケール<br />
| ro = G2S<br />
}}{{Beginner|Mathematics of MOS}}<br />
A '''moment of symmetry''' ('''MOS''' or '''mos'''<ref group="note">The acronym "MOS" is generally pronounced ''em-oh-ess'', while the {{w|anacronym}} "mos", more common in informal and experimental settings, is generally pronounced ''moss''. Sometimes "MOSS" or "moss", standing for "moment of symmetry scale", are used instead, although there is no significant difference in meaning.</ref>) '''scale''' is a [[periodic scale]] where every 2nd (that is, every interval formed by ascending a step) is either small or large with no in-between, and the same goes for 3rds, 4ths, etc. Multiples of the period (which is usually the octave or a fraction thereof), however, come in only one size.<br />
<br />
MOS scales are often referred to as MOSes, thus MOS can be used as either an adjective or a noun.<br />
<br />
== Examples ==<br />
The most widely used MOS scale is the [[5L 2s|diatonic scale]]. It has 7 steps: 5 large ones (major 2nds) and 2 small ones (minor 2nds), and thus is named 5L&nbsp;2s. The major mode is LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode).<br />
<br />
{| class="wikitable"<br />
|+ style="font-size: 105%;" | Interval classes in the 5L&nbsp;2s MOS scale<br />
|-<br />
! rowspan="2" | Interval class<br />
! colspan="2" | Small version<br />
! colspan="2" | Large version<br />
|-<br />
! Quality<br />
! Size<br />
! Quality<br />
! Size<br />
|-<br />
! 2nds (1 step)<br />
| minor<br />
| s<br />
| major<br />
| L<br />
|-<br />
! 3rds (2 steps)<br />
| minor<br />
| {{nowrap|1L + 1s}}<br />
| major<br />
| 2L<br />
|-<br />
! 4ths (3 steps)<br />
| perfect<br />
| {{nowrap|2L + 1s}}<br />
| augmented<br />
| 3L<br />
|-<br />
! 5ths (4 steps)<br />
| diminished<br />
| {{nowrap|2L + 2s}}<br />
| perfect<br />
| {{nowrap|3L + 1s}}<br />
|-<br />
! 6ths (5 steps)<br />
| minor<br />
| {{nowrap|3L + 2s}}<br />
| major<br />
| {{nowrap|4L + 1s}}<br />
|-<br />
! 7ths (6 steps)<br />
| minor<br />
| {{nowrap|4L + 2s}}<br />
| major<br />
| {{nowrap|5L + 1s}}<br />
|-<br />
! 8ves (7 steps)<br />
| perfect<br />
| {{nowrap|5L + 2s}}<br />
| colspan="2" | (only one version)<br />
|}<br />
<br />
Note that the melodic minor scale (LsLLLLs) has only two step sizes, but it is not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.<br />
<br />
Other MOS scales include [[2L&nbsp;3s]], where among its 5 modes are the major pentatonic scale (ssLsL) and the minor pentatonic scale (LssLs), and less commonly [[4L&nbsp;4s]], also known as the octatonic scale in 12edo, which alternates between large and small steps and comes in two modes (LsLsLsLs and sLsLsLsL).<br />
<br />
See the [[catalog of MOS]] for other MOS scales.<br />
<br />
== Periods and generators ==<br />
Every MOS scale can be ''generated'' by stacking a certain interval called the [[generator]] and octave-reducing (or more generally, [[period]]-reducing). For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, 2L&nbsp;3s is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as {{nowrap|C D E F G A C}} does not produces a MOS, because there are more than 2 sizes of each interval class. <br />
<br />
The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53&hellip; However for a quarter-comma meantone fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50&hellip; <br />
<br />
== Step ratio spectrum ==<br />
The [[step ratio]] is the ratio of the larger step size to the smaller step size. MOSes with smaller step ratios sound smooth and soft. MOSes with larger step ratios sound jagged and hard. Different step ratios produce different corresponding potential temperament interpretations. The [[TAMNAMS#Step ratio spectrum|TAMNAMS]] system has names for specific ratios and also ranges of ratios.<br />
<br />
When the step ratio is a rational number, the MOS is tuned to an edo. A counter-example is 5L&nbsp;2s tuned to quarter-comma meantone, which has a step ratio of about 1.649.<br />
<br />
{| class="wikitable"<br />
|+ style="font-size: 105%;" | 5L&nbsp;2s step ratios in various edos<br />
|-<br />
! Example edo<br />
! Step ratio<br />
! TAMNAMS name<br />
! Likely temperament<br />interpretations<br />
|-<br />
! 12<br />
| 2:1<br />
| basic<br />
| [[Meantone]] or [[Schismatic]]<br />
|-<br />
! 19<br />
| 3:2<br />
| soft<br />
| [[Meantone]]<br />
|-<br />
! 22<br />
| 4:1<br />
| superhard<br />
| [[Archy]] or [[Superpyth]]<br />
|}<br />
<br />
== Naming ==<br />
Every MOS can be uniquely specified by giving its [[signature]], i.e. the number of large and small steps, which is typically notated e.g. "5L&nbsp;2s,". Every possible signature corresponds to a valid MOS scale. Sometimes, if one simply wants to talk about step sizes without specifying which is large and small, the notation "5a&nbsp;2b" is used (which could refer to either diatonic or {{nowrap|[[2L 5s|anti-diatonic]] {{=}} 2L 5s}}).<br />
<br />
By default, the [[equave]] of a MOS is assumed to be [[2/1]]. To specify a non-octave equave, "{{angbr|equave}}" is placed after the signature, e.g. {{mos scalesig|4L 5s<3/1>|link=1}}. Using angle brackets (<code>&#x26;#x27E8;</code> and <code>&#x26;#x27E9;</code>) is recommended; using greater-than and less-than signs ("&#x3C;equave&#x3E;") can also be done, but this can conflict with HTML and other uses of these symbols.<br />
<br />
Several naming systems have been proposed for MOSes, which can be seen at [[MOS naming]].<br />
<br />
== History and terminology ==<br />
The term ''MOS'', and the method of scale construction it entails, were invented by [[Erv Wilson]] in 1975. His original paper is archived on Anaphoria.com here: [https://anaphoria.com/mos.pdf ''Moments of Symmetry'']. There is also an introduction by [[Kraig Grady]] here: [https://anaphoria.com/wilsonintroMOS.html ''Introduction to Erv Wilson's Moments of Symmetry''].<br />
<br />
Sometimes, scales are defined with respect to a period and an additional [[equivalence interval]], the interval at which pitch classes repeat. MOSes in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called '''Multi-MOSes'''. For example, a MOS with a half-octave period is called a '''2mos''', with a 1/3-octave period a '''3mos''', and so on. MOSes in which the equivalence interval is equal to the period are sometimes called '''Strict MOSes'''. MOSes in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.<br />
<br />
With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional evenness|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt<ref>Norman Carey and David Clampitt. "Aspects of Well-Formed Scales", ''Music Theory Spectrum'', Vol. 11, No. 2 (Autumn, 1989), pp. 187-206.</ref>. A great deal of work has been done in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilson's concept. They are in a sense the MOS of MOS patterns. This is used to explain the [[Pentatonic|pentatonics]] used in traditional [[Japanese music]] (e.g. {{nowrap|A B C E F A}}), where the 5-tone cycles are derived from a 7-tone MOS, which are not found in the concept of DE.<br />
<br />
== Equivalent definitions and generalizations ==<br />
A scale is a MOS if and only if it satisfies one of the following equivalent criteria:<br />
<br />
# [[Maximum variety]] 2: Ascending by a certain number of steps is equivalent to ascending by one of at most two intervals, and the maximum of two is achieved (i. e. it is not true that ascending by a certain number of steps is always equivalent to ascending by one interval.) <br />
# [[Binary]] and has a [[generator]]: The scale step comes in exactly two sizes, and the scale is formable from stacking some interval called a generator and octave-reducing.<br />
# Mode of a Christoffel word: The scale can be formed by creating a 2D lattice where the period is on the lattice, then taking pitches by travelling vertically and horizontally from the origin, maintaining as close to the line from the origin to the octave as possible without going above it.<br />
<br />
Each definition generalizes to scales with three or more step sizes, but these generalizations are not equivalent. The concepts of [[Balanced word|balance]] and [[distributional evenness]] provide still different generalizations, although defining MOS through these terms is less helpful. For more information, see [[Mathematics of MOS]].<br />
<br />
== Properties ==<br />
=== Basic properties ===<br />
* For every MOS scale with an [[octave]] period (which is usually the [[octave]]), if ''x''-[[edo]] is the [[collapsed]] tuning (where the small step vanishes) and ''y''-[[edo]] is the [[equalized]] tuning (where the large (''L'') step and small (''s'') step are the same size), then by definition it is an {{nowrap|''x''L (''y'' &minus; ''x'')s}} MOS scale, and the [[basic]] tuning where {{nowrap|''L'' {{=}} 2''s''}} is thus {{nowrap|(''x'' + ''y'')}}-[[edo]]. This is also true if the period is 1\''p'', that is, 1 step of ''p''-[[edo]], which implies that ''x'' and ''y'' are divisible by ''p'', though note that in that case (if {{nowrap|''p'' &gt; 1}}) you are considering a "multiperiod" MOS scale.<br />
* More generally, whenever ''px''-[[edo]] and ''py''-[[edo]] are used to define two [[Val|vals]] (usually but not necessarily through taking the [[Patent val|patent vals]]) while simultaneously also being used to define the {{nowrap|''px''L (''py'' &minus; ''px'')s}} MOS scale (where ''p'' is the number of periods per octave), then the ''px'' & ''py'' temperament corresponds to that MOS scale, and adding ''x'' and/or ''y'' corresponds to tuning closer to ''x''-[[edo]] and/or ''y''-[[edo]] respectively. (Optionally, see the below more precise statement for the mathematically-inclined.)<br />
* For the mathematically-inclined, we can say that whenever we consider a MOS with ''X''/''p'' notes per period in the [[collapsed]] tuning and ''Y''/''p'' notes per period in the [[equalized]] tuning and ''p'' periods per [[Octave stretching|tempered octave]] (or more generally tempered [[equave]]), and whenever we want to associate that MOS with the {{nowrap|''X'' &amp; ''Y''}} rank 2 temperament'''*''', we can say that any {{w|natural number|natural}}-coefficient {{w|linear combination}} of vals {{val|''X'' &hellip;}} and {{val|''Y'' &hellip;}} (where {{nowrap|''X'' &lt; ''Y''}}) corresponds uniquely to a tuning of the {{nowrap|''X'' &amp; ''Y''}} rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive) iff {{nowrap|gcd(''a'', ''b'') {{=}} 1}}, because if {{nowrap|''k'' {{=}} gcd(''a'', ''b'') &gt; 1}} then the val {{nowrap|''a''{{val| ''X'' &hellip;}} + ''b''{{val| ''Y'' &hellip;}}}} has a common factor ''k'' in all of its terms, meaning it is guaranteed to be [[contorted]]. The tuning corresponding to the [[wikipedia:Rational number|rational]] ''a''/''b'' is technically only unique up to (discarding of) [[octave stretching]] (or more generally [[equave]]-tempering).<br />
<br />
: The period of this temperament is {{nowrap|1\gcd(''X'', ''Y'')}}, and the rational ''a''/''b'' is very closely related to the [[step ratio]] of the corresponding MOS scale, because {{nowrap|1{{val| ''X'' &hellip;}} + 0{{val| ''Y'' &hellip;}}}} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 0}} tuning while {{nowrap|0{{val| ''X'' &hellip;}} + 1{{val| ''Y'' &hellip;}}}} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 1}} tuning and {{nowrap|1{{val| ''X'' &hellip;}} + 1{{val| ''Y'' &hellip;}}}} is the {{nowrap|''L'' {{=}} 2|''s'' {{=}} 1}} tuning, so that {{nowrap|''L'' {{=}} ''a'' + ''b''}} and {{nowrap|''s'' {{=}} ''b''}} and therefore:<br />
<br />
: {{nowrap|1/([[step ratio]]) {{=}} ''s''/''L''}} {{nowrap|{{=}} ''b''/(''a'' + ''b'')}} implying {{nowrap|[[step ratio]] {{=}} (''a'' + ''b'')/''b'' &ge; 1}} for [[wikipedia:Natural number|natural]] ''a'' and ''b'', where if {{nowrap|''b'' {{=}} 0}} then the step ratio is infinite, corresponding to the [[collapsed]] tuning.<ref group="note">It is '''important to note''' that the correspondence to the {{nowrap|''X'' &amp; ''Y''}} rank 2 temperament only works in all cases if we allow the temperament to be [[contorted]] on its [[subgroup]]; alternatively, it works if we exclude cases where {{nowrap|''X'' &amp; ''Y''}} describe a contorted temperament on the subgroup given. An example is the {{nowrap|5 &amp; 19}} temperament is contorted in the [[5-limit]] (having a generator of a semifourth, corresponding to [[5L&nbsp;14s]]), so we either need to consider the temperament itself to be contorted (generated by something lacking an interpretation in the subgroup given, two of which yielding a meantone-tempered [[~]][[4/3]]) or we exclude it because of its contortion.</ref><br />
<br />
* Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale ''a''L&nbsp;''b''s are {{nowrap|(''a'' + ''b'')L ''a''s}} (generated by generators of soft-of-basic ''a''L ''b''s) and {{nowrap|''a''L (''a'' + ''b'')s}} (generated by generators of hard-of-basic ''a''L''&nbsp;b''s).<br />
* Every MOS scale (with a specified [[equave]] ''&#x190;''&#x200A;), excluding {{nowrap|''a''L ''a''s{{angbr|''&#x190;''&#x200A;}}}}, has a ''parent MOS''. If {{nowrap|''a'' &gt; ''b''}}, the parent of ''a''L&nbsp;''b''s is {{nowrap|''b''L (''a'' &minus; ''b'')s}}; if {{nowrap|''a'' &lt; ''b''}}, the parent of ''a''L&nbsp;''b''s is {{nowrap|''a''L (''b'' &minus; ''a'')s}}.<br />
<br />
=== Advanced discussion ===<br />
See:<br />
* [[Mathematics of MOS]], a more formal definition and a discussion of the mathematical properties.<br />
** [[Recursive structure of MOS scales]], a description of how MOS scales are recursive and how one scale can be converted into a related scale.<br />
** [[MOS scale family tree]], a tree initially described by Erv Wilson that organizes scales by parent-and-child relationship, which also helps illustrate mos recursion.<br />
* [[Generator ranges of MOS]], organized by number of scale steps and quantity of L/s steps.<br />
* [[MOS diagrams]], visualizations of the MOS process.<br />
* [http://x31eq.com/temper/method.html How to Find Linear Temperaments], by [[Graham Breed]]<br />
<br />
== Variations ==<br />
* [[MODMOS scales]] are derived from chromatic alterations of one or more tones of an MOS scale, typically by the interval of {{nowrap|L &minus; s}}, the "chroma".<br />
* [[Muddle]]s are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.<br />
* [[MOS cradle]] is a technique of embedding MOS-like structures inside MOS scales and may or may not produce subsets of MOS scales.<br />
* [[Operations on MOSes]]<br />
<br />
== Listen ==<br />
This is an algorithmically generated recording of every MOS scale that has 14 or fewer notes for a total of 91 scales being showcased here. Each MOS scale played has its simplest step ratio (large step is 2 small step is 1) and therefore is inside the smallest EDO that can support it. Each MOS scale is also in its brightest mode. And rhythmically, each scale is being played with its respective MOS rhythm. Note that changing the mode or step ratio of any of these MOSes may dramatically alter the sound and therefore this recording is not thoroughly representative of each MOS but rather a small taste.<br />
<br />
[[File:Every-MOS-Scale-With-14-Or-Fewer-Notes.mp3|left|800x800px]] {{clear}}<br />
<br />
== See also ==<br />
* [[Diamond-mos notation]], a microtonal [[notation]] system focused on MOS scales<br />
* [[Metallic MOS]], an article focusing on MOS scales based on metallic means, such as [[phi]]<br />
* [[MOS rhythm]]<br />
* [[:Category:MOS scales|Category:MOS scales]], the category including all MOS-related articles on this wiki<br />
* [[Gallery of MOS patterns]]<br />
<br />
== Notes ==<br />
<references group="note" /><br />
<br />
== References ==<br />
<references /><br />
<br />
[[Category:Math]]<br />
[[Category:MOS scale| ]] <!-- Sort order in category: this page shows above A --><br />
[[Category:Scale]]<br />
[[Category:Erv Wilson]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Tetracot&diff=227095Tetracot2026-03-30T02:55:07Z<p>ArrowHead294: </p>
<hr />
<div>{{Interwiki<br />
| en = Tetracot<br />
| de = Tetracot<br />
| es = <br />
| ja = <br />
}}<br />
{{Infobox regtemp<br />
| Title = Tetracot<br />
| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.13<br />
| Comma basis = [[20000/19683]] (2.3.5);<br>[[100/99]], [[243/242]] (2.3.5.11)<br>[[100/99]], [[144/143]], [[243/242]] (2.3.5.11.13)<br />
| Edo join 1 = 7 | Edo join 2 = 27e<br />
| Mapping = 1; 4 9 10 -2<br />
| Generators = 10/9<br />
| Generators tuning = 176.1<br />
| Optimization method = CWE<br />
| MOS scales = [[6L&nbsp;1s]], [[7L&nbsp;6s]], [[7L&nbsp;13s]]<br />
| Pergen = (P8, P5/4)<br />
| Color name = Saquadyo<br />
| Odd limit 1 = 5 | Mistuning 1 = 3.07 | Complexity 1 = 13<br />
| Odd limit 2 = 2.3.5.11.13 15 | Mistuning 2 = 10.9 | Complexity 2 = 20<br />
}}<br />
{{About|the regular temperament|the ploidacot signature|Ploidacot/Tetracot}}<br />
<br />
'''Tetracot''', in this article, is the [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] in the 2.3.5.11.13 [[subgroup]] [[generator|generated]] by a submajor second of about 174–178{{cent}} which represents both [[10/9]] and [[11/10]]. It is so named because the generator is a quarter of fifth: four such generators make a perfect fifth which approximates [[3/2]], which cannot occur in [[12edo]], resulting in [[100/99]], [[144/143]], and [[243/242]] being [[tempering out|tempered out]]. This is in contrast to [[meantone]], where 10/9 is tuned sharper than or equal to just in order to be equated with [[9/8]].<br />
<br />
Tetracot has many [[extension]]s for the 7-, 11-, and 13-limit. See [[Tetracot extensions]]. Equal temperaments that support tetracot include {{EDOs| 27, 34, and 41 }}.<br />
<br />
See [[Tetracot family]] for more technical data.<br />
<br />
== Intervals ==<br />
=== Interval chain ===<br />
In the following table, odd harmonics and subharmonics 1–15 are in '''bold'''. <br />
<br />
{| class="wikitable right-1 right-2"<br />
|-<br />
! #<br />
! Cents*<br />
! Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| 175.8<br />
| 11/10, 10/9<br />
|-<br />
| 2<br />
| 350.6<br />
| 11/9, '''16/13'''<br />
|-<br />
| 3<br />
| 527.4<br />
| 15/11<br />
|-<br />
| 4<br />
| 703.3<br />
| '''3/2'''<br />
|-<br />
| 5<br />
| 879.1<br />
| 5/3<br />
|-<br />
| 6<br />
| 1054.9<br />
| 11/6, 24/13<br />
|-<br />
| 7<br />
| 30.7<br />
| 55/54, 45/44, 40/39<br />
|-<br />
| 8<br />
| 206.5<br />
| '''9/8'''<br />
|-<br />
| 9<br />
| 382.3<br />
| '''5/4'''<br />
|-<br />
| 10<br />
| 558.2<br />
| '''11/8''', 18/13<br />
|-<br />
| 11<br />
| 734.0<br />
| 20/13<br />
|-<br />
| 12<br />
| 909.8<br />
| 22/13<br />
|-<br />
| 13<br />
| 1085.6<br />
| '''15/8'''<br />
|-<br />
| 14<br />
| 61.4<br />
| 33/32, 27/26, 25/24<br />
|-<br />
| 15<br />
| 237.2<br />
| 15/13<br />
|}<br />
<nowiki/>* In 2.3.5.11.13 subgroup CTE tuning<br />
<br />
=== As a detemperament of 7et ===<br />
[[File: Tetracot 7et Detempering.png|thumb|Tetracot as a 34-tone 7et detempering]]<br />
<br />
Tetracot is considered as a [[cluster temperament]] with 7 clusters of notes in an octave, so it is naturally a [[detemperament]] of the [[7edo|7 equal temperament]]. The diagram on the right shows a 34-tone detempered scale, with a generator range of −16 to +17, which covers all the intervals in the no-7 13-odd-limit. Each category is divided into four or five qualities separated by 7 generator steps, which represent [[40/39]], [[45/44]], [[55/54]], [[65/64]], [[66/65]], [[81/80]], and [[121/120]] all at once. <br />
<br />
== Scales ==<br />
* [[Tetracot7]] – [[6L&nbsp;1s]] scale<br />
* [[Tetracot13]] – improper [[7L&nbsp;6s]]<br />
* [[Tetracot20]] – improper [[7L&nbsp;13s]]<br />
<br />
== Tunings ==<br />
=== Tuning spectrum ===<br />
{| class="wikitable center-all left-4"<br />
|-<br />
! Edo<br>generator<br />
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*<br />
! Generator (¢)<br />
! Comments<br />
|-<br />
| <br />
| 11/10<br />
| 165.004<br />
| <br />
|-<br />
| <br />
| 243/200<br />
| 168.574<br />
| 1/2-comma<br />
|-<br />
| 1\7<br />
| <br />
| 171.429<br />
| Lower bound of 2.3.5.11 subgroup 11-odd-limit,<br />2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone<br />
|-<br />
| <br />
| 27/20<br />
| 173.184<br />
| 1/3-comma<br />
|-<br />
| <br />
| 11/9<br />
| 173.704<br />
| <br />
|-<br />
| <br />
| 81/80<br />
| 174.501<br />
| 2/7-comma<br />
|-<br />
| <br />
| 11/6<br />
| 174.894<br />
| <br />
|-<br />
| 7\48<br />
| <br />
| 175.000<br />
| <br />
|-<br />
| <br />
| 11/8<br />
| 175.132<br />
| 2.3.5.11-subgroup 11-odd-limit minimax<br />
|-<br />
| <br />
| 3/2<br />
| 175.489<br />
| 1/4-comma<br />
|-<br />
| 6\41<br />
| <br />
| 175.610<br />
| <br />
|-<br />
| <br />
| 13/11<br />
| 175.899<br />
| 2.3.5.11.13-subgroup 13- and 15-odd-limit minimax<br />
|-<br />
| <br />
| 15/8<br />
| 176.021<br />
| <br />
|-<br />
| <br />
| 5/4<br />
| 176.257<br />
| 5-odd-limit and 5-limit 9-odd-limit minimax, 2/9-comma<br />
|-<br />
| <br />
| 13/9<br />
| 176.338<br />
| <br />
|-<br />
| 5\34<br />
| <br />
| 176.471<br />
| <br />
|-<br />
| <br />
| 15/13<br />
| 176.516<br />
| <br />
|-<br />
| <br />
| 5/3<br />
| 176.872<br />
| 1/5-comma<br />
|-<br />
| <br />
| 13/10<br />
| 176.890<br />
| <br />
|-<br />
| <br />
| 13/12<br />
| 176.905<br />
| <br />
|-<br />
| 4\27<br />
| <br />
| 177.778<br />
| Upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone<br />
|-<br />
| <br />
| 27/25<br />
| 177.794<br />
| 1/6-comma<br />
|-<br />
| <br />
| 243/125<br />
| 178.452<br />
| 1/7-comma<br />
|-<br />
| <br />
| 15/11<br />
| 178.984<br />
| <br />
|-<br />
| <br />
| 13/8<br />
| 179.736<br />
| <br />
|-<br />
| 3\20<br />
| <br />
| 180.000<br />
| Upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone<br />
|-<br />
| <br />
| 9/5<br />
| 182.404<br />
| <br />
|}<br />
<nowiki/>* Besides the octave<br />
<br />
== Music ==<br />
; [[Flora Canou]]<br />
* [https://soundcloud.com/floracanou/october-dieting-plan?in=floracanou/sets/totmc-suite "October Dieting Plan"] from [https://soundcloud.com/floracanou/sets/totmc-suite ''TOTMC Suite''] (2023–2025) – in [[modus]], 34edo tuning<br />
<br />
; [[Zhea Erose]]<br />
* [https://www.youtube.com/watch?v=xYZwye9PWSo ''Modal Studies in Tetracot''] (2021) – in 34edo tuning<br />
<br />
; [[Dustin Schallert]]<br />
* [https://web.archive.org/web/20201127015111/http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Perc-Sitar.mp3 ''Tetracot Perc-Sitar'']<br />
* [https://web.archive.org/web/20201129105050/http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Jam.mp3 ''Tetracot Jam'']<br />
* [https://web.archive.org/web/20201127012230/http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Pump.mp3 ''Tetracot Pump''] – all in modus, 27edo tuning<br />
<br />
; [[Xotla]]<br />
* "Electrostat" from ''Lesser Groove'' (2020) – [https://open.spotify.com/track/5LIPr8n6uQySeLUfM11U2W Spotify] | [https://xotla.bandcamp.com/track/electrostat-tetracot-13 Bandcamp] | [https://www.youtube.com/watch?v=5SAuoyDwpgc YouTube] – ambient electro in Tetracot[13], 34edo tuning<br />
<br />
[[Category:Tetracot| ]] <!-- Main article --><br />
[[Category:Rank-2 temperaments]]<br />
[[Category:Tetracot family]]<br />
[[Category:Rastmic clan]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Xenharmonic_Wiki:Conventions&diff=227062Xenharmonic Wiki:Conventions2026-03-29T01:16:30Z<p>ArrowHead294: /* Quotation marks and apostrophes */</p>
<hr />
<div>A few conventions have evolved in this wiki that would be good to keep consistent with.<br />
<br />
(You may add conventions to this page if you see them in widespread use on the wiki, or if you discuss them first on the talk page and there is broad consensus in favour of them.)<br />
<br />
== Wanted pages ==<br />
Please do not blindly create pages that appear in the [[Special:WantedPages]] list. Most of them are not actually wanted.<br />
<br />
That list is an automatically generated list based on how many other wiki pages point to a page, but it cannot tell the difference between links written by a human, and links that occur as part of a [[:Category:Templates|functionality template]]. Most of them are "false alarms", in other words.<br />
<br />
Here is an example of some list items on "wanted pages" that are actually legitimate, and do warrant page creation:<br />
<small><br />
* Euzenius (3 links)<br />
* James Plamondon (3 links)<br />
* Joseph Ruhf (3 links)<br />
* LIRP-3 (3 links)<br />
* List model (3 links)<br />
</small><br />
<br />
Here is an example of some list items that are machine-generated noise and should be ignored:<br />
<small><br />
* 4L&nbsp;4s (3/1-equivalent) (5 links)<br />
* 4L&nbsp;6s (3/1-equivalent) (5 links)<br />
* 50th-octave temperaments (5 links)<br />
* 51/20 (5 links)<br />
* 512/405 (5 links)<br />
* 517edo (5 links)<br />
* 520edo (5 links)<br />
</small><br />
<br />
Of course, you can still create one of those machine-wanted pages if you actually do have something to say about it. However, we strongly advise against creating just a blank page, or a page with nothing but infoboxes, as this only creates more work for editors without adding new information for readers.<br />
<br />
An easier and more rewarding alternative to browsing "wanted pages" is to instead check out the pages [[Xenharmonic Wiki:Things to do]] and [[Wikifuture]], which list tasks that really do need to be done.<br />
<br />
Also check out [[:Category:Stubs]] and [[:Category:Todo:expand]]. These are short pages that need to be expanded by writing more content on them. Please help us lengthen them with more content.<br />
<br />
== Decimal numbers ==<br />
Use the English convention with decimal point. (Exception: quotes or entire articles in another language)<br />
<br />
{| class="wikitable"<br />
|-<br />
! Correct<br />
! Avoid (example)<br />
|-<br />
| style="background-color: #afa;" | 701.995<br />
| style="background-color: #faa;" | <s>701,995</s><br />
|}<br />
<br />
== Logarithmic interval measures ==<br />
Logarithmic interval measures should generally be given in [[cent]]s since it is the most standard unit for that purpose: it is more easily understood by musicians already familiar with [[12edo]] and by many software programs, such as the [[Scala]] file format. Alternative [[interval size measure]]s are acceptable if they better reflect the idea behind an interval system, scale, etc., but "competing" columns with logarithmic sizes in the same table should usually be avoided. Simple cases like [[degree]]s alongside cents in an interval chain are acceptable.<br />
<br />
[[Cent]] values should be written up to 3 decimal places in most contexts where precision matters, such as [[Technical data guide for regular temperaments|temperament data pages]] and other technical tables. Very small intervals may be written up to 3 significant figures, and in scientific notation if needed. In contexts where high precision does not matter as much, it is generally sufficient to round to 1 decimal place or to the nearest cent.<br />
<br />
== Quotation marks and apostrophes ==<br />
{{w|wp:Manual of Style#Quotation marks|As on Wikipedia}}, please:<br />
# Use {{preferred|<big>"</big>straight quotes<big>"</big>}}, instead of {{avoid|<big>&ldquo;</big>curly quotes<big>&rdquo;</big>}} (the same applies for single quotes: {{preferred|<big>'</big>straight<big>'</big>}} instead of {{avoid|<big>&lsquo;</big>curly<big>&rsquo;</big>}}), and straight apostrophes ({{preferred|<big>'</big>}}) instead of curly apostrophes ({{avoid|<big>&rsquo;</big>}}). Straight quotes and straight apostrophes are much easier to type on most keyboards than their curly (typographic) counterparts and produce fewer encoding problems.<br />
# Avoid using {{avoid|<big>`</big>accent marks<big>´</big>}} (e.g. backticks) and {{avoid|<big>′</big>prime symbols<big>′</big>}} (including, e.g. {{avoid|<big>″</big>double primes<big>″</big>}}) as quotation marks or apostrophes.<br />
# Avoid using {{avoid|<big>&bdquo;</big>low-high quotes<big>&ldquo;</big>}}, {{avoid|<big>「</big>corner brackets<big>」</big>}}, or {{avoid|<big>«</big>guillemets<big>»</big>}} as quotation marks, except in special cases or for non-English text that uses these marks.<br />
<br />
The templates {{tlx|'}} and {{tlx|'s}} are helpful when an apostrophe or single quote appears at the beginning or end of text in '''bold''' or ''italics'', because bold and italic text are themselves indicated by sequences of single quotes. When an apostrophe follows italicized text, {{tlx|`}} and {{tlx|`s}} can be used to additionally prevent the apostrophe from intersecting the last letter.<br />
<br />
== Links ==<br />
Links—both wiki-internal and external ones—should have meaningful text; something like "[[Conventions|here]]" is not very useful for understanding and orientation, often the page title is a good starting point.<br />
<br />
: ''For more info see [[Help: Here-links]]''<br />
<br />
Internal links should be as precise as possible. So, for example, if you follow a link that promises a definition of a term you don't know, it's not helpful if it brings you to an article full of new information and the expected definition hidden in a massive amount of text. (there is still a lot to be done in this area)<br />
<br />
== Original research ==<br />
'''Original research is allowed on the Xenharmonic Wiki.''' There are a number of conventions in place to help readers distinguish between established ideas or terms and emergent ones.<br />
<br />
Below, ''standard framework'' means the set of established ideas, and ''standard terminology'' means the set of established terms.<br />
<br />
# By default, a page in the {{w|WP:MAINSPACE|main namespace}} should be written using the standard framework and terminology whenever possible.<br />
#* References should be provided to support the established ideas and terms, whenever possible, and people who developed ideas and/or coined terms should be credited accordingly.<br />
#* A page in the main namespace may contain original research if it is presented using the standard framework and terminology as much as possible; e.g. [[Generator sequence]].<br />
#* If multiple conflicting terms coexist for a given concept, the most widespread term should be given priority; e.g. [[Superparticular ratio]] (over "epimoric ratio" or "delta-1 ratio"), but [[Delta-N ratio]] (over "superpartient ratio" and "epimeric ratio").<br />
# A page presenting established ideas that contains a few [[:Category:Pages with idiosyncratic terms|idiosyncratic terms]] (i.e. terms used only by a single person or a small group) should have any idiosyncratic term marked as such using [[Template: Idiosyncratic]]; e.g. [[Musical cells]].<br />
#* It is especially important to give proper credit to people who coined idiosyncratic terms, because that helps readers understand that a term is not widely used.<br />
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#* Alternatively, it maybe be placed as a [[Help: Editing #Subpages|subpage]] in the main namespace, under the topic's main page; e.g. [[22edo/Eliora's approach]]. This option is mainly useful for pages with more than one main contributor.<br />
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<br />
What is considered "established" may change over time and should be discussed between editors as needed. In particular, ideas built on established ideas may be considered established even if they are not as widely known as the ideas they are built on; e.g. [[SN scale]].<br />
<br />
In addition, editors should keep in mind that other editors are more likely to edit pages in the main namespace than ones in the user namespace. In short, if you don't want other users interfering too much with your work, stay in the user namespace; and if you don't want other users interfering at all with your work, publish outside the wiki and provide a link to it on an appropriate wiki page.<br />
<br />
== Naming articles ==<br />
{{Main|Xenharmonic Wiki: Article guidelines}}<br />
<br />
== Naming concepts/objects ==<br />
We have outlined a generic rule for naming temperaments and commas. See: [[Temperament naming #Temperament and comma naming conventions]]. <br />
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== See also ==<br />
* [[Help:Editing]]<br />
* [[Help:Help]]<br />
* [[Help:Migration FAQ]]<br />
* [[Wikifuture]]<br />
* [[Xenharmonic Wiki:Things to do]]<br />
<br />
[[Category:Xenharmonic Wiki guidelines]]</div>ArrowHead294https://en.xen.wiki/index.php?title=MOS_scale&diff=227061MOS scale2026-03-29T01:13:19Z<p>ArrowHead294: </p>
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<div>{{interwiki<br />
| en = MOS scale<br />
| de = MOS-Skala<br />
| es =<br />
| ja = MOSスケール<br />
| ro = G2S<br />
}}{{Beginner|Mathematics of MOS}}<br />
A '''moment of symmetry''' ('''MOS''' or '''mos'''<ref group="note">The acronym "MOS" is generally pronounced ''em-oh-ess'', while the {{w|anacronym}} "mos", more common in informal and experimental settings, is generally pronounced ''moss''. Sometimes "MOSS" or "moss", standing for "moment of symmetry scale", are used instead, although there is no significant difference in meaning.</ref>) '''scale''' is a [[periodic scale]] where every 2nd (that is, every interval formed by ascending a step) is either small or large with no in-between, and the same goes for 3rds, 4ths, etc. Multiples of the period (which is usually the octave or a fraction thereof), however, come in only one size.<br />
<br />
MOS scales are often referred to as MOSes, thus MOS can be used as either an adjective or a noun.<br />
<br />
== Examples ==<br />
The most widely used MOS scale is the [[5L 2s|diatonic scale]]. It has 7 steps: 5 large ones (major 2nds) and 2 small ones (minor 2nds), and thus is named 5L&nbsp;2s. The major mode is LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode).<br />
<br />
{| class="wikitable"<br />
|+ style="font-size: 105%;" | Interval classes in the 5L&nbsp;2s MOS scale<br />
|-<br />
! rowspan="2" | Interval class<br />
! colspan="2" | Small version<br />
! colspan="2" | Large version<br />
|-<br />
! Quality<br />
! Size<br />
! Quality<br />
! Size<br />
|-<br />
! 2nds (1 step)<br />
| minor<br />
| s<br />
| major<br />
| L<br />
|-<br />
! 3rds (2 steps)<br />
| minor<br />
| {{nowrap|1L + 1s}}<br />
| major<br />
| 2L<br />
|-<br />
! 4ths (3 steps)<br />
| perfect<br />
| {{nowrap|2L + 1s}}<br />
| augmented<br />
| 3L<br />
|-<br />
! 5ths (4 steps)<br />
| diminished<br />
| {{nowrap|2L + 2s}}<br />
| perfect<br />
| {{nowrap|3L + 1s}}<br />
|-<br />
! 6ths (5 steps)<br />
| minor<br />
| {{nowrap|3L + 2s}}<br />
| major<br />
| {{nowrap|4L + 1s}}<br />
|-<br />
! 7ths (6 steps)<br />
| minor<br />
| {{nowrap|4L + 2s}}<br />
| major<br />
| {{nowrap|5L + 1s}}<br />
|-<br />
! 8ves (7 steps)<br />
| perfect<br />
| {{nowrap|5L + 2s}}<br />
| colspan="2" | (only one version)<br />
|}<br />
<br />
Note that the melodic minor scale (LsLLLLs) has only two step sizes, but it is not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.<br />
<br />
Other MOS scales include [[2L&nbsp;3s]], where among its 5 modes are the major pentatonic scale (ssLsL) and the minor pentatonic scale (LssLs), and less commonly [[4L&nbsp;4s]], also known as the octatonic scale in 12edo, which alternates between large and small steps and comes in two modes (LsLsLsLs and sLsLsLsL).<br />
<br />
See the [[catalog of MOS]] for other MOS scales.<br />
<br />
== Periods and generators ==<br />
Every MOS scale can be ''generated'' by stacking a certain interval called the [[generator]] and octave-reducing (or more generally, [[period]]-reducing). For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, 2L&nbsp;3s is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as {{nowrap|C D E F G A C}} does not produces a MOS, because there are more than 2 sizes of each interval class. <br />
<br />
The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53... However for a quarter-comma meantone fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50... <br />
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== Step ratio spectrum ==<br />
The [[step ratio]] is the ratio of the larger step size to the smaller step size. MOSes with smaller step ratios sound smooth and soft. MOSes with larger step ratios sound jagged and hard. Different step ratios produce different corresponding potential temperament interpretations. The [[TAMNAMS#Step ratio spectrum|TAMNAMS]] system has names for specific ratios and also ranges of ratios.<br />
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When the step ratio is a rational number, the MOS is tuned to an edo. A counter-example is 5L&nbsp;2s tuned to quarter-comma meantone, which has a step ratio of about 1.649.<br />
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{| class="wikitable"<br />
|+ style="font-size: 105%;" | 5L&nbsp;2s step ratios in various edos<br />
|-<br />
! Example edo<br />
! Step ratio<br />
! TAMNAMS name<br />
! Likely temperament<br />interpretations<br />
|-<br />
! 12<br />
| 2:1<br />
| basic<br />
| [[Meantone]] or [[Schismatic]]<br />
|-<br />
! 19<br />
| 3:2<br />
| soft<br />
| [[Meantone]]<br />
|-<br />
! 22<br />
| 4:1<br />
| superhard<br />
| [[Archy]] or [[Superpyth]]<br />
|}<br />
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== Naming ==<br />
Every MOS can be uniquely specified by giving its [[signature]], i.e. the number of large and small steps, which is typically notated e.g. "5L&nbsp;2s,". Every possible signature corresponds to a valid MOS scale. Sometimes, if one simply wants to talk about step sizes without specifying which is large and small, the notation "5a&nbsp;2b" is used (which could refer to either diatonic or {{nowrap|[[2L 5s|anti-diatonic]] {{=}} 2L 5s}}).<br />
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By default, the [[equave]] of a MOS is assumed to be [[2/1]]. To specify a non-octave equave, "{{angbr|equave}}" is placed after the signature, e.g. {{mos scalesig|4L 5s<3/1>|link=1}}. Using angle brackets (<code>&#x26;#x27E8;</code> and <code>&#x26;#x27E9;</code>) is recommended; using greater-than and less-than signs ("&#x3C;equave&#x3E;") can also be done, but this can conflict with HTML and other uses of these symbols.<br />
<br />
Several naming systems have been proposed for MOSes, which can be seen at [[MOS naming]].<br />
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== History and terminology ==<br />
The term ''MOS'', and the method of scale construction it entails, were invented by [[Erv Wilson]] in 1975. His original paper is archived on Anaphoria.com here: [https://anaphoria.com/mos.pdf ''Moments of Symmetry'']. There is also an introduction by [[Kraig Grady]] here: [https://anaphoria.com/wilsonintroMOS.html ''Introduction to Erv Wilson's Moments of Symmetry''].<br />
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Sometimes, scales are defined with respect to a period and an additional [[equivalence interval]], the interval at which pitch classes repeat. MOSes in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called '''Multi-MOSes'''. For example, a MOS with a half-octave period is called a '''2mos''', with a 1/3-octave period a '''3mos''', and so on. MOSes in which the equivalence interval is equal to the period are sometimes called '''Strict MOSes'''. MOSes in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.<br />
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With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional evenness|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt<ref>Norman Carey and David Clampitt. "Aspects of Well-Formed Scales", ''Music Theory Spectrum'', Vol. 11, No. 2 (Autumn, 1989), pp. 187-206.</ref>. A great deal of work has been done in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilson's concept. They are in a sense the MOS of MOS patterns. This is used to explain the [[Pentatonic|pentatonics]] used in traditional [[Japanese music]] (e.g. {{nowrap|A B C E F A}}), where the 5-tone cycles are derived from a 7-tone MOS, which are not found in the concept of DE.<br />
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== Equivalent definitions and generalizations ==<br />
A scale is a MOS if and only if it satisfies one of the following equivalent criteria:<br />
<br />
# [[Maximum variety]] 2: Ascending by a certain number of steps is equivalent to ascending by one of at most two intervals, and the maximum of two is achieved (i. e. it is not true that ascending by a certain number of steps is always equivalent to ascending by one interval.) <br />
# [[Binary]] and has a [[generator]]: The scale step comes in exactly two sizes, and the scale is formable from stacking some interval called a generator and octave-reducing.<br />
# Mode of a Christoffel word: The scale can be formed by creating a 2D lattice where the period is on the lattice, then taking pitches by travelling vertically and horizontally from the origin, maintaining as close to the line from the origin to the octave as possible without going above it.<br />
<br />
Each definition generalizes to scales with three or more step sizes, but these generalizations are not equivalent. The concepts of [[Balanced word|balance]] and [[distributional evenness]] provide still different generalizations, although defining MOS through these terms is less helpful. For more information, see [[Mathematics of MOS]].<br />
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== Properties ==<br />
=== Basic properties ===<br />
<br />
* For every MOS scale with an [[octave]] period (which is usually the [[octave]]), if ''x''-[[edo]] is the [[collapsed]] tuning (where the small step vanishes) and ''y''-[[edo]] is the [[equalized]] tuning (where the large (''L'') step and small (''s'') step are the same size), then by definition it is an {{nowrap|''x''L (''y'' &minus; ''x'')s}} MOS scale, and the [[basic]] tuning where {{nowrap|''L'' {{=}} 2''s''}} is thus {{nowrap|(''x'' + ''y'')}}-[[edo]]. This is also true if the period is 1\''p'', that is, 1 step of ''p''-[[edo]], which implies that ''x'' and ''y'' are divisible by ''p'', though note that in that case (if {{nowrap|''p'' &gt; 1}}) you are considering a "multiperiod" MOS scale.<br />
* More generally, whenever ''px''-[[edo]] and ''py''-[[edo]] are used to define two [[Val|vals]] (usually but not necessarily through taking the [[Patent val|patent vals]]) while simultaneously also being used to define the {{nowrap|''px''L (''py'' &minus; ''px'')s}} MOS scale (where ''p'' is the number of periods per octave), then the ''px'' & ''py'' temperament corresponds to that MOS scale, and adding ''x'' and/or ''y'' corresponds to tuning closer to ''x''-[[edo]] and/or ''y''-[[edo]] respectively. (Optionally, see the below more precise statement for the mathematically-inclined.)<br />
* For the mathematically-inclined, we can say that whenever we consider a MOS with ''X''/''p'' notes per period in the [[collapsed]] tuning and ''Y''/''p'' notes per period in the [[equalized]] tuning and ''p'' periods per [[Octave stretching|tempered octave]] (or more generally tempered [[equave]]), and whenever we want to associate that MOS with the {{nowrap|''X'' &amp; ''Y''}} rank 2 temperament'''*''', we can say that any {{w|natural number|natural}}-coefficient {{w|linear combination}} of vals {{val|''X'' ...}} and {{val|''Y'' ...}} (where {{nowrap|''X'' &lt; ''Y''}}) corresponds uniquely to a tuning of the {{nowrap|''X'' &amp; ''Y''}} rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive) iff {{nowrap|gcd(''a'', ''b'') {{=}} 1}}, because if {{nowrap|''k'' {{=}} gcd(''a'', ''b'') &gt; 1}} then the val {{nowrap|''a''{{val| ''X'' ...}} + ''b''{{val| ''Y'' ...}}}} has a common factor ''k'' in all of its terms, meaning it is guaranteed to be [[contorted]]. The tuning corresponding to the [[wikipedia:Rational number|rational]] ''a''/''b'' is technically only unique up to (discarding of) [[octave stretching]] (or more generally [[equave]]-tempering).<br />
<br />
: The period of this temperament is {{nowrap|1\gcd(''X'', ''Y'')}}, and the rational ''a''/''b'' is very closely related to the [[step ratio]] of the corresponding MOS scale, because {{nowrap|1{{val| ''X'' ...}} + 0{{val| ''Y'' ...}}}} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 0}} tuning while {{nowrap|0{{val| ''X'' ...}} + 1{{val| ''Y'' ...}}}} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 1}} tuning and {{nowrap|1{{val| ''X'' ...}} + 1{{val| ''Y'' ...}}}} is the {{nowrap|''L'' {{=}} 2|''s'' {{=}} 1}} tuning, so that {{nowrap|''L'' {{=}} ''a'' + ''b''}} and {{nowrap|''s'' {{=}} ''b''}} and therefore:<br />
<br />
: {{nowrap|1/([[step ratio]]) {{=}} ''s''/''L''}} {{nowrap|{{=}} ''b''/(''a'' + ''b'')}} implying {{nowrap|[[step ratio]] {{=}} (''a'' + ''b'')/''b'' &ge; 1}} for [[wikipedia:Natural number|natural]] ''a'' and ''b'', where if {{nowrap|''b'' {{=}} 0}} then the step ratio is infinite, corresponding to the [[collapsed]] tuning.<ref group="note">It is '''important to note''' that the correspondence to the {{nowrap|''X'' &amp; ''Y''}} rank 2 temperament only works in all cases if we allow the temperament to be [[contorted]] on its [[subgroup]]; alternatively, it works if we exclude cases where {{nowrap|''X'' &amp; ''Y''}} describe a contorted temperament on the subgroup given. An example is the {{nowrap|5 &amp; 19}} temperament is contorted in the [[5-limit]] (having a generator of a semifourth, corresponding to [[5L&nbsp;14s]]), so we either need to consider the temperament itself to be contorted (generated by something lacking an interpretation in the subgroup given, two of which yielding a meantone-tempered [[~]][[4/3]]) or we exclude it because of its contortion.</ref><br />
<br />
* Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale ''a''L&nbsp;''b''s are {{nowrap|(''a'' + ''b'')L ''a''s}} (generated by generators of soft-of-basic ''a''L ''b''s) and {{nowrap|''a''L (''a'' + ''b'')s}} (generated by generators of hard-of-basic ''a''L''&nbsp;b''s).<br />
* Every MOS scale (with a specified [[equave]] ''&#x190;''&#x200A;), excluding {{nowrap|''a''L ''a''s{{angbr|''&#x190;''&#x200A;}}}}, has a ''parent MOS''. If {{nowrap|''a'' &gt; ''b''}}, the parent of ''a''L&nbsp;''b''s is {{nowrap|''b''L (''a'' &minus; ''b'')s}}; if {{nowrap|''a'' &lt; ''b''}}, the parent of ''a''L&nbsp;''b''s is {{nowrap|''a''L (''b'' &minus; ''a'')s}}.<br />
<br />
=== Advanced discussion ===<br />
See:<br />
* [[Mathematics of MOS]], a more formal definition and a discussion of the mathematical properties.<br />
** [[Recursive structure of MOS scales]], a description of how MOS scales are recursive and how one scale can be converted into a related scale.<br />
** [[MOS scale family tree]], a tree initially described by Erv Wilson that organizes scales by parent-and-child relationship, which also helps illustrate mos recursion.<br />
* [[Generator ranges of MOS]], organized by number of scale steps and quantity of L/s steps.<br />
* [[MOS diagrams]], visualizations of the MOS process.<br />
* [http://x31eq.com/temper/method.html How to Find Linear Temperaments], by [[Graham Breed]]<br />
<br />
== Variations ==<br />
* [[MODMOS scales]] are derived from chromatic alterations of one or more tones of an MOS scale, typically by the interval of {{nowrap|L &minus; s}}, the "chroma".<br />
* [[Muddle]]s are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.<br />
* [[MOS cradle]] is a technique of embedding MOS-like structures inside MOS scales and may or may not produce subsets of MOS scales.<br />
* [[Operations on MOSes]]<br />
<br />
== Listen ==<br />
This is an algorithmically generated recording of every MOS scale that has 14 or fewer notes for a total of 91 scales being showcased here. Each MOS scale played has its simplest step ratio (large step is 2 small step is 1) and therefore is inside the smallest EDO that can support it. Each MOS scale is also in its brightest mode. And rhythmically, each scale is being played with its respective MOS rhythm. Note that changing the mode or step ratio of any of these MOSes may dramatically alter the sound and therefore this recording is not thoroughly representative of each MOS but rather a small taste.<br />
<br />
[[File:Every-MOS-Scale-With-14-Or-Fewer-Notes.mp3|left|800x800px]] {{clear}}<br />
<br />
== See also ==<br />
* [[Diamond-mos notation]], a microtonal [[notation]] system focused on MOS scales<br />
* [[Metallic MOS]], an article focusing on MOS scales based on metallic means, such as [[phi]]<br />
* [[MOS rhythm]]<br />
* [[:Category:MOS scales|Category:MOS scales]], the category including all MOS-related articles on this wiki<br />
* [[Gallery of MOS patterns]]<br />
<br />
== Notes ==<br />
<references group="note" /><br />
<br />
== References ==<br />
<references /><br />
<br />
[[Category:Math]]<br />
[[Category:MOS scale| ]] <!-- Sort order in category: this page shows above A --><br />
[[Category:Scale]]<br />
[[Category:Erv Wilson]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Riemann_zeta_function&diff=225487Riemann zeta function2026-03-09T23:31:22Z<p>ArrowHead294: </p>
<hr />
<div>{{Texops}}<br />
{{Wikipedia|Riemann zeta function}}<br />
The Riemann zeta function is a famous mathematical function, best known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem involving the distribution of the prime numbers. However, it also has an intriguing musical interpretation: the zeta function shows how "well" a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size. <br />
<br />
As a result, although the zeta function is best known for its use in analytic number theory, the zeta function is present in the background of some tuning theory—the [[harmonic entropy]] model of [[concordance]] can be shown to be related to the Fourier transform of the zeta function, and several tuning-theoretic metrics, if extended to the infinite-limit, yield expressions that are related to the zeta function. Sometimes these are in terms of the "prime zeta function", which is closely related and can also be derived as a simple expression of the zeta function.<br />
<br />
If you look for a filter to quickly sort all the equal temperaments into those that approximate JI well and those that do not, the [[#Zeta edo lists|edo lists]] below can be useful. The caveat is that it collapses the variety of characteristics of a temperament to a one-dimensional rating, with little capacity to show the nuances of each system. It is therefore best to keep in mind that judging the temperaments by zeta is no replacement for investigating each temperament in detail. <br />
<br />
There are other metrics besides zeta for other definitions of "approximating well", which you can find in: [[:Category:Regular temperament tuning|optimised regular temperament tunings]].<br />
<br />
Much of the below is thanks to the insights of [[Gene Ward Smith]]. Below is the original derivation as he presented it, followed by a different derivation from [[Mike Battaglia]] below which extends some of the results.<br />
<br />
== Terminology ==<br />
; Riemann zeta function ("zeta") : A mathematical function which is tied to the harmonic series and to prime numbers, used in tuning theory as an "edo goodness" function to evaluate how close to JI an edo is.<br />
<br />
; Zeta record edo : An equal tuning that sets some kind of record in regards to the zeta function compared to all smaller equal tunings.<br />
<br />
; Record zeta peak : An equal tuning which is closer to JI than any previous tuning, and is usually an edo with compressed or stretched octaves, evaluated by the absolute "goodness" of the edo according to the zeta function.<br />
<br />
; Record zeta peak integer : A zeta record edo by absolute "goodness", when compared only to other edos (i.e. ignoring stretched/compressed equivalences).<br />
<br />
; Zero : A point where the Riemann zeta function is equal to zero, such as ~2.759edo, representing an equal tuning that does not represent JI much at all. Edos close to zeroes are called ''zeta valley edos''; all known zeroes are on the "critical line" used to obtain tuning information. <br />
<br />
; Record z gap : A zeta record edo by the size of the gap between its surrounding zeroes, adjusted for the fact that zeroes generally become more dense with larger inputs.<br />
<br />
; Record zeta integral : A zeta record edo by the size of the area enclosed by the shape of the function between the edo's surrounding zeroes.<br />
<br />
== Quick info: zeta peak edos ==<br />
These lists give the best [[equal divisions of the octave]] for their size according to the zeta metric.<br />
<br />
Zeta peak edos (tempered octaves): {{EDOs|1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, … }}<br />
<br />
Zeta peak integer edos (pure octaves): {{EDOs|1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, … }}<br />
<br />
See the [[#Zeta edo lists|section below]] for more information.<br />
<br />
== Graph links ==<br />
A link to the graph of zeta can be found at [https://samuelj.li/complex-function-plotter/#abs(zeta(i*2*pi*real(z)%2Fln(2)%2Bimag(z))) Zeta in Samuelj Plotter]. (In the top left menu, make sure that "Enable Checkerboard" is unticked and "Invert Gradient" and "Continuous Gradient" are ticked.) The function has been reoriented to place edo size along the horizontal axis and weight along the vertical axis, and also scaled by {{sfrac|2π|ln(2)}} to ensure that the real number line aligns with edos. One can see that with higher weights, the function approaches a cyclic function with a period of 1; this corresponds to the prime 2 dominating more and more extremely as other harmonics are weighted less with higher weights. You can see this easier by raising the entire expression to an absurdly high power, such as 100. Note, however, that this visualization is inaccurate beyond a couple hundred: around 146.5, 324.5 and 473.5, and in many cases after, there appear to be zeroes that are not on the critical line; this is an artifact of the way the function is approximated and is the ultimate reason why the Riemann hypothesis remains unsolved. These actually correspond to zeroes that are very close together but on the critical line. <br />
<br />
You may also view the graph of zeta along the critical line on Desmos: [https://www.desmos.com/calculator/dstp7wnidf Zeta in Desmos]. This makes it easier to see peaks, but only works for {{nowrap| σ {{=}} {{sfrac|1|2}} }}. <br />
<br />
=== Plots ===<br />
Below are some demonstrative plots of the zeta function (strictly speaking, the [[#The Z function: a mathematically convenient version of zeta|Z function]]) on the critical line.<br />
<br />
Using the [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ online plotter] we can plot Z in the regions corresponding to scale divisions, using the conversion factor {{nowrap|''t'' {{=}} {{sfrac|2π|ln(2)}}''x''}}, for ''x'' a number near or at an edo number. Hence, for instance, to plot 12 plot around 108.777, to plot 31 plot around 281.006, and so forth. An alternative plotter is the applet [http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html here].<br />
<br />
If you have access to {{w|Mathematica}}, which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematica-generated plot of Z{{pars|{{sfrac|2π''x''|ln(2)}}|1.8}} in the region around [[12edo]]:<br />
<br />
[[File:plot12.png|alt=plot12.png|plot12.png]]<br />
<br />
The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an approximation of JI. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/''x'', and hence the size of the octave in the zeta peak value tuning for ''N''edo is ''N''/''x''; if ''x'' is slightly larger than ''N'' as here with {{nowrap|''N'' {{=}} 12}}, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with ''x'' less than ''N'', we have stretched octaves.<br />
<br />
For larger edos, the width of the peak narrows, but for strong edos the height more than compensates, measured in terms of the area under the peak (the absolute value of the integral of Z between two zeros.) Note how [[270edo]] completely dominates its neighbors:<br />
<br />
[[File:plot270.png|alt=plot270.png|plot270.png]]<br />
<br />
Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +∞. This can be determined by looking at the absolute value of zeta along other ''s'' values, such as {{nowrap|''s'' {{=}} 1}} or {{nowrap|''s'' {{=}} {{sfrac|3|4}}}}, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3.<br />
<br />
To generate this plot using the free version of Wolfram Cloud, you can run <code>Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9, 12.1}]</code> and then in the menu select '''Evaluation &gt; Evaluate Cells'''. Change "'''11.9'''" and "'''12.1'''" to whatever values you want, e.g. to view the curve around 15edo you might use the values "'''14.9'''" and "'''15.1'''".<br />
<br />
You can also view the plot using [https://www.desmos.com/calculator/dstp7wnidf Zeta in Desmos].<br />
<br />
== Gene Smith's original derivation ==<br />
Suppose ''x'' is a variable representing some equal division of the octave. For example, if {{nowrap|''x'' {{=}} 80}}, ''x'' corresponds to [[80edo]] with a step size of 15 [[cent]]s and with pure octaves. Suppose that ''x'' can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well, so that it ranges over all possible [[equal temperament]]s. The [[Bohlen–Pierce scale]], 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of {{nowrap|''x'' {{=}} 8.202}}.<br />
<br />
Now suppose that [https://www.desmos.com/calculator/krigk43int {{rr|''x''}}] denotes the difference between ''x'' and the integer nearest to ''x'':<br />
<br />
<math><br />
\lfloor x \rceil = \left| x - \left\lfloor x + \frac{1}{2} \right\rfloor \right|<br />
</math><br />
<br />
For example, {{nowrap|{{rr|8.202}} {{=}} 0.202}}, since it is the difference between 8.202 and the nearest integer, which is 8. Meanwhile, {{nowrap|{{rr|7.95}} {{=}} 0.05}}, which is the difference between 7.95 and the nearest integer, which is 8. This represents the absolute relative error of the octave in equal tuning ''x'', or alternatively how much x is detuned from an edo.<br />
<br />
For any value of ''x'', we can construct a ''p''-[[limit]] [[val]] by rounding {{nowrap|''x'' log<sub>2</sub>(''q'')}} to the nearest integer for each prime ''q'' up to ''p''. (More technically, this corresponds to what's known as a [[generalized patent val]].) For example, for {{nowrap|''x'' {{=}} 12}}, we find 2 at 12, 3 at 19, 5 at 28, etc. Now consider [https://www.desmos.com/calculator/4uamhon9tt the function]<br />
<br />
<math><br />
\displaystyle \xi_p(x)<br />
= \sum_{\substack{2 \leq q \leq p \\ q \text{ prime}}} \left(\frac{\lfloor x \log_2 q \rceil}{\log_2 q}\right)^2<br />
</math><br />
<br />
Here, the numerator represents the relative error on each prime, and the denominator represents a weight factor of the logarithm of each prime<ref group="note">Specifically, one reason we use the weighting 1/log<sub>2</sub>(''p'') is because of certain desirable properties it has that singles it out as of unique interest: if the complexity of a prime ''p'' is {{nowrap| log<sub>2</sub>(''p'') }}, then ''p''<sup>''n''</sup> is ''n'' times as complex as ''p''. Using {{nowrap| log<sub>2</sub>(''p'') }} as the complexity also means that the complexity of a harmonic, according to its prime factorization, exactly matches where it's found in the harmonic series, so that e.g. {{nowrap| 25 {{=}} 5 × 5 }} is slightly less complex than {{nowrap| 26 {{=}} 2 × 13 }} is slightly less complex than {{nowrap| 27 {{=}} 3 × 3 × 3 }}. Therefore, the {{nowrap| 1/log<sub>2</sub>(''p'') }} weighting is a kind of natural inverse-complexity weighting, that is, a simplicity weighting.</ref>, so the function represents a ''p''-limit badness metric. Also, for those unfamiliar, squaring the error is commonly done because it solves the flaws of two alternative ways of measuring error. Specifically, if you look only at the maximum error, you miss opportunities to make the tuning much better ''overall'' by allowing slightly more damage on the most damaged intervals, while if you look only at the average error, then it may be that you are unnecessarily damaging a few intervals a lot just to get intervals that are already in-tune slightly more in-tune, so both extremes have pathological behaviours, and using the squared error counters both of these behaviours so that it represents a more balanced approach to optimization that is used in a variety of disciplines.<br />
<br />
This function has local minima, corresponding to associated generalized patent vals. The minima occur for values of ''x'' which are the [[Tenney–Euclidean tuning]]s of the octaves of the associated vals, while ξ<sub>''p''</sub> for these minima is the square of the [[Tenney–Euclidean relative error]] of the val—equal to the TE error times the TE complexity, and sometimes known as "TE simple badness."<br />
<br />
Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could [https://www.desmos.com/calculator/0qhhewlsaz change the weighting factor to a power] so that it does converge:<br />
<br />
<math>\displaystyle \xi_\infty(x) = \sum_{\substack{q \geq 2 \\ q \text{ prime}}} \frac{\lfloor x \log_2 q \rceil^2}{q^s}</math><br />
<br />
Importantly, when {{nowrap|''s'' {{=}} 1}}, the weighting is 1/''p'' for a prime ''p'', so that it's very similar to a {{sfrac|1|''p'' − 1}} weighting. This latter weighting is equal to the average number of times prime ''p'' occurs as a factor in the harmonic series, counting repetition, so is of interest because it represents how many harmonics will feel damage from this prime being mistuned. Therefore, the weighting {{nowrap| 1/''p'' }} corresponds to under-prioritizing small primes slightly (with the effect being less slight the smaller the prime, so that at the most extreme, at prime 2 we have 1/2 instead of 1/1 weighting and at prime 3 we have 1/3 instead of 1/2 weighting).<br />
<br />
Seeing that we notate the power as ''s'', it might become apparent where the Riemann zeta function will eventually show up.<br />
<br />
If ''s'' is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting cancels this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure—in fact, the primary intuition behind Tenney weighting is that it is the weighting pattern that values 25, 27, and 29 approximately evenly in importance despite being different powers. We can go ahead and include them by adding a factor of {{sfrac|1|''n''}} for each prime power ''p''<sup>''n''</sup>. A somewhat peculiar but useful way to write the result of doing this is in terms of the {{w|von Mangoldt function}}, an {{w|arithmetic function}} on positive integers which is equal to ln(''p'') on prime powers ''p''<sup>''n''</sup>, and is zero elsewhere. This is written using a capital lambda, as Λ(''n''), and in terms of it we can include prime powers in our error function as<br />
<br />
<math>\displaystyle \xi_\infty(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\lfloor x \log_2 n \rceil^2}{n^s}</math><br />
<br />
where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.<br />
<br />
Another consequence of the above definition which might be objected to is that it results in a function with a {{w|Continuous function#Relation to differentiability and integrability|discontinuous derivative}}, whereas a smooth function be preferred. The function ⌊''x''⌉<sup>2</sup> is quadratically increasing near integer values of ''x'', and is periodic with period 1. Another function with these same properties is {{nowrap|1 − cos(2π''x'')}}, which is a smooth and in fact an {{w|entire function}}. Let us therefore now define for any {{nowrap|''s'' &gt; 1}}:<br />
<br />
<math>\displaystyle E_s(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x \log_2 n)}{n^s}</math><br />
<br />
For any fixed {{nowrap|''s'' &gt; 1}} this gives a real {{w|analytic function}} defined for all ''x'', and hence with all the smoothness properties we could desire.<br />
<br />
We can clean up this definition to get essentially the same function:<br />
<br />
<math>\displaystyle F_s(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x \log_2 n)}{n^s}</math><br />
<br />
This new function has the property that {{nowrap|F<sub>s</sub>(''x'') {{=}} F<sub>s</sub>(0) − E<sub>s</sub>(''x'')}}, so that all we have done is flip the sign of E<sub>s</sub>(''x'') and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum.<br />
<br />
Of more interest is the fact that it is a known mathematical function. The logarithm of the {{w|Riemann zeta function}} function {{subpage|appendix|u|s=Dirichlet series for the von Mangoldt function|text=can be expressed}} in terms of a {{w|Dirichlet series}} involving the von Mangoldt function:<br />
<br />
<math>\displaystyle<br />
\ln \zeta(s)=\sum_{n=2}^\infty \frac{\Lambda(n)}{\ln(n)}\,\frac{1}{n^s}<br />
</math><br />
<br />
We can rewrite the cosine term in <math>F_s(x)</math> using the real part of an exponential:<br />
<br />
<math>\displaystyle<br />
\cos(2 \pi x \log_2 n)<br />
= \mathrm{Re}\left( \exp{(- 2 \pi i x \log_2 n)} \right)<br />
= \mathrm{Re}\left( n^{- \frac{2 \pi i}{\ln 2} x} \right)<br />
</math><br />
<br />
Substituting back into <math>F_s(x)</math> gives:<br />
<br />
<math>\displaystyle <br />
F_s(x)<br />
= \mathrm{Re} \left( \sum_{n=2}^\infty \frac{\Lambda(n)}{\ln n} \frac{n^{- \frac{2 \pi i}{\ln 2} x}}{n^s} \right)<br />
= \mathrm{Re} \left( \ln \zeta \left(s + \frac{2 \pi i}{\ln 2}x \right) \right)<br />
</math><br />
<br />
If we take exponentials of both sides, then<br />
<br />
<math>\displaystyle \exp(F_s(x)) = \left| \zeta\left(s + \frac{2 \pi i}{\ln 2}x\right) \right|</math><br />
<br />
so that we see that the absolute value of the zeta function serves to measure the relative error of an equal division.<br />
<br />
There is an open question as to whether the analytic continuation preserves the properties we are interested in, though empirically it seems to. However, if you do not want to trust the analytic continuation, then the {{nowrap| ''s'' {{=}} 1 }} line is of interest for the reasons discussed, as though the infinite sum does not converge, every sum for {{nowrap| ''s'' > 1 }} ''does'' converge, so that we can consider the line at {{nowrap| ''s'' {{=}} 1 }} as being the "limit" as ''s'' approaches 1 from above. This can also be used to sanity-check results at {{nowrap| ''s'' {{=}} 1/2 }} by seeing where they agree, e.g. as done in the section on [[#Absolute zeta peak edos]], which looks at getting more practical tuning information out of the zeta function via an adjustment for considering absolute mistuning rather than relative error.<br />
<br />
== Mike Battaglia's expanded results ==<br />
=== Zeta yields relative error over all rationals ===<br />
Above, Gene proves that the zeta function measures the [[Tenney–Euclidean relative error]], sometimes called ''Tenney–Euclidean simple badness'', of any edo, taken over all prime powers. The relative error is simply equal to the tuning error times the size of the edo, so we can easily get the raw absolute tuning error from this as well by simply dividing by the size of the edo.<br />
<br />
Here, we strengthen that result to show that the zeta function additionally measures weighted relative error over all rational numbers, relative to the size of the edo.<br />
<br />
Let's dive in!<br />
<br />
First, let's take the zeta function, expressed as a Dirichlet series:<br />
<br />
<math> \displaystyle<br />
\zeta(s) = \sum_n n^{-s}</math><br />
<br />
Now let's do two things: we're going to expand {{nowrap|''s'' {{=}} σ + ''it''}}, and we're going to multiply <math>\zeta(s)</math> by its complex conjugate <math>\overline{\zeta(s)}</math>, noting that <math>\overline{\zeta(s)}=\zeta(\overline{s})</math> and <math>\left| \zeta(s) \right|^{2} = \zeta(s)\overline{\zeta(s)}</math>.<br />
We get:<br />
<br />
<math> \displaystyle<br />
\left| \zeta(s) \right| ^2 = \left[\sum_n n^{-(\sigma+it)}\right] \cdot \left[\sum_d d^{-(\sigma-it)}\right]</math><br />
<br />
where d is a new variable used internally in the second summation.<br />
<br />
Now, let's focus on {{nowrap|σ &gt; 1}}, so that both series are absolutely convergent. The following rearrangement of terms is then justified:<br />
<br />
<math> \displaystyle<br />
\left| \zeta(s) \right|^2 = \sum_{n,d} \left[n^{-(\sigma+it)} \cdot d^{-(\sigma-it)}\right] = \sum_{n,d} \frac{\left({\tfrac{n}{d}}\right)^{-it}}{(nd)^{\sigma}}</math><br />
<br />
<span style="line-height: 1.5;">Now let's do a bit of algebra with the exponential function, and use Euler's identity:</span><br />
<br />
<math> \displaystyle<br />
\left| \zeta(s) \right|^2 = \sum_{n,d} \frac{e^{-it \ln\left({\tfrac{n}{d}}\right)}}{(nd)^{\sigma}}<br />
= \sum_{n,d} \frac{\cos\left(-t \ln\left({\tfrac{n}{d}}\right)\right) + i\sin\left(-t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}}<br />
= \sum_{n,d} \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right) - i\sin\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}}</math><br />
<br />
where the last equality makes use of the fact that {{nowrap|cos(−''x'') {{=}} cos(''x'')}} and {{nowrap|sin(−''x'') {{=}} −sin(''x'')}}.<br />
<br />
Now, let's decompose the sum into three parts: {{nowrap|''n'' {{=}} ''d''|''n'' &gt; ''d''|and ''n'' &lt; ''d''}}. Here's what we get:<br />
<br />
<math> \displaystyle<br />
\left| \zeta(s) \right|^2 = \sum_{n=d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right) - i\sin\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right] +<br />
\sum_{n>d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right) - i\sin\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right] +<br />
\sum_{n< d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right) - i\sin\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right]</math><br />
<br />
We'll deal with each of these separately.<br />
<br />
First, in the leftmost summation, we can see that {{nowrap|''n'' {{=}} ''d''}} implies {{nowrap|ln{{pars|{{sfrac|''n''|''d''}}|1.8}} {{=}} 0}}. Since {{nowrap|sin(0) {{=}} 0}}, the sin term in the numerator cancels out, yielding:<br />
<br />
<math> \displaystyle<br />
\left| \zeta(s) \right|^2 = \sum_{n=d} \left[ \frac{\cos\left( t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right] +<br />
\sum_{n>d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right) - i\sin\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right] +<br />
\sum_{n< d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right) - i\sin\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right]</math><br />
<br />
We will not simplify the cosine term further right now, the reasons for which will become apparent below.<br />
<br />
Now, let's handle the two summations on the right. The key thing to note here is that we can pair up every term in the second summation with a corresponding term in the third summation that interchanges ''n'' and ''d''. To make this clear, let ''p'' and ''q'' be two integers, and assume without loss of generality that {{nowrap|''p'' &gt; ''q''}}. The term corresponding to {{nowrap|''n'' {{=}} ''p''|''d'' {{=}} ''q''}} will then appear in the second summation, and the term {{nowrap|''n'' {{=}} ''q''|''d'' {{=}} ''p''}} will appear in the third summation. Juxtaposing those together, we get the following:<br />
<br />
<math> \displaystyle<br />
\frac{\cos\left(t \ln\left({\tfrac{p}{q}}\right)\right) - i\sin\left(t \ln\left({\tfrac{p}{q}}\right)\right)}{(pq)^{\sigma}} +<br />
\frac{\cos\left(t \ln\left({\tfrac{q}{p}}\right)\right) - i\sin\left(t \ln\left({\tfrac{q}{p}}\right)\right)}{(pq)^{\sigma}}</math><br />
<br />
Now, noting that {{nowrap|ln{{pars|{{sfrac|''p''|''q''}}|1.8}} {{=}} −ln{{pars|{{sfrac|''q''|''p''}}|1.8}}}} and that sin is an odd function, we can see that the sin terms cancel out, leaving<br />
<br />
<math> \displaystyle<br />
\frac{\cos\left(t \ln\left({\tfrac{p}{q}}\right)\right)}{(pq)^{\sigma}} +<br />
\frac{\cos\left(t \ln\left({\tfrac{q}{p}}\right)\right)}{(pq)^{\sigma}}</math><br />
<br />
Now, since every term in these two summations has a pair like this, and since we've done nothing but continued rearrangements of an absolutely convergent series, we can modify the original three-part summation to cancel the sin terms out as follows:<br />
<br />
<math> \displaystyle<br />
\left| \zeta(s) \right|^2 = \sum_{n=d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right] +<br />
\sum_{n>d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right] +<br />
\sum_{n< d} \left[ \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}} \right]</math><br />
<br />
Putting the whole thing back into one series, we get<br />
<br />
<math> \displaystyle<br />
\left| \zeta(s) \right|^2 = \sum_{n,d} \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}}</math><br />
<br />
Finally, by making the mysterious substitution {{nowrap|''t'' {{=}} {{sfrac|2π|ln(2)}}''x''}}, the musical implications of the above will start to reveal themselves:<br />
<br />
<math> \displaystyle<br />
\left| \zeta(s) \right|^2 = \sum_{n,d} \frac{\cos\left(2\pi x \log_2\left(\tfrac{n}{d}\right)\right)}{(nd)^{\sigma}}</math><br />
<br />
Let's take a breather and see what we've got.<br />
<br />
=== Interpretation of results: cosine relative error ===<br />
For every strictly positive rational ''n''/''d'', there is a cosine with period {{nowrap|log<sub>2</sub>{{pars|{{sfrac|''n''|''d''}}|1.8}}}}. This cosine peaks at {{nowrap|''x'' {{=}} {{sfrac|''N''|log<sub>2</sub>(''n''/''d'')}}}} for all integers ''N'', or in other words, the ''N''-th equal division of the rational number {{frac|''n''|''d''}}, and hits troughs midway between.<br />
<br />
Our mysterious substitution above was chosen to set the units for this up nicely. The variable ''x'' now happens to be measured in divisions of the octave. (The original variable ''t'', which was the imaginary part of the zeta argument ''s'', can be thought of as the number of divisions of the interval {{nowrap|''e''<sup>2π</sup> ≈ 535.49}}, or what [[Keenan Pepper]] has called the "[[zetave|natural interval]].")<br />
<br />
As mentioned in Gene's original zeta derivation, these cosine functions can be thought of as good approximations to the terms in the TE error computation, which are all the squared errors for the different primes. Rather than taking the square of the error, we instead put the error through the function {{sfrac|1 − cos(''x'')|2}}, which is close enough for small values of ''x''. Since we are always rounding off to the best mapping, this error is never more 0.5 steps of the edo, so since we have {{nowrap| −0.5 < ''x'' < 0.5 }} we have a decent enough approximation.<br />
<br />
We will call this '''cosine (relative) error''', by analogy with '''TE (relative) error'''. It is easy to see that the cosine error is approximately equal to the TE error when the error is small, and only diverges slightly for large errors.<br />
<br />
There are three major differences between our cosine error functions, and the way we are incorporating them into the result, and what TE is doing:<br />
# First, the function here is flipped upside down—that is, we are measuring accuracy rather than error—as well as shifted vertically down along the ''y''-axis. Since it is trivial to convert between the two, and since we only care about the relative rankings of edos, it is clear that we're measuring essentially the same thing.<br />
# Instead of weighting each interval by {{sfrac|1|log(''nd'')}}, we weight it by {{sfrac|1|(''nd'')<sup>σ</sup>}}.<br />
# Instead of only looking at the primes, as we do in TE, we are now looking at ''all'' intervals, and in particular looking at the best mapping for each interval.<br />
<br />
The last one is nontrivial, and we will go into detail below.<br />
<br />
There are also a few notes we will only write in passing, for now, perhaps to build on later:<br />
# If we do want {{sfrac|1|log(''nd'')}} weighting, we can derive this kind of weighting from an antiderivative of the zeta function.<br />
# If we only want the primes, rather than all intervals, we can use something called the ''prime zeta function'' to get those kinds of summations.<br />
# If we do want the true TE squared error rather than our cosine error, then we would end up getting something called ''parabolic waves'' rather than cosine waves for each interval. A parabolic wave is the antiderivative of a sawtooth wave, and as it is a periodic signal, it has a Fourier series and can be expressed as a sum of sinusoids. We can use this to get a derivation of the squared error as an infinite sum of zeta functions.<br />
<br />
For now, though, we will focus only on the basic zeta result that we have.<br />
<br />
Going back to the infinite summation above, we note that these cosine error (or really cosine accuracy) functions are being weighted by {{sfrac|1|(''nd'')<sup>σ</sup>}}. Note that ''σ'', which is the real part of the zeta argument ''s'', serves as sort of a complexity weighting—it determines how quickly complex rational numbers become irrelevant. Framed another way, we can think of it as the degree of rolloff formed by the resultant (musical, not mathematical) harmonic series formed by those rationals with {{nowrap| ''d'' {{=}} 1 }}. Note that this rolloff is much stronger than the usual {{sfrac|1|log(''nd'')}} rolloff exhibited by TE error, which is one reason that zeta converges to something coherent for all rational numbers, whereas TE fails to converge as the limit increases. We will use the term ''rolloff'' to identify the variable σ below.<br />
<br />
Putting this all together, we can take the approach to fix ''σ'', specifying a rolloff, and then let ''x'' (or ''t'') vary, specifying an edo. The resulting function gives us the measured accuracy of edos across all unreduced rational numbers with respect to the chosen rolloff. Taking it all together, we get a Tenney-weighted sum of cosine accuracy over all unreduced rationals. QED.<br />
<br />
It is extremely noteworthy to mention how composite rationals are treated differently than with TE error. In addition to our usual error metric on the primes, we also go to each rational, look for the best [[direct approximation]] of that rational within the edo, and add ''that'' to the edo's score. In particular, we do this even when the best mapping for some rational does not match up with the mapping you would get from it just looking at the primes.<br />
<br />
So, for instance, in 16edo, the best mapping for 3/2 is 9 steps out of 16, and using that mapping, we get that 9/8 is 2 steps, since {{nowrap| 9 × 2 − 16 {{=}} 2 }}. However, there is a better mapping for 9/8 at 3 steps—one which ignores the fact that it is no longer equal to two 3/2's. This can be particularly useful for playing chords: 16edo's direct approximation for 9 is useful when playing the chord 4:5:7:9, and the val mapping for 9 is useful when playing the major ninth chord 8:10:12:15:18. We can think of the zeta function as rewarding equal temperaments not just for having a good approximation of the primes, but also for having good extra approximations of rationals which can be used in this way. And although 16edo is pretty high error, similar phenomena can be found for any edo which becomes [[inconsistent]] for some chord of interest.<br />
<br />
One way to frame this in the usual group-theoretic paradigm is to consider the group in which each strictly positive rational number is given its own linearly independent basis element. In other words, look at the {{w|free group}} over the strictly positive rationals, which we'll call ''meta-JI''. The zeta function can then be thought of as yielding an error for all meta-JI [[generalized patent val]]s. Whether this can be extended to all meta-JI vals, or modified to yield something nice like a norm on the group of meta-JI vals, is an open question. Regardless, this may be a useful conceptual bridge to understand how to relate the zeta function to ordinary regular temperament theory.<br />
<br />
Now, one nitpick to notice above is that this expression technically involves all unreduced rationals, e.g. there will be a cosine error term not just for 3/2, but also for 6/4, 9/6, etc. However, we can easily show that the same expression also measures the cosine relative error for reduced rationals:<br />
<br />
=== From unreduced rationals to reduced rationals ===<br />
Let us go back to this expression here:<br />
<br />
<math> \displaystyle<br />
\left| \zeta(s) \right|^2 = \sum_{n,d} \frac{\cos\left(t \ln\left({\tfrac{n}{d}}\right)\right)}{(nd)^{\sigma}}</math><br />
<br />
Note that since there is no restriction that ''n'' and ''d'' be coprime, the rationals we are using here do not have to be reduced. So this shows that zeta yields an error metric over all unreduced rationals, but leaves open the question of how reduced rationals are handled. It turns out that the same function also measures the error of reduced rationals, scaled only by a rolloff-dependent constant factor across all edos.<br />
<br />
To see this, let us first note that every unreduced rational {{sfrac|''n''|''d''}} can be decomposed into the product of a reduced rational {{sfrac|''n''{{``}}|''d''{{-`}}}} and a common factor {{sfrac|''c''|''c''}}. Furthermore, note that for any reduced rational {{sfrac|''n''{{``}}|''d''{{-`}}}}, we can generate all unreduced rationals {{sfrac|''n''|''d''}} corresponding to it by multiplying it by all such common factors {{sfrac|''c''|''c''}}, where ''c'' is a strictly positive natural number.<br />
<br />
This allows us to change our original summation so that it's over three variables, ''n''{{``}}, ''d''{{-`}}, and ''c''{{-`}}, where ''n''{{``}} and ''d''{{-`}} are coprime, and ''c'' is a strictly positive natural number:<br />
<br />
<math> \displaystyle<br />
\left| \zeta(s) \right|^2 = \sum_{n',d',c} \frac{\cos\left(t \ln\left({\tfrac{cn'}{cd'}}\right)\right)}{(cn' \cdot cd')^{\sigma}}</math><br />
<br />
Now, the common factor {{sfrac|''c''|''c''}} cancels out inside the log in the numerator. However, in the denominator, we get an extra factor of ''c''<sup>2</sup> to contend with. This yields<br />
<br />
<math> \displaystyle<br />
\left| \zeta(s) \right|^2 = \sum_{n',d',c} \frac{\cos\left(t \ln\left({\tfrac{n'}{d'}}\right)\right)}{(c^2 \cdot n'd')^{\sigma}}<br />
= \sum_{n',d',c} \left[ \frac{1}{c^{2\sigma}} \cdot \frac{\cos\left(t \ln\left({\tfrac{n'}{d'}}\right)\right)}{(n'd')^{\sigma}} \right]</math><br />
<br />
Now, since we are still assuming that {{nowrap| ''σ'' > 1 }} and everything is absolutely convergent, we can decompose this into a product of series as follows<br />
<br />
<math> \displaystyle<br />
\left| \zeta(s) \right|^2 = \left[ \sum_c \frac{1}{c^{2\sigma}} \right] \cdot \left[ \sum_{n',d'} \frac{\cos\left(t \ln\left({\tfrac{n'}{d'}}\right)\right)}{(n'd')^{\sigma}} \right]</math><br />
<br />
Finally, we note that on the left summation we simply have another zeta series, yielding<br />
<br />
<math> \displaystyle<br />
\left| \zeta(s) \right|^2 = \zeta(2\sigma) \cdot \left[ \sum_{n',d'} \frac{\cos\left(t \ln\left({\tfrac{n'}{d'}}\right)\right)}{(n'd')^{\sigma}} \right]</math><br />
<br />
<math> \displaystyle<br />
\frac{\left| \zeta(s) \right|^2}{\zeta(2\sigma)} = \sum_{n',d'} \frac{\cos\left(t \ln\left({\tfrac{n'}{d'}}\right)\right)}{(n'd')^{\sigma}}</math><br />
<br />
Now, since we are fixing ''σ'' and letting ''t'' vary, the left zeta term is constant for all edos. This demonstrates that the zeta function also measures cosine error over all the reduced rationals, up to a constant factor. QED.<br />
<br />
=== Measuring error on harmonics only ===<br />
So far we have shown the following:<br />
<br />
* Error on prime powers: <math>\log\,\left| \zeta(\sigma+it) \right|</math><br />
* Error on unreduced rationals: <math>\left| \zeta(\sigma+it) \right|^2</math><br />
* Error on reduced rationals: <math>\frac{\left| \zeta(\sigma+it) \right|^2}{\zeta(2\sigma)}</math><br />
<br />
Since the second is a simple monotonic transformation of the first, we can see that the same function basically measures both the relative error on just the prime powers, and also on all unreduced rationals, at least in the sense that edos will be ranked identically by both measures. The third function is really just the second function divided by a constant, since we only really care about letting ''t'' vary—we instead typically set ''σ'' to some value which represents the weighting rolloff on rationals. So, all three of these functions will rank edos identically.<br />
<br />
We also note that, above, Gene tended to look at things in terms of the Z(''t'') function, which is defined so that we have {{nowrap|{{abs|Z(''t'')}} {{=}} {{abs|ζ(''t'')}}}}. So, the absolute value of the Z function is also monotonically equivalent to the above set of expressions, so that any one of these things will produce the same ranking on edos.<br />
<br />
It turns out that using the same principles of derivation above, we can also derive another expression, this time for the relative error on only the harmonics—i.e. those intervals of the form 1/1, 2/1, 3/1, …, ''n''/1, …. This was studied in a paper by Peter Buch called [[:File:Zetamusic5.pdf|"Favored cardinalities of scales"]]. The expression is:<br />
<br />
Error on harmonics only: <math>\mathrm{Re}\left(\zeta(\sigma + it)\right)</math><br />
<br />
Note that, although the last four expressions were all monotonic transformations of one another, this one is not—this is the ''real part'' of the zeta function, whereas the others were all some simple monotonic function of the ''absolute value'' of the zeta function. The results, however, are very similar—in particular, the peaks are approximately to one another, shifted by only a small amount (at least for reasonably-sized edos up to a few hundred).<br />
<br />
=== Relationship to harmonic entropy ===<br />
The expression<br />
<br />
<math>\displaystyle{\left| \zeta\left(\frac{1}{2} + it\right) \right|^2 \cdot \overline {\phi(t)}}</math><br />
<br />
is, up to a flip in sign, the Fourier transform of the unnormalized Harmonic Shannon Entropy for {{nowrap| ''N'' {{=}} &infin; }}, where φ(''t'') is the characteristic function (aka Fourier transform) of the spreading distribution and {{overline|φ(''t'')}} denotes complex conjugation.<br />
<br />
Note that in the most common case where the spreading distribution is symmetric (as in the case of the Gaussian and Laplace distributions), the characteristic function is purely real and hence the conjugate is unnecessary. In particular, when the spreading distribution is a Gaussian, the characteristic function is also a Gaussian.<br />
<br />
More can be found at the page on [[Harmonic entropy #Extending_HE_to_.5Bmath.5DN.3D.5Cinfty.5B.2Fmath.5D:_zeta-HE|harmonic entropy]], including a generalization to Renyi entropy for arbitrary <math>a</math>.<br />
<br />
== The matter of sigma: the critical strip, zeta peaks, and Gram points ==<br />
So long as {{nowrap| ''s'' ≥ 1 }}, the absolute value of the zeta function can be seen as a relative error measurement. However, the rationale for that view of things departs when {{nowrap| ''s'' < 1 }}, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when {{nowrap| 0 < ''s'' < 1 }}. As s approaches the value {{nowrap|''s'' {{=}} {{sfrac|1|2}}}} of the [http://mathworld.wolfram.com/CriticalLine.html critical line], the "information content" of the zeta function concerning higher primes increases and it behaves increasingly like a badness measure (or more correctly, since we have inverted it, like a goodness measure.) The quasi-symmetric [https://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html functional equation] of the zeta function tells us that past the critical line the information content starts to decrease again, with {{nowrap| 1 − ''s'' }} and ''s'' having the same information content; that is, for ''s'' > {{sfrac|1|2}}, {{nowrap|1 − ''s''}} essentially multiplies the zeta function at ''s'' by a fixed, monotonic increasing function. Hence it is the zeta function between {{nowrap|''s'' {{=}} {{sfrac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, which is of the most interest.<br />
<br />
=== Introduction to Gram points ===<br />
As {{nowrap| ''s'' > 1 }} gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply {{nowrap|1 + 2<sup>−''z''</sup>}}, which approaches 1 as {{nowrap|''s'' {{=}} Re(''z'')}} becomes larger. When {{nowrap|''s'' ≫ 1}} and ''x'' is an integer, the real part of zeta is approximately {{nowrap|1 + 2<sup>−''s''</sup>}}, and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from {{nowrap|''s'' {{=}} +&infin;}} with ''x'' an integer, we can trace a line back towards the critical strip on which zeta is real. Since when {{nowrap|''s'' ≫ 1}} the derivative is approximately −{{sfrac|ln(2)|2<sup>''s''</sup>}}, it is negative on this line of real values for zeta, meaning that the real value for zeta increases as ''s'' decreases. The zeta function approaches 1 uniformly as ''s'' increases to infinity, so as ''s'' decreases, the real-valued zeta function along this line of real values continues to increase though all real values from 1 to infinity monotonically. When it crosses the critical line where {{nowrap| ''s'' {{=}} {{sfrac|1|2}} }}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap| ζ({{frac|1|2}} + ''ig'') }} are real are called "Gram points", after {{w|Jørgen Pedersen Gram}}. We thus have associated pure-octave edos, where ''x'' is an integer, to a value near to the pure octave, at the special sorts of Gram points which corresponds to edos.<br />
<br />
=== Gram points and zeta peaks ===<br />
Because the value of zeta increased continuously as it made its way from +&infin; to the critical line, we might expect the values of zeta at these special Gram points to be relatively large. This would be especially true if −ζ′(''z'') is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to {{w|Bernhard Riemann}} which he failed to publish because it was so nasty becomes a bit simpler when used at a Gram point. It is named the {{w|Riemann–Siegel formula}} since {{w|Carl Ludwig Siegel}} went looking for it and was able to reconstruct it after rooting industriously around in Riemann's unpublished papers. From this formula, it is apparent that when x corresponds to a good edo, the value of {{nowrap|ζ({{frac|1|2}} + ''ig'')}} at the corresponding Gram point should be especially large.<br />
<br />
=== The Z function: a mathematically convenient version of zeta ===<br />
The absolute value of {{nowrap|ζ{{pars|{{sfrac|1|2}} + ''ig''}}|1.8}} at a Gram point corresponding to an edo is near to a local maximum, but not actually at one. At the local maximum, of course, the partial derivative of {{nowrap|ζ{{pars|{{sfrac|1|2}} + ''it''}}|1.8}} with respect to ''t'' will be zero; however this does not mean its derivative there will be zero. In fact, the {{w|Riemann hypothesis}} is equivalent to the claim that all zeros of {{nowrap|ζ′(''s'' + ''it'')}} occur when {{nowrap|''s'' &gt; {{sfrac|1|2}}}}, which is where all known zeros lie. These do not have values of ''t'' corresponding to good edos. For this and other reasons, it is helpful to have a function which is real for values on the critical line but whose absolute value is the same as that of zeta. This is provided by the {{w|''Z'' function}}, which is defined (in terms of the {{subpage|appendix|u|s=Z function and Riemann-Siegel theta function|text=Riemann-Siegel theta function}}) as:<br />
<br />
<math>Z(t) = \exp(i \theta(t)) \zeta\left(\frac{1}{2} + it\right)</math>.<br />
<br />
The factor of <math>\exp(i \theta(t))</math> simply modifies zeta by a complex phase, and so the absolute value of Z along the real axis is the same as the absolute value of ζ at the corresponding place on the critical line, and the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip, and since θ is holomorphic on the strip with imaginary part between −{{sfrac|1|2}} and {{sfrac|1|2}}, so is Z. And Z is a real even function of the real variable ''t'', since theta was defined so as to give precisely this property.<br />
<br />
== Zeta edo lists ==<br />
=== Record edos ===<br />
The prime-approximating strength of an edo can be determined by the magnitude of Z(''x''). Since a higher {{nowrap|{{abs|Z(''x'')}}}} correlates to a stronger tuning, we would like to find a sequence with successively larger {{nowrap|{{abs|Z(''x'')}}}}-associated values satisfying some property.<br />
<br />
==== Zeta peak edos ====<br />
If we examine the increasingly larger peak values of {{nowrap|{{abs|Z(''x'')}}}}, we find they occur with values of ''x'' such that {{nowrap|Z′(''x'') {{=}} 0}} near to integers, so that there is a sequence of [[edo]]s {{EDOs| 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664, 14348, 16808, 28742, 34691, 36269, 57578, 58973, 95524, 102557, 112985, 148418, 212147, 241200,}} … of '''zeta peak edos'''. This is listed in the On-Line Encyclopedia of Integer Sequences as {{OEIS|A117536}}. Note that these peaks occur close to integer values, but are never exactly located at an integer; this can be interpreted as the zeta function suggesting detuned ([[stretched and compressed tuning|stretched or compressed]]) octaves for the edo in question, similar to the [[TOP tuning]] (although the two tunings are in general not the same). As a result, this list can also be thought of as "tempered-octave zeta peak edos."<br />
<br />
==== Zeta peak integer edos ====<br />
Alternatively (as [[groundfault]] has found), if we do not allow octave detuning and instead look at only the record {{nowrap|{{abs|Z(''x'')}}}} zeta scores corresponding to exact edos with pure octaves, we get {{EDOs| 1, 2, 3, 5, 7, 10, 12, 19, 22, 31, 41, 53, 87, 118, 130, 171, 224, 270, 311, 472, 494, 742, 1065, 1106, 1395, 1578, 2460, 2684, 3566, 4231, 4973, 5585, 8269, 8539, 14124, 14348, 16808, 28742, 30631, 34691, 36269, 57578, 58973,}} … of '''zeta peak integer edos'''. Edos not present in the previous list but present here include {{EDOs| 87, 311, 472, 1065, 3566, 4231, 4973, 14124, 30631,}} … and edos present in the previous list but not present here include {{EDOs| 4, 27, 72, 99, 152, 217, 342, 422, 441, 764, 935, 954, 1012, 1178, 1236, 1448, 3395, 6079, 7033, 11664,}} … with 72's removal perhaps being the most surprising, showing the strength of 53 in that 72 does not improve on 53's peak. This definition may be better for measuring how accurate edos are without detuned octaves, whereas the previous list assumes that the octave is tempered along with all other intervals. This list can thus also be thought of as "pure-octave zeta peak edos."<br />
<br />
==== Zeta integral edos ====<br />
Similarly, if we take the integral of {{nowrap|{{abs|Z(''x'')}}}} between successive zeros, and use this to define a sequence of increasing values for this integral, these again occur near integers and define an edo. This sequence, the '''zeta integral edos''', goes {{EDOs| 2, 5, 7, 12, 19, 31, 41, 53, 72, 130, 171, 224, 270, 764, 954, 1178, 1395, 1578, 2684, 3395, 7033, 8269, 8539, 14348, 16808, 36269, 58973,}} … This is listed in the OEIS as {{OEIS|A117538}}. The zeta integral edos seem to be, on the whole, the best of the zeta function sequences, but the other two should not be discounted; the peak values seem to give more weight to the lower primes, and the zeta gap sequence discussed below to the higher primes.<br />
<br />
==== Zeta gap edos ====<br />
Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of '''zeta gap edos'''. These are {{EDOs| 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190, 8539, 14348, 58973, 95524,}} … Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as {{OEIS|A117537}}. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.<br />
<br />
==== Strict zeta edos ====<br />
We may define the '''strict zeta edos''' to be the edos that are in all four of the above lists. The list of strict zeta edos begins {{EDOs| 2, 5, 7, 12, 19, 31, 53, 270, 1395, 1578, 8539, 14348, 58973,}} ….<br />
<br />
This, however, requires that an edo's zeta peak be record-holding with both pure and detuned octaves. It is actually debatable whether or not this constraint is a good idea, as it does not account for tuning tendencies that are skewed sharpward or flatward. If detuned octaves are allowed, the two smallest additional edos that are strict zeta edos would be [[72edo]] and [[954edo]].<br />
<br />
==== List of record zeta edos ====<br />
{| class="wikitable" style="text-align: center; margin: auto auto auto auto;"<br />
|+ style="font-size: 105%;" | Zeta record edos up to 1000<br />
|-<br />
! colspan="2" | Edo<br />
| [[1edo|1]]<br />
| [[2edo|2]]<br />
| [[3edo|3]]<br />
| [[4edo|4]]<br />
| [[5edo|5]]<br />
| [[7edo|7]]<br />
| [[10edo|10]]<br />
| [[12edo|12]]<br />
| [[19edo|19]]<br />
| [[22edo|22]]<br />
| [[27edo|27]]<br />
| [[31edo|31]]<br />
| [[41edo|41]]<br />
| [[46edo|46]]<br />
| [[53edo|53]]<br />
| [[72edo|72]]<br />
| [[87edo|87]]<br />
| [[99edo|99]]<br />
| [[118edo|118]]<br />
| [[130edo|130]]<br />
| [[152edo|152]]<br />
| [[171edo|171]]<br />
| [[217edo|217]]<br />
| [[224edo|224]]<br />
| [[270edo|270]]<br />
| [[311edo|311]]<br />
| [[342edo|342]]<br />
| [[422edo|422]]<br />
| [[441edo|441]]<br />
| [[472edo|472]]<br />
| [[494edo|494]]<br />
| [[742edo|742]]<br />
| [[764edo|764]]<br />
| [[935edo|935]]<br />
| [[954edo|954]] <br />
|-<br />
! rowspan="2" | Record<br>peak !! Detuned<br>octaves<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| <br />
| &#x2605;<br />
| &#x2605;<br />
| <br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| <br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| <br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
|-<br />
! Pure<br>octaves<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| <br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
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| &#x2605;<br />
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| &#x2605;<br />
| &#x2605;<br />
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| &#x2605;<br />
| <br />
| &#x2605;<br />
| &#x2605;<br />
| &#x2605;<br />
| <br />
| <br />
| <br />
| &#x2605;<br />
| &#x2605;<br />
| <br />
| <br />
| <br />
| <br />
|-<br />
! colspan="2" | Integral<br />
|<br />
| &#x2605;<br />
| <br />
| <br />
| &#x2605;<br />
| &#x2605;<br />
| <br />
| &#x2605;<br />
| &#x2605;<br />
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| &#x2605;<br />
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| <br />
| <br />
| &#x2605;<br />
| <br />
| &#x2605;<br />
| <br />
| &#x2605;<br />
| &#x2605;<br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| &#x2605;<br />
| <br />
| &#x2605; <br />
|-<br />
! colspan="2" | Gap<br />
|<br />
| &#x2605;<br />
| &#x2605;<br />
| <br />
| &#x2605;<br />
| &#x2605;<br />
| <br />
| &#x2605;<br />
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| <br />
| <br />
| &#x2605;<br />
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| <br />
| <br />
| &#x2605;<br />
| &#x2605;<br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| <br />
| &#x2605;<br />
|}<br />
<br />
=== Anti-record edos ===<br />
==== Zeta valley edos ====<br />
Just like with zeta peak edos which have progressively higher {{nowrap|{{abs|Z(x)}}}} scores, we can also look at edos with progressively ''lower'' {{nowrap|{{abs|Z(x)}}}} for integer values of ''x''. This gives us {{EDOs| 1, 8, 18, 39, 55, 64, 79, 5941, 8294,}}… which correspond to ''zeta valley edos''. Zeta valley edos can be thought of as pure-octave tunings that tend to deviate from ''p''-limit JI as much as possible while still preserving octaves, and can serve as "more xenharmonic" tunings. Zeta valley edos are only measured with pure octaves, since "tempered-octave zeta valley edos" would simply be any zero of Z(x). Keep in mind, however, that the ''most'' xenharmonic tunings (essentially, tuning systems that avoid ''all'' ''p''-limit JI as much as possible) would not contain octaves at all.<br />
<br />
Notice that there is a very large jump from [[79edo]] to [[5941edo]]. We know that record {{nowrap|{{abs|Z(x)}}}} scores, both with tempered octaves and pure octaves, grow logarithmically on average. If we assume the scores of integer edos are uniformly distributed on the interval {{nowrap|[0, ''c'' log(''x'')]}}, the probability for the next edo to have a zeta score less than a given small value is also very small, so we would expect valley edos to be rarer than peak edos. So, it would be more productive to find edos which zeta score is simply less than a given threshold.<br />
<br />
=== Other lists ===<br />
{{Idiosyncratic terms|"Absolute zeta peak edos" was coined by {{u|Godtone}}, "''k''-ary-peak edos" was coined by {{u|Akselai}}, the types of "local zeta" edos and "indecisive edos" were coined by [[Budjarn Lambeth]].}}<br />
<br />
==== Absolute zeta peak edos ====<br />
If we consider that zeta is a measure of relative error (that is, error measured relative to the step size), we realize that plenty of equal temperaments<ref group="note">Note importantly that we speak of ''equal temperaments'' rather than ''edos'' because generally a record peak ''does not'' correspond to an edo, which can have tangible consequences (a significant example is discussed in the next section).</ref> are excluded simply because, though practically speaking they have great tuning properties, they are not as "efficient" with their number of tones as the last record peak. Arguably what we are interested in is a sequence of edos that generally do increasingly better at tuning JI in terms of lowering the average cent error. Therefore, it suffices to multiply the score by the size of the equal temperament. Surprisingly, the list for ''s'' = 1/2 — which is supposedly where high-limit information is maximized — is ''almost identical'' to the one for ''s'' = 1 — which is the smallest value of ''s'' that we can assume to be meaningful without assuming that the analytic continuation preserves the tuning properties we are interested in — so that we have reassurance from the ''s'' = 1 list that the ''s'' = 1/2 list is meaningful wherever they agree. This is important because surprisingly, the two lists of equal temperament are ''identical up to [[311edo|311et]]'', with only one edo, [[8edo]], omitted from the list for {{nowrap| ''s'' {{=}} 1 }}. This list is {{EDOs| 1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 17, 19, 22, 24, 26, 27, 31, 34, 41, 46, 53, 58, 65, 68, 72, 84, 87, 94, 99, 111, 118, 130, 140, 152, 171, 183, 198, 212, 217, 224, 243, 270, 311, … }}.<br />
<br />
==== Extended list of absolute zeta peak edos ====<br />
If you look at the graph of zeta (for any zeta graph of interest), another issue quickly becomes evident: many equal temperaments of interest fail to have peaks of record height by only small amounts, so that we intuitively want to include them in a more comprehensive list. However, trying to "fix" this issue quickly leads into another issue: how many "nearly record" edos should we include, and why? The smallest alteration we can make is to allow an equal temperament that does better than the second-best-scoring equal temperament so far. But sometimes we have two very strong equal temperaments appear in quick succession, and given the motivation is to find a more comprehensive list anyways, here we'll include any equal temperament that does better than the third-best-scoring equal temperament so far. The motivation for this cutoff is that you intuitively might expect that the three best equal temperaments found so far represent roughly how good we can do in a given range of step sizes, so that they define what is "normal" for that range, that is, it's the heuristic of the "rule of three". Again, the list for ''s'' = 1/2 is almost identical to ''s'' = 1 for equal temperaments up to 311et, though this time the differences are less trivial: [[176edo|176et]] and [[202edo|202et]] only appear for ''s'' = 1/2, so are put in brackets. The list is {{EDOs| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 34, 36, 38, 39*, 41, 43, 45, 46, 48, 50, 53, 56, 58, 60, 63, 65, 68, 72, 77, 80, 84, 87, 89, 94, 96, 99, 103, 106, 111, 113, 118, 121, 125, 130, 137, 140, 145, 149, 152, 159, 161, 166, 171, (176,) 183, 190, 193, 198, (202,) 212, 217, 224, 229, 239, 243, 248, 255, 270, 277, 282, 289, 301, 311, … }}.<br />
<br />
The equal temperaments added relative to the non-extended list of only things that are records proper are: {{EDOs| 6, 8, 11, 13, 16, 21, 29, 36, 38, 39, 43, 45, 48, 50, 56, 60, 63, 77, 80, 89, 96, 103, 106, 113, 121, 125, 137, 145, 149, 159, 161, 166, (176,) 190, 193, (202,) 229, 239, 248, 255, 277, 282, 289, 301 }}<ref group="note">39et is a notable example because 39edo corresponds to a zeta valley, so it is surprising that it would be included here; the reason that it is included is because this is ''not'' 39edo, but 39 ''equal temperament'', corresponding to a 3.8{{cent}} flat-tempered octave so that it is actually ~39.124edo, that is, it corresponds to the 173rd zeta peak, known by the shorthand 173zpi (where i stands for index). Therefore, this may prove a good testcase for investigating the effects of zeta-based octave-tempering, though given the size of the stretch, the difference is likely to be subtle, but the fact that it "changes zeta's mind" this much is itself interesting. You can also interpret this result differently, which is as evidence that you should not include equal temperaments worse than the third-best-scoring equal temperament so far, given the somewhat dubious inclusion of 39et, however it should be noted that this is more to do with that at the very beginning of the list there are not many equal temperaments to "beat" so that beating the third-best-scoring equal temperament so far is easy, though arguably this is not a flaw because people are often more likely to try a smaller equal temperament. It is also perhaps worth noting that 37et almost makes this extended list, but the omission of 37et is much better addressed by no-3's zeta.</ref>. <br />
<br />
==== ''k''-ary-peak edos ====<br />
If we want to find the second-best edos ranked by zeta peaks, then given a full list of zeta peaks, we can remove the successively higher peaks to get another sequence of succesively higher peaks, which correspond to edos called '''2-ary peak edos''': defined as non-zeta-peak edos with a higher zeta peak than any smaller non-zeta-peak edo. <br />
<br />
This list can be used finding an alternative to any given zeta peak edo of similar size and still-okay accuracy, but with different regular temperament properties (e.g. 9 as alternative to 10, 17 as alternative to 19).<br />
<br />
{{EDOs| 6, 8, 9, 14, 15, 17, 24, 34, 46, 58, 65, 77, 87, 111, 140, 183, 243, 301, 311, 460, 472, 525, 571, 581, 814, 836, 882, 1205,}} …<br />
<br />
We can then remove those secondary peaks again to get '''3-ary peak edos''': non-zeta-peak edos with a higher zeta peak than any smaller edo that is neither a zeta peak nor a 2-ary peak.<br />
<!-- add list here --><br />
<br />
We can repeat this process as many times as we want, resulting in '''''k''-ary-peak edos'''. The ordinary peak edos are 1-ary peak edos, then there are 2-ary peak edos, 3-ary peak edos, and so on. However keep in mind that the higher ''k'' gets, the less meaningful the peaks will get, especially for smaller edos (less than about 100).<br />
<br />
==== Local zeta peak edos ====<br />
We may define ''local zeta peak'' edos as a generalization of the ''zeta peak'' edos as those that do not necessarily have successively higher zeta peaks but simply have a higher zeta peak than the edos on either side of them. This lists a wide variety of edos that approximate primes well for their size, even when they aren't better than every smaller edo. It could be helpful for:<br />
* finding edos with plenty of consonances in size ranges that lack any record-holding zeta edos (e.g. between 60 and 70 tones)<br />
* finding edos for composers who enjoy exploring new territory, because it lists those edos (including undiscovered ones) that have lots of new consonances to explore relative to their size, and are ripe for exploration<br />
<br />
{{EDOs| 5, 7, 10, 12, 15, 17, 19, 22, 24, 27, 29, 31, 34, 36, 38, 41, 43, 46, 48, 50, 53, 56, 58, 60, 63, 65, 68, 72, 75, 77, 80, 82, 84, 87, 89, 91, 94, 96, 99,}} …<br />
<br />
==== Local zeta peak integer edos ====<br />
Similarly, we may define ''local zeta peak integer'' edos as a generalization of the ''zeta peak integer'' edos, i.e. those that have a higher zeta peak than the edos on either side of them ''with pure octaves'':<br />
<br />
{{EDOs| 3, 5, 7, 10, 12, 15, 17, 19, 22, 24, 26, 29, 31, 34, 36, 41, 43, 46, 48, 50, 53, 56, 58, 63, 65, 68, 70, 72, 74, 77, 80, 82, 84, 87, 89, 94, 96, 99,}} …<br />
<br />
Edos not present in the previous list but present here include 3, 26, 70, 74, …<br />
<br />
Edos present in the previous list but not present here include 27, 38, 60, 75, 91, …<br />
<br />
==== Local zeta valley edos ====<br />
We may define ''local zeta valley'' edos as those with a ''lower'' best nearby zeta peak than the edos on either side of them. This is helpful for finding edos that force the use of methods other than traditional concordant harmony, or for composers seeking a challenge/limitation to inspire creativity.<br />
<br />
{{EDOs| 6, 8, 11, 13, 16, 18, 20, 23, 25, 28, 30, 33, 35, 37, 40, 42, 44, 47, 49, 52, 54, 57, 59, 61, 64, 66, 69, 71, 73, 76, 78, 81, 83, 86, 88, 90, 92, 95, 97, }} …<br />
<br />
==== Indecisive edos ====<br />
Finally, ''indecisive'' edos can be defined as edos which are ''neither'' local zeta peak, nor local zeta valley. For some, these tunings might narrow down the range of compositional choices available so as to be not so many to promote indecision, but not so few as to promote frustration.<br />
<br />
{{EDOs| 9, 14, 21, 26, 32, 39, 45, 51, 55, 62, 67, 70, 74, 79, 85, 93, 98, }} …<br />
<br />
=== Further lists ===<br />
See [[The Riemann zeta function and tuning/Record lists|the record lists]].<br />
<br />
== Optimal octave stretch ==<br />
Another use for the Riemann zeta function is to determine the optimal tuning for an edo, meaning the optimal octave stretch. This is because the zeta peaks are typically not integers. The fractional part can give us the degree to which the generator diverges from what you would need to have the octave be a perfect 1200 cents. <br />
<br />
For all edos 1 through 100, and for a list of successively higher zeta peaks, taken to five decimal places, see [[table of zeta-stretched edos]].<br />
<br />
=== Zeta peak index ===<br />
These octave-stretched edos are not the only tunings which can be produced from zeta peaks. They are only one type of tuning within a larger family of equal-step tunings called zeta peak indices. They have their own article here, with a table of the first 500 or so: [[ZPI|zeta peak index (ZPI)]].<br />
<br />
== Removing primes ==<br />
An [http://mathworld.wolfram.com/EulerProduct.html Euler product] formula for the Riemann zeta function {{subpage|appendix|u|s=Euler product expression for the zeta function|text=can be easily derived}}:<br />
<br />
<math>\displaystyle{<br />
\zeta(s) = \prod_p \left(1 - p^{-s}\right)^{-1}<br />
}</math><br />
<br />
where the product is over all primes ''p''. The product converges for values of ''s'' with real part greater than one, while at {{nowrap|''s'' {{=}} 1}} it diverges to infinity. We may remove a finite list of primes from consideration by multiplying ζ(''s'') by the corresponding factors {{nowrap|(1 − ''p''<sup>−''s''</sup>)}} for each prime ''p'' we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for ''s'' with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, {{nowrap|(1 − 2<sup>−''s''</sup>)ζ(''s'')}} is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.<br />
<br />
Along any line of constant <math>\sigma</math>, {{subpage|appendix|u|s=Conversion factor for removing primes|text=it can be shown that}}:<br />
<br />
<math>\displaystyle{<br />
\left| 1 - p^{-\sigma - it} \right| = \sqrt{1 + \frac{1}{p^{2\sigma}} - \frac{2 \cos(t \ln p)}{p^\sigma}}<br />
}</math>;<br />
<br />
in particular, on the critical line,<br />
<br />
<math>\displaystyle{<br />
\left| 1 - p^{-\frac{1}{2} - it} \right| = \sqrt{1 + \frac{1}{p} - \frac{2 \cos(t \ln p)}{\sqrt{p}}}<br />
}</math>.<br />
<br />
Multiplying the Z-function by this factor of adjustment gives a Z-function with the prime ''p'' removed from consideration. Zeta peak and zeta integral tunings may then be found as before. Note that multiplying this factor is technically only accurate for sums whose result is related to the Z function rather than the real part of the zeta function.<br />
<br />
For example, if we want to find zeta peak [[EDT]]s (division of the [[3/1|{{ordinal|3}}]] harmonic, or "tritave")—noting that here we must substitute <math>t = \frac{2\pi x}{\ln(3)}</math> instead of <math>\frac{2\pi x}{\ln(2)}</math>—in the no-twos subgroup, our modified Z function is:<br />
<br />
<math><br />
\displaystyle Z\left(\frac{2\pi}{\ln(3)}x\right)\sqrt{\frac{3}{2}-\sqrt{2}\cos\left(\frac{2\pi\ln(2)}{\ln(3)}x\right)}<br />
</math>.<br />
<br />
Removing 2 leads to increasing adjusted peak values corresponding to edts into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 75, 88, 131, 245, 316,…}} parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen–Pierce]] division of the tritave, but the multiples 26 and 39 also.<br />
<br />
== Further information ==<br />
* {{subpage|appendix|u|s=Black magic formulas|text=How it can be shown what ETs have a sharp vs. flat tendency?}}<br />
* {{subpage|appendix|u|s=Computing zeta|text=How do you actually compute zeta?}}<br />
<br />
== Links ==<br />
* [https://arxiv.org/abs/math/0309433 X-Ray of Riemann zeta-function] by Juan Arias-de-Reyna<br />
* [http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/ Selberg's limit theorem] by Terence Tao [http://www.webcitation.org/5xrvgjW6T Permalink]<br />
* [[:File:Zetamusic5.pdf|Favored cardinalities of scales]] by Peter Buch<br />
* [http://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01568-0/S0025-5718-03-01568-0.pdf Computational estimation of the order of {{nowrap|ζ({{frac|1|2}} + ''it'')}}] by Tadej Kotnik<br />
* [https://www-users.cse.umn.edu/~odlyzko/zeta_tables/index.html Andrew Odlyzko: Tables of zeros of the Riemann zeta function]<br />
* [https://www-users.cse.umn.edu/~odlyzko/doc/zeta.html Andrew Odlyzko: Papers on Zeros of the Riemann Zeta Function and Related Topics]<br />
* [https://www.lmfdb.org/zeros/zeta/?N=1&t=&limit=100 Zeros of Zeta]<br />
<br />
== Notes ==<br />
<references group="note"/><br />
<br />
[[Category:Zeta| ]] <!-- Main article --><br />
[[Category:Math]]<br />
[[Category:Tuning]]<br />
[[Category:Number theory]]<br />
[[Category:Pages with proofs]]<br />
{{Todo| increase applicability | simplify }}</div>ArrowHead294https://en.xen.wiki/index.php?title=55edo&diff=22548655edo2026-03-09T23:30:10Z<p>ArrowHead294: /* Theory */ Fix link</p>
<hr />
<div>{{interwiki<br />
| de = 55-EDO<br />
| en = 55edo<br />
| es = 55 EDO<br />
| ja = <br />
}}<br />
{{Infobox ET}}<br />
{{ED intro}}<br />
<br />
== Theory ==<br />
55edo is a [[The Riemann zeta function and tuning#Zeta valley edos|zeta valley edo]], so it does not approximate the harmonic series very well for its size. Despite this, it can be used as a [[meantone]] tuning, and is close to [[1/6-comma meantone]] (and is almost exactly 10/57-comma meantone). {{w|Georg Philipp Telemann|Telemann}} suggested it as a theoretical basis for analyzing the [[meantone intervals|intervals of meantone]]. {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}} recommended 55edo or something close to it, with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.<ref>Chesnut, John (1977) ''Mozart's Teaching of Intonation'', '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]</ref> It can also be used for [[Meantone_family#Mohajira|Mohajira]] and [[Meantone_family#Liese|Liese]] temperaments. It also supports an extremely sharp tuning of [[huygens|Huygens/undecimal meantone]] using the 55de [[val]], meaning that primes 7 and 11 are mapped very sharply to their second-best mapping.<br />
<br />
=== Odd harmonics ===<br />
{{Harmonics in equal|55}}<br />
<br />
=== Subsets and supersets ===<br />
Since 55 factors into {{factorization|55}}, 55edo contains [[5edo]] and [[11edo]] as its subsets.<br />
<br />
== Intervals ==<br />
{| class="wikitable center-1 right-2 left-3"<br />
|-<br />
! [[Degree|&#35;]]<br />
! [[Cent]]s<br />
! Approximate ratios<br />
! colspan="3" | [[Ups and downs notation]]<br />
|-<br />
| 0<br />
| 0.0<br />
| [[1/1]]<br />
| P1<br />
| perfect 1sn<br />
| D<br />
|-<br />
| 1<br />
| 21.8<br />
| [[65/64]], [[78/77]], [[99/98]], ''[[128/125]]''<br />
| ^1<br />
| up 1sn<br />
| ^D<br />
|-<br />
| 2<br />
| 43.6<br />
| [[36/35]], ''[[64/63]]''<br />
| ^^1<br />
| dup 1sn<br />
| ^^D<br />
|-<br />
| 3<br />
| 65.5<br />
| [[28/27]]<br />
| vvm2<br />
| dudminor 2nd<br />
| vvEb<br />
|-<br />
| 4<br />
| 87.3<br />
| [[21/20]], ''[[18/17]]'', ''[[25/24]]''<br />
| vm2<br />
| downminor 2nd<br />
| vEb<br />
|-<br />
| 5<br />
| 109.1<br />
| [[16/15]], [[17/16]]<br />
| m2<br />
| minor 2nd<br />
| Eb<br />
|-<br />
| 6<br />
| 130.9<br />
| [[13/12]], [[14/13]]<br />
| ^m2<br />
| upminor 2nd<br />
| ^Eb<br />
|-<br />
| 7<br />
| 152.7<br />
| [[12/11]], ''[[11/10]]''<br />
| ~2<br />
| mid 2nd<br />
| vvE<br />
|-<br />
| 8<br />
| 174.5<br />
| <br />
| vM2<br />
| downmajor 2nd<br />
| vE<br />
|-<br />
| 9<br />
| 196.4<br />
| [[9/8]], ''[[10/9]]''<br />
| M2<br />
| major 2nd<br />
| E<br />
|-<br />
| 10<br />
| 218.2<br />
| [[17/15]]<br />
| ^M2<br />
| upmajor 2nd<br />
| ^E<br />
|-<br />
| 11<br />
| 240.0<br />
| [[8/7]]<br />
| ^^M2<br />
| dupmajor 2nd<br />
| ^^E<br />
|-<br />
| 12<br />
| 261.8<br />
| [[7/6]]<br />
| vvm3<br />
| dudminor 3rd<br />
| vvF<br />
|-<br />
| 13<br />
| 283.6<br />
| [[13/11]]<br />
| vm3<br />
| downminor 3rd<br />
| vF<br />
|-<br />
| 14<br />
| 305.5<br />
| [[6/5]]<br />
| m3<br />
| minor 3rd<br />
| F<br />
|-<br />
| 15<br />
| 327.3<br />
| <br />
| ^m3<br />
| upminor 3rd<br />
| ^F<br />
|-<br />
| 16<br />
| 349.1<br />
| [[11/9]], [[27/22]]<br />
| ~3<br />
| mid 3rd<br />
| ^^F<br />
|-<br />
| 17<br />
| 370.9<br />
| [[26/21]], ''[[16/13]]''<br />
| vM3<br />
| downmajor 3rd<br />
| vF#<br />
|-<br />
| 18<br />
| 392.7<br />
| [[5/4]]<br />
| M3<br />
| major 3rd<br />
| F#<br />
|-<br />
| 19<br />
| 414.5<br />
| [[14/11]]<br />
| ^M3<br />
| upmajor 3rd<br />
| ^F#<br />
|-<br />
| 20<br />
| 436.4<br />
| [[9/7]]<br />
| ^^M3<br />
| dupmajor 3rd<br />
| ^^F#<br />
|-<br />
| 21<br />
| 458.2<br />
| ''[[21/16]]''<br />
| vv4<br />
| dud 4th<br />
| vvG<br />
|-<br />
| 22<br />
| 480.0<br />
| <br />
| v4<br />
| down 4th<br />
| vG<br />
|-<br />
| 23<br />
| 501.8<br />
| [[4/3]], ''[[27/20]]''<br />
| P4<br />
| perfect 4th<br />
| G<br />
|-<br />
| 24<br />
| 523.6<br />
| <br />
| ^4<br />
| up 4th<br />
| ^G<br />
|-<br />
| 25<br />
| 545.5<br />
| [[11/8]], [[15/11]]<br />
| ~4<br />
| mid 4th<br />
| ^^G<br />
|-<br />
| 26<br />
| 567.3<br />
| [[7/5]], [[18/13]]<br />
| vA4<br />
| downaug 4th<br />
| vG#<br />
|-<br />
| 27<br />
| 589.1<br />
| [[24/17]]<br />
| A4, vd5<br />
| aug 4th, downdim 5th<br />
| G#, vAb<br />
|-<br />
| 28<br />
| 610.9<br />
| [[17/12]]<br />
| ^A4, d5<br />
| upaug 4th, dim 5th<br />
| ^G#, Ab<br />
|-<br />
| 29<br />
| 632.7<br />
| [[10/7]], [[13/9]]<br />
| ^d5<br />
| updim 5th<br />
| ^Ab<br />
|-<br />
| 30<br />
| 654.5<br />
| [[16/11]], [[22/15]]<br />
| ~5<br />
| mid 5th<br />
| vvA<br />
|-<br />
| 31<br />
| 676.4<br />
| <br />
| v5<br />
| down 5th<br />
| vA<br />
|-<br />
| 32<br />
| 698.2<br />
| [[3/2]], ''[[40/27]]''<br />
| P5<br />
| perfect 5th<br />
| A<br />
|-<br />
| 33<br />
| 720.0<br />
| <br />
| ^5<br />
| up 5th<br />
| ^A<br />
|-<br />
| 34<br />
| 741.8<br />
| ''[[32/21]]''<br />
| ^^5<br />
| dup 5th<br />
| ^^A<br />
|-<br />
| 35<br />
| 763.6<br />
| [[14/9]]<br />
| vvm6<br />
| dudminor 6th<br />
| vvBb<br />
|-<br />
| 36<br />
| 785.5<br />
| [[11/7]]<br />
| vm6<br />
| downminor 6th<br />
| vBb<br />
|-<br />
| 37<br />
| 807.3<br />
| [[8/5]]<br />
| m6<br />
| minor 6th<br />
| Bb<br />
|-<br />
| 38<br />
| 829.1<br />
| [[21/13]], ''[[13/8]]''<br />
| ^m6<br />
| upminor 6th<br />
| ^Bb<br />
|-<br />
| 39<br />
| 850.9<br />
| [[18/11]], [[44/27]]<br />
| ~6<br />
| mid 6th<br />
| vvB<br />
|-<br />
| 40<br />
| 872.7<br />
| <br />
| vM6<br />
| downmajor 6th<br />
| vB<br />
|-<br />
| 41<br />
| 894.5<br />
| [[5/3]]<br />
| M6<br />
| major 6th<br />
| B<br />
|-<br />
| 42<br />
| 916.4<br />
| [[22/13]]<br />
| ^M6<br />
| upmajor 6th<br />
| ^B<br />
|-<br />
| 43<br />
| 938.2<br />
| [[12/7]]<br />
| ^^M6<br />
| dupmajor 6th<br />
| ^^B<br />
|-<br />
| 44<br />
| 960.0<br />
| [[7/4]]<br />
| vvm7<br />
| dudminor 7th<br />
| vvC<br />
|-<br />
| 45<br />
| 981.8<br />
| [[30/17]]<br />
| vm7<br />
| downminor 7th<br />
| vC<br />
|-<br />
| 46<br />
| 1003.6<br />
| [[16/9]], ''[[9/5]]''<br />
| m7<br />
| minor 7th<br />
| C<br />
|-<br />
| 47<br />
| 1025.5<br />
| <br />
| ^m7<br />
| upminor 7th<br />
| ^C<br />
|-<br />
| 48<br />
| 1047.3<br />
| [[11/6]], ''[[20/11]]''<br />
| ~7<br />
| mid 7th<br />
| ^^C<br />
|-<br />
| 49<br />
| 1069.1<br />
| [[13/7]], [[24/13]]<br />
| vM7<br />
| downmajor 7th<br />
| vC#<br />
|-<br />
| 50<br />
| 1090.9<br />
| [[15/8]], ''[[32/17]]''<br />
| M7<br />
| major 7th<br />
| C#<br />
|-<br />
| 51<br />
| 1112.7<br />
| [[40/21]], ''[[17/9]]'', ''[[48/25]]''<br />
| ^M7<br />
| upmajor 7th<br />
| ^C#<br />
|-<br />
| 52<br />
| 1134.5<br />
| [[56/27]]<br />
| ^^M7<br />
| dupmajor 7th<br />
| ^^C#<br />
|-<br />
| 53<br />
| 1156.4<br />
| [[35/18]], ''[[63/32]]''<br />
| vv8<br />
| dud 8ve<br />
| vvD<br />
|-<br />
| 54<br />
| 1178.2<br />
| [[128/65]], [[77/39]], [[196/99]], ''[[125/64]]''<br />
| v8<br />
| down 8ve<br />
| vD<br />
|-<br />
| 55<br />
| 1200.0<br />
| [[2/1]]<br />
| P8<br />
| perfect 8ve<br />
| D<br />
|}<br />
<nowiki />* 55f val (tending flat), inconsistent intervals labeled in ''italic''<br />
<br />
== Notation ==<br />
=== Ups and downs notation ===<br />
55edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.<br />
{{Ups and downs sharpness}}<br />
<br />
[[Alternative symbols for ups and downs notation]] uses sharps, flats, half- and sesquisharps, and half- and sesquiflats with arrows, borrowed from extended [[Helmholtz–Ellis notation]] and [[24edo#Stein-Zimmerman Accidentals|Stein-Zimmerman accidental set]]:<br />
{{Sharpness-sharp4}}<br />
<br />
=== Sagittal notation ===<br />
==== Evo flavor ====<br />
<imagemap><br />
File:55-EDO_Evo_Sagittal.svg<br />
desc none<br />
rect 80 0 300 50 [[Sagittal_notation]]<br />
rect 300 0 615 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]<br />
rect 20 80 160 106 [[896/891]]<br />
rect 160 80 280 106 [[33/32]]<br />
default [[File:55-EDO_Evo_Sagittal.svg]]<br />
</imagemap><br />
<br />
==== Revo flavor ====<br />
<imagemap><br />
File:55-EDO_Revo_Sagittal.svg<br />
desc none<br />
rect 80 0 300 50 [[Sagittal_notation]]<br />
rect 300 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]<br />
rect 20 80 160 106 [[896/891]]<br />
rect 160 80 280 106 [[33/32]]<br />
default [[File:55-EDO_Revo_Sagittal.svg]]<br />
</imagemap><br />
<br />
==== Evo-SZ flavor ====<br />
<imagemap><br />
File:55-EDO_Evo-SZ_Sagittal.svg<br />
desc none<br />
rect 80 0 300 50 [[Sagittal_notation]]<br />
rect 300 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]<br />
rect 20 80 160 106 [[896/891]]<br />
rect 160 80 280 106 [[33/32]]<br />
default [[File:55-EDO_Evo-SZ_Sagittal.svg]]<br />
</imagemap><br />
<br />
=== 31-tone subset ===<br />
The 31-out-of-55edo subset can be notated entirely with the standard notation of 7 each of naturals/sharps/flats, and 5 each of doublesharps/doubleflats, as a 31-tone chain-of-5ths from Gbb to Ax.<br />
<br />
[[File:Monzo55Notation.jpeg|400px|frameless|alt=Diagram of 31-tone subset of 55edo using plain Western notation, by Joe Monzo.|Diagram of 31-tone subset of 55edo using plain Western notation, by [[Joe Monzo]].]]<br />
<br />
== Approximation to JI ==<br />
[[File:55ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 55edo]]<br />
<br />
=== Selected just intervals by error ===<br />
{{Q-odd-limit intervals|55}}<br />
{{Q-odd-limit intervals|55.05|apx=val|header=none|tag=none|title=15-odd-limit intervals by 55d val mapping}}<br />
<br />
== Regular temperament properties ==<br />
{| class="wikitable center-4 center-5 center-6"<br />
|-<br />
! rowspan="2" | [[Subgroup]]<br />
! rowspan="2" | [[Comma list]]<br />
! rowspan="2" | [[Mapping]]<br />
! rowspan="2" | Optimal<br>8ve stretch (¢)<br />
! colspan="2" | Tuning error<br />
|-<br />
! [[TE error|Absolute]] (¢)<br />
! [[TE simple badness|Relative]] (%)<br />
|-<br />
| 2.3<br />
| {{monzo| -87 55 }}<br />
| {{mapping| 55 87 }}<br />
| +1.31<br />
| 1.19<br />
| 7.21<br />
|-<br />
| 2.3.5<br />
| 81/80, {{monzo| 31 1 -14 }}<br />
| {{mapping| 55 87 128 }}<br />
| −0.13<br />
| 2.10<br />
| 9.63<br />
|}<br />
<br />
=== Uniform maps ===<br />
{{Uniform map|edo=55}}<br />
<br />
=== Commas ===<br />
{{Todo|cleanup|inline=true}}<br />
<br />
'''5-limit commas''': [[81/80]], [[Quintosec_family|{{monzo| 47 -15 -10 }}]], {{monzo| 31 1 -14 }}, {{monzo| 27 5 -15 }}<br />
<br />
'''7-limit commas''': 31104/30625, [[6144/6125]], 81648/78125, 16128/15625, 28672/28125, 33075/32768, 83349/80000, 1029/1000, [[686/675]], [[10976/10935]], [[Cloudy comma|16807/16384]], 84035/82944<br />
<br />
'''11-limit commas''': 59049/58564, 74088/73205, 46656/46585, 21609/21296, 12005/11979, 19683/19360, [[243/242]], 3087/3025, 5488/5445, 19683/19250, 1944/1925, 45927/45056, 2835/2816, 35721/34375, 7056/6875, 12544/12375, 7203/7040, 2401/2376, 24057/24010, 72171/70000, 891/875, [[176/175]], 2079/2048, [[385/384]], 3234/3125, 17248/16875, 26411/25600, 26411/2592, 26411/262404, 88209/87808, 30976/30625, 3267/3200, [[121/120]], 81312/78125, 41503/40000, 41503/40500, 35937/35000, 2662/2625, 42592/42525, 83853/81920, 9317/9216, 65219/62500, 43923/43904, 14641/14400, [[14641/14580]]<br />
<br />
'''13-limit commas''': 59535/57122, 29400/28561, 29568/28561, 29645/28561, 24576/24167, 99225/96668, 24500/24167, 50421/48334, 45927/43940, 2268/2197, 2240/2197, 57624/54925, 61875/61516, 57024/54925, 11264/10985, 72765/70304, 13475/13182, 22869/21970, 6776/6591, 20736/20449, 20480/20449, 84035/81796, 91125/91091, 65536/65065, 15309/14872, 1890/1859, 5600/5577, 9604/9295, 59049/57967, 58320/57967, 4374/4225, 864/845, [[512/507]], 11025/10816, 6125/6084, 21952/21125, 16807/16224, 84035/82134, 66825/66248, 90112/88725, 56133/54080, 693/676, 1540/1521, 26411/25350, 58806/57967, 58080/57967, 88209/84500, 4356/4225, 7744/7605, 88935/86528, 33275/33124, 27951/27040, 9317/9126, 58564/57967, 43923/42250, 17496/17303, 87808/86515, 55296/55055, 25515/25168, [[1575/1573]], 64827/62920, 4802/4719, 98415/98098, 59049/57200, 729/715, [[144/143]], 18375/18304, 18522/17875, 10976/10725, 84035/82368, 59049/56875, 11664/11375, 2304/2275, [[4096/4095]], 1701/1664, [[105/104]], 42336/40625, 25088/24375, 21609/20800, 2401/2340, 9604/9477, 72171/71344, 2673/2600, [[66/65]], [[352/351]], 13475/13312, 33957/32500, 15092/14625, 81675/81536, 58806/56875, 11616/11375, 61952/61425, 68607/66560, 847/832, 4235/4212, 35937/35672, 1331/1300, 5324/5265, 58564/56875, 85293/85184, 13377/13310, 85293/84700, 15288/15125, 31213/30976, 67392/67375, 28431/28160, 34944/34375, 4459/4400, 4459/4455, 28431/28000, [[351/350]], 79872/78125, 66339/65536, 51597/50000, 637/625, 10192/10125, 31213/30720, [[31213/31104]], 30888/30625, 1287/1280, 81081/78125, 16016/15625, 49049/48000, 49049/48600, 14157/14000, 33033/32768, 77077/75000, 51909/51200, 17303/17280, 75712/75625, 8281/8250, 41067/40960, 31941/31250, 9464/9375, 57967/57600, 91091/90000, 61347/61250, 79092/78125<br />
<br />
=== Rank-2 temperaments ===<br />
{| class="wikitable center-all left-5"<br />
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator<br />
|-<br />
! Periods<br>per 8ve<br />
! Generator*<br />
! Cents*<br />
! Associated<br>ratio*<br />
! Temperament<br />
|-<br />
| 1<br />
| 6\55<br />
| 130.9<br />
| 14/13<br />
| [[Twothirdtonic]] (55f)<br />
|-<br />
|1<br />
|8\55<br />
|174.5<br />
|[[10/9]]~[[11/10]]<br />
|[[Tetracot]] (55c)<br />
|-<br />
| 1<br />
| 16\55<br />
| 349.1<br />
| 11/9<br />
| [[Mohaha]]<br />
|-<br />
| 1<br />
| 23\55<br />
| 501.8<br />
| 4/3<br />
| [[Meantone]] (55d)<br />
|-<br />
| 1<br />
| 26\55<br />
| 567.3<br />
| 7/5<br />
| [[Liese]] (55)<br />
|-<br />
| 1<br />
| 27\55<br />
| 589.1<br />
| 45/32<br />
| [[Untriton]] (55d) / [[aufo]] (55)<br />
|-<br />
| 5<br />
| 17\55<br>(5\55)<br />
| 370.9<br>(109.1)<br />
| 99/80<br>(16/15)<br />
| [[Quintosec]]<br />
|-<br />
| 11<br />
| 23\55<br>(3\55)<br />
| 501.8<br>(65.5)<br />
| 4/3<br>(36/35)<br />
| [[Hendecatonic]] (55)<br />
|}<br />
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct<br />
<br />
== Scales ==<br />
; Subsets of twothirdtonic[37]<br />
* Undecimal otonal-like pentatonic: 17 8 7 12 11<br />
<br />
; Subsets of hendecatonic[33]<br />
* Septimal pentatonic-like: 10 13 9 13 10<br />
* Septimal minor blues-like: 13 10 4 5 13 10<br />
* Septimal heptatonic blues-like: 13 10 4 5 8 5 10<br />
<br />
; Others<br />
* Sakura-like scale containing [[phi]]: 9 6 18 5 17<br />
* Quasi-[[equiheptatonic]] scale: 8 8 7 9 7 9 7<br />
<br />
== Instruments ==<br />
* [[Lumatone mapping for 55edo]]<br />
<br />
== Music ==<br />
=== Modern renderings ===<br />
; {{W|Johann Sebastian Bach}}<br />
* [https://www.youtube.com/watch?v=oymJKnYzzOw "Jesus bleibet meine Freude" from ''Herz und Mund und Tat und Leben'', BWV 147] (1723) – arranged for two organs, rendered by Claudi Meneghin (2021)<br />
* [https://www.youtube.com/watch?v=xoCNOIsjfeU "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by [[Claudi Meneghin]] (2024)<br />
* "Ricercar a 6" from ''The Musical Offering'', BWV 1079 (1747) – rendered by Claudi Meneghin<br />
** [https://www.youtube.com/watch?v=OkRVNo19guo harpsichord rendition] (2025)<br />
** [https://www.youtube.com/watch?v=X_qROPtHf9g fortepiano rendition] (2025)<br />
** [https://www.youtube.com/watch?v=X9SexO03MTw organ rendition] (2026)<br />
* [https://www.youtube.com/watch?v=Y5sIjh_Te40 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)<br />
* [https://www.youtube.com/watch?v=QOPxqNgkVWM "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)<br />
<br />
; {{W|Nicolaus Bruhns}}<br />
* [https://www.youtube.com/watch?v=OfOt3nOp-f8 ''Prelude in E Minor "The Great"''] – rendered by [[Claudi Meneghin]] (2023)<br />
* [https://www.youtube.com/watch?v=tuIPIhSxUPs ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)<br />
<br />
; {{W|Georg Frideric Handel}}<br />
* [https://www.youtube.com/watch?v=rDvKPuzsno8 ''Fugue'' from "Suite in E minor", HWV 429] (1720) – arranged for Baroque ensemble and drums, rendered by Claudi Meneghin (2025) <br />
<br />
; {{W|Scott Joplin}}<br />
* ''Maple Leaf Rag'' (1899) – arranged for harpsichord and rendered by [[Claudi Meneghin]] ([https://www.youtube.com/watch?v=GbhpuoIJgxk 2024 version]; [https://www.youtube.com/shorts/3Y9y9I6q1as 2026 version])<br />
<br />
; {{W|Wolfgang Amadeus Mozart}}<br />
* [https://www.youtube.com/watch?v=C_AML6XW-2g ''Rondo alla Turca'' from the Piano Sonata No. 11, KV 331] (1778) – rendered by Francium (2023)<br />
* [https://www.youtube.com/watch?v=XgRksdk6zyQ ''Fugue in G minor'', KV 401] (1782) – rendered by Francium (2023)<br />
* [http://www.seraph.it/dep/int/AdagioKV540.mp3 ''Adagio in B minor'', KV 540] (1788) – rendered by Carlo Serafini (2011) ([http://www.seraph.it/blog_files/706c4662272db7703def4d57edfcb955-119.html blog entry])<br />
* [https://www.youtube.com/watch?v=pFjJCj2MBTM ''Allegro'' from the Piano Sonata No. 16, KV 545] (1788) – rendered by Francium (2023)<br />
* [https://www.youtube.com/watch?v=p88MWgdio14&list=PLC6ZSKWKnVz0mOTLQkCUi9ydWGLpBP8gZ&index=2 ''Mozart's Gigue KV 574, Arranged for Fortepiano (55-edo)''] – rendered by [[Claudi Meneghin]] (2025)<br />
<br />
; {{W|Keiichi Okabe}}<br />
* [https://www.youtube.com/watch?v=L24G4Y7tZgI ''Yuutsu no Yuutsu''] (2006) – rendered by MortisTheneRd (2024)<br />
<br />
=== 21st century ===<br />
; [[Bryan Deister]]<br />
* [https://www.youtube.com/shorts/l62rb8ULCXs ''55edo improv''] (2025)<br />
* [https://www.youtube.com/watch?v=kVmToKkZU88 ''Waltz in 55edo''] (2025)<br />
* [https://www.youtube.com/shorts/eQHpMFLrjjQ ''55edo prelude''] (2025)<br />
<br />
; [[James Kukula]]<br />
* ''[https://app.box.com/s/8hq89cb3rqqkrhvkxgvqtppa255kcqrq?fbclid=IwY2xjawISjSlleHRuA2FlbQIxMAABHcl5t8n_C7QUJqdEnwSaWBc5u3BpldmcAjhQQljsQIPl1qJ-zdCr9T8NMw_aem_Ez0m-Ls_ZqI0-c0Ld-28Yg 55edo Melted Syntonic]'' (2025)<br />
<br />
; [[Budjarn Lambeth]]<br />
* ''[https://www.youtube.com/watch?v=9c5MtrZFNhA Improvisation One in 55edo]'' (2025)<br />
* ''[https://www.youtube.com/watch?v=ggFGUn1Ya2A Improvisation Two in 55edo]'' (2025)<br />
<br />
; [[Claudi Meneghin]]<br />
* [https://www.youtube.com/watch?v=AgsJCTyxqiM ''Double Fugue on "We Wish You a Merry Christmas" for String Quartet''] (2020)<br />
* [https://www.youtube.com/watch?v=rAbbvyotIr4 ''Canon at the Diatonic Semitone on an Ancient Lombard Theme''] (2021)<br />
* [https://www.youtube.com/watch?v=hCUIx1RzvEk ''Chacony "Lament & Deception"'' for Two Violins and Cello] (2021), [https://www.youtube.com/watch?v=abJP4euMlsg for Baroque Wind Ensemble] (2023)<br />
* [https://www.youtube.com/watch?v=9zfWeO0eJdA Fantasy "Almost a Fugue" on a Theme by Giuliani, for String Quartet] (2021)<br />
* [https://www.youtube.com/watch?v=jOiub14Cskw ''Double Fugue on "Old McDonald" + "Shave & a Haircut"''] (2024)<br />
<br />
; [[Herman Miller]]<br />
* ''[https://soundcloud.com/morphosyntax-1/road-trip-to-nowhere Road Trip to Nowhere]'' (2021)<br />
* ''[https://soundcloud.com/morphosyntax-1/migration Migration]'' (2025)<br />
<br />
== External links ==<br />
* ''[http://tonalsoft.com/monzo/55edo/55edo.aspx Mozart's tuning: 55-edo and its close relative, 1/6-comma meantone]'' (containing another listening example) on [[Tonalsoft Encyclopedia]]<br />
<br />
== References ==<br />
<references /><br />
<br />
[[Category:Meantone]]<br />
[[Category:Historical]]<br />
[[Category:Listen]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Template:Infobox_interval_region&diff=225483Template:Infobox interval region2026-03-09T21:13:32Z<p>ArrowHead294: </p>
<hr />
<div><includeonly>{{#invoke: Infobox_interval_region | infobox_IR<br />
| regionname={{{Name|{{PAGENAME}}}}}<br />
| complement={{{Complement|}}}<br />
| centsLowerWide={{{Cents lower wide|}}}<br />
| centsLower={{{Cents lower|}}}<br />
| centsUpperWide={{{Cents upper wide|}}}<br />
| centsUpper={{{Cents upper|}}}<br />
| equave={{{Equave|}}}<br />
| ji3={{{3-limit intervals|}}}<br />
| ji5={{{5-limit intervals|}}}<br />
| ji7={{{7-limit intervals|}}}<br />
| ji11={{{11-limit intervals|}}}<br />
| ji13={{{13-limit intervals|}}}<br />
| jiHigh={{{Other JI intervals|}}}<br />
| ji = {{{JI intervals|}}}<br />
| subregions = {{{Subregions|}}}<br />
| superregions = {{{Superregions|}}}<br />
| prevRegion = {{{Lower region|}}}<br />
| nextRegion = {{{Higher region|}}}<br />
| MOSes={{{MOSes|}}}<br />
}}<br />
{{#if: {{#ifeq: {{lc: {{ARTICLEROOTPAGENAME}}}}|template:infobox interval region|true}}{{#if: {{{debug|}}}|true}}||[[Category:Interval regions]]}}<br />
</includeonly><noinclude><br />
{{documentation}}<br />
<br />
[[Category:Draft templates]]<br />
[[Category:Infoboxes]]<br />
</noinclude></div>ArrowHead294https://en.xen.wiki/index.php?title=Template:Infobox_interval_region&diff=225482Template:Infobox interval region2026-03-09T21:12:26Z<p>ArrowHead294: </p>
<hr />
<div><includeonly>{{#invoke: Infobox_interval_region | infobox_IR<br />
| regionname={{{Name|{{PAGENAME}}}}}<br />
| complement={{{Complement|}}}<br />
| centsLowerWide={{{Cents lower wide|}}}<br />
| centsLower={{{Cents lower|}}}<br />
| centsUpperWide={{{Cents upper wide|}}}<br />
| centsUpper={{{Cents upper|}}}<br />
| equave={{{Equave|}}}<br />
| ji3={{{3-limit intervals|}}}<br />
| ji5={{{5-limit intervals|}}}<br />
| ji7={{{7-limit intervals|}}}<br />
| ji11={{{11-limit intervals|}}}<br />
| ji13={{{13-limit intervals|}}}<br />
| jiHigh={{{Other JI intervals|}}}<br />
| ji = {{{JI intervals|}}}<br />
| subregions = {{{Subregions|}}}<br />
| superregions = {{{Superregions|}}}<br />
| prevRegion = {{{Lower region|}}}<br />
| nextRegion = {{{Higher region|}}}<br />
| MOSes={{{MOSes|}}}<br />
}}<br />
{{#if: {{#ifeq: {{lc:{{ARTICLEROOTPAGENAME}}}}|template:infobox interval region|True}}{{#if: {{{debug|}}}|True}}||[[Category:Interval regions]]}}<br />
</includeonly><noinclude><br />
{{documentation}}<br />
<br />
[[Category:Draft templates]]<br />
[[Category:Infoboxes]]<br />
</noinclude></div>ArrowHead294https://en.xen.wiki/index.php?title=Template:Escape_template_list&diff=225481Template:Escape template list2026-03-09T18:48:30Z<p>ArrowHead294: <nowiki&nbsp;/>* works fine</p>
<hr />
<div>{| class="wikitable" style="text-align: center; <noinclude>margin: auto auto auto auto;</noinclude>"<br />
<includeonly><br />
|+ style="font-size: 105%;" | Template calls for escaping characters<br />
</includeonly><br />
|- style="white-space: nowrap;"<br />
! Template call<includeonly>*</includeonly> !! Output<includeonly>**</includeonly> !! HTML alternative<includeonly>***</includeonly><br />
|-<br />
| {{tlx|!|plaincode}} '''(m)''' || <nowiki>|</nowiki> || {{plaincode|&amp;#124;}} or {{tlx|pipe|plaincode}}<br />
|-<br />
| {{tlx|{{=}}|plaincode}} '''(m)''' || <nowiki>=</nowiki> || {{plaincode|&amp;#61;}}<br />
|-<br />
| {{tlx|!!|plaincode}} || <nowiki>||</nowiki> || {{plaincode|&amp;#124;&amp;#124;}}<br />
|-<br />
| {{tlx|!(|plaincode}} || <nowiki>[</nowiki> || {{plaincode|&amp;#91;}} or {{tlx|lbrack|plaincode}}<br />
|-<br />
| {{tlx|)!|plaincode}} || <nowiki>]</nowiki> || {{plaincode|&amp;#93;}} or {{tlx|rbrack|plaincode}}<br />
|-<br />
| {{tlx|!((|plaincode}} || <nowiki>[[</nowiki> || {{plaincode|&amp;#91;&amp;#91;}}<br />
|-<br />
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|-<br />
| {{tlx|!-|plaincode}} || <nowiki>|-</nowiki> || {{plaincode|&amp;#124;&amp;#45;}}<br />
|-<br />
| {{tlx|!)|plaincode}} || <nowiki>|}</nowiki> || {{plaincode|&amp;#124;&amp;#125;}}<br />
|-<br />
| {{tlx|&#39;|plaincode}} || &#39; || {{plaincode|&amp;#39;}}<br />
|-<br />
| {{tlx|colon|plaincode}} || <nowiki>:</nowiki> || {{plaincode|&amp;#58;}}<br />
|-<br />
| {{tlx|^(|plaincode}} || &lt; || {{plaincode|&amp;#60;}} or {{plaincode|&amp;lt;}}<br />
|-<br />
| {{tlx|)^|plaincode}} || &gt; || {{plaincode|&amp;#62;}} or {{plaincode|&amp;gt;}}<br />
<includeonly><br />
{{Navbar table|cols=3|Template:Escape template list}}<br />
</includeonly><br />
|}<br />
<includeonly><br />
<nowiki/>* '''m''' for [[mw:Help:Magic words|magic word]]<br />
<br />
<nowiki/>** Delayed interpretation as Wiki markup<br />
<br />
<nowiki/>*** Never interpreted as Wiki markup<br />
</includeonly><noinclude><br />
{{documentation}}<br />
<br />
[[Category:Character templates]]<br />
[[Category:Tables]]<br />
</noinclude></div>ArrowHead294https://en.xen.wiki/index.php?title=User:ArrowHead294&diff=225443User:ArrowHead2942026-03-08T19:35:55Z<p>ArrowHead294: </p>
<hr />
<div>{{subpage|sandbox|l|text=Hi!}} I'm primarily a music theorist and singer nowadays, though most of my musical training is in classical piano and cello.<br />
__NOTOC__<br />
== My musical experience ==<br />
Piano and keyboard: 2004–2017 (K–12), 2022–present<br />
<br />
Cello: 2009–2017 (Grades 5–12)<br />
<br />
Voice: 2020–present<br />
: Voice type: Baritone / bass-baritone<br />
:: Chest voice: E♭2 to E4<br />
::: Can go down to D2 if needed<br />
::: Can belt up to ~G4, A♭4 on occasion<br />
:: Mixed voice: C4–C5<br />
:: Falsetto and head voice: D4 to E♭5<br />
::: Can reach E5 and F5<br />
<br />
I've also been told I have "perfect pitch" since I acquired the ability to recognise notes (in {{nowrap|A {{=}} 440 Hz}} and 12edo) from a young age, though I have grown increasingly disdainful towards the terms "perfect pitch" and "absolute pitch" since late 2022 (the start of my last term of college) since it locked me into 12edo with {{nowrap|A {{=}} 440 Hz}} and anything else sounded "wrong" to me. This is even more true nowadays since I now prefer other tunings over 12edo in a lot of cases.<br />
<br />
== Favourite tunings within Western music ==<br />
{{Main|User:ArrowHead294/EDO impressions|l1=EDO impressions}}<br />
Most Western musicians only know 12edo, and my musical background is mainly classical, so my main interests have been in Pythagorean and meantone. I'm quite active in church despite being non-religious, and most of the alternative tunings I introduce to others are flatter-than-12 meantones as they're relatively easy to get into.<br />
<br />
I've known about quarter tones ([[24edo]]) since I was very young. I've even dabbled around with it before 2022, and used its absolute frequencies with {{nowrap|A {{=}} 440 Hz}} as a way to compare just intonation and equal temperament, though for most of my life that was the only microtonal tuning I knew about. As a result, my knowledge of just intonation was very much limited to the 2.3.5.11.37 subgroup.<br />
<br />
Since late 2022, I've also gotten into sixth tones ([[36edo]]) for exploring septimal harmonies.<br />
<br />
[[31edo]] is the first alternative tuning I learned about which helped me break out of 12-TET's walled garden, and [[Sevish]]'s song ''Better Left Unanswered'' is the first microtonal work I've ever heard that wasn't in standard quarter tones. I learned about it mainly after exploring quarter-comma meantone in classical music, and I think it's the most practical alternate tuning for most Western musicians to get into. It has excellent 7-limit and 11-limit harmonies as well (even better than 36edo), so it could also be useful for blues and jazz.<br />
<br />
This was followed by [[19edo]], after hearing ''Sunsrise'' by Supahstar Saga. 19 has its unique quirks that make it a good tuning for a lot of Western music, though I think its sound is best suited to songs with largely pentatonic melodies since the diatonic scale sounds quite loose to me. Augmented and diminished chords sound very weird, though, even more jarring than 31.<br />
<br />
=== Beyond traditional Western music ===<br />
For xenharmony and music beyond meantone, I've been mainly interested in [[22edo]] for [[superpyth]]agorean and [[Porcupine]] temperament, [[Orwell]] in 31edo, [[34edo]] for [[Tetracot]] (including [[68edo]] for [[Octacot]] by extension), and [[46edo]] for [[Sensi]] and 5-limit harmony beyond meantone.<br />
<br />
== Miscellaneous ==<br />
{| class="mw-collapsible mw-collapsed"<br />
|-<br />
! <br />
|-<br />
| The contemporary version of the {{w|Lord's Prayer}} goes:<br />
<br />
<pre><br />
Our Father in heaven,<br />
hallowed be your name,<br />
your kingdom come,<br />
your will be done,<br />
on earth as in heaven.<br />
Give us today our daily bread.<br />
Forgive us our sins<br />
as we forgive those who sin against us.<br />
Save us from the time of trial<br />
and deliver us from evil.<br />
For the kingdom, the power, and the glory are yours<br />
now and for ever.<br />
Amen.<br />
</pre><br />
|}<br />
<br />
When encoded in {{w|hexadecimal}} with CRLF (<code>\r\n</code>) newlines, this becomes:<br />
<br />
<pre><br />
4f 75 72 20 46 61 74 68 65 72 20 69 6e 20 68 65 61 76 65 6e 2c 0d 0a 68 61 6c 6c 6f 77 65 64 20 62 65 20 79 6f 75 72 20 6e 61 6d 65 2c 0d 0a 79 6f 75 72 20 6b 69 6e 67 64 6f 6d 20 63 6f 6d 65 2c 0d 0a 79 6f 75 72 20 77 69 6c 6c 20 62 65 20 64 6f 6e 65 2c 0d 0a 6f 6e 20 65 61 72 74 68 20 61 73 20 69 6e 20 68 65 61 76 65 6e 2e 0d 0a 47 69 76 65 20 75 73 20 74 6f 64 61 79 20 6f 75 72 20 64 61 69 6c 79 20 62 72 65 61 64 2e 0d 0a 46 6f 72 67 69 76 65 20 75 73 20 6f 75 72 20 73 69 6e 73 0d 0a 61 73 20 77 65 20 66 6f 72 67 69 76 65 20 74 68 6f 73 65 20 77 68 6f 20 73 69 6e 20 61 67 61 69 6e 73 74 20 75 73 2e 0d 0a 53 61 76 65 20 75 73 20 66 72 6f 6d 20 74 68 65 20 74 69 6d 65 20 6f 66 20 74 72 69 61 6c 0d 0a 61 6e 64 20 64 65 6c 69 76 65 72 20 75 73 20 66 72 6f 6d 20 65 76 69 6c 2e 0d 0a 46 6f 72 20 74 68 65 20 6b 69 6e 67 64 6f 6d 2c 20 74 68 65 20 70 6f 77 65 72 2c 20 61 6e 64 20 74 68 65 20 67 6c 6f 72 79 20 61 72 65 20 79 6f 75 72 73 0d 0a 6e 6f 77 20 61 6e 64 20 66 6f 72 20 65 76 65 72 2e 0d 0a 41 6d 65 6e 2e<br />
</pre><br />
<br />
When encoded in {{w|Base64}} with the RFC&nbsp;4648 and numeral-first alphabets, this becomes (respectively):<br />
<br />
<pre style="word-break: break-all;"><br />
T3VyIEZhdGhlciBpbiBoZWF2ZW4sDQpoYWxsb3dlZCBiZSB5b3VyIG5hbWUsDQp5b3VyIGtpbmdkb20gY29tZSwNCnlvdXIgd2lsbCBiZSBkb25lLA0Kb24gZWFydGggYXMgaW4gaGVhdmVuLg0KR2l2ZSB1cyB0b2RheSBvdXIgZGFpbHkgYnJlYWQuDQpGb3JnaXZlIHVzIG91ciBzaW5zDQphcyB3ZSBmb3JnaXZlIHRob3NlIHdobyBzaW4gYWdhaW5zdCB1cy4NClNhdmUgdXMgZnJvbSB0aGUgdGltZSBvZiB0cmlhbA0KYW5kIGRlbGl2ZXIgdXMgZnJvbSBldmlsLg0KRm9yIHRoZSBraW5nZG9tLCB0aGUgcG93ZXIsIGFuZCB0aGUgZ2xvcnkgYXJlIHlvdXJzDQpub3cgYW5kIGZvciBldmVyLg0KQW1lbi4=<br />
</pre><br />
<br />
<pre style="word-break: break-all;"><br />
JtLo84PXT6XbSY1fRY1ePM5sPMui3GfeOMniRtTbP21YPI1vRtLo86vXRMKi3GfvRtLo86jfRcTaRsqWOszjPImD2dblTN8WTsbiR21YPI1aRsvbB0qARsuWPM5oT6WWONCWQMuWQ6LXTcLkBWqAHsbsPI1rSo1qRsHXUI1lTN8WP65fR7aWOd9bOMGk3Gf6Rt9dQNPb87Lp86zrSY1pQMvp3GfXSo1tPI1cRt9dQNPb87HeRtDb87TeRo1pQMuWOMTXQMvpT21rSouD2bDXTcKWTNCWPd9lRI1qQ6KWT6bjPI1lPY1qScbXR0qAOMva86HbR6bsPN8WTNCWPd9lRI1bTcbiBWqAHczo87HePI1hQMvdP6zjB21qQ6KWS6ztPN8i865kP21qQ6KWPsnlSdaWON9b87blTN9p3GfkRtSWOMva86PlSY1bTcLoBWqAGMrbRYu=<br />
</pre><br />
<br />
== See also ==<br />
* {{subpage|Purely consistent EDOs by odd limit}}<br />
* {{subpage|Regex snippets}}<br />
* {{subpage|The Star-Spangled Banner}}<br />
* {{subpage|UTF-8 extensions}}<br />
<br />
[[Category:User en-N]]<br />
[[Category:User zh-3]]<br />
[[Category:User de-1]]<br />
[[Category:User on Discord]]<br />
[[Category:User in EST/EDT]]</div>ArrowHead294https://en.xen.wiki/index.php?title=User:ArrowHead294&diff=225442User:ArrowHead2942026-03-08T19:35:04Z<p>ArrowHead294: </p>
<hr />
<div>{{subpage|sandbox|l|text=Hi!}} I'm primarily a music theorist and singer nowadays, though most of my musical training is in classical piano and cello.<br />
__NOTOC__<br />
== My musical experience ==<br />
Piano and keyboard: 2004–2017 (K–12), 2022–present<br />
<br />
Cello: 2009–2017 (Grades 5–12)<br />
<br />
Voice: 2020–present<br />
: Voice type: Baritone / bass-baritone<br />
:: Chest voice: E♭2 to E4<br />
::: Can go down to D2 if needed<br />
::: Can belt up to ~G4, A♭4 on occasion<br />
:: Mixed voice: C4–C5<br />
:: Falsetto and head voice: D4 to E♭5<br />
::: Can reach E5 and F5<br />
<br />
I've also been told I have "perfect pitch" since I acquired the ability to recognise notes (in {{nowrap|A {{=}} 440 Hz}} and 12edo) from a young age, though I have grown increasingly disdainful towards the terms "perfect pitch" and "absolute pitch" since late 2022 (the start of my last term of college) since it locked me into 12edo with {{nowrap|A {{=}} 440 Hz}} and anything else sounded "wrong" to me. This is even more true nowadays since I now prefer other tunings over 12edo in a lot of cases.<br />
<br />
== Favourite tunings within Western music ==<br />
{{Main|User:ArrowHead294/EDO impressions|l1=EDO impressions}}<br />
Most Western musicians only know 12edo, and my musical background is mainly classical, so my main interests have been in Pythagorean and meantone. I'm quite active in church despite being non-religious, and most of the alternative tunings I introduce to others are flatter-than-12 meantones as they're relatively easy to get into.<br />
<br />
I've known about quarter tones ([[24edo]]) since I was very young. I've even dabbled around with it before 2022, and used its absolute frequencies with {{nowrap|A {{=}} 440 Hz}} as a way to compare just intonation and equal temperament, though for most of my life that was the only microtonal tuning I knew about. As a result, my knowledge of just intonation was very much limited to the 2.3.5.11.37 subgroup.<br />
<br />
Since late 2022, I've also gotten into sixth tones ([[36edo]]) for exploring septimal harmonies.<br />
<br />
[[31edo]] is the first alternative tuning I learned about which helped me break out of 12-TET's walled garden, and [[Sevish]]'s song ''Better Left Unanswered'' is the first microtonal work I've ever heard that wasn't in standard quarter tones. I learned about it mainly after exploring quarter-comma meantone in classical music, and I think it's the most practical alternate tuning for most Western musicians to get into. It has excellent 7-limit and 11-limit harmonies as well (even better than 36edo), so it could also be useful for blues and jazz.<br />
<br />
This was followed by [[19edo]], after hearing ''Sunsrise'' by Supahstar Saga. 19 has its unique quirks that make it a good tuning for a lot of Western music, though I think its sound is best suited to songs with largely pentatonic melodies since the diatonic scale sounds quite loose to me. Augmented and diminished chords sound very weird, though, even more jarring than 31.<br />
<br />
=== Beyond traditional Western music ===<br />
For xenharmony and music beyond meantone, I've been mainly interested in [[22edo]] for [[superpyth]]agorean and [[Porcupine]] temperament, [[Orwell]] in 31edo, [[34edo]] for [[Tetracot]] (including [[68edo]] for [[Octacot]] by extension), and [[46edo]] for [[Sensi]] and 5-limit harmony beyond meantone.<br />
<br />
<br />
== Miscellaneous ==<br />
{| class="mw-collapsible mw-collapsed"<br />
|-<br />
! <br />
|-<br />
| The contemporary version of the {{w|Lord's Prayer}} goes:<br />
<br />
<pre><br />
Our Father in heaven,<br />
hallowed be your name,<br />
your kingdom come,<br />
your will be done,<br />
on earth as in heaven.<br />
Give us today our daily bread.<br />
Forgive us our sins<br />
as we forgive those who sin against us.<br />
Save us from the time of trial<br />
and deliver us from evil.<br />
For the kingdom, the power, and the glory are yours<br />
now and for ever.<br />
Amen.<br />
</pre><br />
|}<br />
<br />
When encoded in {{w|hexadecimal}} with CRLF (<code>\r\n</code>) newlines, this becomes:<br />
<br />
<pre><br />
4f 75 72 20 46 61 74 68 65 72 20 69 6e 20 68 65 61 76 65 6e 2c 0d 0a 68 61 6c 6c 6f 77 65 64 20 62 65 20 79 6f 75 72 20 6e 61 6d 65 2c 0d 0a 79 6f 75 72 20 6b 69 6e 67 64 6f 6d 20 63 6f 6d 65 2c 0d 0a 79 6f 75 72 20 77 69 6c 6c 20 62 65 20 64 6f 6e 65 2c 0d 0a 6f 6e 20 65 61 72 74 68 20 61 73 20 69 6e 20 68 65 61 76 65 6e 2e 0d 0a 47 69 76 65 20 75 73 20 74 6f 64 61 79 20 6f 75 72 20 64 61 69 6c 79 20 62 72 65 61 64 2e 0d 0a 46 6f 72 67 69 76 65 20 75 73 20 6f 75 72 20 73 69 6e 73 0d 0a 61 73 20 77 65 20 66 6f 72 67 69 76 65 20 74 68 6f 73 65 20 77 68 6f 20 73 69 6e 20 61 67 61 69 6e 73 74 20 75 73 2e 0d 0a 53 61 76 65 20 75 73 20 66 72 6f 6d 20 74 68 65 20 74 69 6d 65 20 6f 66 20 74 72 69 61 6c 0d 0a 61 6e 64 20 64 65 6c 69 76 65 72 20 75 73 20 66 72 6f 6d 20 65 76 69 6c 2e 0d 0a 46 6f 72 20 74 68 65 20 6b 69 6e 67 64 6f 6d 2c 20 74 68 65 20 70 6f 77 65 72 2c 20 61 6e 64 20 74 68 65 20 67 6c 6f 72 79 20 61 72 65 20 79 6f 75 72 73 0d 0a 6e 6f 77 20 61 6e 64 20 66 6f 72 20 65 76 65 72 2e 0d 0a 41 6d 65 6e 2e<br />
</pre><br />
<br />
When encoded in {{w|Base64}} with the RFC&nbsp;4648 and numeral-first alphabets, this becomes (respectively):<br />
<br />
<pre style="word-break: break-all;"><br />
T3VyIEZhdGhlciBpbiBoZWF2ZW4sDQpoYWxsb3dlZCBiZSB5b3VyIG5hbWUsDQp5b3VyIGtpbmdkb20gY29tZSwNCnlvdXIgd2lsbCBiZSBkb25lLA0Kb24gZWFydGggYXMgaW4gaGVhdmVuLg0KR2l2ZSB1cyB0b2RheSBvdXIgZGFpbHkgYnJlYWQuDQpGb3JnaXZlIHVzIG91ciBzaW5zDQphcyB3ZSBmb3JnaXZlIHRob3NlIHdobyBzaW4gYWdhaW5zdCB1cy4NClNhdmUgdXMgZnJvbSB0aGUgdGltZSBvZiB0cmlhbA0KYW5kIGRlbGl2ZXIgdXMgZnJvbSBldmlsLg0KRm9yIHRoZSBraW5nZG9tLCB0aGUgcG93ZXIsIGFuZCB0aGUgZ2xvcnkgYXJlIHlvdXJzDQpub3cgYW5kIGZvciBldmVyLg0KQW1lbi4=<br />
</pre><br />
<br />
<pre style="word-break: break-all;"><br />
JtLo84PXT6XbSY1fRY1ePM5sPMui3GfeOMniRtTbP21YPI1vRtLo86vXRMKi3GfvRtLo86jfRcTaRsqWOszjPImD2dblTN8WTsbiR21YPI1aRsvbB0qARsuWPM5oT6WWONCWQMuWQ6LXTcLkBWqAHsbsPI1rSo1qRsHXUI1lTN8WP65fR7aWOd9bOMGk3Gf6Rt9dQNPb87Lp86zrSY1pQMvp3GfXSo1tPI1cRt9dQNPb87HeRtDb87TeRo1pQMuWOMTXQMvpT21rSouD2bDXTcKWTNCWPd9lRI1qQ6KWT6bjPI1lPY1qScbXR0qAOMva86HbR6bsPN8WTNCWPd9lRI1bTcbiBWqAHczo87HePI1hQMvdP6zjB21qQ6KWS6ztPN8i865kP21qQ6KWPsnlSdaWON9b87blTN9p3GfkRtSWOMva86PlSY1bTcLoBWqAGMrbRYu=<br />
</pre><br />
<br />
== See also ==<br />
* {{subpage|Purely consistent EDOs by odd limit}}<br />
* {{subpage|Regex snippets}}<br />
* {{subpage|The Star-Spangled Banner}}<br />
* {{subpage|UTF-8 extensions}}<br />
<br />
[[Category:User en-N]]<br />
[[Category:User zh-3]]<br />
[[Category:User de-1]]<br />
[[Category:User on Discord]]<br />
[[Category:User in EST/EDT]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Kleismic&diff=225272Kleismic2026-03-06T18:16:09Z<p>ArrowHead294: </p>
<hr />
<div>{{Interwiki<br />
| en = Kleismic<br />
| de = Hanson-Kleismisch<br />
}}<br />
{{Infobox regtemp<br />
| Title = Kleismic<br />
| Subgroups = 2.3.5, 2.3.5.13<br />
| Comma basis = [[15625/15552]] (2.3.5); <br>[[325/324]], [[625/624]] (2.3.5.13)<br />
| Edo join 1 = 15 | Edo join 2 = 19<br />
| Mapping = 1; 6 5 14<br />
| Generators = 6/5 | Generators tuning = 317.1 | Optimization method = CWE<br />
| MOS scales = [[3L&nbsp;1s]], [[4L&nbsp;3s]], [[4L&nbsp;7s]], [[4L&nbsp;11s]], [[15L&nbsp;4s]]<br />
| Pergen = (P8, P12/6)<br />
| Color name = Tribiyoti<br />
| Odd limit 1 = 5 | Mistuning 1 = 1.35 | Complexity 1 = 7<br />
| Odd limit 2 = 2.3.5.13 15 | Mistuning 2 = 2.35 | Complexity 2 = 15<br />
}}<br />
'''Kleismic''', alternatively called '''hanson''' in the [[5-limit]], is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[kleismic family]], [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which stacked are equated to the [[3/1|perfect twelfth (3/1)]], and thereby characterized by the vanishing of the [[15625/15552|kleisma]] ([[ratio]]: 15625/15552, {{monzo|legend=1| -6 -5 6 }}).<br />
<br />
Another useful interpretation of the kleisma as a comma is that it makes the classical chromatic semitone, [[25/24]], into a third-tone by equating three of this interval to [[9/8]]. As {{nowrap| 9/8 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S|25}}) and the marveltwin comma [[325/324]] ([[S-expression|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5. <br />
<br />
Extensions with prime 7 include [[catakleismic]] (which adds [[225/224]], finding 7 at 22 generators up), [[countercata]] (which adds [[5120/5103]], finding 7 at 31 generators down), [[metakleismic]] (which adds [[179200/177147]], finding 7 at 56 generators up), [[keemun]] (which adds [[49/48]], finding 7 at 3 generators up), anakleismic (which adds [[2240/2187]], finding 7 at 37 generators up), and [[catalan]] (which adds [[64/63]], finding 7 at 12 generators down). Of these, catakleismic can perhaps be considered the canonical extension, as it makes an intuitive further equivalence of 25/24~26/25~27/26 to [[28/27]] (by tempering out the [[square superparticular]] [[729/728]] ({{S|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, identifying [[64/63]] with [[65/64]] by tempering out [[4096/4095]]. Catakleismic and countercata merge in [[53edo]], as the former finds 7 at 22 generators up while the latter finds it at 31 generators down (22 + 31 = 53).<br />
<br />
Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[just intonation|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).<br />
<br />
For technical data, see [[Kleismic family #Kleismic a.k.a. hanson]].<br />
<br />
== Interval chain ==<br />
In the following table, odd harmonics 1–15 are labeled in '''bold'''. <br />
<br />
{| class="wikitable sortable center-1 right-2"<br />
! #<br />
! Cents*<br />
! class="unsortable" | Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| 317.1<br />
| 6/5<br />
|-<br />
| 2<br />
| 634.2<br />
| 13/9, 36/25<br />
|-<br />
| 3<br />
| 951.3<br />
| 26/15<br />
|-<br />
| 4<br />
| 68.4<br />
| 25/24, 26/25, 27/26<br />
|-<br />
| 5<br />
| 385.5<br />
| '''5/4'''<br />
|-<br />
| 6<br />
| 702.6<br />
| '''3/2'''<br />
|-<br />
| 7<br />
| 1019.6<br />
| 9/5<br />
|-<br />
| 8<br />
| 136.7<br />
| 13/12, 27/25<br />
|-<br />
| 9<br />
| 453.8<br />
| 13/10<br />
|-<br />
| 10<br />
| 770.9<br />
| 25/16, 39/25<br />
|-<br />
| 11<br />
| 1088.0<br />
| '''15/8'''<br />
|-<br />
| 12<br />
| 205.1<br />
| '''9/8'''<br />
|-<br />
| 13<br />
| 522.2<br />
| 27/20<br />
|-<br />
| 14<br />
| 839.3<br />
| '''13/8'''<br />
|-<br />
| 15<br />
| 1156.4<br />
| 39/20<br />
|-<br />
| 16<br />
| 273.5<br />
| 75/64<br />
|-<br />
| 17<br />
| 590.6<br />
| 45/32<br />
|-<br />
| 18<br />
| 907.7<br />
| 27/16<br />
|-<br />
| 19<br />
| 24.7<br />
| 65/64, 81/80<br />
|}<br />
<nowiki/>* In 2.3.5.13-subgroup [[CWE tuning]], octave reduced<br />
<br />
== Tunings ==<br />
[[File:Kleismic.png|thumb|alt=Kleismic.png|A chart of the tuning spectrum of hanson and cata, showing the offsets of odd harmonics 3, 5, 9, 13, and 15, as a function of the generator; all edo tunings are shown with vertical lines whose length indicates the edo's tolerance, i.e. half of its step size in either direction of just, and some small edos supporting the temperament are labeled. Comma fractions with corresponding eigenmonzos also labeled.]]<br />
<br />
=== Optimized tunings ===<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings<br />
|-<br />
! rowspan="2" | !! colspan="2" | Euclidean<br />
|-<br />
! Constrained !! Destretched<br />
|-<br />
! Tenney<br />
| CTE: ~6/5 = 317.0308{{c}} || POTE: ~6/5 = 317.007{{c}}<br />
|-<br />
! Equilateral<br />
| CEE: ~6/5 = 317.1033{{c}}<br>(11/61-kleisma)<br />
|}<br />
<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.13-subgroup norm-based tunings<br />
|-<br />
! rowspan="2" | !! colspan="2" | Euclidean<br />
|-<br />
! Constrained !! Destretched<br />
|-<br />
! Tenney<br />
| CTE: ~6/5 = 317.1110{{c}} || POTE: ~6/5 = 317.0756{{c}}<br />
|}<br />
<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings<br />
|-<br />
! Optimized chord !! Generator value !! Polynomial !! Further notes<br />
|-<br />
| 3:4:5 (+1 +1) || ~6/5 = 317.1496 || ''g''<sup>6</sup> + 2''g''<sup>5</sup> &minus; 8 = 0 || {{dash|1, 3, 5|med}} equal-beating tuning, close to 8/43-kleisma<br />
|-<br />
| 4:5:6 (+1 +1) || ~6/5 = 317.9593 || ''g''<sup>6</sup> &minus; 2''g''<sup>5</sup> + 2 = 0 || {{dash|1, 3, 5|med}} equal-beating tuning, close to 2/7-kleisma<br />
|-<br />
| 10:12:15 (+2 +3) || ~6/5 = 317.6675 || ''g''<sup>6</sup> &minus; 5''g'' + 3 = 0 || Close to 1/4-kleisma<br />
|-<br />
| 9:13:15 (+2 +1) || ~6/5 = 317.5679 || 3''g''<sup>3</sup> + 4''g'' &minus; 10 = 0 || Close to 13/36-marveltwin comma<br />
|-<br />
| 13:15:18 (+2 +3) || ~6/5 = 317.0010 || 3''g''<sup>3</sup> &minus; ''g'' &minus; 4 = 0 || Close to 13/51-marveltwin comma<br />
|}<br />
<br />
=== Other tunings ===<br />
* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.0000{{c}}, ~6/5 = 317.1983{{c}}<br />
<br />
=== Tuning spectrum ===<br />
{| class="wikitable center-all left-4"<br />
|-<br />
! Edo<br>generator<br />
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*<br />
! Generator (¢)<br />
! Comments<br />
|-<br />
| <br />
| [[6/5]]<br />
| 315.6413<br />
| Untempered tuning, lower bound of 5-odd-limit diamond tradeoff<br />
|-<br />
| '''[[19edo|5\19]]'''<br />
| <br />
| '''315.7895'''<br />
| '''Lower bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone'''<br />
|-<br />
| <br />
| [[27/26]]<br />
| 316.3343<br />
| 1/4-[[625/624|tunbarsma]]<br />
|-<br />
| [[110edo|29\110]]<br />
| <br />
| 316.3636<br />
| 110ff val<br />
|-<br />
| [[91edo|24\91]]<br />
| <br />
| 316.4835<br />
| 91f val<br />
|-<br />
| <br />
| [[27/25]]<br />
| 316.6547<br />
| 1/8-kleisma<br />
|-<br />
| [[72edo|19\72]]<br />
| <br />
| 316.6667<br />
| <br />
|-<br />
| <br />
| [[9/5]]<br />
| 316.7995<br />
| 1/7-kleisma<br />
|-<br />
| [[125edo|33\125]]<br />
| <br />
| 316.8000<br />
| 125f val<br />
|-<br />
| <br />
| [[26/25]]<br />
| 316.9750<br />
| 1/4-[[325/324|marveltwin comma]]<br />
|-<br />
| [[53edo|14\53]]<br />
| <br />
| 316.9811<br />
| <br />
|-<br />
| <br />
| [[3/2]]<br />
| 316.9925<br />
| 1/6-kleisma; 5- and 9-odd-limit minimax tuning<br />
|-<br />
| [[246edo|65\246]]<br />
| <br />
| 317.0732<br />
| <br />
|-<br />
| [[193edo|51\193]]<br />
| <br />
| 317.0984<br />
| <br />
|-<br />
| <br />
| [[15/8]]<br />
| 317.1153<br />
| 2/11-kleisma<br />
|-<br />
| <br />
| [[13/10]]<br />
| 317.1349<br />
| 13- and 15-odd-limit minimax tuning<br />
|-<br />
| [[140edo|37\140]]<br />
| <br />
| 317.1429<br />
| <br />
|-<br />
| <br />
| [[13/8]]<br />
| 317.1805<br />
| <br />
|-<br />
| [[227edo|60\227]]<br />
| <br />
| 317.1807<br />
| <br />
|-<br />
| [[87edo|23\87]]<br />
| <br />
| 317.2414<br />
| <br />
|-<br />
| <br />
| [[5/4]]<br />
| 317.2627<br />
| 1/5-kleisma, upper bound of 5-odd-limit diamond tradeoff<br />
|-<br />
| <br />
| [[13/12]]<br />
| 317.3216<br />
| <br />
|-<br />
| [[121edo|32\121]]<br />
| <br />
| 317.3554<br />
| <br />
|-<br />
| [[155edo|41\155]]<br />
| <br />
| 317.4194<br />
| <br />
|-<br />
| <br />
| [[15/13]]<br />
| 317.4197<br />
| 1/3-marveltwin comma<br />
|-<br />
| [[34edo|9\34]]<br />
| <br />
| 317.6471<br />
| <br />
|-<br />
| <br />
| [[25/24]]<br />
| 317.6681<br />
| 1/4-kleisma, virtually [[Delta-rational chord|DR]] 10:12:15<br />
|-<br />
| [[83edo|22\83]]<br />
| <br />
| 318.0723<br />
| 83f val<br />
|-<br />
| <br />
| [[13/9]]<br />
| 318.3088<br />
| 1/2-marveltwin comma, upper bound of 2.3.5.13-subgroup 15-odd-limit diamond tradeoff<br />
|-<br />
| <br />
| [[125/72]]<br />
| 318.3437<br />
| 1/3-kleisma<br />
|-<br />
| [[49edo|13\49]]<br />
| <br />
| 318.3673<br />
| 49f val<br />
|-<br />
| <br />
| [[625/432]]<br />
| 319.6949<br />
| 1/2-kleisma<br />
|-<br />
| '''[[15edo|4\15]]'''<br />
| <br />
| '''320.0000'''<br />
| '''Upper bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone'''<br />
|}<br />
<nowiki/>* Besides the octave<br />
<br />
== Scales ==<br />
* [[Cata7]] ([[4L 3s]])<br />
* [[Cata11]] ([[4L 7s]])<br />
* [[Cata15]] ([[4L 11s]])<br />
* [[Cata19]] ([[15L 4s]])<br />
<br />
== Music ==<br />
; [[Petr Pařízek]]<br />
* [https://web.archive.org/web/20201127013042/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Parizek/Hanson%20%20Improv.mp3 ''Hanson Improv'']<br />
<br />
; [[Chris Vaisvil]]<br />
* [http://clones.soonlabel.com/public/micro/Hanson/daily20110127-in-hanson11.mp3 ''In Hanson11'']<br />
<br />
== External links ==<br />
* [http://dkeenan.com/Music/ChainOfMinor3rds.htm ''11 note chain-of-minor-thirds scale''], by [[David Keenan]]<br />
<br />
[[Category:Kleismic| ]] <!-- Main article --><br />
[[Category:Rank-2 temperaments]]<br />
[[Category:Kleismic family]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Sensi&diff=225271Sensi2026-03-06T18:14:53Z<p>ArrowHead294: </p>
<hr />
<div>{{interwiki<br />
| en = Sensi<br />
| de = Sensi<br />
}}<br />
{{Infobox regtemp<br />
| Title = Sensi<br />
| Subgroups = 2.3.5.7, 2.3.5.7.13<br />
| Comma basis = [[126/125]], [[245/243]] (7-limit); <br>[[91/90]], [[126/125]], [[169/168]] (2.3.5.7.13)<br />
| Edo join 1 = 19 | Edo join 2 = 27<br />
| Mapping = 1; 7 9 13 10<br />
| Generators = 9/7 | Generators tuning = 443.3 | Optimization method = CWE<br />
| MOS scales = [[3L&nbsp;2s]], [[3L&nbsp;5s]], [[8L&nbsp;3s]], [[8L&nbsp;11s]]<br />
| Pergen = (P8, ccP5/7)<br />
| Odd limit 1 = 7 | Mistuning 1 = 7.5 | Complexity 1 = 19<br />
| Odd limit 2 = 2.3.5.7.13 21 | Mistuning 2 = 11.1 | Complexity 2 = 27<br />
}}<br />
'''Sensi''' is a [[rank-2 temperament|rank-2]] [[regular temperament]] that is [[generator|generated]] by an extremely sharp major third of between 442 and 445{{cent}}, which is taken in the [[7-limit]] to represent a sharpened [[9/7]]. The most important equivalence in sensi (i.e. [[tempering out]] the comma [[245/243]]) is known as ''sensamagic'', by which two of these thirds stack to a major sixth which approximates [[5/3]]. Sensi then makes the additional tempering of [[126/125]], through which three of these major sixths approximate [[7/6]], two octaves up. The [[6/1|6th harmonic]] is therefore split into seven, and [[5/4]] is divided into three parts, each identified with [[15/14]]. Furthermore, since the supermajor third is tempered so sharply, it makes sense to have it represent both 9/7 and [[13/10]], which means [[91/90]] is tempered out in the 2.3.5.7.13 [[subgroup]]. There the 15/14 interval also represents [[14/13]] and [[13/12]], which results in [[169/168]] and [[196/195]] being tempered out.<br />
<br />
The structure whereby 5/3 is split into two supermajor thirds is obviously xenharmonic as this cannot occur in [[12edo]]. But particularly, as the simplest [[EDO]]s with similar structures are [[8edo]] and [[11edo]] (hence the 8-note ([[3L&nbsp;5s]], checkertonic) and 11-note ([[8L&nbsp;3s]], flanatonic) [[MOS scale]]s), sensi has a very xenmelodic character compared to many other ways of organizing the 7-limit (such as [[superpyth]], which is based on the familiar [[chain of fifths]], and even [[porcupine]], which is fundamentally heptatonic).<br />
<br />
Equal temperaments that support sensi include [[19edo]] (generator 7\19; [[soft]] checkertonic), [[27edo]] (generator 10\27; [[supersoft]] checkertonic), as well as [[46edo]] (generator 17\46; {{nowrap| L/s {{=}} 7/5 }}, more optimized for sensi temperament) and [[65edo]] (generator 24\65; {{nowrap| L/s {{=}} 10/7 }}) using the 65f [[val]] with a flat 13.<br />
<br />
See [[Sensipent family #Sensi]] for more technical data, [[sensi extensions]] for extensions of sensi to include the [[11/1|11th]] and [[17/1|17th]] harmonics, and [[#Related temperaments]] for alternative interpretations of similar structures to sensi.<br />
<br />
== Theory ==<br />
=== Interval chain ===<br />
In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''. <br />
<br />
{| class="wikitable right-1 right-2 sortable"<br />
|-<br />
! #<br />
! Cents*<br />
! class="unsortable" | Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| 443.4<br />
| 9/7, 13/10<br />
|-<br />
| 2<br />
| 886.7<br />
| 5/3, 42/25<br />
|-<br />
| 3<br />
| 130.1<br />
| 13/12, 14/13, 15/14, 27/25<br />
|-<br />
| 4<br />
| 573.4<br />
| 7/5, 18/13, 25/18<br />
|-<br />
| 5<br />
| 1016.8<br />
| 9/5<br />
|-<br />
| 6<br />
| 260.1<br />
| 7/6, 15/13<br />
|-<br />
| 7<br />
| 703.5<br />
| '''3/2'''<br />
|-<br />
| 8<br />
| 1146.9<br />
| 27/14, 35/18<br />
|-<br />
| 9<br />
| 390.2<br />
| '''5/4'''<br />
|-<br />
| 10<br />
| 833.6<br />
| '''13/8''', 21/13<br />
|-<br />
| 11<br />
| 76.9<br />
| 21/20, 25/24<br />
|-<br />
| 12<br />
| 520.3<br />
| 27/20<br />
|-<br />
| 13<br />
| 963.7<br />
| '''7/4'''<br />
|-<br />
| 14<br />
| 207.0<br />
| '''9/8'''<br />
|-<br />
| 15<br />
| 650.4<br />
| 35/24 (sensor '''16/11''', sensus 22/15)<br />
|-<br />
| 16<br />
| 1093.7<br />
| '''15/8''' (sensor '''32/17''', sensus 17/9)<br />
|-<br />
| 17<br />
| 337.1<br />
| 39/32 (sensus 17/14)<br />
|-<br />
| 18<br />
| 780.4<br />
| 25/16<br />
|-<br />
| 19<br />
| 23.8<br />
| 49/48, 65/64, 81/80<br />
|-<br />
| 20<br />
| 467.2<br />
| '''21/16'''<br />
|}<br />
<nowiki/>* In 2.3.5.7.13 CWE tuning<br />
<br />
=== Intervals of Sensi[8] ===<br />
Sensi[8] is a [[mos scale]] with a [[3L&nbsp;5s]] pattern. See [[3L&nbsp;5s #Modes]] to see which modes have which qualities for each interval size.<br />
<br />
Sortable table of Sensi[8]'s major and minor intervals in various sensi tunings:<br />
<br />
{| class="wikitable right-2 right-3 right-4 sortable"<br />
|-<br />
! class="unsortable" | Degree<br />
! Size in [[19edo]] (soft)<br />
! Size in [[27edo]] (supersoft)<br />
! Size in [[46edo]]<br />
! class="unsortable" | Approximate ratios<br />
! &#35; generators up<br />
|-<br />
| Unison<br />
| 0\19, 0.0<br />
| 0\27, 0.0<br />
| 0\46, 0.0<br />
| 1/1<br />
| 0<br />
|-<br />
| Min. sen2nd<br />
| 2\19, 126.3<br />
| 3\27, 133.3<br />
| 5\46, 130.4<br />
| 14/13<br />
| +3<br />
|-<br />
| Maj. sen2nd<br />
| 3\19, 189.5<br />
| 4\27, 177.8<br />
| 7\46, 182.6<br />
| 10/9<br />
| −5<br />
|-<br />
| Min. sen3rd<br />
| 4\19, 252.6<br />
| 6\27, 266.7<br />
| 10\46, 260.9<br />
| 7/6<br />
| +6<br />
|-<br />
| Maj. sen3rd<br />
| 5\19, 315.8<br />
| 7\27, 311.1<br />
| 12\46, 313.0<br />
| 6/5<br />
| −2<br />
|-<br />
| Perf. sen4th<br />
| 7\19, 442.1<br />
| 10\27, 444.4<br />
| 17\46, 443.5<br />
| 9/7, 13/10<br />
| +1<br />
|-<br />
| Aug. sen4th<br />
| 8\19, 505.3<br />
| 11\27, 488.9<br />
| 19\46, 495.7<br />
| 4/3<br />
| −7<br />
|-<br />
| Min. sen5th<br />
| 9\19, 568.4<br />
| 13\27, 577.8<br />
| 22\46, 573.9<br />
| 7/5, 18/13<br />
| +4<br />
|-<br />
| Maj. sen5th<br />
| 10\19, 631.6<br />
| 14\27, 622.2<br />
| 24\46, 626.1 <br />
| 10/7, 13/9<br />
| −4<br />
|-<br />
| Dim. sen6th<br />
| 11\19, 694.7<br />
| 16\27, 711.1<br />
| 27\46, 704.3<br />
| 3/2<br />
| +7<br />
|-<br />
| Perf. sen6th<br />
| 12\19, 757.9<br />
| 17\27, 755.6<br />
| 20\46, 756.5<br />
| 14/9, 20/13<br />
| −1<br />
|-<br />
| Min. sen7th<br />
| 14\19, 884.2<br />
| 20\27, 888.9<br />
| 34\46, 887.0<br />
| 5/3<br />
| +2<br />
|-<br />
| Maj. sen7th<br />
| 15\19, 947.4<br />
| 21\27, 933.3<br />
| 36\46, 939.1<br />
| 12/7<br />
| −6<br />
|-<br />
| Min. sen8th<br />
| 16\19, 1010.5<br />
| 23\27, 1022.2<br />
| 39\46, 1017.4<br />
| 9/5<br />
| +5<br />
|-<br />
| Maj. sen8th<br />
| 17\19, 1073.7<br />
| 24\27, 1066.7<br />
| 41\46, 1069.6<br />
| 13/7<br />
| −3<br />
|}<br />
<br />
=== Related temperaments ===<br />
It is worth noting that sensi distinguishes itself from other structures, the [[sensamagic clan|sensamagic temperaments]], based around 245/243 (whose basic form in the 2.9/7.5/3 subgroup is known as [[sentry]]) by virtue of its minor third (6/5) being ''flattened'' from just rather than sharpened. This results in the supermajor third being sharpened even more than is typical, so much so that it is tuned [[interseptimal]]ly and may not fulfill all the functions that [[~]]9/7 is intended to have.<br />
<br />
One way around this is to eschew the generator's interpretation as 9/7 altogether, and focus on the [[5-limit]] part of sensi, which is known as [[sensipent]] (whose comma is [[78732/78125]]). From there, an interpretation of the generator as {{nowrap|[[31/24]]~[[40/31]]}} is apparent. Beyond the 2.3.5.31 subgroup, more accurate interpretations (in comparison to sensi) of sensipent's extended harmony are given by [[sensible]] (adding primes 11, 17, and 23) and [[sendai]] (adding 23 and 29). There are also alternative mappings of 7, including [[sensei]] (+32 generators, with a tuning flat of 65edo) and [[warrior]] (−33 generators, with a tuning between 65edo and 46edo); warrior combines well with the mapping of sensible, and sensei with sendai.<br />
<br />
==== BPS ====<br />
: ''Main article: [[Relationship between Bohlen–Pierce and octave-ful temperaments#Relationship of rank-2 Bohlen.E2.80.93Pierce.E2.80.93Stearns temperament to octave-ful temperaments|Relationship between Bohlen–Pierce and octave-ful temperaments]].<br />
<br />
Since the sensamagic comma, 245/243, contains no 2 in its [[monzo|factorization]], only primes 3, 5, and 7, it can be tempered out in a [[3/1|tritave (3/1)]]-repeating, [[3.5.7 subgroup]] context, where the generator (9/7) is now the tritave-reduced 7th subharmonic, two of which give the 5th harmonic. This is known as [[BPS|Bohlen–Pierce–Stearns (BPS)]] temperament, and it generates a [[4L 5s (3/1-equivalent)|4L&nbsp;5s]] scale against the tritave (sometimes known as ''Lambda''). Where this temperament connects to sensi is that, at 7 generators, BPS reaches an interval that it identifies with [[125/63]], which is rather close to the octave; sensi is obtained by treating this interval as the mapping of 2/1, which provides the interesting notion of using sensi in a 3/1-periodic 3.5.7.2 setting.<br />
<br />
== Chords and harmony ==<br />
{{See also| Chords of sensus }}<br />
<br />
The fundamental otonal consonance of sensi is 4:5:6:7:9:13. However, the full chord is only available in the 19-note mos.<br />
<br />
One of the most common consonant triads in sensi is the 6:10:13 triad, which spans 3 generators. Sensi[8] has five 6:10:13 triads, four 7:9:13 triads, three 5:6:7:9 tetrads and one 5:6:7:9:13 pentad. Having many diminished triads, it is similar to the 12edo diminished scale in some ways. Sensi is interesting mainly because it gives new 13-limit interpretations to fairly familiar (in the sense of extended meantone-like) intervals. Restricted to the 8-note MOS, it is essentially a [[non-over-1 temperament]].<br />
<br />
Melodically, Sensi[8] sounds fairly familiar because many intervals are either 5-limit or have familiar categorical interpretations, being represented in the meantone tuning [[19edo]]. For example, the small step of about 130{{c}} categorizes pretty well as a large semitone (except at places in the scale where two of them make a flat subminor third); the large step is a small whole tone representing 10/9.<br />
<br />
The root-sen5th-sen8th chords in Sensi[8] usually spell 5:7:9 (root-minor sen5th-minor sen8th) and 7:10:13 (root-major sen5th-major sen8th) chords (shown in the Anti-Dylathian mode QJKLMNOPQ = ssLssLsL):<br />
* Q M P = ssLs sLs L ≈ 5:7:9<br />
* J N Q = sLss LsL s is the odd one out<br />
* K O J = LssL sLs s ≈ 7:10:13<br />
* L P K = ssLs Lss L ≈ 5:7:9<br />
* M Q L = sLsL ssL s ≈ 7:10:13<br />
* N J M = LsLs sLs s ≈ 7:10:13<br />
* O K N = sLss Lss L ≈ 5:7:9<br />
* P L O = LssL ssL s ≈ 7:10:13<br />
<br />
Other otonal chords approximated in the 8-note mos include:<br />
<br />
* {{dash|Root, maj. sen7th, maj. sen8th ≈ 7:12:13|s=space}}<br />
* {{dash|Root, maj. sen2nd, maj. sen5th ≈ 9:10:13|s=space}}<br />
* {{dash|Root, min. sen3rd, dim. sen6th ≈ 6:7:9|s=space}}<br />
* {{dash|Root, perf. sen4th, dim. sen6th ≈ 10:13:15 (ultramajor triad)|s=space}}<br />
* {{dash|Root, perf. sen4th, maj. sen7th ≈ 7:9:13|s=space}}<br />
* {{dash|Root, perf. sen4th, maj. sen5th, maj. sen7th ≈ 7:9:10:13|s=space}}<br />
* {{dash|Root, perf. sen4th, min. sen7th ≈ 10:13:18|s=space}}<br />
* {{dash|Root, perf. sen4th, min. sen5th, min. sen7th ≈ 10:13:14:18|s=space}}<br />
* {{dash|Root, min. sen7th, min. sen3rd (+ octave) ≈ 3:5:7|s=space}}<br />
* {{dash|Root, min. sen7th, min. sen2nd (+ octave) ≈ 6:10:13|s=space}}<br />
* {{dash|Root, dim. sen6th, min. sen7th ≈ 6:9:10|s=space}}<br />
* {{dash|Root, dim. sen6th, min. sen2nd (+octave) ≈ 6:9:13|s=space}}<br />
<br />
== Scales ==<br />
* [[Sensi5]]<br />
* [[Sensi8]]<br />
* [[Sensi11]]<br />
* [[Sensi19]]<br />
* [[Sensi27]]<br />
<br />
== Tunings ==<br />
=== Norm-based tunings ===<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings<br />
|-<br />
! rowspan="2" | <br />
! colspan="3" | Euclidean<br />
|-<br />
! Constrained<br />
! Constrained & skewed<br />
! Destretched<br />
|-<br />
! Tenney<br />
| CTE: ~9/7 = 443.3166{{c}}<br />
| CWE: ~9/7 = 443.3493{{c}}<br />
| POTE: ~9/7 = 443.3827{{c}}<br />
|}<br />
<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.13-subgroup norm-based tunings<br />
|-<br />
! rowspan="2" | <br />
! colspan="3" | Euclidean<br />
|-<br />
! Constrained<br />
! Constrained & skewed<br />
! Destretched<br />
|-<br />
! Tenney<br />
| CTE: ~9/7 = 443.4016{{c}}<br />
| CWE: ~9/7 = 443.3581{{c}}<br />
| POTE: ~9/7 = 443.3220{{c}}<br />
|}<br />
<br />
=== Target tunings ===<br />
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"<br />
|+ style="white-space: nowrap;" | Target tunings<br />
|-<br />
! rowspan="2" | Target<br />
! colspan="2" | Minimax<br />
|-<br />
! Generator<br />
! Eigenmonzo*<br />
|-<br />
| 7-odd-limit<br />
| ~9/7 = 443.756{{c}}<br />
| 7/4<br />
|-<br />
| 9-odd-limit<br />
| ~9/7 = 443.519{{c}}<br />
| 9/5<br />
|-<br />
| no-11 13-odd-limit<br />
| ~9/7 = 443.519{{c}}<br />
| 9/5<br />
|-<br />
| no-11 15-odd-limit<br />
| ~9/7 = 443.136{{c}}<br />
| 3/2<br />
|}<br />
<br />
=== Tuning spectrum ===<br />
{| class="wikitable center-all left-4"<br />
|-<br />
! Edo<br>generators<br />
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*<br />
! Generator (¢)<br />
! Comments<br />
|-<br />
| <br />
| 9/7<br />
| 435.084<br />
| <br />
|-<br />
| [[11edo|4\11]]<br />
| <br />
| 436.364<br />
| 11cdf val<br />
|-<br />
| <br />
| 15/14<br />
| 439.814<br />
| <br />
|-<br />
| <br />
| 13/9<br />
| 440.846<br />
| <br />
|-<br />
| <br />
| 15/13<br />
| 441.290<br />
| <br />
|-<br />
| [[19edo|7\19]]<br />
| <br />
| 442.105<br />
| Lower bound of 7- and 9-odd-limit, <br>2.3.5.7.13-subgroup 13-, 15-, and 21-odd-limit diamond monotone<br />
|-<br />
| <br />
| 5/3<br />
| 442.179<br />
| <br />
|-<br />
| <br />
| 13/7<br />
| 442.766<br />
| <br />
|-<br />
| <br />
| 5/4<br />
| 442.924<br />
| 5-odd-limit minimax<br />
|-<br />
| <br />
| 15/8<br />
| 443.017<br />
| <br />
|-<br />
| <br />
| 21/13<br />
| 443.025<br />
| <br />
|-<br />
| [[65edo|24\65]]<br />
| <br />
| 443.077<br />
| 65f val<br />
|-<br />
| <br />
| 3/2<br />
| 443.136<br />
| 2.3.5.7.13-subgroup 15- and 21-odd-limit minimax<br />
|-<br />
| [[46edo|17\46]]<br />
| <br />
| 443.478<br />
|<br />
|-<br />
| <br />
| 9/5<br />
| 443.519<br />
| 9-odd-limit and 2.3.5.7.13-subgroup 13-odd-limit minimax<br />
|-<br />
| <br />
| 21/16<br />
| 443.539<br />
| <br />
|-<br />
| <br />
| 7/4<br />
| 443.756<br />
| 7-odd-limit minimax<br />
|-<br />
| [[73edo|27\73]]<br />
| <br />
| 443.836<br />
|<br />
|-<br />
| <br />
| 21/20<br />
| 444.042<br />
| <br />
|-<br />
| <br />
| 13/8<br />
| 444.053<br />
|<br />
|-<br />
| [[27edo|10\27]]<br />
| <br />
| 444.444<br />
| Upper bound of 9-odd-limit, <br>2.3.5.7.13-subgroup 13-, 15-, and 21-odd-limit diamond monotone<br />
|-<br />
| <br />
| 7/6<br />
| 444.478<br />
|<br />
|-<br />
| <br />
| 117/70<br />
| 444.649<br />
| Exact geometric mean of 9/7 and 13/10<br />
|-<br />
| <br />
| 7/5<br />
| 445.628<br />
|<br />
|-<br />
| <br />
| 13/12<br />
| 446.191<br />
|<br />
|-<br />
| [[8edo|3\8]]<br />
| <br />
| 450.000<br />
| 8d val, upper bound of 7-odd-limit diamond monotone<br />
|-<br />
| <br />
| 13/10<br />
| 454.214<br />
|<br />
|}<br />
<nowiki/>* Besides the octave<br />
<br />
== Visualizations ==<br />
=== Steps of sensi ===<br />
This diagram shows Sensi[5], [8], [11], and [19] with intervals named in relation to the L and s of Sensi[8]. <br />
<br />
[[File:steps_of_sensi.png|Steps of sensi|alt=steps_of_sensi.png]]<br />
<br />
Note that X, M and Z are not standard, but d and A are; they are short for "diminished" and "augmented".<br />
<br />
=== Map of sensi ===<br />
These diagrams relate the sensi generator chain (horizontal axis) to the steps within the octave (vertical axis) for Sensi[8] and [11]. <br />
<br />
[[File:map_of_sensi-8-.png|Map of Sensi[8]|alt=map_of_sensi[8].png]]<br />
[[File:map_of_sensi-11-_correction2.png|Map of Sensi[11]|alt=map_of_sensi[11]_correction2.png]]<br />
<br />
=== Isomorphic layout ===<br />
{{See also| Lumatone mapping for sensi }}<br />
<br />
This diagram shows a layout for playing sensi temperament on an [[isomorphic keyboard]]. <br />
<br />
[[File:sensi_isomorphic_layout.png|sensi_isomorphic_layout.png|alt=sensi_isomorphic_layout.png]]<br />
<br />
The darkest hexagons represent the same note (eg. C), but offset by octaves. The next-darkest hexagons show the notes of Sensi[5]. Imagine stepping from hex to hex as you move across the keyboard from left to right, landing only on the darkest and next-darkest hexes. The light red hexagons show additional notes needed to play Sensi[8]. The Large step of Sensi[8] is represented by a move straight down, so this pattern is a little more zig-zaggy than the pattern for Sensi[5]. Add the white hexes and you have Sensi[11]. The small step of Sensi[11] (indicated in the diagram as "c" for chroma), is represented by a move straight down and down-left. This pattern actually involves moving backward in the horizontal direction, and is therefore more zig-zaggy.<br />
<br />
=== Sensi[19] guitar ===<br />
[[File:sensi-19-in46.jpg|sensi[19]in46.jpg|alt=sensi[19]in46.jpg]]<br />
<br />
== Music ==<br />
; [[Andrew Heathwaite]]<br />
* [[Technical Notes for Newbeams #Tumbledown Stew|"Tumbledown Stew" from ''Newbeams'']]<br />
* [[Technical Notes for Newbeams #Hypnocloudsmack 3|"Hypnocloudsmack 3" from ''Newbeams'']]<br />
<br />
; [[Budjarn Lambeth]]<br />
* [https://www.youtube.com/watch?v=qc0CkUKj7t4 Music in Sensi Temperament (+ Tempered Octaves) – Mar 2024]<br />
<br />
[[Category:Sensi| ]] <!-- Main article --><br />
[[Category:Rank-2 temperaments]]<br />
[[Category:Sensipent family]]<br />
[[Category:Sensamagic clan]]<br />
[[Category:Starling temperaments]]<br />
[[Category:Sengic temperaments]]<br />
[[Category:Naiadic]]</div>ArrowHead294https://en.xen.wiki/index.php?title=User:ArrowHead294&diff=225166User:ArrowHead2942026-03-04T19:42:10Z<p>ArrowHead294: /* Miscellaneous */</p>
<hr />
<div>{{subpage|sandbox|l|text=Hi!}} I'm primarily a music theorist and singer nowadays, though most of my musical training is in classical piano and cello.<br />
<br />
== My musical experience ==<br />
Piano and keyboard: 2004–2017 (K–12), 2022–present<br />
<br />
Cello: 2009–2017 (Grades 5–12)<br />
<br />
Voice: 2020–present<br />
: Voice type: Baritone / bass-baritone<br />
:: Chest voice: E♭2 to E4<br />
::: Can go down to D2 if needed<br />
::: Can belt up to ~G4, A♭4 on occasion<br />
:: Mixed voice: C4–C5<br />
:: Falsetto and head voice: D4 to E♭5<br />
::: Can reach E5 and F5<br />
<br />
I've also been told I have "perfect pitch" since I acquired the ability to recognise notes (in {{nowrap|A {{=}} 440 Hz}} and 12edo) from a young age, though I have grown increasingly disdainful towards the terms "perfect pitch" and "absolute pitch" since late 2022 (the start of my last term of college) since it locked me into 12edo with {{nowrap|A {{=}} 440 Hz}} and anything else sounded "wrong" to me. This is even more true nowadays since I now prefer other tunings over 12edo in a lot of cases.<br />
<br />
== Favourite tunings within Western music ==<br />
{{Main|User:ArrowHead294/EDO impressions|l1=EDO impressions}}<br />
Most Western musicians only know 12edo, and my musical background is mainly classical, so my main interests have been in Pythagorean and meantone. I'm quite active in church despite being non-religious, and most of the alternative tunings I introduce to others are flatter-than-12 meantones as they're relatively easy to get into.<br />
<br />
I've known about quarter tones ([[24edo]]) since I was very young. I've even dabbled around with it before 2022, and used its absolute frequencies with {{nowrap|A {{=}} 440 Hz}} as a way to compare just intonation and equal temperament, though for most of my life that was the only microtonal tuning I knew about. As a result, my knowledge of just intonation was very much limited to the 2.3.5.11.37 subgroup.<br />
<br />
Since late 2022, I've also gotten into sixth tones ([[36edo]]) for exploring septimal harmonies.<br />
<br />
[[31edo]] is the first alternative tuning I learned about which helped me break out of 12-TET's walled garden, and [[Sevish]]'s song ''Better Left Unanswered'' is the first microtonal work I've ever heard that wasn't in standard quarter tones. I learned about it mainly after exploring quarter-comma meantone in classical music, and I think it's the most practical alternate tuning for most Western musicians to get into. It has excellent 7-limit and 11-limit harmonies as well (even better than 36edo), so it could also be useful for blues and jazz.<br />
<br />
This was followed by [[19edo]], after hearing ''Sunsrise'' by Supahstar Saga. 19 has its unique quirks that make it a good tuning for a lot of Western music, though I think its sound is best suited to songs with largely pentatonic melodies since the diatonic scale sounds quite loose to me. Augmented and diminished chords sound very weird, though, even more jarring than 31.<br />
<br />
=== Beyond traditional Western music ===<br />
For xenharmony and music beyond meantone, I've been mainly interested in [[22edo]] for [[superpyth]]agorean and [[Porcupine]] temperament, [[Orwell]] in 31edo, [[34edo]] for [[Tetracot]] (including [[68edo]] for [[Octacot]] by extension), and [[46edo]] for [[Sensi]] and 5-limit harmony beyond meantone.<br />
<br />
== Miscellaneous ==<br />
=== The Lord's Prayer ===<br />
The contemporary version of the {{w|Lord's Prayer}} goes:<br />
<br />
<pre><br />
Our Father in heaven,<br />
hallowed be your name,<br />
your kingdom come,<br />
your will be done,<br />
on earth as in heaven.<br />
Give us today our daily bread.<br />
Forgive us our sins<br />
as we forgive those who sin against us.<br />
Save us from the time of trial<br />
and deliver us from evil.<br />
For the kingdom, the power, and the glory are yours<br />
now and for ever.<br />
Amen.<br />
</pre><br />
<br />
When encoded in {{w|hexadecimal}} with CRLF (<code>\r\n</code>) newlines, this becomes:<br />
<br />
<pre><br />
4f 75 72 20 46 61 74 68 65 72 20 69 6e 20 68 65 61 76 65 6e 2c 0d 0a 68 61 6c 6c 6f 77 65 64 20 62 65 20 79 6f 75 72 20 6e 61 6d 65 2c 0d 0a 79 6f 75 72 20 6b 69 6e 67 64 6f 6d 20 63 6f 6d 65 2c 0d 0a 79 6f 75 72 20 77 69 6c 6c 20 62 65 20 64 6f 6e 65 2c 0d 0a 6f 6e 20 65 61 72 74 68 20 61 73 20 69 6e 20 68 65 61 76 65 6e 2e 0d 0a 47 69 76 65 20 75 73 20 74 6f 64 61 79 20 6f 75 72 20 64 61 69 6c 79 20 62 72 65 61 64 2e 0d 0a 46 6f 72 67 69 76 65 20 75 73 20 6f 75 72 20 73 69 6e 73 0d 0a 61 73 20 77 65 20 66 6f 72 67 69 76 65 20 74 68 6f 73 65 20 77 68 6f 20 73 69 6e 20 61 67 61 69 6e 73 74 20 75 73 2e 0d 0a 53 61 76 65 20 75 73 20 66 72 6f 6d 20 74 68 65 20 74 69 6d 65 20 6f 66 20 74 72 69 61 6c 0d 0a 61 6e 64 20 64 65 6c 69 76 65 72 20 75 73 20 66 72 6f 6d 20 65 76 69 6c 2e 0d 0a 46 6f 72 20 74 68 65 20 6b 69 6e 67 64 6f 6d 2c 20 74 68 65 20 70 6f 77 65 72 2c 20 61 6e 64 20 74 68 65 20 67 6c 6f 72 79 20 61 72 65 20 79 6f 75 72 73 0d 0a 6e 6f 77 20 61 6e 64 20 66 6f 72 20 65 76 65 72 2e 0d 0a 41 6d 65 6e 2e<br />
</pre><br />
<br />
When encoded in {{w|Base64}} with the RFC&nbsp;4648 and numeral-first alphabets, this becomes (respectively):<br />
<br />
<pre style="word-break: break-all;"><br />
T3VyIEZhdGhlciBpbiBoZWF2ZW4sDQpoYWxsb3dlZCBiZSB5b3VyIG5hbWUsDQp5b3VyIGtpbmdkb20gY29tZSwNCnlvdXIgd2lsbCBiZSBkb25lLA0Kb24gZWFydGggYXMgaW4gaGVhdmVuLg0KR2l2ZSB1cyB0b2RheSBvdXIgZGFpbHkgYnJlYWQuDQpGb3JnaXZlIHVzIG91ciBzaW5zDQphcyB3ZSBmb3JnaXZlIHRob3NlIHdobyBzaW4gYWdhaW5zdCB1cy4NClNhdmUgdXMgZnJvbSB0aGUgdGltZSBvZiB0cmlhbA0KYW5kIGRlbGl2ZXIgdXMgZnJvbSBldmlsLg0KRm9yIHRoZSBraW5nZG9tLCB0aGUgcG93ZXIsIGFuZCB0aGUgZ2xvcnkgYXJlIHlvdXJzDQpub3cgYW5kIGZvciBldmVyLg0KQW1lbi4=<br />
</pre><br />
<br />
<pre style="word-break: break-all;"><br />
JtLo84PXT6XbSY1fRY1ePM5sPMui3GfeOMniRtTbP21YPI1vRtLo86vXRMKi3GfvRtLo86jfRcTaRsqWOszjPImD2dblTN8WTsbiR21YPI1aRsvbB0qARsuWPM5oT6WWONCWQMuWQ6LXTcLkBWqAHsbsPI1rSo1qRsHXUI1lTN8WP65fR7aWOd9bOMGk3Gf6Rt9dQNPb87Lp86zrSY1pQMvp3GfXSo1tPI1cRt9dQNPb87HeRtDb87TeRo1pQMuWOMTXQMvpT21rSouD2bDXTcKWTNCWPd9lRI1qQ6KWT6bjPI1lPY1qScbXR0qAOMva86HbR6bsPN8WTNCWPd9lRI1bTcbiBWqAHczo87HePI1hQMvdP6zjB21qQ6KWS6ztPN8i865kP21qQ6KWPsnlSdaWON9b87blTN9p3GfkRtSWOMva86PlSY1bTcLoBWqAGMrbRYu=<br />
</pre><br />
<br />
== See also ==<br />
* {{subpage|Purely consistent EDOs by odd limit}}<br />
* {{subpage|Regex snippets}}<br />
* {{subpage|The Star-Spangled Banner}}<br />
* {{subpage|UTF-8 extensions}}<br />
<br />
[[Category:User en-N]]<br />
[[Category:User zh-3]]<br />
[[Category:User de-1]]<br />
[[Category:User on Discord]]<br />
[[Category:User in EST/EDT]]</div>ArrowHead294https://en.xen.wiki/index.php?title=User:ArrowHead294&diff=225165User:ArrowHead2942026-03-04T19:41:02Z<p>ArrowHead294: </p>
<hr />
<div>{{subpage|sandbox|l|text=Hi!}} I'm primarily a music theorist and singer nowadays, though most of my musical training is in classical piano and cello.<br />
<br />
== My musical experience ==<br />
Piano and keyboard: 2004–2017 (K–12), 2022–present<br />
<br />
Cello: 2009–2017 (Grades 5–12)<br />
<br />
Voice: 2020–present<br />
: Voice type: Baritone / bass-baritone<br />
:: Chest voice: E♭2 to E4<br />
::: Can go down to D2 if needed<br />
::: Can belt up to ~G4, A♭4 on occasion<br />
:: Mixed voice: C4–C5<br />
:: Falsetto and head voice: D4 to E♭5<br />
::: Can reach E5 and F5<br />
<br />
I've also been told I have "perfect pitch" since I acquired the ability to recognise notes (in {{nowrap|A {{=}} 440 Hz}} and 12edo) from a young age, though I have grown increasingly disdainful towards the terms "perfect pitch" and "absolute pitch" since late 2022 (the start of my last term of college) since it locked me into 12edo with {{nowrap|A {{=}} 440 Hz}} and anything else sounded "wrong" to me. This is even more true nowadays since I now prefer other tunings over 12edo in a lot of cases.<br />
<br />
== Favourite tunings within Western music ==<br />
{{Main|User:ArrowHead294/EDO impressions|l1=EDO impressions}}<br />
Most Western musicians only know 12edo, and my musical background is mainly classical, so my main interests have been in Pythagorean and meantone. I'm quite active in church despite being non-religious, and most of the alternative tunings I introduce to others are flatter-than-12 meantones as they're relatively easy to get into.<br />
<br />
I've known about quarter tones ([[24edo]]) since I was very young. I've even dabbled around with it before 2022, and used its absolute frequencies with {{nowrap|A {{=}} 440 Hz}} as a way to compare just intonation and equal temperament, though for most of my life that was the only microtonal tuning I knew about. As a result, my knowledge of just intonation was very much limited to the 2.3.5.11.37 subgroup.<br />
<br />
Since late 2022, I've also gotten into sixth tones ([[36edo]]) for exploring septimal harmonies.<br />
<br />
[[31edo]] is the first alternative tuning I learned about which helped me break out of 12-TET's walled garden, and [[Sevish]]'s song ''Better Left Unanswered'' is the first microtonal work I've ever heard that wasn't in standard quarter tones. I learned about it mainly after exploring quarter-comma meantone in classical music, and I think it's the most practical alternate tuning for most Western musicians to get into. It has excellent 7-limit and 11-limit harmonies as well (even better than 36edo), so it could also be useful for blues and jazz.<br />
<br />
This was followed by [[19edo]], after hearing ''Sunsrise'' by Supahstar Saga. 19 has its unique quirks that make it a good tuning for a lot of Western music, though I think its sound is best suited to songs with largely pentatonic melodies since the diatonic scale sounds quite loose to me. Augmented and diminished chords sound very weird, though, even more jarring than 31.<br />
<br />
=== Beyond traditional Western music ===<br />
For xenharmony and music beyond meantone, I've been mainly interested in [[22edo]] for [[superpyth]]agorean and [[Porcupine]] temperament, [[Orwell]] in 31edo, [[34edo]] for [[Tetracot]] (including [[68edo]] for [[Octacot]] by extension), and [[46edo]] for [[Sensi]] and 5-limit harmony beyond meantone.<br />
<br />
== Miscellaneous ==<br />
=== The Lord's Prayer ===<br />
The contemporary version of the {{w|Lord's Prayer}} goes:<br />
<br />
<pre><br />
Our Father in heaven,<br />
hallowed be your name,<br />
your kingdom come,<br />
your will be done,<br />
on earth as in heaven.<br />
Give us today our daily bread.<br />
Forgive us our sins<br />
as we forgive those who sin against us.<br />
Save us from the time of trial<br />
and deliver us from evil.<br />
For the kingdom, the power, and the glory are yours<br />
now and for ever.<br />
Amen.<br />
</pre><br />
<br />
When encoded in {{w|hexadecimal}} with CRLF (<code>\r\n</code>) newlines, this becomes:<br />
<br />
<pre><br />
4f 75 72 20 46 61 74 68 65 72 20 69 6e 20 68 65 61 76 65 6e 2c 0d 0a 68 61 6c 6c 6f 77 65 64 20 62 65 20 79 6f 75 72 20 6e 61 6d 65 2c 0d 0a 79 6f 75 72 20 6b 69 6e 67 64 6f 6d 20 63 6f 6d 65 2c 0d 0a 79 6f 75 72 20 77 69 6c 6c 20 62 65 20 64 6f 6e 65 2c 0d 0a 6f 6e 20 65 61 72 74 68 20 61 73 20 69 6e 20 68 65 61 76 65 6e 2e 0d 0a 47 69 76 65 20 75 73 20 74 6f 64 61 79 20 6f 75 72 20 64 61 69 6c 79 20 62 72 65 61 64 2e 0d 0a 46 6f 72 67 69 76 65 20 75 73 20 6f 75 72 20 73 69 6e 73 0d 0a 61 73 20 77 65 20 66 6f 72 67 69 76 65 20 74 68 6f 73 65 20 77 68 6f 20 73 69 6e 20 61 67 61 69 6e 73 74 20 75 73 2e 0d 0a 53 61 76 65 20 75 73 20 66 72 6f 6d 20 74 68 65 20 74 69 6d 65 20 6f 66 20 74 72 69 61 6c 0d 0a 61 6e 64 20 64 65 6c 69 76 65 72 20 75 73 20 66 72 6f 6d 20 65 76 69 6c 2e 0d 0a 46 6f 72 20 74 68 65 20 6b 69 6e 67 64 6f 6d 2c 20 74 68 65 20 70 6f 77 65 72 2c 20 61 6e 64 20 74 68 65 20 67 6c 6f 72 79 20 61 72 65 20 79 6f 75 72 73 0d 0a 6e 6f 77 20 61 6e 64 20 66 6f 72 20 65 76 65 72 2e 0d 0a 41 6d 65 6e 2e<br />
</pre><br />
<br />
When encoded in {{w|Base64}} with the RFC&nbsp;4648 and numerical-first alphabets, this becomes (respectively):<br />
<br />
<pre style="word-break: break-all;"><br />
T3VyIEZhdGhlciBpbiBoZWF2ZW4sDQpoYWxsb3dlZCBiZSB5b3VyIG5hbWUsDQp5b3VyIGtpbmdkb20gY29tZSwNCnlvdXIgd2lsbCBiZSBkb25lLA0Kb24gZWFydGggYXMgaW4gaGVhdmVuLg0KR2l2ZSB1cyB0b2RheSBvdXIgZGFpbHkgYnJlYWQuDQpGb3JnaXZlIHVzIG91ciBzaW5zDQphcyB3ZSBmb3JnaXZlIHRob3NlIHdobyBzaW4gYWdhaW5zdCB1cy4NClNhdmUgdXMgZnJvbSB0aGUgdGltZSBvZiB0cmlhbA0KYW5kIGRlbGl2ZXIgdXMgZnJvbSBldmlsLg0KRm9yIHRoZSBraW5nZG9tLCB0aGUgcG93ZXIsIGFuZCB0aGUgZ2xvcnkgYXJlIHlvdXJzDQpub3cgYW5kIGZvciBldmVyLg0KQW1lbi4=<br />
</pre><br />
<br />
<pre style="word-break: break-all;"><br />
JtLo84PXT6XbSY1fRY1ePM5sPMui3GfeOMniRtTbP21YPI1vRtLo86vXRMKi3GfvRtLo86jfRcTaRsqWOszjPImD2dblTN8WTsbiR21YPI1aRsvbB0qARsuWPM5oT6WWONCWQMuWQ6LXTcLkBWqAHsbsPI1rSo1qRsHXUI1lTN8WP65fR7aWOd9bOMGk3Gf6Rt9dQNPb87Lp86zrSY1pQMvp3GfXSo1tPI1cRt9dQNPb87HeRtDb87TeRo1pQMuWOMTXQMvpT21rSouD2bDXTcKWTNCWPd9lRI1qQ6KWT6bjPI1lPY1qScbXR0qAOMva86HbR6bsPN8WTNCWPd9lRI1bTcbiBWqAHczo87HePI1hQMvdP6zjB21qQ6KWS6ztPN8i865kP21qQ6KWPsnlSdaWON9b87blTN9p3GfkRtSWOMva86PlSY1bTcLoBWqAGMrbRYu=<br />
</pre><br />
<br />
== See also ==<br />
* {{subpage|Purely consistent EDOs by odd limit}}<br />
* {{subpage|Regex snippets}}<br />
* {{subpage|The Star-Spangled Banner}}<br />
* {{subpage|UTF-8 extensions}}<br />
<br />
[[Category:User en-N]]<br />
[[Category:User zh-3]]<br />
[[Category:User de-1]]<br />
[[Category:User on Discord]]<br />
[[Category:User in EST/EDT]]</div>ArrowHead294https://en.xen.wiki/index.php?title=User:ArrowHead294&diff=225164User:ArrowHead2942026-03-04T19:35:42Z<p>ArrowHead294: /* See also */</p>
<hr />
<div>{{subpage|sandbox|l|text=Hi!}} I'm primarily a music theorist and singer nowadays, though most of my musical training is in classical piano and cello.<br />
<br />
== My musical experience ==<br />
Piano and keyboard: 2004–2017 (K–12), 2022–present<br />
<br />
Cello: 2009–2017 (Grades 5–12)<br />
<br />
Voice: 2020–present<br />
: Voice type: Baritone / bass-baritone<br />
:: Chest voice: E♭2 to E4<br />
::: Can go down to D2 if needed<br />
::: Can belt up to ~G4, A♭4 on occasion<br />
:: Mixed voice: C4–C5<br />
:: Falsetto and head voice: D4 to E♭5<br />
::: Can reach E5 and F5<br />
<br />
I've also been told I have "perfect pitch" since I acquired the ability to recognise notes (in {{nowrap|A {{=}} 440 Hz}} and 12edo) from a young age, though I have grown increasingly disdainful towards the terms "perfect pitch" and "absolute pitch" since late 2022 (the start of my last term of college) since it locked me into 12edo with {{nowrap|A {{=}} 440 Hz}} and anything else sounded "wrong" to me. This is even more true nowadays since I now prefer other tunings over 12edo in a lot of cases.<br />
<br />
== Favourite tunings within Western music ==<br />
{{Main|User:ArrowHead294/EDO impressions|l1=EDO impressions}}<br />
Most Western musicians only know 12edo, and my musical background is mainly classical, so my main interests have been in Pythagorean and meantone. I'm quite active in church despite being non-religious, and most of the alternative tunings I introduce to others are flatter-than-12 meantones as they're relatively easy to get into.<br />
<br />
I've known about quarter tones ([[24edo]]) since I was very young. I've even dabbled around with it before 2022, and used its absolute frequencies with {{nowrap|A {{=}} 440 Hz}} as a way to compare just intonation and equal temperament, though for most of my life that was the only microtonal tuning I knew about. As a result, my knowledge of just intonation was very much limited to the 2.3.5.11.37 subgroup.<br />
<br />
Since late 2022, I've also gotten into sixth tones ([[36edo]]) for exploring septimal harmonies.<br />
<br />
[[31edo]] is the first alternative tuning I learned about which helped me break out of 12-TET's walled garden, and [[Sevish]]'s song ''Better Left Unanswered'' is the first microtonal work I've ever heard that wasn't in standard quarter tones. I learned about it mainly after exploring quarter-comma meantone in classical music, and I think it's the most practical alternate tuning for most Western musicians to get into. It has excellent 7-limit and 11-limit harmonies as well (even better than 36edo), so it could also be useful for blues and jazz.<br />
<br />
This was followed by [[19edo]], after hearing ''Sunsrise'' by Supahstar Saga. 19 has its unique quirks that make it a good tuning for a lot of Western music, though I think its sound is best suited to songs with largely pentatonic melodies since the diatonic scale sounds quite loose to me. Augmented and diminished chords sound very weird, though, even more jarring than 31.<br />
<br />
=== Beyond traditional Western music ===<br />
For xenharmony and music beyond meantone, I've been mainly interested in [[22edo]] for [[superpyth]]agorean and [[Porcupine]] temperament, [[Orwell]] in 31edo, [[34edo]] for [[Tetracot]] (including [[68edo]] for [[Octacot]] by extension), and [[46edo]] for [[Sensi]] and 5-limit harmony beyond meantone.<br />
<br />
== See also ==<br />
* {{subpage|Purely consistent EDOs by odd limit}}<br />
* {{subpage|Regex snippets}}<br />
* <s>{{subpage|The Star-Spangled Banner}}</s><br />
* {{subpage|UTF-8 extensions}}<br />
<br />
[[Category:User en-N]]<br />
[[Category:User zh-3]]<br />
[[Category:User de-1]]<br />
[[Category:User on Discord]]<br />
[[Category:User in EST/EDT]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Parapyth&diff=225102Parapyth2026-03-03T18:22:12Z<p>ArrowHead294: </p>
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<div>'''Parapyth''', also known as '''parapythagorean''', is the rank-3 [[temperament]] tempering out [[352/351]] and [[364/363]] in the 2.3.7.11.13 [[subgroup]]. <br />
<br />
Inspired by [[Secor29htt|George Secor's 29-tone high tolerance temperament]], parapyth was found by [[Margo Schulter]] in 2002, and it continued to be developed as part of her ''neoclassical tuning theory'' (NTT), although a [[regular temperament]] perspective is as viable. <br />
<br />
In the early prototype, there was only a single chain of fifths, tuned slightly sharp such that: <br />
* The minor third (−3 fifths) is [[13/11]], tempering out 352/351; <br />
* The major third (+4 fifths) hits [[14/11]], tempering out [[896/891]]; <br />
* The augmented unison (+7 fifths) hits [[14/13]], tempering out [[28672/28431]].<br />
<br />
This temperament is now known as [[pepperoni]]. Parapyth encapsulates pepperoni and adds a {{nowrap| 28/27 ~ 33/32 }} spacer interval such that harmonics 7, 11, and 13 are all made available simply by using two chains of fifths. <br />
<br />
See [[Pentacircle clan#Parapyth]] for technical data.<br />
<br />
== Interval lattice ==<br />
{| class="wikitable" style="margin: 20px auto 20px auto;"<br />
|-<br />
| [[File:Lattice Parapyth RTT.png|1000px]]<br />
|-<br />
| In CTE tuning and lattice basis {{nowrap|{~2, ~3, ~7/4}<nowiki/>}}<br />
|- style="border-top: double;"<br />
| [[File:Lattice Parapyth NTT.png|1000px]]<br />
|-<br />
| In MET-24 tuning and lattice basis {{nowrap|{~2, ~3, ~33/32}<nowiki/>}}<br />
|}<br />
<br />
These diagrams differ by lattice bases and tunings. The first diagram is generated by {{nowrap|{~2, ~3, ~7/4}<nowiki/>}}, corresponding to the octave-reduced form of the mapping, and tuned to the 2.3.7.11.13 subgroup CTE tuning. The second diagram shows the preferred settings in Margo Schulter's neoclassical tuning theory, where it is generated by {{nowrap|{~2, ~3, ~33/32}<nowiki/>}}, and tuned to MET-24.<br />
<br />
== Scales ==<br />
* [[Parapyth12]] – 12-tone Fokker block in 2.3.7.11.13 TOP tuning<br />
* [[Pepperoni7]] – 7-tone single chain of fifths in 271edo tuning<br />
* [[Pepperoni12]] – 12-tone single chain of fifths in 271edo tuning<br />
* [[MET-24]] – 24-tone double chain of fifths in 2048edo tuning<br />
<br />
== Tunings ==<br />
The most important tuning for parapyth is that given by MET-24 (''milder extended temperament''): <br />
* ~2/1 = 1200.000{{c}}, ~3/2 = 703.711{{c}}, ~33/32 = 57.422{{c}}. <br />
<br />
Another tuning derives from a 24-tone subset of George Secor's 29-HTT, thus a "24-HTT": <br />
* ~2/1 = 1200.000{{c}}, ~3/2 = 703.579{{c}}, ~33/32 = 58.090{{c}}. <br />
<br />
The fifth is in the 9th-secorian-comma tuning, which makes the augmented second of [[63/52]] pure. This fifth leads to an equal 3.247-cent error in [[9/8]] and 14/13 ({{nowrap| 63/52 {{=}} (9/8)⋅(14/13) }}) and thus a possible minimax tuning for the no-5 13-odd-limit. The minor third is extremely close to just 13/11, only off by 1/3 harmonisma. The spacer is determined such that [[7/4]] is pure. <br />
<br />
Yet another possible tuning is that given by [[Peppermint-24]]: <br />
* ~2/1 = 1200.000{{c}}, ~3/2 = 704.096{{c}}, ~33/32 = 58.680{{c}}. <br />
<br />
The fifth leads to [[step ratio]] φ for the [[5L 7s|chromatic scale]] and the spacer tunes the 7/6 pure. <br />
<br />
=== Edo tunings ===<br />
The parapyth edos below 311 that are not contorted in 2.3.7.11.13 are {{EDOs| 17, 22, 24, 29, 41, 46, 58, 63, 65, 80, 87, 104, 109, 121, 128, 133, 145, 150, 167, 172, 184, 191, 196, 213, 230, 232, 237, 254, 259, 271, 278, 283, and 295 }}.<br />
<br />
[[87edo]] is special for being the smallest "strict parapyth edo" (tempers out 352/351 and 364/363 and maps all of 121/120, 144/143, and 169/168 positively, meeting [[Margo Schulter]]'s criterion for "middle parapyth in the strict sense"). The following are strict parapyth edos below 311 that are not contorted in the 13-limit: {{Optimal ET sequence| 87, 104, 121, 128, 133, 145, 150, 167, 184, 191, 196, ''208'', 213, 230, 232, 237, 254, 259, 271, 278, 283, 295 }}. (Note: 208edo is contorted in 2.3.7.11.13 subgroup but not in the full 13-limit.)<br />
<br />
If we instead mean "parapyth" to refer to [[etypyth]] – its most elegant extension to the no-5's 17-limit (so we ignore [[100/99|S10]] and [[121/120|S11]]) – then the minimal strict etypyth (a.k.a. [[etypyth|17-limit parapyth]]) is [[46edo]], although this requires accepting its [[21/17]] as standing in for ~[[16/13]] and ~[[26/21]], corresponding roughly to (the [[octave complement]] of) [[acoustic phi]] so that stacking this interval gives a ~17:21:26:32 chord. The benefit of taking this no-5's interpretation is you do not deal with any conceptual issues arising from an out-of-tune [[15/13]] in 46edo, but you could deal with this alternately by interpreting simply only in the [[13-odd-limit]] adding odds 17, 21 and 23, which highlights that a benefit of 46edo is a fairly accurate [[23/16]] in the usual parapyth mapping of a tritone (C–F♯), tempering out {{nowrap| ([[23/16]])/[[729/512|(9/8)<sup>3</sup>]] {{=}} [[736/729]] }}. Alternatively, if you want a more accurate [[9/7]], [[7/6]], [[13/11]], [[104edo]] is an excellent etypyth tuning. 104edo is a dual-5 system that supports both the [[sensamagic]] (104) and [[pele]] (104c) mappings of 5, so that the combined [[25/16]] is very accurate (tempered together with the 81/52 (C–vG♯), distinguished from [[11/7]] (C–A♭) and [[14/9]] (C–^G) simultaneously). Pele may be preferable as a default due to it observing [[100/99|S10]] and [[121/120|S11]]. Sensamagic has the capacity to observe them too, but in the specific case of 104edo it tempers out S10.<br />
<br />
== See also ==<br />
* [[Leapday]] – a rank-2 reduction of parapyth with additional extensions for approximating harmonics 17 and 23<br />
<br />
== External links ==<br />
* [https://www.xenharmonikon.org/2022/07/15/met-24-a-milder-extended-temperament/ Xenharmonikon | ''MET-24: A Milder Extended Temperament''] by Margo Schulter<br />
* [https://www.bestii.com/~mschulter/tn101812-3degrees.txt ''A Friendly Introduction to "Rank-3" Temperaments: Designing a System with Three Degrees of Freedom''] by Margo Schulter<br />
* [https://www.bestii.com/~mschulter/met24-partage.txt ''The MET-24 temperament for Maqam music: Partitions or divisions of the apotome in context''] by Margo Schulter<br />
<br />
[[Category:Parapyth| ]] <!-- Main article --><br />
[[Category:Rank-3 temperaments]]<br />
[[Category:Pentacircle clan]]</div>ArrowHead294https://en.xen.wiki/index.php?title=2.3.5.7.13_subgroup&diff=2250212.3.5.7.13 subgroup2026-03-02T12:20:53Z<p>ArrowHead294: Fix this up a bit</p>
<hr />
<div>The '''2.3.5.7.13 subgroup''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1|2]], [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[13/1|13]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5, 7 and 13; this makes it a rank-5 system. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[7/4]], [[13/8]], [[13/7]], [[13/10]], [[39/32]], and so on.<br />
<br />
It can be thought out as an extension of the [[7-limit]] with a tridecimal xenharmonic touch, or as a retraction of the full 13-limit obtained by removing 11. It can be similar to the [[11-limit]], specially considering neutral interval pairs such as {{nowrap|39/32~11/9}} and {{nowrap|16/13~27/22}}, which are connected by the small comma of [[352/351]].<br />
<br />
The subgroup can be very easily rank-reduced into the 7-limit through the [[schismina]], an unnoticeable comma which connects ratios of 35 to 13, such that for example {{nowrap|[[36/35]]~[[1053/1024]]}}, or {{nowrap|[[45/32]]~[[128/91]]}}. The same can be said with the [[pontigailimma]], an atomic comma which is harder to visualize but entails significantly more accuracy. See article for comma equivalences.<br />
<br />
== Regular temperaments ==<br />
=== Rank-1 temperaments (edos) ===<br />
The 2.3.5.13 subgroup is relatively well approximated by the following edos (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs|'''7''', '''10''', 12, '''19''', '''53''', 72*, 130, 140, 171*, '''224''', 243, '''270''', '''441''', 494...}}<br />
<br />
<nowiki/>* Very accurate 7-limit but relatively innacurate prime 13.<br />
<br />
=== Rank-2 temperaments ===<br />
[[Catakleismic]] provides a low badness approximation to the subgroup, using a slightly sharp ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, 7/4 at +22 gens and ~13/8 at +14 gens. It is coarsely represented by [[19edo]], and well represented by [[53edo]] and [[72edo]], with [[125edo]] and [[197edo]] making for much better approximations.<br />
<br />
No-11 [[cassandra]] provides a more complex temperament using a [[chain of fifths]], well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. It is also decent in [[147edo]], though inconsistent. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (−8 fifths) [[8192/6561]], the doubly diminished octave 8388608/4782969 and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound very close to 5/4, 7/4, and 13/8 respectively. This is not so much a temperament as it is a relabelling of the 3-limit, which offers 5 and 7 and 13 with −1.954{{c}} and +3.804{{c}} and +1.428{{c}} of error respectively. <br />
<br />
Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at −16 gens, 7/4 at +25 gens, and 13/8 at −13 gens. [[Pontiac]] reaches 7/4 through +39 fifths, and 13/8 through −33 fifths, and it makes for a much better mapping, which is very well represented in [[171edo|171]] and [[224edo]].<br />
<br />
For those searching higher accuracy temperaments, [[Wizmic microtemperaments#Gariwizmic|Gariwizmic]] also keeps the chain of fifths, spliting the octave in half, but does not temper the schisma. It finds 5/4 at 39 fifths minus one semioctave, 7/4 at −14 fifths, and 13/8 at −27 fifths plus a semioctave. This is a much worse mapping, but it ends at [[270edo]], which is known for its astounding accuracy in the 13-limit.<br />
<br />
Another non-chain-of-fifths temperaments that converge in 270edo, and are thus great candidates for the 2.3.5.7.13 subgroup are [[buzzard]], [[cotoneum]], [[newt]], and [[ennealimmal]]. Ennealimmal is extremely accurate and well represented, as it can be naturally extended to the subgroup by adding the schismina, equating the [[36/35]] generator to the [[1053/1024]]. The pontigailimma is by extension tempered out too.<br />
<br />
=== Rank-3 temperaments ===<br />
{4375/4374, 4096/4095} ({{nowrap|270 & 441 & 935}}) is very accurate and has very low badness. As the pontigailimma is the difference between the ragisma and schismina, it is tempered out too.<br />
<br />
{[[1990656/1990625]], [[140625/140608]]}, the temperament that tempers the pontigailimma and the catasma, is also extremely accurate, orders of magnitude more than the last one.<br />
<br />
=== Rank-4 temperaments ===<br />
{[[1990656/1990625]]}, the temperament that tempers the pointigailimma alone is an unfathomably accurate nanotemperament, due to the extremely tiny size of the pontigailimma.</div>ArrowHead294https://en.xen.wiki/index.php?title=22edo&diff=22487822edo2026-02-27T16:36:58Z<p>ArrowHead294: /* Theory */</p>
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<div>{{interwiki<br />
| de = 22-EDO<br />
| en = 22edo<br />
| es = 22 EDO<br />
| ja = 22平均律<br />
}}<br />
{{Infobox ET}}<br />
{{Wikipedia|22 equal temperament}}<br />
{{ED intro}} Because it distinguishes [[10/9]] and [[9/8]], it is not a [[meantone]] system.<br />
<br />
== History ==<br />
The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}}. Inspired by the supposed division of the octave into 22 unequal parts in the [[Indian music|music theory of India]], Bosanquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy. In this he was followed in the twentieth century by theorist José Würschmidt, who noted it as a possible next step after [[19edo]], and {{w|James Murray Barbour|J. Murray Barbour}} in his classic survey of tuning history, ''Tuning and Temperament''.<br />
<br />
== Theory ==<br />
22edo is the third edo, after 12 and 19, which is capable of approximating the [[5-limit]] to within a [[Tenney–Euclidean temperament measures #TE error|Tenney–Euclidean error]] of 4 cents. Moreover, it does well beyond just the 5-limit; unlike 12 or 19, it is able to approximate the [[7-limit|7-]] and [[11-limit]] to within 3 cents of error, and in fact 22 is the smallest edo to represent the [[11-odd-limit]] [[consistent]]ly, though [[31edo]] is more accurate.<br />
<br />
Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' [[tempering out|temper out]] [[81/80]] (the syntonic comma), and instead maps it to one step. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory; yet it is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.<br />
<br />
22edo's approximation to the [[7/1|7th harmonic]] is about 13 cents sharp, somewhat similar to 12edo's approximation to the [[5/1|5th harmonic]]. Because of this and the sharp fifth, 22edo tempers out [[64/63]], equating the pythagorean minor seventh with [[7/4]], and [[support]]ing [[superpyth]]. In that manner, 22edo can be thought of as widening the gap of [[49/48]] between septimal intervals like [[7/6]] and [[8/7]] to a full quarter-tone. However, the opposite effect consequentially occurs in the 5-limit: while 5/4 and 6/5 are closer to JI than in 12edo, 5/4 is flat and 6/5 is sharp, resulting in [[25/24]] being narrowed to a quarter tone. An important reason for this contrast is that 22edo tempers out [[50/49]], so the [[7/5]] and [[10/7]] are equated to the 600{{c}} half-octave tritone, and 5/4 and 7/4 are separated by a semioctave, as well as 6/5 and [[12/7]]. Reasonably, [[36/35]] is also tempered to 1 step just like 25/24 and 49/48.<br />
<br />
22edo's approximation of the 11-limit is somewhat contentious: While it represents 11/8 well (about 5–6{{c}} flat) and maps 14/11 to a supermajor third (albeit inaccurately sharp), it lacks a [[neutral third]] dividing the perfect fifth in two, which means 11-limit harmony that is dependent upon neutral intervals does not work very well. This is partially because of its fifth, which is about 7{{c}} sharp, but also because 22edo's step is just short of being small enough to include 5 categories of seconds and thirds (subminor, minor, neutral, major, and supermajor, which [[24edo]], [[27edo]], and 31edo both include fully). Because 22edo does not contain "neutral" intervals, [[11/9]] is mapped to the same interval as 6/5 and [[12/11]] is mapped to the submajor second, inflating [[243/242]] to a full step.<br />
<br />
Since 22edo's fifth is sharp of just by approximately one quarter of the septimal comma ([[64/63]]), and since it tunes the septimal supermajor third ([[9/7]]) almost exactly just, it can be treated, for all practical purposes, as an extended "quarter-comma superpyth", in the same way that 31edo can be treated as an extended [[quarter-comma meantone]].<br />
<br />
22edo is also the third-smallest edo (after [[10edo]] and [[15edo]]) that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].<br />
<br />
=== Prime harmonics === <br />
{{Harmonics in equal|22}}<br />
<br />
=== As a tuning of other temperaments ===<br />
==== Observance of 81/80 ====<br />
22edo, unlike 12 and 19, is not a system of [[meantone]] temperament, and as such it distinguishes a number of [[3-limit]] and [[5-limit]] intervals that meantone tunings (most notably [[12edo]], [[19edo]], 31edo, and 43edo) do not distinguish, such as the two whole tones of 9/8 and 10/9. Indeed, these distinctions are significantly exaggerated in 22edo in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]], and [[53edo]], allowing many opportunities for alternate interpretations of these intervals. As a result of the observance of 81/80, the standard 5-limit diatonic scale does not collapse to the [[5L&nbsp;2s]] [[mos]] as in meantone systems. Instead, it is a ternary scale, having the [[nicetone]] pattern.<br />
<br />
==== Superpyth temperament ====<br />
The 5L&nbsp;2s diatonic (LLsLLLs) in 22edo is instead derived from [[superpyth]] temperament. Despite having the same melodic structure as meantone's diatonic scale, 22edo's diatonic mos has subminor and supermajor thirds of 7/6 and 9/7, rather than classical minor and major thirds of 6/5 and 5/4. This means that the septimal comma 64/63 is tempered out rather than the syntonic comma of 81/80, which one of 22et's core features. <br />
<br />
Superpyth temperament equates the Pythagorean sevenths (such as A–G and C–B♭ in [[chain-of-fifths notation]]) to ''harmonic'' sevenths instead of 5-limit minor sevenths (approximating [[7/4]] instead of [[9/5]]). Due to the sharper fifths, the diatonic scale is more uneven than in meantone systems and 12edo. In addition to the more uneven diatonic scale, 22edo has a quasi-equal pentatonic scale (the major whole tone and subminor third are rather close in size). The step patterns of the pentatonic and diatonic scales in 22et are {{dash|4, 4, 5, 4, 5}} and {{dash|4, 4, 1, 4, 4, 4, 1}} respectively. In superpyth (and thus in 22edo and technically 12edo), the [[36:45:54:64|1–5/4–3/2–16/9]] dominant seventh chord and an otonal tetrad are represented by the same chord.<br />
<br />
==== Porcupine temperament ====<br />
22edo additionally tempers out the porcupine comma or maximal diesis of [[250/243]] ([[S-expression|S10<sup>2</sup>⋅S11]]), which means that 22edo [[support]]s [[porcupine]] temperament. The generator for porcupine is a very flat minor whole tone of ~[[10/9]] (usually tuned slightly flat of [[11/10]]), two of which is a sharp ~[[6/5]], and three of which is a slightly flat ~[[4/3]], implying the existence of an equal-step tetrachord, which is characteristic of porcupine. <br />
<br />
Porcupine temperament allows the 5-limit diatonic scale (the [[zarlino]] scale), present as {{nowrap|{{dash|4, 3, 2, 4, 3, 4, 2}}}} and tuned particularly accurately in 22edo, to be notated with only 1 set of accidentals (conventionally sharps and flats) representing both the syntonic comma and the classical chromatic semitone, as the difference between them (250/243) is tempered out.<br />
<br />
It can be observed that the tuning damage that porcupine tempering implies (the ones just described) is highly characteristic of the tuning properties of 22edo and as such represents one excellent point of departure for examining the harmonic properties of 22edo. Porcupine's generator forms mos scales of 7 and 8, which in 22edo are tuned respectively as {{dash|4, 3, 3, 3, 3, 3, 3}} and {{dash|1, 3, 3, 3, 3, 3, 3, 3}} (and their respective modes).<br />
<br />
==== Pajara temperament ====<br />
A third important temperament that 22edo supports is [[pajara]]. In the 5-limit, [[2048/2025]] (diaschisma) is tempered out, meaning that the 5-limit tritones are equated to one another and to the [[semioctave]]. This means that 3/2 is a semioctave away from 16/15, and 5/4 is a semioctave away from 16/9. In the 7-limit, [[50/49]] (jubilisma) is tempered out, meaning that the tritones [[7/5]] and [[10/7]] are equated to the semioctave, and consequently 64/63 is tempered out as in superpyth—5/4 is a semioctave away from 7/4. Since 50/49 is tempered out, the 25/24 and 49/48 intervals are equated to a single interval, and it functions as a chroma in the [[2L&nbsp;8s]] mos. This suggests the use of a decatonic notation system, where 7/6 and 8/7 are the same number of scale degrees, and 7/4 is a major interval. Thus the [[4:5:6:7|1–5/4–3/2–7/4]] major tetrad has 5/4 and 7/4 as major intervals, and replacing them with the corresponding minor intervals gives us the [[70:84:105:120|1–6/5–3/2–12/7]] subharmonic sixth chord or minor tetrad. Pajara temperament is also supported by [[12edo]], as it also tempers out 50/49 and 64/63.<br />
<br />
The decatonic scales of pajara have been considered by many to be a system in the 7-limit analogous to the diatonic scale of meantone temperament in the 5-limit, as described in Paul Erlich's paper [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf Tuning, Tonality and 22-Tone Temperament].<br />
<br />
==== Additional commas ====<br />
Both 22edo and 12edo also temper out {{nowrap|(50/49)/(64/63) {{=}} 225/224}} ({{S|15}}, [[marvel comma]]), so that the marvel augmented triad is a chord of 22et. A 7-limit comma not tempered out by 12et which 22et does temper out is [[1728/1715]], the orwell comma; therefore, the [[orwell tetrad]] is also a chord of 22et. The [[orwell]] temperament uses the septimal subminor third (5 degrees) as a generator, and forms mos scales with step patterns {{dash|2, 3, 2, 3, 2, 3, 2, 3, 2}} and {{dash|2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2}}. While orwell can be tuned more accurately in other temperaments, such as [[31edo]], [[53edo]], and [[84edo]], 22edo has a leg-up on the others melodically, as the large and small steps of Orwell[9] are easier to distinguish.<br />
<br />
=== Subsets, supersets, and inheritances ===<br />
As 22 is divisible by 11, a 22edo instrument can play any music in [[11edo]], in the same way that [[12edo]] can play [[6edo]] (the whole tone scale). 11edo is interesting for sounding melodically very similar to 12edo (whole steps, half steps and minor thirds in the familiar 1:2:3 ratio), but harmonically very different, in particular because it lacks perfect fifths/fourths and 5-limit major thirds/minor sixths. Similarly, 22edo is melodically similar to [[24edo]] as both contain quartertones and minor, neutral, and major seconds; but 22edo offers much better all-around harmonies than 24. In particular, 22edo can be roughly conceptualized as 24 but with only two types of thirds rather than three. In [[Sagittal notation]], 11 can be notated as every other note of 22.<br />
<br />
22 inherits 11edo's [[11/8]] and [[7/4]], and inherits [[2edo]]'s tritone, which is mapped in both systems to [[7/5]] and [[10/7]].<br />
<br />
=== Other features ===<br />
The 163.6{{c}} "flat minor whole tone" or "submajor second" is a key interval in 22edo, in part because it functions as no less than three different consonant ratios in the 11-limit: 10/9, 11/10, and 12/11. It is thus extremely ambiguous and flexible. The trade-off is that it is very much in the cracks of the 12-equal piano, and so for most 12-equal listeners, it takes some getting used to. Simple translations of 5-limit music into 22edo can sound very different, with a more complex harmonic quality inevitably arising. 22edo does not contain a neutral third, but both the 5-limit thirds have a "neutral-like" quality since they are tempered closer together rather than farther apart as in 12edo.<br />
<br />
=== Higher-limit interpretations ===<br />
22edo can also be treated as adding harmonics 3 and 5 to [[11edo]]'s 2.9.15.7.11.17 subgroup, making it a rather accurate 2.3.5.7.11.17 [[subgroup]] temperament. Also note that its approximation of the 31st harmonic is within half a cent, which is very accurate. It also approximates some intervals involving the 29th harmonic well, especially 29/24, which is also matched within half a cent. This leaves us with the 2.3.5.7.11.17.29.31 subgroup.<br />
<br />
== Intervals ==<br />
{{See also|22edo solfege}}<br />
{{See also|SKULO interval names#Alternatives}}<br />
<br />
{| class="wikitable center-all right-2 left-3"<br />
|-<br />
! Degree<br />
! Cents<br />
! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.7.11.17 subgroup}}</ref><br />
! colspan="3" | [[Ups and downs notation|Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and ^^d2)<br />
! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(K {{=}} 1)}}<br />
! Audio<br />
|-<br />
| 0<br />
| 0.0<br />
| [[1/1]]<br />
| perfect unison<br />
| P1<br />
| D<br />
| perfect unison<br />
| P1<br />
| D<br />
| [[File:0-0.000c_P1.mp3]]<br />
|-<br />
| 1<br />
| 54.5<br />
| [[36/35]], [[34/33]], [[33/32]], [[32/31]]<br />
| up-unison, minor 2nd<br />
| ^1, m2<br />
| ^D, Eb<br />
| comma-wide unison, minor 2nd<br />
| K1, m2<br />
| KD, Eb<br />
| [[File:0-54.545c_22edo.mp3]]<br />
|-<br />
| 2<br />
| 109.1<br />
| [[18/17]], [[17/16]], [[16/15]], [[15/14]]<br />
| downaug 1sn, upminor 2nd<br />
| vA1, ^m2<br />
| vD#, ^Eb<br />
| classic minor 2nd<br />
| Km2<br />
| KEb<br />
| [[File:0-109.091c_11edo.mp3]]<br />
|-<br />
| 3<br />
| 163.6<br />
| [[12/11]], [[11/10]], [[10/9]]<br />
| aug 1sn, downmajor 2nd<br />
| A1, vM2<br />
| D#, vE<br />
| classic/comma-narrow major 2nd<br />
| kM2<br />
| kE<br />
| [[File:0-163.636c_22edo.mp3]]<br />
|-<br />
| 4<br />
| 218.2<br />
| [[9/8]], [[17/15]], [[8/7]]<br />
| major 2nd<br />
| M2<br />
| E<br />
| major 2nd<br />
| M2<br />
| E<br />
| [[File:0-218.182c_11edo.mp3]]<br />
|-<br />
| 5<br />
| 272.7<br />
| [[20/17]], [[7/6]]<br />
| minor 3rd<br />
| m3<br />
| F<br />
| minor 3rd<br />
| m3<br />
| F<br />
| [[File:0-272.727c_22edo.mp3]]<br />
|-<br />
| 6<br />
| 327.3<br />
| [[6/5]], [[17/14]], [[11/9]]<br />
| upminor 3rd<br />
| ^m3<br />
| ^F<br />
| classic minor 3rd<br />
| Km3<br />
| KF<br />
| [[File:0-327.273c_11edo.mp3]]<br />
|-<br />
| 7<br />
| 381.8<br />
| [[5/4]], [[96/77]]<br />
| downmajor 3rd<br />
| vM3<br />
| vF#<br />
| classic major 3rd<br />
| kM3<br />
| kF#<br />
| [[File:0-381.818c_22edo.mp3]]<br />
|-<br />
| 8<br />
| 436.4<br />
| [[14/11]], [[9/7]], [[22/17]]<br />
| major 3rd<br />
| M3<br />
| F#<br />
| major 3rd<br />
| M3<br />
| F#<br />
| [[File:0-436.364c_11edo.mp3]]<br />
|-<br />
| 9<br />
| 490.9<br />
| [[4/3]]<br />
| perfect 4th<br />
| P4<br />
| G<br />
| perfect 4th<br />
| P4<br />
| G<br />
| [[File:0-490.909c_22edo.mp3]]<br />
|-<br />
| 10<br />
| 545.5<br />
| [[15/11]], [[11/8]]<br />
| up-4th, dim 5th<br />
| ^4, d5<br />
| ^G, Ab<br />
| comma-wide 4th<br />
| K4<br />
| KG<br />
| [[File:0-545.455c_11edo.mp3]]<br />
|-<br />
| 11<br />
| 600.0<br />
| [[7/5]], [[24/17]], [[17/12]], [[10/7]]<br />
| downaug 4th, updim 5th<br />
| vA4, ^d5<br />
| vG#, ^Ab<br />
| comma-narrow augmented 4th<br />comma-wide diminished 5th<br />
| kA4<br />Kd5<br />
| kG#, KAb<br />
| [[File:0-600.000c_2edo.mp3]]<br />
|-<br />
| 12<br />
| 654.5<br />
| [[16/11]], [[22/15]]<br />
| aug 4th, down-5th<br />
| A4, v5<br />
| G#, vA<br />
| comma-narrow 5th<br />
| k5<br />
| kA<br />
| [[File:0-654.545c_11edo.mp3]]<br />
|-<br />
| 13<br />
| 709.1<br />
| [[3/2]]<br />
| perfect 5th<br />
| P5<br />
| A<br />
| perfect 5th<br />
| P5<br />
| A<br />
| [[File:0-709.091c_22edo.mp3]]<br />
|-<br />
| 14<br />
| 763.6<br />
| [[17/11]], [[14/9]], [[11/7]]<br />
| minor 6th<br />
| m6<br />
| Bb<br />
| minor 6th<br />
| m6<br />
| Bb<br />
| [[File:0-763.636c_11edo.mp3]]<br />
|-<br />
| 15<br />
| 818.2<br />
| [[8/5]], [[77/48]]<br />
| upminor 6th<br />
| ^m6<br />
| ^Bb<br />
| classic minor 6th<br />
| Km6<br />
| KBb<br />
| [[File:0-818.182c_22edo.mp3]]<br />
|-<br />
| 16<br />
| 872.7<br />
| [[18/11]], [[28/17]], [[5/3]]<br />
| downmajor 6th<br />
| vM6<br />
| vB<br />
| classic major 6th<br />
| kM6<br />
| kB<br />
| [[File:0-872.727c_11edo.mp3]]<br />
|-<br />
| 17<br />
| 927.3<br />
| [[17/10]], [[12/7]]<br />
| major 6th<br />
| M6<br />
| B<br />
| major 6th<br />
| M6<br />
| B<br />
| [[File:0-927.273c_22edo.mp3]]<br />
|-<br />
| 18<br />
| 981.8<br />
| [[7/4]], [[30/17]], [[16/9]]<br />
| minor 7th<br />
| m7<br />
| C<br />
| minor 7th<br />
| m7<br />
| C<br />
| [[File:0-981.818c_11edo.mp3]]<br />
|-<br />
| 19<br />
| 1036.4<br />
| [[9/5]], [[11/6]], [[20/11]]<br />
| upminor 7th, dim 8ve<br />
| ^m7, d8<br />
| ^C, Db<br />
| classic minor 7th<br />
| Km7<br />
| kC<br />
| [[File:0-1036.364c_22edo.mp3]]<br />
|-<br />
| 20<br />
| 1090.9<br />
| [[28/15]], [[15/8]], [[32/17]], [[17/9]]<br />
| downmajor 7th, updim 8ve<br />
| vM7, ^d8<br />
| vC#, ^Db<br />
| classic major 7th<br />
| kM7<br />
| kC#<br />
| [[File:0-1090.909c_11edo.mp3]]<br />
|-<br />
| 21<br />
| 1145.5<br />
| [[31/16]], [[64/33]], [[33/17]], [[35/18]]<br />
| major 7th, down 8ve<br />
| M7, v8<br />
| C#, vD<br />
| major 7th / comma-narrow 8ve<br />
| M7 / k8<br />
| C#, kD<br />
| [[File:0-1145.455c_22edo.mp3]]<br />
|-<br />
| 22<br />
| 1200.0<br />
| [[2/1]]<br />
| perfect octave<br />
| P8<br />
| D<br />
| perfect 8ve<br />
| P8<br />
| D<br />
| [[File:0-1200.000c_P8.mp3]]<br />
|}<br />
<br />
== Notation ==<br />
=== Ups and downs notation ===<br />
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.<br />
{{sharpness-sharp3a}}<br />
<br />
Standard Pythagorean [[chain-of-fifths notation]] can be used alongside ups (^) and downs (v), where a single up or down alters the pitch of a note by 1 EDOstep (1\22). Note that E&#x266D; and D&#x266F; are different notes and that E&#x266D; is significantly lower in pitch than D&#x266F;.<br />
<br />
{| class="wikitable right-1 right-2 center-3 center-4"<br />
|+ style="font-size: 105%;" | Notation of 22edo<br />
|-<br />
! rowspan="2" | [[Degree]]<br />
! rowspan="2" | [[Cent]]s<br />
! colspan="2" | [[Ups and downs notation|Ups and downs notation]]<br />
|-<br />
! [[5L 2s|Diatonic Interval Names]]<br />
! Note Names<br />
|-<br />
| 0<br />
| 0.0<br />
| '''Perfect unison (P1)'''<br />
| '''D'''<br />
|-<br />
| 1<br />
| 54.5<br />
| Minor second (m2)<br />Up unison (^1)<br />
| Eb<br />^D<br />
|-<br />
| 2<br />
| 109.1<br />
| Upminor second (^m2)<br />Downaugmented unison (vA1)<br />Diminished third (d3)<br />
| ^Eb<br />vD#<br />Fb<br />
|-<br />
| 3<br />
| 163.6<br />
| Downmajor second (vM2)<br />Augmented unison (A1)<br />
| vE<br />D#<br />
|-<br />
| 4<br />
| 218.2<br />
| '''Major second (M2)'''<br />Upaugmented unison (^A1)<br />Downminor third (vm3)<br />
| '''E'''<br />^D#<br />vF<br />
|-<br />
| 5<br />
| 272.7<br />
| Upmajor second (^M2)<br />'''Minor third (m3)'''<br />
| ^E<br />'''F'''<br />
|-<br />
| 6<br />
| 327.3<br />
| '''Upminor third (^m3)'''<br />Diminished fourth (d4)<br />
| '''^F'''<br />Gb<br />
|-<br />
| 7<br />
| 381.8<br />
| '''Downmajor third (vM3)'''<br />Augmented second (A2)<br />Updiminished fourth (^d4)<br />
| '''vF#'''<br />E#<br />^Gb<br />
|-<br />
| 8<br />
| 436.4<br />
| '''Major third (M3)'''<br />Upaugmented second (^A2)<br />Down fourth (v4)<br />
| '''F#'''<br />^E#<br />vG<br />
|-<br />
| 9<br />
| 490.9<br />
| '''Perfect fourth (P4)'''<br />
| '''G'''<br />
|-<br />
| 10<br />
| 545.5<br />
| Up fourth (^4)<br />Diminished fifth (d5)<br />
| ^G<br />Ab<br />
|-<br />
| 11<br />
| 600.0<br />
| Downaugmented fourth (vA4)<br />Updiminished fifth (^d5)<br />
| vG#<br />^Ab<br />
|-<br />
| 12<br />
| 654.5<br />
| Augmented fourth (A4)<br />Down fifth (v5)<br />
| G#<br />vA<br />
|-<br />
| 13<br />
| 709.1<br />
| '''Perfect fifth (P5)'''<br />
| '''A'''<br />
|-<br />
| 14<br />
| 763.6<br />
| Up fifth (^5)<br />Minor sixth (m6)<br />
| ^A<br />Bb<br />
|-<br />
| 15<br />
| 818.2<br />
| Downaugmented fifth (vA5)<br />Upminor sixth (^m6)<br />
| vA#<br />^Bb<br />
|-<br />
| 16<br />
| 872.7<br />
| Augmented fifth (A5)<br />'''Downmajor sixth (vM6)'''<br />
| A#<br />'''vB'''<br />
|-<br />
| 17<br />
| 927.3<br />
| '''Major sixth (M6)'''<br />Upaugmented fifth (^A5)<br />Downminor seventh (vm7)<br />
| '''B'''<br />^A#<br />vC<br />
|-<br />
| 18<br />
| 981.8<br />
| '''Minor seventh (m7)'''<br />Upmajor sixth (^M6)<br />Downdiminished octave (vd8)<br />
| '''C'''<br />^B<br />vDb<br />
|-<br />
| 19<br />
| 1036.4<br />
| '''Upminor seventh (^m7)'''<br />Diminished octave (d8)<br />
| '''^C'''<br />Db<br />
|-<br />
| 20<br />
| 1090.9<br />
| Downmajor seventh (vM7)<br />Updiminished octave (^d8)<br />Augmented sixth (A6)<br />
| vC#<br />^Db<br />B#<br />
|-<br />
| 21<br />
| 1145.5<br />
| Major seventh (M7)<br />Down octave (v8)<br />
| C#<br />vD<br />
|-<br />
| 22<br />
| 1200.0<br />
| '''Perfect octave (P8)'''<br />
| '''D'''<br />
|}<br />
<br />
Treating [[Ups and downs notation|ups and downs]] as "fused" with sharps and flats, and never appearing separately:<br />
<br />
[[File:Tibia_22edo_ups_and_downs_guide_1.png|alt=Tibia 22edo ups and downs guide 1.png|800x147px|Tibia 22edo ups and downs guide 1.png]]<br />
<br />
Treating ups and downs as independent of sharps and flats, and sometimes appearing separately:<br />
<br />
[[File:Tibia_22edo_ups_and_downs_guide_2.png|alt=Tibia 22edo ups and downs guide 2.png|800x150px|Tibia 22edo ups and downs guide 2.png]]<br />
<br />
A D downmajor scale with mandatory accidentals (no key signature), with minimal accidentals (only when needed to override the key signature), and with independent ups and downs.<br />
<br />
[[File:Tibia_22edo_guide_D_major.png|alt=Tibia 22edo guide D major.png|800x68px|Tibia 22edo guide D major.png]]<br />
<br />
Alternatively, arrow accidentals from [[Helmholtz–Ellis notation]] can be used instead of independent ups and downs:<br />
<br />
{{Sharpness-sharp3}}<br />
<br />
If arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with double arrows.<br />
<br />
Shown below is [[Paul Erlich]]'s "Tibia" in G, with independent ups and downs.<br />
<br />
<gallery mode="slideshow"><br />
File:Tibia in G CORRECTED-1.png|alt=Tibia in G CORRECTED-1.png|Tibia in G (page 1)<br />
File:Tibia in G CORRECTED-2.png|alt=Tibia in G CORRECTED-2.png|Tibia in G (page 2)<br />
</gallery><br />
<br />
=== Sagittal notation ===<br />
This notation uses the same sagittal sequence as EDOs [[15edo#Sagittal notation|15]] and [[29edo#Sagittal notation|29]], is a subset of the notations for EDOs [[44edo#Sagittal notation|44]] and [[66edo#Sagittal notation|66]], and is a superset of the notation for [[11edo#Sagittal notation|11-EDO]].<br />
<br />
==== Evo flavor ====<br />
{{Sagittal chart|Evo}}<br />
<br />
==== Revo flavor ====<br />
{{Sagittal chart}}<br />
<br />
When 22edo is treated as generated by a cycle of its fifths, the natural notes {{nowrap|F C G D A E B}} represent a chain of those 13\22 fifths; consequently, the whole tone comes out to four degrees and the apotome (Pythagorean sharp/flat) comes out to three degrees. Three pairs of sagittal symbols, dividing that apotome into three parts, are all that is necessary, and offer plenty of enharmonic equivalents:<br />
<br />
[[File:22edo.png|alt=22edo.png|22edo.png]]<br />
<br />
This notation is consistent with Sagittal's notation of 5-limit JI harmony: "major" 3rds and 6ths appear as (super)pythagorean intervals flattened by a syntonic comma.<br />
<br />
The division of the apotome into three syntonic commas also indicates 22's tempering out of the [[250/243|porcupine comma]] (which is equivalent to three syntonic commas minus a Pythagorean apotome).<br />
<br />
We also have, from the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], this diagram of how to notate 22-EDO in the Revo flavor of Sagittal:<br />
<br />
[[File:22edo Sagittal.png|800px]]<br />
<br />
=== Superpyth/Porcupine notation ===<br />
Superpyth/Porcupine notation is a system arising from both superpyth and porcupine temperament. It categorizes each 22edo interval as major and minor of one or both of those temperaments. s indicates superpyth and p indicates porcupine. Because p now represents porcupine and not perfect, P in perfect intervals is no longer used in this system. Instead the number is used without P and is read as either just the number or "Natural". Example: P5 becomes 5 or N5 = Perfect fifth becomes Natural fifth.<br />
<br />
=== Porcupine notation ===<br />
Porcupine notation uses the porcupine generator to generate the notation as well. The 2nd and 7th are perfect, and the 4th and 5th are imperfect like the 3rd and 6th. The natural notes represent a chain of 2nds ABCDEFG. This is the only way to use a heptatonic notation without additional accidentals.<br />
<br />
The keyboard runs {{nowrap|D * * E * * F * * G * * * A * * B * * C * * D}}.<br />
<br />
A score video demonstrating this type of notation using redefined sharp and flat symbols is available: [https://www.youtube.com/watch?v=se79rdp705Y ''Study #1 in Porcupine Temperament: "Flying Straight Down" (Microtonal/Xenharmonic)''] (2020) by [[John Moriarty]]. Note that the sharp of one note is lower than the flat of the next note, in contrast to sharps and flats in the diatonic notation with ups and downs described above.<br />
<br />
=== Pentatonic notation ===<br />
In Pentatonic notation, the degrees are unison, subthird, fourthoid, fifthoid, subseventh and octoid. The natural notes represent a chain of 5ths FCGDA. This is the only way to use a chain-of-fifths notation without additional accidentals. <br />
<br />
The keyboard runs {{nowrap|D * * * * F * * * G * * * A * * * * C * * * D}}.<br />
<br />
=== Decatonic notation ===<br />
The Decatonic notation is based on Paul Erlich's decatonic scales. Unlike typical notation, the decatonic system is based on a scale of 10 tones rather than 7. This approach requires an entire re-learning of chords, intervals, and notation, but it allows 22EDO to be notated using only one pair of accidentals, and gives the opportunity to escape a heptatonic thinking pattern. The system is based on two chains of fifths: one represented by Latin letters, the other by Greek. The two chains can be looked at as two juxtaposed pentatonic scales.<br />
<br />
Chain 1: {{nowrap|C G D A E}}<br />
<br />
Chain 2: {{nowrap|γ δ α ε β}}<br />
<br />
The alphabet is, in ascending order: {{nowrap|C δ D ε E γ G α A β C}}<br />
<br />
In this alphabet, a chain of fifths is preserved because equivalent Greek letters also represent fifths if they are the same as their Latin counterparts. For example G&ndash;D is a fifth, and so is γ&ndash;δ.<br />
<br />
=== Comparison of 22edo notation systems ===<br />
{| class="wikitable center-all right-2 mw-collapsible mw-collapsed"<br />
|-<br />
! [[Degree]]<br />
! [[Cent]]s<br />
! colspan="2" | Superpyth/Porcupine <br />
! colspan="3" | Porcupine (Onyx)<br />
! colspan="3" |Porcupine (Zarlino)<br />
! colspan="3" | Pentatonic<br />
! colspan="3" | Decatonic<br />
! colspan="3" | [[Ups and downs notation|Ups and Downs]]<br />
! colspan="3" | [[SKULO interval names]]<br />
|-<br />
| 0<br />
| 0<br />
| Natural Unison<br />
| 1<br />
| perfect unison<br />
| P1<br />
| D<br />
|perfect unison<br />
|P1<br />
|C<br />
| perfect unison<br />
| P1<br />
| D<br />
| natural 1st<br />
| N1<br />
| C<br />
| perfect unison<br />
| P1<br />
| D<br />
| perfect unison<br />
| P1<br />
| D<br />
|-<br />
| 1<br />
| 55<br />
| s-minor second<br />
| sm2<br />
| aug unison<br />
| A1<br />
| D#<br />
|augmented unison<br />
|A1<br />
|C#<br />
| aug unison<br />
| A1<br />
| D#<br />
| flat 2nd<br />
| f2<br />
| C#, δb<br />
| up-unison, minor 2nd<br />
| ^1, m2<br />
| ^D, Eb<br />
| comma-wide unison, minor 2nd<br />
| K1, m2<br />
| KD, Eb<br />
|-<br />
| 2<br />
| 109<br />
| p-diminished second<br />
| pd2<br />
| dim 2nd<br />
| d2<br />
| Eb<br />
|minor second<br />
|m2<br />
|Db<br />
| double-aug unison,<br />double-dim sub3rd<br />
| AA1,<br />dds3<br />
| Dx,<br />Fb<span style="vertical-align: super;">3</span><br />
| natural 2nd<br />
| N2<br />
| δ<br />
| downaug 1sn, upminor 2nd<br />
| vA1, ^m2<br />
| vD#, ^Eb<br />
| classic minor 2nd<br />
| Km2<br />
| KEb<br />
|-<br />
| 3<br />
| 164<br />
| p-minor second<br />
| pm2<br />
| perfect 2nd<br />
| P2<br />
| E<br />
|narrow major second<br />
|nM2<br />
|D<br />
| dim sub3rd<br />
| ds3<br />
| Fbb<br />
| sharp 2nd, flat 3rd<br />
| s2, f3<br />
| δ#, Db<br />
| aug 1sn, downmajor 2nd<br />
| A1, vM2<br />
| D#, vE<br />
| classic/comma-narrow major 2nd<br />
| kM2<br />
| kE<br />
|-<br />
| 4<br />
| 218<br />
| (s/p) Major second<br />
| M2<br />
| aug 2nd<br />
| A2<br />
| E#<br />
|wide major second<br />
|WM2<br />
|D#<br />
| minor sub3rd<br />
| ms3<br />
| Fb<br />
| natural 3rd<br />
| N3<br />
| D<br />
| major 2nd<br />
| M2<br />
| E<br />
| major 2nd<br />
| M2<br />
| E<br />
|-<br />
| 5<br />
| 273<br />
| s-minor third<br />
| sm3<br />
| dim 3rd<br />
| d3<br />
| Fb<br />
|wolf third<br />
|w3<br />
|Ebb<br />
| major sub3rd<br />
| Ms3<br />
| F<br />
| sharp 3rd<br />
| s3<br />
| D#<br />
| minor 3rd<br />
| m3<br />
| F<br />
| minor 3rd<br />
| m3<br />
| F<br />
|-<br />
| 6<br />
| 327<br />
| p-minor third<br />
| pm3<br />
| minor 3rd<br />
| m3<br />
| F<br />
|minor third<br />
|m3<br />
|Eb<br />
| aug sub3rd<br />
| As3<br />
| F#<br />
| flat 4th<br />
| f4<br />
| εb<br />
| upminor 3rd<br />
| ^m3<br />
| ^F<br />
| classic minor 3rd<br />
| Km3<br />
| KF<br />
|-<br />
| 7<br />
| 382<br />
| p-Major third<br />
| pM3<br />
| major 3rd<br />
| M3<br />
| F#<br />
|major third<br />
|M3<br />
|E<br />
| double-aug sub3rd,<br />double-dim 4thoid<br />
| AAs3,<br />dd4d<br />
| Fx,<br />Gbb<br />
| natural 4th<br />
| N4<br />
| ε<br />
| downmajor 3rd<br />
| vM3<br />
| vF#<br />
| classic major 3rd<br />
| kM3<br />
| kF#<br />
|-<br />
| 8<br />
| 436<br />
| s-Major third<br />
| sM3<br />
| aug 3rd, dim 4th<br />
| A3, d4<br />
| Fx, Gb<br />
|augmented third<br />
|A3<br />
|E#<br />
| dim 4thoid<br />
| d4d<br />
| Gb<br />
| sharp 4th, flat 5th<br />
| s4, f5<br />
| ε#, Eb<br />
| major 3rd<br />
| M3<br />
| F#<br />
| major 3rd<br />
| M3<br />
| F#<br />
|-<br />
| 9<br />
| 491<br />
| Natural Fourth<br />
| 4, N4<br />
| minor 4th<br />
| m4<br />
| G<br />
|perfect fourth<br />
|P4<br />
|F<br />
| perfect 4thoid<br />
| P4d<br />
| G<br />
| natural 5th<br />
| N5<br />
| E<br />
| perfect 4th<br />
| P4<br />
| G<br />
| perfect 4th<br />
| P4<br />
| G<br />
|-<br />
| 10<br />
| 545<br />
| p-Major fourth, s-dim fifth<br />
| pM4, sd5<br />
| major 4th<br />
| M4<br />
| G#<br />
|wolf fourth<br />
|w4<br />
|F#<br />
| aug 4thoid<br />
| A4d<br />
| G#<br />
| sharp 5th, flat 6th<br />
| s5, f6<br />
| E#, γb<br />
| up-4th, dim 5th<br />
| ^4, d5<br />
| ^G, Ab<br />
| comma-wide 4th<br />
| K4<br />
| KG<br />
|-<br />
| 11<br />
| 600<br />
| p-Augmented Fourth,<br />p-diminished Fifth,<br />Half-Octave<br />
| A4, HO<br />
| aug 4th, <br />dim 5th<br />
| A4, d5<br />
| Gx, <br />Abb<br />
|augmented fourth, diminished fifth<br />
|A4, d5<br />
|F##, Gbb<br />
| double-aug 4thoid,<br />double-dim 5thoid<br />
| AA4d, <br />dd5d<br />
| Gx, <br />Abb<br />
| natural 6th<br />
| N6<br />
| γ<br />
| downaug 4th, updim 5th<br />
| vA4, ^d5<br />
| vG#, ^Ab<br />
| comma-narrow augmented 4th<br />comma-wide diminished 5th<br />
| kA4<br />Kd5<br />
| kG#, KAb<br />
|-<br />
| 12<br />
| 655<br />
| p-minor Fifth, s-aug Fourth<br />
| pm5, sA4<br />
| minor 5th<br />
| m5<br />
| Ab<br />
|wolf fifth<br />
|w5<br />
|Gb<br />
| dim 5thoid<br />
| d5d<br />
| Ab<br />
| sharp 6th, flat 7th<br />
| s6, f7<br />
| γ#, Gb<br />
| aug 4th, down-5th<br />
| A4, v5<br />
| G#, vA<br />
| comma-narrow 5th<br />
| k5<br />
| kA<br />
|-<br />
| 13<br />
| 709<br />
| Natural Fifth<br />
| 5, N5<br />
| major 5th<br />
| M5<br />
| A<br />
|perfect fifth<br />
|P5<br />
|G<br />
| perfect 5thoid<br />
| P5d<br />
| A<br />
| natural 7th<br />
| N7<br />
| G<br />
| perfect 5th<br />
| P5<br />
| A<br />
| perfect 5th<br />
| P5<br />
| A<br />
|-<br />
| 14<br />
| 764<br />
| s-minor sixth<br />
| sm6<br />
| aug 5th, dim 6th<br />
| A5, d6<br />
| A#, Bbb<br />
|diminished sixth<br />
|d6<br />
|Abb<br />
| aug 5thoid<br />
| A5d<br />
| A#<br />
| sharp 7th<br />
| s7<br />
| G#<br />
| minor 6th<br />
| m6<br />
| Bb<br />
| minor 6th<br />
| m6<br />
| Bb<br />
|-<br />
| 15<br />
| 818<br />
| p-minor sixth<br />
| pm6<br />
| minor 6th<br />
| m6<br />
| Bb<br />
|minor sixth<br />
|m6<br />
|Ab<br />
| double-aug 5thoid,<br />double-dim sub7th<br />
| AA5d,<br />dds7<br />
| Ax,<br />Cb<span style="vertical-align: super;">3</span><br />
| flat 8th<br />
| f8<br />
| αb<br />
| upminor 6th<br />
| ^m6<br />
| ^Bb<br />
| classic minor 6th<br />
| Km6<br />
| KBb<br />
|-<br />
| 16<br />
| 873<br />
| p-Major sixth<br />
| pM6<br />
| major 6th<br />
| M6<br />
| B<br />
|major sixth<br />
|M6<br />
|A<br />
| dim sub7th<br />
| ds7<br />
| Cbb<br />
| natural 8th<br />
| N8<br />
| α<br />
| downmajor 6th<br />
| vM6<br />
| vB<br />
| classic major 6th<br />
| kM6<br />
| kB<br />
|-<br />
| 17<br />
| 927<br />
| s-Major sixth<br />
| sM6<br />
| aug 6th<br />
| A6<br />
| B#<br />
|wolf sixth<br />
|w6<br />
|A#<br />
| minor sub7th<br />
| ms7<br />
| Cb<br />
| sharp 8th, flat 9th<br />
| s8, f9<br />
| α#, Ab<br />
| major 6th<br />
| M6<br />
| B<br />
| major 6th<br />
| M6<br />
| B<br />
|-<br />
| 18<br />
| 982<br />
| (s/p) minor seventh<br />
| m7<br />
| dim 7th<br />
| d7<br />
| Cb<br />
|narrow minor seventh<br />
|nm7<br />
|Bbb<br />
| major sub7th<br />
| Ms7<br />
| C<br />
| natural 9th<br />
| N9<br />
| A<br />
| minor 7th<br />
| m7<br />
| C<br />
| minor 7th<br />
| m7<br />
| C<br />
|-<br />
| 19<br />
| 1036<br />
| p-Major seventh<br />
| pM7<br />
| perfect 7th<br />
| P7<br />
| C<br />
|wide minor seventh<br />
|Wm7<br />
|Bb<br />
| aug sub7th<br />
| As7<br />
| C#<br />
| sharp 9th, flat 10th<br />
| s9, f10<br />
| A#, βb<br />
| upminor 7th, dim 8ve<br />
| ^m7, d8<br />
| ^C, Db<br />
| classic minor 7th<br />
| Km7<br />
| kC<br />
|-<br />
| 20<br />
| 1091<br />
| p-Augmented seventh<br />
| pA7<br />
| aug 7th<br />
| A7<br />
| C#<br />
|major seventh<br />
|M7<br />
|B<br />
| double-aug sub7th,<br />double-dim octave<br />
| AAs7,<br />dd8<br />
| Cx,<br />Dbb<br />
| natural 10th<br />
| N10<br />
| β<br />
| downmajor 7th, updim 8ve<br />
| vM7, ^d8<br />
| vC#, ^Db<br />
| classic major 7th<br />
| kM7<br />
| kC#<br />
|-<br />
| 21<br />
| 1145<br />
| s-Major seventh<br />
| sM7<br />
| dim 8ve<br />
| d8<br />
| Db<br />
|diminished octave<br />
|d8<br />
|Cb<br />
| dim octave<br />
| d8<br />
| Db<br />
| sharp 10th<br />
| s10<br />
| β#, Cb<br />
| major 7th, down 8ve<br />
| M7, v8<br />
| C#, vD<br />
| major 7th / comma-narrow 8ve<br />
| M7 / k8<br />
| C#, kD<br />
|-<br />
| 22<br />
| 1200<br />
| Octave<br />
| 8<br />
| perfect octave<br />
| P8<br />
| D<br />
|perfect octave<br />
|P8<br />
|C<br />
| perfect octave<br />
| P8<br />
| D<br />
| natural 11th<br />
| N11<br />
| C<br />
| perfect octave<br />
| P8<br />
| D<br />
| perfect 8ve<br />
| P8<br />
| D<br />
|}<br />
<br />
== Approximation to JI ==<br />
[[File:22ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 17-limit intervals approximated in 22edo]]<br />
<br />
=== Interval mappings ===<br />
{{Q-odd-limit intervals|22}}<br />
<br />
== Regular temperament properties ==<br />
{| class="wikitable center-4 center-5 center-6"<br />
|-<br />
! rowspan="2" | [[Subgroup]]<br />
! rowspan="2" | [[Comma list]]<br />
! rowspan="2" | [[Mapping]]<br />
! rowspan="2" | Optimal<br>8ve stretch (¢)<br />
! colspan="2" | Tuning error<br />
|-<br />
! [[TE error|Absolute]] (¢)<br />
! [[TE simple badness|Relative]] (%)<br />
|-<br />
| 2.3<br />
| {{monzo| 35 -22 }}<br />
| {{mapping| 22 35 }}<br />
| −2.25<br />
| 2.25<br />
| 4.12<br />
|-<br />
| 2.3.5<br />
| 250/243, 2048/2025<br />
| {{mapping| 22 35 51 }}<br />
| −0.86<br />
| 2.70<br />
| 4.94<br />
|-<br />
| 2.3.5.7<br />
| 50/49, 64/63, 245/243<br />
| {{mapping| 22 35 51 62 }}<br />
| −1.80<br />
| 2.85<br />
| 5.23<br />
|-<br />
| 2.3.5.7.11<br />
| 50/49, 55/54, 64/63, 99/98<br />
| {{mapping| 22 35 51 62 76 }}<br />
| −1.11<br />
| 2.90<br />
| 5.33<br />
|-<br />
| 2.3.5.7.11.17<br />
| 50/49, 55/54, 64/63, 85/84, 99/98<br />
| {{mapping| 22 35 51 62 76 90 }}<br />
| −1.09<br />
| 2.65<br />
| 4.87<br />
|}<br />
* 22et is lower in relative error than any previous equal temperaments in the 11-limit. The next equal temperament that does better in this subgroup is [[31edo|31]]. <br />
* 22et does best in the 2.3.5.7.11.17 subgroup, and the next equal temperament that does better in this subgroup is [[46edo|46]]. <br />
<br />
=== Uniform maps ===<br />
{{Uniform map|edo=22}}<br />
<br />
=== Commas ===<br />
22et [[tempering out|tempers out]] the following [[commas]]. This assumes the [[val]] {{val| 22 35 51 62 76 81 }}.<br />
<br />
{| class="commatable wikitable center-all left-3 right-4 left-6"<br />
|-<br />
! [[Harmonic limit|Prime<br>limit]]<br />
! [[Ratio]]<ref group="note">{{rd}}</ref><br />
! [[Monzo]]<br />
! [[Cents]]<br />
! [[Color name]]<br />
! Name<br />
|-<br />
| 3<br />
| <abbr title="34359738368/31381059609">(22 digits)</abbr><br />
| {{monzo| 35 -22 }}<br />
| 156.98<br />
| Trisawa<br />
| 22-comma<br />
|-<br />
| 5<br />
| [[20480/19683]]<br />
| {{monzo| 12 -9 1 }}<br />
| 68.72<br />
| Sayo<br />
| Superpyth comma<br />
|-<br />
| 5<br />
| [[250/243]]<br />
| {{monzo| 1 -5 3 }}<br />
| 49.17<br />
| Triyo<br />
| Porcupine comma<br />
|-<br />
| 5<br />
| [[3125/3072]]<br />
| {{monzo|-10 -1 5 }}<br />
| 29.61<br />
| Laquinyo<br />
| Magic comma<br />
|-<br />
| 5<br />
| [[2048/2025]]<br />
| {{monzo| 11 -4 -2 }}<br />
| 19.55<br />
| Sagugu<br />
| Diaschisma<br />
|-<br />
| 5<br />
| [[2109375/2097152| (14 digits)]]<br />
| {{monzo|-21 3 7 }}<br />
| 10.06<br />
| Lasepyo<br />
| [[Semicomma]]<br />
|-<br />
| 5<br />
| <abbr title="4294967296/4271484375">(20 digits)</abbr><br />
| {{monzo| 32 -7 -9 }}<br />
| 9.49<br />
| Sasa-tritrigu<br />
| [[Escapade comma]]<br />
|-<br />
| 5<br />
| <abbr title="9010162353515625/9007199254740992">(32 digits)</abbr><br />
| {{monzo|-53 10 16 }}<br />
| 0.57<br />
| Quadla-quadquadyo<br />
| [[Kwazy comma]]<br />
|-<br />
| 7<br />
| [[50/49]]<br />
| {{monzo| 1 0 2 -2 }}<br />
| 34.98<br />
| Biruyo<br />
| Jubilisma<br />
|-<br />
| 7<br />
| [[64/63]]<br />
| {{monzo| 6 -2 0 -1 }}<br />
| 27.26<br />
| Ru<br />
| Septimal comma<br />
|-<br />
| 7<br />
| [[875/864]]<br />
| {{monzo|-5 -3 3 1 }}<br />
| 21.90<br />
| Zotriyo<br />
| Keema<br />
|-<br />
| 7<br />
| [[2430/2401]]<br />
| {{monzo| 1 5 1 -4 }}<br />
| 20.79<br />
| Quadru-ayo<br />
| Nuwell comma<br />
|-<br />
| 7<br />
| [[245/243]]<br />
| {{monzo| 0 -5 1 2 }}<br />
| 14.19<br />
| Zozoyo<br />
| Sensamagic comma<br />
|-<br />
| 7<br />
| [[1728/1715]]<br />
| {{monzo| 6 3 -1 -3 }}<br />
| 13.07<br />
| Triru-agu<br />
| Orwellisma<br />
|-<br />
| 7<br />
| [[225/224]]<br />
| {{monzo|-5 2 2 -1 }}<br />
| 7.71<br />
| Ruyoyo<br />
| Marvel comma<br />
|-<br />
| 7<br />
| [[10976/10935]]<br />
| {{monzo| 5 -7 -1 3 }}<br />
| 6.48<br />
| Trizo-agu<br />
| Hemimage comma<br />
|-<br />
| 7<br />
| [[6144/6125]]<br />
| {{monzo| 11 1 -3 -2 }}<br />
| 5.36<br />
| Saruru-atrigu<br />
| Porwell comma<br />
|-<br />
| 7<br />
| [[65625/65536]]<br />
| {{monzo|-16 1 5 1 }}<br />
| 2.35<br />
| Lazoquinyo<br />
| Horwell comma<br />
|-<br />
| 7<br />
| <abbr title="420175/419904">(12 digits)</abbr><br />
| {{monzo|-6 -8 2 5 }}<br />
| 1.12<br />
| Quinzo-ayoyo<br />
| [[Wizma]]<br />
|-<br />
| 11<br />
| [[99/98]]<br />
| {{monzo|-1 2 0 -2 1 }}<br />
| 17.58<br />
| Loruru<br />
| Mothwellsma<br />
|-<br />
| 11<br />
| [[100/99]]<br />
| {{monzo| 2 -2 2 0 -1 }}<br />
| 17.40<br />
| Luyoyo<br />
| Ptolemisma<br />
|-<br />
| 11<br />
| [[121/120]]<br />
| {{monzo|-3 -1 -1 0 2 }}<br />
| 14.37<br />
| Lologu<br />
| Biyatisma<br />
|-<br />
| 11<br />
| [[176/175]]<br />
| {{monzo| 4 0 -2 -1 1 }}<br />
| 9.86<br />
| Lorugugu<br />
| Valinorsma<br />
|-<br />
| 11<br />
| [[896/891]]<br />
| {{monzo| 7 -4 0 1 -1 }}<br />
| 9.69<br />
| Saluzo<br />
| Pentacircle comma<br />
|-<br />
| 11<br />
| [[65536/65219]]<br />
| {{monzo| 16 0 0 -2 -3 }}<br />
| 8.39<br />
| Satrilu-aruru<br />
| Orgonisma<br />
|-<br />
| 11<br />
| [[385/384]]<br />
| {{monzo|-7 -1 1 1 1 }}<br />
| 4.50<br />
| Lozoyo<br />
| Keenanisma<br />
|-<br />
| 11<br />
| [[540/539]]<br />
| {{monzo| 2 3 1 -2 -1 }}<br />
| 3.21<br />
| Lururuyo<br />
| Swetisma<br />
|-<br />
| 11<br />
| [[4000/3993]]<br />
| {{monzo| 5 -1 3 0 -3 }}<br />
| 3.03<br />
| Triluyo<br />
| Wizardharry comma<br />
|-<br />
| 11<br />
| [[9801/9800]]<br />
| {{monzo|-3 4 -2 -2 2 }}<br />
| 0.18<br />
| Bilorugu<br />
| Kalisma<br />
|-<br />
| 13<br />
| [[65/64]]<br />
| {{monzo|-6 0 1 0 0 1 }}<br />
| 26.84<br />
| Thoyo<br />
| Wilsorma<br />
|-<br />
| 13<br />
| [[78/77]]<br />
| {{monzo| 1 1 0 -1 -1 1 }}<br />
| 22.34<br />
| Tholuru<br />
| Negustma<br />
|-<br />
| 13<br />
| [[91/90]]<br />
| {{monzo|-1 -2 -1 1 0 1 }}<br />
| 19.13<br />
| Thozogu<br />
| Superleap comma, biome comma<br />
|-<br />
| 13<br />
| [[31213/31104]]<br />
| {{monzo|-7 -5 0 4 0 1 }}<br />
| 6.06<br />
| Thoquadzo<br />
| Praveensma<br />
|-<br />
| 31<br />
| [[125/124]]<br />
| {{monzo|-2 0 3 0 0 0 0 0 0 0 -1 }}<br />
| 13.91<br />
| Thiwutriyo<br />
| Twizzler comma<br />
|}<br />
<br />
=== Rank-2 temperaments ===<br />
* [[List of 22et rank two temperaments by badness]]<br />
* [[List of 22et rank two temperaments by complexity]]<br />
* [[List of edo-distinct 22et rank two temperaments]]<br />
<br />
{| class="wikitable center-1 center-2"<br />
|-<br />
! Periods<br>per 8ve<br />
! Generator<br />
! Temperaments<br />
|-<br />
| 1<br />
| 1\22<br />
| [[Escapade]] / [[escaped]]<br>[[Chromo]]<br>[[Ceratitid]]<br />
|-<br />
| 1<br />
| 3\22<br />
| [[Porcupine]]<br />
|-<br />
| 1<br />
| 5\22<br />
| [[Orwell]] (22) / blair (22) / winston (22f)<br />
|-<br />
| 1<br />
| 7\22<br />
| [[Magic]] / telepathy<br />
|-<br />
| 1<br />
| 9\22<br />
| [[Superpyth]] / [[suprapyth]]<br />
|-<br />
| 2<br />
| 1\22<br />
| [[Shrutar]] / hemipaj<br>[[Comic]]<br />
|-<br />
| 2<br />
| 2\22<br />
| [[Srutal]] / [[pajara]] / pajarous<br />
|-<br />
| 2<br />
| 3\22<br />
| [[Hedgehog]] / [[echidna]]<br />
|-<br />
| 2<br />
| 4\22<br />
| [[Astrology]]<br>[[Antikythera]]<br>[[Wizard]]<br />
|-<br />
| 2<br />
| 5\22<br />
| [[Doublewide]] / fleetwood<br />
|-<br />
| 11<br />
| 1\22<br />
| [[Undeka]]<br>[[Hendecatonic]]<br />
|}<br />
<br />
== Octave stretch or compression ==<br />
22edo can benefit from slightly compressing the octave, especially when using it as an 7-limit equal temperament. With the right amount of compression we can find a slightly better 3rd harmonic and significantly better 7th harmonic at the expense of somewhat less accurate approximations of 5 and 11. <br />
<br />
Good compressed-22 options include: [[ZPI|80zpi]] or [[57ed6]].<br />
<br />
== Scales ==<br />
{{Main|22edo modes}}<br />
{{See also|List of MOS scales in 22edo}}<br />
<br />
== Tetrachords ==<br />
{{Main|22edo tetrachords}}<br />
<br />
== Chords ==<br />
{{Main|22edo chords}}<br />
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:<br />
<br />
{| class="wikitable center-all"<br />
|-<br />
! Quality<br />
! [[Color name]]<br />
! [[Monzo]] Format<br />
! Examples<br />
|-<br />
| rowspan="2" | minor<br />
| zo<br />
| {{monzo| a b 0 1 }}<br />
| 7/6, 7/4<br />
|-<br />
| fourthward wa<br />
| {{monzo| a b }} where {{nowrap|b &lt; −1}}<br />
| 32/27, 16/9<br />
|-<br />
| upminor<br />
| gu<br />
| {{monzo| a b −1 }}<br />
| 6/5, 9/5<br />
|-<br />
| downmajor<br />
| yo<br />
| {{monzo| a b 1 }}<br />
| 5/4, 5/3<br />
|-<br />
| rowspan="2" | major<br />
| fifthward wa<br />
| {{monzo| a b }} where {{nowrap|b &gt; 1}}<br />
| 9/8, 27/16<br />
|-<br />
| ru<br />
| {{monzo| a b 0 −1 }}<br />
| 9/7, 12/7<br />
|}<br />
<br />
All 22edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).Here are the zo, gu, yo, and ru triads:<br />
<br />
{| class="wikitable center-all"<br />
|-<br />
! [[Kite's color notation|Color of the 3rd]]<br />
! JI Chord<br />
! Notes as edosteps<br />
! Notes of C chord<br />
! Written name<br />
! Spoken name<br />
|-<br />
| zo<br />
| 6:7:9<br />
| 0-5-13<br />
| C Eb G<br />
| Cm<br />
| C minor<br />
|-<br />
| gu<br />
| 10:12:15<br />
| 0-6-13<br />
| C ^Eb G<br />
| C^m<br />
| C upminor<br />
|-<br />
| yo<br />
| 4:5:6<br />
| 0-7-13<br />
| C vE G<br />
| Cv<br />
| C downmajor or C down<br />
|-<br />
| ru<br />
| 14:18:21<br />
| 0-8-13<br />
| C E G<br />
| C<br />
| C major or C<br />
|}<br />
<br />
Examples:<br />
<br />
* 0-4-13 = C D G = C2<br />
* 0-9-13 = C F G = C4<br />
* 0-10-13 = C ^F G = C^4 or C(^4)<br />
* 0-5-10 = C Eb Gb = Cd = Cdim<br />
* 0-5-11 = C Eb ^Gb = Cd(^5)<br />
* 0-5-12 = C Eb vG = Cm(v5)<br />
<br />
== Instruments ==<br />
=== Keyboards ===<br />
[[File:22-tone halberstadt layout.png|alt=|frameless]]<br />
<br />
A potential layout for a 22edo keyboard with both split black and white keys.<br />
<br />
[[Lumatone mapping for 22edo|Lumatone mappings for 22edo]] are available.<br />
<br />
== Music ==<br />
{{Main| 22edo/Music }}<br />
{{Catrel|22edo tracks}}<br />
<br />
== See also ==<br />
* [[User:Unque/22edo Composition Theory|Unque's approach]]<br />
* [[William Lynch's thoughts on septimal harmony and 22edo|William Lynch's approach]]<br />
* [[22edo/Eliora's approach|Eliora's approach]]<br />
<br />
== Further reading ==<br />
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.<br />
* [http://lumma.org/tuning/erlich/erlich-decatonic.pdf Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament'']<br />
* [http://porcupinemusic.weebly.com/ "Porcupine Music" - Website Focused on the Development of 22 EDO music]<br />
* [https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]<br />
* [https://www.youtube.com/playlist?list=PLWl3gB1BGAwX4sPnbFc5L3gU_IoyUDQ9V Joseph Monzo's visualizations of 22edo scale generation from temperaments]<br />
<br />
== Notes ==<br />
<references group="note" /><br />
<br />
== References ==<br />
# Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]<br />
# Bosanquet, R.H.M. [https://www.webcitation.org/5kjJcrhEx ''On the Hindoo division of the octave, with additions to the theory of higher orders''], Proceedings of the Royal Society of London vol. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, ''Hindu Music from Various Authors'', Chowkhamba Sanskrit Series, Varanasi, India, 1965<br />
<br />
[[Category:Twentuning]]<br />
[[Category:Alpharabian]]<br />
[[Category:Superpyth]]<br />
[[Category:Pajara]]<br />
[[Category:Orwell]]<br />
[[Category:Porcupine]]<br />
[[Category:Magic]]<br />
[[Category:Quartismic]]<br />
[[Category:Todo:complete table]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Template:Alert&diff=224870Template:Alert2026-02-27T15:50:21Z<p>ArrowHead294: </p>
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</noinclude></div>ArrowHead294https://en.xen.wiki/index.php?title=7L_8s&diff=2246967L 8s2026-02-25T03:43:26Z<p>ArrowHead294: </p>
<hr />
<div>{{Infobox MOS}}<br />
{{MOS intro}}<br />
<br />
It is notable for supporting [[Porcupine]], of the [[Porcupine_family|porcupine family]].<br />
<br />
== Name ==<br />
Leriendil uses the name "roklotic" for 7L&nbsp;8s, due to its similarity to the Roklotian scale used in Famanan music theory.<br />
<br />
== Scale properties ==<br />
{{TAMNAMS use}}<br />
<br />
=== Intervals ===<br />
{{MOS intervals}}<br />
<br />
=== Generator chain ===<br />
{{MOS genchain}}<br />
<br />
=== Modes ===<br />
{{MOS mode degrees}}<br />
<br />
== Scale tree ==<br />
{{todo|inline=1|complete table|text=There was previously octachord info in the old scale tree, in the form of the step pattern LsLsLsL. Please add it to the new scale tree.}}<br />
{{MOS tuning spectrum<br />
| Depth = 6<br />
| 3/2 = Optimal rank range ({{nowrap|L/s {{=}} 3/2}}) porcupine<br />
| 13/8 = Golden porcupine {{nowrap|L/s {{=}} φ}}<br />
}}<br />
<br />
{{stub}}<br />
<br />
[[Category:Porcupine]]<br />
[[Category:Abstract MOS patterns]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Xen_concepts_for_beginners&diff=224695Xen concepts for beginners2026-02-25T03:39:55Z<p>ArrowHead294: /* Edos */</p>
<hr />
<div>{{Texmap}}<br />
{{Beginner}}<br />
<br />
== Interval math ==<br />
Xen discussion uses two kinds of units:<br />
* ''Frequency ratios''<br />
** The [[frequency]] is the absolute pitch of any given tone, usually measured in [[hertz]] (Hz). The ratio between frequencies is just a number. Equal intervals have the same [[frequency ratio]].<br />
* ''Logarithmic units'' such as [[cents]] and [[edo]] steps that treat intervals we hear as equal as the same additive unit<br />
<br />
To stack two intervals, we use different types of operations for the two kinds of units. To stack two intervals written as ratios, we ''multiply'', whereas to stack two intervals written as cents or edo steps, we ''add'' the intuitive way. To "unstack" an interval from another interval, we ''divide'' the respective ratios and ''subtract'' logarithmic units. To convert between cents and ratios we use the following formulas:<br />
<br />
<math><br />
\begin{align}<br />
\text{cents} &= 1200 \cdot \log_{2} \left( \text{ratio} \right) \\<br />
\text{ratio} &= 2^{\left( \text{cents}/1200 \right)}<br />
\end{align}<br />
</math><br />
<br />
The unison has frequency ratio 1/1 and is 0 cents. The octave has frequency ratio 2/1 and is exactly 1200 cents. A standard semitone (in 12edo) has frequency ratio <math>\sqrt[12]{2}</math> and is exactly 100 cents (by definition).<br />
<br />
The notation ''m''\''n'' means ''m'' steps of ''n''-edo. For example, 12edo's perfect fifth can be denoted as 7\12, meaning "7 steps of 12-tone equal temperament".<br />
<br />
A common operation in xen math is the [[mediant]]. The mediant of two fractions, ''a''/''b'' and ''c''/''d'', is the "freshman sum" {{sfrac|''a'' + ''b''|''c'' + ''d''}}, which is always between ''a''/''b'' and ''c''/''d''. For example, the mediant of 4/3, the just perfect fourth, and 5/4, the just major third, is 9/7, the supermajor third. If two fractions are in lowest terms, their mediant is the simplest fraction that is strictly between both. The mediant is commonly used for both JI ratios and edo intervals.<br />
<br />
Another important operation is [[octave reduction|reduction]]. To reduce an interval a by an interval b means to stack or "unstack" ''b'' from ''a'' until ''a'' is at least the unison and less than ''b''. For example, 3/1 reduced by 2/1 is 3/2.<br />
<br />
== Basic JI ==<br />
[[Just intonation]] (JI) is the set of intervals that are tuned to rational frequency ratios, ones can be written as fractions of whole numbers.<br />
<br />
The easiest way to get concordance (smoothness, blending and buzzing) is to use low-numbered JI ratios in your interval or chord, for example the just perfect fifth [[3/2]], the just major third [[5/4]], and the lesser septimal tritone [[7/5]]. When pure JI ratios are used, a psychoacoustic effect called JI buzz occurs. When the overall chord is low number JI, such as 8:9:10:11:12:13:14, the result is very concordant.<br />
<br />
No edo interval except for the octave (2/1) and stacks of it is exact JI. A JI ratio might be far from a 12edo interval; for example 7/4 is 969 cents. This is another reason why JI is a common approach to xen.<br />
<br />
As stacking JI ratios involves multiplying, primes are important as the simplest building blocks of arbitrary JI ratios. So we can write every ratio as a vector called a ''monzo'', a list of powers for primes. We can visualize each ratio as living in some JI lattice (the set of all intervals built by stacking a finite set of basic intervals).<br />
<br />
There are many approaches to JI music: lattice-based JI, constant structure scales, free JI, primodality, tonality diamonds, combination product sets…<br />
<br />
JI is usually less mathy than RTT.<br />
<br />
The approach that RTT cares about the most is lattice-based JI. A JI lattice, or a subgroup, is built by stacking a finite set of JI intervals, usually primes such as 2, 3, 5, and 7.<br />
<br />
There are two ways the term ''[[limit]]'' is used.<br />
* The ''[[harmonic limit|p-prime-limit]]'' is the lattice built by multiplying the primes at most ''p'', possibly multiple times. We write a JI lattice by writing the basic intervals separated by periods. For example, {{nowrap|6/5 {{=}} 2 × 3 / 5}} is in the 5-limit, or the 2.3.5 subgroup, and so is {{nowrap|45/32 {{=}} (3 × 3 × 5) / (2 × 2 × 2 × 2 × 2)}}.<br />
* The ''[[odd limit|q-odd-limit]]'' is the set of all JI ratios where the larger of the numerator and denominator after removing factors of 2 from the JI ratios is at most the odd number ''q''. For example, 7/6, 13/5, 11/10, and 16/15 are all in the 15-odd-limit.<br />
<br />
== Basic RTT ==<br />
''Assuming several things from common 12edo practice'', JI has several disadvantages. To get infinite modulation and have exactly the same chords on every note, we need infinitely many notes unlike the finitely many notes of 12edo. JI with such infinite modulation and regularity also has small intervals that may be undesirable, called commas. This is the problem that [[regular temperament theory]] (RTT) exists to solve. Regular temperaments equate certain intervals by considering the difference between them as a comma and "[[tempering out]]" the difference. However, note that one need not treat JI like one would an edo, and that some regular temperament tunings are infinite and don't provide the advantages of finiteness.<br />
<br />
From the perspective of an edo user, another problem RTT solves is that there are very few small edos and they do not constitute that wide a palette. Especially in larger edos, RTT provides a way of not being overwhelmed with dozens of notes.<br />
<br />
RTT views edos as regular temperaments. Under this view, edos simplify the infinite JI space to a finite set, deforming the intervals so that certain chosen intervals vanish. We can also approach simplifying JI ratios from edos themselves, namely how edos approximate each prime. This is a vector called a [[val]]. Vals map primes to a set number of edo steps and thus tell us how many edo steps each interval in JI is mapped to. The usual 12edo val (called the 12edo [[patent val]]) in the 5-limit is {{val| 12 19 28 }}, as the 12edo intervals that are closest to 2/1, 3/1 and 5/1 are 12, 19 and 28 steps respectively.<br />
<br />
There are various temperaments in xen with varying levels of practicality. The most important one to know is probably [[meantone]] temperament, which equates four fifths ({{nowrap|(3/2)<sup>4</sup> {{=}} 81/16}}) with a major third plus two octaves ({{nowrap|(5/4) × 4 = 5 {{=}} 80/16}}), which is encoded by tempering out the syntonic comma [[81/80]] (monzo {{monzo| -4 4 -1 }}). <br />
<br />
A val tempers out a comma if, when you construct the comma from primes according to their tunings in the val, the result is 0 cents or the unison. For example, 12edo is a meantone edo because:<br />
* The patent val for 12edo in the 5-limit is {{val| 12 19 28 }}.<br />
* The comma 81/80 has monzo {{monzo| -4 4 -1 }}.<br />
* Constructing the tuning of a comma from mappings of primes involves multiplying each entry in the val to a corresponding entry in the comma's monzo, and then adding the resulting numbers together; this operation is called a "dot product".<br />
** {{nowrap|12 × (−4) {{=}} −48}}, corresponding to going down 4 octaves.<br />
** {{nowrap|19 × 4 {{=}} 76}}, corresponding to going up 4 perfect twelfths (or, to going up 4 octaves and 4 fifths).<br />
** {{nowrap|28 × (−1) = −28}}, corresponding to dividing by 5 (going down two octaves and a major third).<br />
** {{nowrap|(76 − 48) − 28 {{=}} 0}}<br />
* Since the result is 0, 12edo supports meantone. In RTT math, this can be written as:<br />
<br />
<math>\vmp{12 & 19 & 28}{-4 & 4 & -1} = 12 \times \left(-4\right) + 19 \times 4 + 28 \times \left(-1\right) = -48 + 76 - 28 = 0.</math><br />
<br />
== MOS scales ==<br />
[[Mos]] (moment of symmetry) scales are one way to generalize the diatonic scale; the diatonic scale is a mos scale. They are scales with two step sizes (large (L) and small (s)) with a uniquely elegant combination of properties:<br />
<br />
For every number of steps, the scale has at most two interval sizes with that number of steps. The scale can be made by stacking a certain fixed interval called the ''[[periods and generators|generator]]'' (and reducing by an interval called the ''[[periods and generators|period]]'', usually the octave or some equal division of it such as 1\2 or 1\3), over and over, stopping at some point where there are two step sizes distributed as evenly as possible.<br />
<br />
Every mos scale with ''m'' large steps and ''n'' small steps is a mode of some pattern. This is why you only need to write ''m''L&nbsp;''n''s for an octave-equivalent mos scale and specify the mode (using [[UDP]] for example). For example, every 5L&nbsp;3s mos scale is a mode of the pattern LLsLLsLs.<br />
<br />
An important way that mos scales vary is [[hardness]], defined as the size (in cents) of the L divided by the size (in cents) of the s step. Hardness can range from 1 to infinity. The larger the hardness, the harder the mos tuning; the smaller (closer to 1) the hardness, the softer the tuning. The two extremes are where the mos pattern no longer holds; 1 is where L and s steps are equal, and infinity is where s is so small that it disappears.<br />
<br />
Any given mos pattern is available in more than one edo, and the basic tuning of a mos pattern gives the smallest edo that provides that mos pattern. To adjust the hardness of a mos provided by an edo, we can add two edos, obtaining an edo where the hardness is the mediant of the two original edos'. For a diatonic example, 12edo has basic ({{nowrap|L/s {{=}} 2/1}}) diatonic, 17edo has hard ({{nowrap|L/s {{=}} 3/1}}) diatonic, and 19edo has soft ({{nowrap|L/s {{=}} 3/2}}) diatonic. {{nowrap|12 + 19 {{=}} 31}}, and 31edo diatonic has hardness {{nowrap|{{sfrac|2 + 3|1 + 2}} {{=}} 5/3}}.<br />
<br />
The generator size and the period thus determine the mos scales that can be obtained. Hardness varies with generator size within a mos's range.<br />
<br />
Every mos scale pattern has a generator range. Since the familiar diatonic scale is a mos 5L&nbsp;2s, here is an important fact to know: If the period is the octave and the generator is a fifth between 4\7 (686{{c}}) and 3\5 (720{{c}}), the resulting pattern is the diatonic mos.<br />
<br />
[[TAMNAMS]] is a common method for naming intervals of a mos scale.<br />
<br />
The table below shows the tuning spectrum for the diatonic scale and the temperaments each subset is associated with:<br />
<br />
{| class="wikitable" style="margin: auto auto auto auto;"<br />
|+ style="font-size: 105%;" | Tuning ranges of the diatonic mos<br />
|-<br />
! colspan="3" | Range !! rowspan="2" colspan="2" | Temperaments encompassed<br />
|-<br />
! Edo !! Cents !! Hardness<br />
|-<br />
| 7edo to 33edo || 685.714 to 690.909 || 1/1 to 5/4 || colspan="2" | [[Deeptone]]<br />
|-<br />
| 33edo to 19edo || 690.909 to 694.737 || 5/4 to 3/2 || rowspan="2" | Meantone || [[Flattertone]], [[flattone]]<br />
|-<br />
| 19edo to 12edo || 694.737 to 700.000 || 3/2 to 2/1 || [[Septimal meantone]]<br />
|-<br />
| 12edo to 29edo || 700.000 to 703.448 || 2/1 to 5/2 || colspan="2" | [[Schismic]]<br />
|-<br />
| 29edo to 17edo || 703.448 to 705.882 || 5/2 to 3/1 || colspan="2" | [[Pepperoni]], [[leapday]]<br />
|-<br />
| 17edo to 22edo || 705.882 to 709.091 || 3/1 to 4/1 || rowspan="3" | Superpyth || [[Quasisuper]]<br />
|-<br />
| 22edo to 27edo || 709.091 to 711.111 || 4/1 to 5/1 || [[Superpyth]]<br />
|-<br />
| 27edo to 5edo || 711.111 to 720.000 || 5/1 to &infin; || [[Ultrapyth]]<br />
|}<br />
<br />
== Edos ==<br />
* [[5edo]]: Equalized pentatonic ("equipentatonic").<br />
* [[7edo]]: Equalized diatonic ("equiheptatonic").<br />
* [[9edo]]: The simplest edo with a [[2L&nbsp;5s]] mos (sssLssL). This mos is of interest because it can be viewed as a tuning of the diatonic scale where whole steps are smaller than half steps (an "antidiatonic" scale). The corresponding temperament is [[mavila]], which is llike meantone except major and minor intervals are swapped. Some larger edos like 16edo and 23edo tune it better, though mavila has poor accuracy in general since the fifth is very flat.<br />
* [[11edo]]: Stretched 12edo, has [[4L&nbsp;3s]] mos (LLsLsLs) which is a stretched diatonic.<br />
* [[13edo]]: Compressed 12edo having the [[5L&nbsp;3s]] mos (LLsLLsLs) which is a compressed version of the diatonic scale.<br />
* [[15edo]]: The smallest edo with a [[5L&nbsp;5s]] mos (LsLsLsLsLs) commonly called the blackwood scale. Also the smallest with a [[7L&nbsp;1s]] mos (LLLLsLLL). Both scales are known for supporting relatively familiar major and minor chords with relatively unfamiliar melodic structures.<br />
* [[16edo]]: Has 2L&nbsp;5s (sssLssL) and [[7L&nbsp;2s]] (LLLsLLLLs), generated by the mavila temperament, for which it is a more accurate tuning than 9edo.<br />
* [[17edo]]: The smallest edo after 12edo with a diatonic scale, and the smallest after 12edo to provide perfect fifths which are consonant for most purposes. Its major intervals are sharper and its minor intervals flatter than in 12edo, so it is often said to have a dramatic sound. First neutral diatonic edo (providing neutral seconds, thirds, sixths, and sevenths).<br />
* [[18edo]]: Has two fifths, 733{{c}} and 667{{c}}, that are nearly equally off from [[3/2]].<br />
* [[19edo]]: The smallest edo after 12edo which supports [[meantone]]. Just major and minor thirds are better approximated than in 12edo, but perfect fifths are represented significantly worse, at 7.2{{c}} flat instead of 2{{c}}. First [[interordinal]] diatonic edo (interordinals are semifourths, semisixths, semitenths, and semitwelfths). Diminished and augmented seconds, thirds, sixths, and sevenths are now distinct intervals with entirely new functions, whereas in 12edo they are conflated with simpler intervals.<br />
* [[22edo]]: Diatonic mos with a fifth significantly sharper than just, so that four fifths and three fourths give supermajor and subminor thirds (approximately [[9/7]] and [[7/6]]) instead of major and minor thirds. Has a 5-limit major third (approximate [[5/4]]) which is ''not'' be reached by stacking four fifths. Supports [[superpyth]] along with [[7L&nbsp;1s]] and [[7L&nbsp;8s]] [[Porcupine]] scales.<br />
* [[23edo]]: The largest edo without a diatonic, 5edo, or 7edo fifth. Supports mavila, just like 9edo and 16edo, with the flat fifth.<br />
* [[24edo]]: Has both neutral thirds (and other neutral intervals) and semifourths (and other interordinals), each of these lending itself to different harmony. Has 12edo mos scales as well as new ones, such as [[5L&nbsp;4s]] and [[4L&nbsp;3s]].<br />
* [[26edo]]: Has a fifth even flatter than that of 19edo, at 9.6{{c}} flat, and an even softer diatonic mos than 19edo, so much that the diatonic major third is nearly exactly [[26/21]] and the diatonic minor second is nearly exactly [[13/12]]. The [[7/4]] is also nearly exact, and the edo also has a good [[10/9]], [[14/11]], and [[11/8]].<br />
* [[27edo]]: Even harder diatonic mos than 22edo; at 9.2{{c}} sharp of just, the fifth is approximately about as sharp as 26edo's is flat. It has 12edo's [[5/4]], a near-exact [[7/6]], and an approximate [[16/13]] neutral third. Four fifths give a supermajor thirdand three fourths give a subminor third, just like in 22edo.<br />
* [[29edo]]: First edo with a perfect fifth closer to just intonation than 12edo. The minor third is extremely close to just [[13/11]]. It offers a tuning of 7L&nbsp;1s with more consonant fifths than 15edo or 22edo before it. Its diatonic scale has similar melodic properties to 17edo, although subtler.<br />
* [[31edo]]: One of the most popular meantone edos. Close to historical [[quarter-comma meantone]]. Not only is its major third close to just [[5/4]], it also matches the harmonic seventh [[7/4]] well.<br />
* [[34edo]]: Good for the 5-limit (2.3.5), as it does not temper out 81/80 and has a good 5/4. Also contains all notes of 17edo.<br />
* [[36edo]]: Good for primes [[3/2|3]] and [[7/4|7]].<br />
* [[37edo]]: Good for primes [[5/4|5]], [[7/4|7]], [[11/8|11]] and [[13/8|13]], but renders 3/2 sharp, even more so than 27edo.<br />
* [[41edo]]: Often considered remarkably good for the primes up to 11. Good 3; flat 5 and 7; sharp 11 and 13. Known for the [[Kite guitar]].<br />
* [[43edo]]: Possibly the most optimal tuning for meantone, with 5 tuned sharp and 3 tuned flat by almost exactly the same amount. Has better approximations of 11 and 13 than 19edo and 31edo, though the 7/4 is noticeably sharp.<br />
* [[46edo]]: Neogothic 3; sharp 5; flat 7, 11, and 13; good 17. Supports [[parapyth]]. Often compared to 41edo; some favor one, some the other.<br />
* [[53edo]]: Is a stack of near-just 3/2's which also approximates primes 5, 7, 13, and 19.<br />
* [[72edo]]: A notable subdivision of 12edo that is a very strong 11-limit (primes 2, 3, 5, 7, 11) temperament for its size.<br />
* [[87edo]]: Even better in 2.3.5.11.13 than 72edo is in the 11-limit, and a consistent and precise edo for approximating harmonics 8 to 16, but ratios with 7 suffer due to the 7 being flat and the 3 being sharp.<br />
* [[311edo]]: An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41. Essentially, the final boss of RTT edos.<br />
<br />
[[Category:Overview]]</div>ArrowHead294https://en.xen.wiki/index.php?title=User:ArrowHead294/UTF-8_extensions&diff=224629User:ArrowHead294/UTF-8 extensions2026-02-23T18:39:30Z<p>ArrowHead294: </p>
<hr />
<div>This is a table illustrating how Unicode code points are converted into UTF-8, specifically, what code points correspond to two-, three-, four-, five-, and six-byte sequences. A break is made in the four-byte sequences as UTF-8 is currently restricted to U+10FFFF to match the constraints of UTF-16, but extensions are shown anyways to show how UTF-8 is capable of encoding up to {{nowrap|2<sup>31</sup> − 1}} =&nbsp;0x7FFFFFFF without using <code>FE</code> and <code>FF</code>.<br />
<br />
{| class="wikitable"<br />
|+ style="font-size: 105%;" | Code point ↔ UTF-8 conversion<br />
|-<br />
! colspan="2" style="border-right: 4px solid black;" | Code point<br />
! colspan="6" style="border-right: 4px solid black;" | Bytes<br />
! colspan="2" | Eight-byte UTF-8<br />
|-<br />
! First<br />
! style="border-right: 4px solid black;" | Last<br />
! 1<br />
! 2<br />
! 3<br />
! 4<br />
! 5<br />
! style="border-right: 4px solid black;" | 6<br />
! First<br />
! Last<br />
|-<br />
| style="text-align: right;" | {{plaincode|U+0000}}<br />
| style="border-right: 4px solid black; text-align: right;" | {{plaincode|U+007F}}<br />
| {{plaincode|0''yyyzzzz''}}<br />
| colspan="5" style="background: darkgray; border-right: 4px solid black;" | <br />
| {{plaincode|00}}<br />
| {{plaincode|7F}}<br />
|-<br />
| style="text-align: right;" | {{plaincode|U+0080}}<br />{{plaincode|U+0400}}<br />
| style="border-right: 4px solid black; text-align: right;" | {{plaincode|U+03FF}}<br />{{plaincode|U+07FF}}<br />
| {{plaincode|110''xxxyy''}}<br />
| {{plaincode|10''yyzzzz''}}<br />
| colspan="4" style="background: darkgray; border-right: 4px solid black;" | <br />
| {{plaincode|C2&nbsp;80}}<br />{{plaincode|D0&nbsp;80}}<br />
| {{plaincode|CF&nbsp;BF}}<br />{{plaincode|DF&nbsp;BF}}<br />
|-<br />
| style="text-align: right;" | {{plaincode|U+0800}}<br />
| style="border-right: 4px solid black; text-align: right;" | {{plaincode|U+FFFF}}<br />
| {{plaincode|1110''wwww''}}<br />
| {{plaincode|10''xxxxyy''}}<br />
| {{plaincode|10''yyzzzz''}}<br />
| colspan="3" style="background: darkgray; border-right: 4px solid black;" | <br />
| {{plaincode|E0&nbsp;A0&nbsp;80}}<br />
| {{plaincode|EF&nbsp;BF&nbsp;BF}}<br />
|-<br />
| style="text-align: right;" | {{plaincode|U+010000}}<br />{{plaincode|U+110000}}<br />
| style="border-right: 4px solid black; text-align: right;" | {{plaincode|U+10FFFF}}<br />{{plaincode|U+1FFFFF}}<br />
| {{plaincode|11110''uvv''}}<br />
| {{plaincode|10''vvwwww''}}<br />
| {{plaincode|10''xxxxyy''}}<br />
| {{plaincode|10''yyzzzz''}}<br />
| colspan="2" style="background: darkgray; border-right: 4px solid black;" | <br />
| {{plaincode|F0&nbsp;90&nbsp;80&nbsp;80}}<br />{{plaincode|F4&nbsp;90&nbsp;80&nbsp;80}}<br />
| {{plaincode|F4&nbsp;8F&nbsp;BF&nbsp;BF}}<br />{{plaincode|F7&nbsp;BF&nbsp;BF&nbsp;BF}}<br />
|-<br />
| style="text-align: right;" | {{plaincode|U+200000}}<br />
| style="border-right: 4px solid black; text-align: right;" | {{plaincode|U+3FFFFFF}}<br />
| {{plaincode|111110''tt''}}<br />
| {{plaincode|10''uuuuvv''}}<br />
| {{plaincode|10''vvwwww''}}<br />
| {{plaincode|10''xxxxyy''}}<br />
| {{plaincode|10''yyzzzz''}}<br />
| style="background: darkgray; border-right: 4px solid black;" | <br />
| {{plaincode|F8&nbsp;88&nbsp;80&nbsp;80&nbsp;80}}<br />
| {{plaincode|FB&nbsp;BF&nbsp;BF&nbsp;BF&nbsp;BF}}<br />
|-<br />
| style="text-align: right;" | {{plaincode|U+4000000}}<br />
| style="border-right: 4px solid black; text-align: right;" | {{plaincode|U+7FFFFFFF}}<br />
| {{plaincode|1111110''s''}}<br />
| {{plaincode|10''sstttt''}}<br />
| {{plaincode|10''uuuuvv''}}<br />
| {{plaincode|10''vvwwww''}}<br />
| {{plaincode|10''xxxxyy''}}<br />
| style="border-right: 4px solid black;" | {{plaincode|10''yyzzzz''}}<br />
| {{plaincode|FC&nbsp;84&nbsp;80&nbsp;80&nbsp;80&nbsp;80}}<br />
| {{plaincode|FD&nbsp;BF&nbsp;BF&nbsp;BF&nbsp;BF&nbsp;BF}}<br />
|}</div>ArrowHead294https://en.xen.wiki/index.php?title=Template:Infobox_interval_region/doc&diff=224622Template:Infobox interval region/doc2026-02-23T12:24:58Z<p>ArrowHead294: Requesting documentation</p>
<hr />
<div>{{dochead}}{{substitute|no}}{{todo|inline=1| documentation }}</div>ArrowHead294https://en.xen.wiki/index.php?title=Geometric_mean&diff=224401Geometric mean2026-02-20T19:52:01Z<p>ArrowHead294: </p>
<hr />
<div>: ''"Mean" redirects here. For other types, see [[Pythagorean means]].''<br />
<br />
In tuning, the '''geometric mean''', '''pitch mean''', or simply '''mean''' generates new pitch materials by taking the mean in the [[Wikipedia: Logarithmic scale|logarithmic scale]] of pitch i.e. the scale proportional to the logarithm of frequency, such as [[cent]]s. It can be said with respect to pitches in frequency as well as intervals in frequency ratios on a certain common fundamental. The idea of treating [[quarter-comma meantone]] as the "strict" meantone is backed by this type of mean. <br />
<br />
The geometric mean ''f'' of two frequencies ''f''<sub>1</sub> and ''f''<sub>2</sub> is <br />
<br />
<math>\displaystyle f = \sqrt {f_1 f_2}</math><br />
<br />
Similarly, the geometric mean ''r'' of two frequency ratios ''r''<sub>1</sub> and ''r''<sub>2</sub> on a common fundamental is<br />
<br />
<math>\displaystyle r = \sqrt {r_1 r_2}</math><br />
<br />
Unlike [[mediant]], how the ratios are written out has no effect on their geometric mean. <br />
<br />
== Examples ==<br />
The geometric mean of [[1/1]] and [[3/2]] is sqrt (3/2): sqrt (1 × 3/2) = sqrt (3/2). <br />
<br />
The geometric mean of [[5/4]] and [[6/5]] is sqrt (3/2): sqrt ((5/4)(6/5)) = sqrt (6/4) = sqrt (3/2). <br />
<br />
The geometric mean of [[9/8]] and [[10/9]] is sqrt (5/4): sqrt ((9/8)(10/9)) = sqrt (10/8) = sqrt (5/4). <br />
<br />
== Generalizations ==<br />
=== To more frequencies or frequency ratios ===<br />
<br />
The geometric mean ''f'' of ''m'' frequencies ''f''<sub>1</sub>, ''f''<sub>2</sub>, …, ''f''<sub>''m''</sub> is <br />
<br />
<math>\displaystyle f = \left(\prod_{i = 1}^{m} f_i\right)^{1/m}</math><br />
<br />
The geometric mean ''r'' of ''m'' frequency ratios ''r''<sub>1</sub>, ''r''<sub>2</sub>, …, ''r''<sub>''m''</sub> on a common fundamental is<br />
<br />
<math>\displaystyle r = \left(\prod_{i = 1}^{m} r_i\right)^{1/m}</math><br />
<br />
=== To an equally spaced sequence ===<br />
This generalization connects the operation to [[equal tuning]]s. <br />
<br />
The ''m'' equal sequence of two frequencies ''f''<sub>1</sub> and ''f''<sub>2</sub> is <br />
<br />
<math>\displaystyle \left\lbrace i \in \mathbb {Z} \mid f_1^{i/m} \cdot f_2^{1 - i/m} \right\rbrace</math><br />
<br />
The ''m'' equal sequence of two frequency ratios ''r''<sub>1</sub> and ''r''<sub>2</sub> on a common fundamental is<br />
<br />
<math>\displaystyle \left\lbrace i \in \mathbb {Z} \mid r_1^{i/m} \cdot r_2^{1 - i/m} \right\rbrace</math><br />
<br />
The geometric mean is found by setting {{nowrap|''i'' {{=}} 1}} and {{nowrap|''m'' {{=}} 2}}. <br />
<br />
== Terminology ==<br />
The term ''geometric mean'' comes from math. See [[Wikipedia: Geometric mean]]. It would have made sense to call it ''logarithmic mean'' but for its established usage in math to mean something else. See [[Wikipedia: Logarithmic mean]]. <br />
<br />
== See also ==<br />
* [[Pythagorean means]]<br />
** [[Arithmetic mean]]<br />
** [[Inverse-arithmetic mean]]<br />
* [[Mediant]]<br />
<br />
[[Category:Pythagorean means]]<br />
[[Category:Terms]]<br />
[[Category:Elementary math]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Template:Uniform_map/doc&diff=224399Template:Uniform map/doc2026-02-20T16:50:34Z<p>ArrowHead294: </p>
<hr />
<div>{{dochead}}{{substitute|no}}<br />
The '''Uniform map''' templates creates a table of [[prime limit|''p''-prime-limit]] [[uniform map]]s between two size boundaries.<br />
<br />
=== Usage ===<br />
This template accepts the following named arguments:<br />
; <code>edo</code><br />
: The central EDO for the range.<br />
<br />
; <code>min</code><br />
: Minimum size, given as 1\size (write only the decimal number).<br />
<br />
; <code>max</code><br />
: Maximum size, given as 1\size (write only the decimal number).<br />
<br />
; <code>limit</code><br />
: Prime limit (write only the prime number); defaults to 13-limit.<br />
<br />
If no min/max are provided, the range wil be <code>edo</code> &#177;0.2 (see first example below).<br />
<br />
==== Examples ====<br />
Preferably, stick to just providing the EDO for the page:<br />
<pre>{{Uniform map|edo=7}}</pre><br />
<br />
gives:<br />
{{Uniform map|edo=7}}<br />
<br />
You can customize the table by providing more arguments:<br />
<pre>{{Uniform map|min=6.5|max=7.5|limit=5}}</pre><br />
<br />
gives:<br />
{{Uniform map|min=6.5|max=7.5|limit=5}}</div>ArrowHead294https://en.xen.wiki/index.php?title=33edo&diff=22439833edo2026-02-20T16:50:01Z<p>ArrowHead294: /* Regular temperament properties */</p>
<hr />
<div>{{Infobox ET}}<br />
{{ED intro}}<br />
<br />
== Theory ==<br />
=== Structural properties ===<br />
While relatively uncommon, 33edo is actually quite an interesting system. As a multiple of [[11edo]], it approximates the 7th and 11th harmonics via [[orgone]] temperament (see [[26edo]]). 33edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having a [[3L&nbsp;7s]] with {{nowrap|L {{=}} 4|s {{=}} 3}}. The 33c ({{val| 33 52 76 93 }}) and 33cd ({{val| 33 52 76 92 }}) mappings temper out [[81/80]] and can be used to represent [[1/2-comma meantone]], a "[[flattertone]]" tuning where the whole tone is [[10/9]] in size. Indeed, the perfect fifth is tuned about 11{{c}} flat, and two stacked fifths fall only 0.6{{c}} flat of 10/9. Leaving the scale be would result in the standard diatonic scale ([[5L&nbsp;2s]]) having minor seconds of four steps and whole tones of five steps. This also results in common practice minor and major chords becoming more supraminor and submajor in character, making everything sound almost neutral in quality. The 33cd val also tempers out [[49/48]], which along with the tempering of 81/80 means it supports [[godzilla]].<br />
<br />
Besides the 33cd val, one may also consider the patent val. This val maps 5/4 and 7/4 much more accurately (though still somewhat questionable), but 6/5 and 7/6, and especially 10/9 and 9/7 are much more damaged. Notable commas this val tempers out include 128/125, 36/35, and 225/224, supporting [[august]].<br />
<br />
33edo maps both the [[4:5:6]] and [[6:7:8]] chords inconsistently, with the third harmonic being about a third of a step flat and the 5th and 7th harmonics being about a third of a step sharp. It is thus reasonable to use the second-best approximation of [[3/1|3]], [[5/1|5]], or [[7/1|7]] in either chord, but in any case, the worst of the three intervals in the chord is detuned by over 22 cents, meaning 33edo is near-maximally bad for its size for tonal harmony. From this reasoning, 33edo's triple, [[99edo]], would be a very strong 7-limit system, and it indeed is.<br />
<br />
Instead of the flat 19-step fifth you may use the 20-step sharp fifth, over 25{{c}} sharp. Two of these lead to a 9/8 of 7\33, which is about 22/19 in size and may be counted as a small third. Between the flat 5\33 version of 9/8 and the sharp 7\33 version there is, of course, a {{nowrap|6\33 {{=}} 2\[[11edo|11]]}} interval of 218{{c}}. Together, these add up to {{nowrap|6\33 + 5\33 {{=}} 11\33 {{=}} 1\3}}, or 400{{c}}, the same major third as 12edo. We also have both a 327{{c}} minor third ({{nowrap|9\33 {{=}} 6\22 {{=}} 3\11}}), the same as that of [[22edo]], and a flatter 8\33 third of 291{{c}}, which if you like could also be called a flat 19th harmonic, but much more accurately a 13/11 sharp by 1.7{{c}} (if you use the patent val it is an extremely inaccurate 6/5). Another talent it has is that 7/5 is tuned quite accurately by 16\33, and we may put two 8\33 versions of 13/11 together to produce the [[cuthbert triad]]. The 8\33 generator, with MOS of size 5, 9, and 13, gives plenty of scope for these, as well as the 11th, 13th, and 19th harmonics (taking the generator as a 19/16) which are relatively well in tune.<br />
<br />
33edo contains an accurate approximation of the [[Bohlen–Pierce]] scale with 4\33 near [[13edt|1\13edt]].<br />
<br />
Other notable 33edo scales are [[diasem]] with {{nowrap|L:m:s {{=}} 5:3:1}} and [[5L&nbsp;4s]] with {{nowrap|L:s {{=}} 5:2}}. This step ratio for 5L&nbsp;4s is great for its semitone size of 72.7{{c}}.<br />
<br />
=== Odd harmonics ===<br />
{{Harmonics in equal|33}}<br />
<br />
33edo is not especially good at representing all rational intervals in the [[7-limit]], but it does very well on the 7-limit [[k*N subgroups|3*33 subgroup]] 2.27.15.21.11.13. On this subgroup it tunes things to the same tuning as [[99edo]], and as a subgroup patent val it tempers out the same commas. The 99 equal temperaments hemififths, amity, parakleismic, hemiwuerschmidt, ennealimmal and hendecatonic can be reduced to this subgroup and give various possibilities for MOS scales, etc. In particular, the [[terrain]] 2.7/5.9/5 subgroup temperament can be tuned via the 5\33 generator. The full system of harmony provides the optimal patent val for [[slurpee]] temperament in the 5-, 7-, 11-, and 13-limits.<br />
<br />
While it might not be the most harmonically accurate temperament, it is structurally quite interesting, and it approximates the full 19-limit consort in its own way. You could even say it tunes the 23rd and 29th harmonics ten cents flat if you were so inclined; as well as getting within two cents of the 37th.<br />
<br />
=== Miscellany ===<br />
33 is also the number of years in the Iranian calendar's leap cycle, where leap year is inserted once every 4 or 5 years. This corresponds to the [[1L&nbsp;7s]] with the step ratio of 5:4.<br />
<br />
== Intervals ==<br />
{| class="wikitable center-all"<br />
|-<br />
! rowspan="2" |Step #<br />
! ET<br />
! colspan="2" | Just<br />
! rowspan="2" | Difference<br>(ET minus Just)<br />
! rowspan="2" colspan="3" | Extended Pythagorean notation<br />
|-<br />
! Cents<br />
! Interval<br />
! Cents<br />
|-<br />
| 0<br />
| 0<br />
| [[1/1]]<br />
| 0<br />
| 0<br />
| Perfect Unison<br />
| P1<br />
| D<br />
|-<br />
| 1<br />
| 36.364<br />
| [[48/47]]<br />
| 36.448<br />
| −0.085<br />
| Augmented Unison<br />
| A1<br />
| D#<br />
|-<br />
| 2<br />
| 72.727<br />
| [[24/23]]<br />
| 73.681<br />
| −0.953<br />
| Double-aug 1sn<br />
| AA1<br />
| Dx<br />
|-<br />
| 3<br />
| 109.091<br />
| [[16/15]]<br />
| 111.731<br />
| −2.640<br />
| Diminished 2nd<br />
| d2<br />
| Ebb<br />
|-<br />
| 4<br />
| 145.455<br />
| [[12/11]]<br />
| 150.637<br />
| −5.183<br />
| Minor 2nd<br />
| m2<br />
| Eb<br />
|-<br />
| 5<br />
| 181.818<br />
| [[10/9]]<br />
| 182.404<br />
| −0.586<br />
| Major 2nd<br />
| M2<br />
| E<br />
|-<br />
| 6<br />
| 218.182<br />
| [[17/15]]<br />
| 216.687<br />
| +1.495<br />
| Augmented 2nd<br />
| A2<br />
| E#<br />
|-<br />
| 7<br />
| 254.545<br />
| [[15/13]]<br />
| 247.741<br />
| +6.804<br />
| Double-aug 2nd/Double-dim 3rd<br />
| AA2/dd3<br />
| Ex/Fbb<br />
|-<br />
| 8<br />
| 290.909<br />
| [[13/11]]<br />
| 289.210<br />
| +1.699<br />
| Diminished 3rd<br />
| d3<br />
| Fb<br />
|-<br />
| 9<br />
| 327.273<br />
| [[6/5]]<br />
| 315.641<br />
| +11.631<br />
| Minor 3rd<br />
| m3<br />
| F<br />
|-<br />
| 10<br />
| 363.636<br />
| [[16/13]]<br />
| 359.472<br />
| +4.164<br />
| Major 3rd<br />
| M3<br />
| F#<br />
|-<br />
| 11<br />
| 400.000<br />
| [[5/4]]<br />
| 386.314<br />
| +13.686<br />
| Augmented 3rd<br />
| A3<br />
| Fx<br />
|-<br />
| 12<br />
| 436.364<br />
| [[9/7]]<br />
| 435.084<br />
| +1.280<br />
| Double-dim 4th<br />
| dd4<br />
| Gbb<br />
|-<br />
| 13<br />
| 472.727<br />
| [[21/16]]<br />
| 470.781<br />
| +1.946<br />
| Diminished 4th<br />
| d4<br />
| Gb<br />
|-<br />
| 14<br />
| 509.091<br />
| [[4/3]]<br />
| 498.045<br />
| +11.046<br />
| Perfect 4th<br />
| P4<br />
| G<br />
|-<br />
| 15<br />
| 545.455<br />
| [[11/8]]<br />
| 551.318<br />
| −5.863<br />
| Augmented 4th<br />
| A4<br />
| G#<br />
|-<br />
| 16<br />
| 581.818<br />
| [[7/5]]<br />
| 582.513<br />
| −0.694<br />
| Double-aug 4th<br />
| AA4<br />
| Gx<br />
|-<br />
| 17<br />
| 618.182<br />
| [[10/7]]<br />
| 617.488<br />
| +0.694<br />
| Double-dim 5th<br />
| dd5<br />
| Abb<br />
|-<br />
| 18<br />
| 654.545<br />
| [[16/11]]<br />
| 648.682<br />
| +5.863<br />
| Diminished 5th<br />
| d5<br />
| Ab<br />
|-<br />
| 19<br />
| 690.909<br />
| [[3/2]]<br />
| 701.955<br />
| −11.046<br />
| Perfect 5th<br />
| P5<br />
| A<br />
|-<br />
| 20<br />
| 727.273<br />
| [[32/21]]<br />
| 729.219<br />
| -1.946<br />
| Augmented 5th<br />
| A5<br />
| A#<br />
|-<br />
| 21<br />
| 763.636<br />
| [[14/9]]<br />
| 764.916<br />
| −1.280<br />
| Double-aug 5th<br />
| AA5<br />
| Ax<br />
|-<br />
| 22<br />
| 800.000<br />
| [[8/5]]<br />
| 813.686<br />
| −13.686<br />
| Double-dim 6th<br />
| d6<br />
| Bbb<br />
|-<br />
| 23<br />
| 836.364<br />
| [[13/8]]<br />
| 840.528<br />
| −4.164<br />
| Minor 6th<br />
| m6<br />
| Bb<br />
|-<br />
| 24<br />
| 872.727<br />
| [[5/3]]<br />
| 884.359<br />
| −11.631<br />
| Major 6th<br />
| M6<br />
| B<br />
|-<br />
| 25<br />
| 909.091<br />
| [[22/13]]<br />
| 910.790<br />
| −1.699<br />
| Augmented 6th<br />
| A6<br />
| B#<br />
|-<br />
| 26<br />
| 945.455<br />
| [[12/7]]<br />
| 933.129<br />
| +12.325<br />
| Double-aug 6th/Double-dim 7th<br />
| AA6/dd7<br />
| Bx/Cbb<br />
|-<br />
| 27<br />
| 981.818<br />
| [[30/17]]<br />
| 983.313<br />
| −1.495<br />
| Diminished 7th<br />
| d7<br />
| Cb<br />
|-<br />
| 28<br />
| 1018.182<br />
| [[9/5]]<br />
| 1017.596<br />
| +0.586<br />
| Minor 7th<br />
| m7<br />
| C<br />
|-<br />
| 29<br />
| 1054.545<br />
| [[11/6]]<br />
| 1049.363<br />
| +5.183<br />
| Major 7th<br />
| M7<br />
| C#<br />
|-<br />
| 30<br />
| 1090.909<br />
| [[15/8]]<br />
| 1088.268<br />
| +2.640<br />
| Augmented 7th<br />
| A7<br />
| Cx<br />
|-<br />
| 31<br />
| 1127.273<br />
| [[23/12]]<br />
| 1126.319<br />
| −0.953<br />
| Double-dim 8ve<br />
| dd8<br />
| Dbb<br />
|-<br />
| 32<br />
| 1163.636<br />
| [[47/24]]<br />
| 1163.551<br />
| +0.085<br />
| Diminished 8ve<br />
| d8<br />
| Db<br />
|-<br />
| 33<br />
| 1200<br />
| [[2/1]]<br />
| 1200<br />
| 0<br />
| Perfect Octave<br />
| P8<br />
| D<br />
|}<br />
<br />
== Notation ==<br />
=== Standard notation ===<br />
Because the [[chromatic semitone]] in 33edo is 1 step, 33edo can be notated using only naturals, sharps, and flats. However, many key signatures will require double- and triple-sharps and flats, which means that notation in distant keys can be very unwieldy.<br />
<br />
{{sharpness-sharp1}}<br />
<br />
=== Sagittal notation ===<br />
This notation uses the same sagittal sequence as EDOs [[23edo#Sagittal notation|23]] and [[28edo#Sagittal notation|28]].<br />
<br />
<imagemap><br />
File:33-EDO_Sagittal.svg<br />
desc none<br />
rect 80 0 300 50 [[Sagittal_notation]]<br />
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]<br />
rect 20 80 399 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]<br />
default [[File:33-EDO_Sagittal.svg]]<br />
</imagemap><br />
<br />
== Approximation to JI ==<br />
{{Q-odd-limit intervals}}<br />
{{Q-odd-limit intervals|32.87|apx=val|header=none|tag=none|title=15-odd-limit intervals by 33cd val mapping}}<br />
<br />
== Regular temperament properties ==<br />
{| class="wikitable center-4 center-5 center-6"<br />
|-<br />
! rowspan="2" | [[Subgroup]]<br />
! rowspan="2" | [[Comma list]]<br />
! rowspan="2" | [[Mapping]]<br />
! rowspan="2" | Optimal<br>8ve stretch (¢)<br />
! colspan="2" | Tuning error<br />
|-<br />
! [[TE error|Absolute]] (¢)<br />
! [[TE simple badness|Relative]] (%)<br />
|-<br />
| 2.3<br />
| {{monzo| -52 33 }}<br />
| {{mapping| 33 52 }}<br />
| +3.48<br />
| 3.49<br />
| 9.59<br />
|-<br />
| 2.3.5<br />
| 81/80, 1171875/1048576<br />
| {{mapping| 33 52 76 }} (33c)<br />
| +5.59<br />
| 4.13<br />
| 11.29<br />
|-<br />
| 2.3.5.7<br />
| 49/48, 81/80, 1875/1792<br />
| {{mapping| 33 52 76 92 }} (33cd)<br />
| +6.29<br />
| 3.77<br />
| 10.31<br />
|-<br />
| 2.3.5.7.11<br />
| 45/44, 49/48, 81/80, 1375/1344<br />
| {{mapping| 33 52 76 92 114 }} (33cd)<br />
| +5.36<br />
| 3.84<br />
| 10.50<br />
|-<br />
| 2.3.5.7.11.13<br />
| 45/44, 49/48, 65/64, 81/80, 275/273<br />
| {{mapping| 33 52 76 92 114 122 }} (33cd)<br />
| +4.65<br />
| 3.84<br />
| 10.52<br />
|}<br />
<br />
=== Rank-2 temperaments ===<br />
{| class="wikitable center-all left-5"<br />
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator<br />
|-<br />
! Periods<br>per 8ve<br />
! Generator*<br />
! Cents*<br />
! Associated<br>ratio*<br />
! Temperaments<br />
|-<br />
| 1<br />
| 2\33<br />
| 72.73<br />
| 21/20<br />
| [[Slurpee]] (33)<br />
|-<br />
| 1<br />
| 4\33<br />
| 145.45<br />
| 12/11<br />
| [[Bohpier]] (33cd)<br />
|-<br />
| 1<br />
| 7\33<br />
| 254.55<br />
| 8/7<br />
| [[Godzilla]] (33cd)<br />
|-<br />
| 1<br />
| 8\33<br />
| 290.91<br />
| 25/21<br />
| [[Quasitemp]] (33b)<br />
|-<br />
| 1<br />
| 10\33<br />
| 363.64<br />
| 49/40<br />
| [[Submajor]] (33ee) / [[interpental]] (33e)<br />
|-<br />
| 1<br />
| 14\33<br />
| 509.09<br />
| 4/3<br />
| [[Flattertone]] (33cd)<br>[[Deeptone]] a.k.a. tragicomical (33)<br />
|-<br />
| 1<br />
| 16\33<br />
| 581.82<br />
| 7/5<br />
| [[Tritonic]] (33)<br />
|-<br />
| 3<br />
| 7\33<br>(4\33)<br />
| 254.55<br>(145.45)<br />
| 8/7<br>(12/11)<br />
| [[Triforce]] (33d)<br />
|-<br />
| 3<br />
| 13\33<br>(2\33)<br />
| 472.73<br>(72.73)<br />
| 4/3<br>(25/24)<br />
| [[Inflated]] (33bcddd)<br />
|-<br />
| 3<br />
| 14\33<br>(3\33)<br />
| 509.09<br>(98.09)<br />
| 4/3<br>(16/15)<br />
| [[August]] (33)<br />
|}<br />
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct<br />
<br />
=== Uniform maps ===<br />
{{Uniform map|min=32.8|max=33.2}}<br />
<br />
== Octave stretch or compression ==<br />
33edo is nearby to many other [[equal tuning]]s which can act as stretched or compressed versions of 33edo, improving some of its harmonics at the expense of others.<br />
<br />
Useful options include:<br />
* Stretched: [[ed5|76ed5]], [[ed7|92ed7]], [[52edt]], [[zpi|138zpi]]<br />
* Compressed: [[ed7|93ed7]], [[ed5|77ed5]], [[equal tuning|115ed11]]<br />
<br />
[[File:33edo.png|alt=33edo.png|966x199px|33edo.png]]<br />
<br />
== Scales ==<br />
* {{main|List of MOS scales in {{ROOTPAGENAME}}}}<br />
* Approximate [[12afdo]], 4 3 4 3 3 2 3 2 3 2 2 2<br />
* August[12], 3 2 3 3 3 2 3 3 3 2 3 3<br />
* [[Diasem]], 5 3 5 1 5 3 5 1 5 (*right-handed)<br />
* Diasem, 5 1 5 3 5 1 5 3 5 (*left-handed)<br />
* [[Diaslen]] (4sR), 1 5 1 5 2 5 1 5 1 5 2<br />
* Diaslen (4sL), 2 5 1 5 1 5 2 5 1 5 1<br />
* Diaslen (4sC), 1 5 2 5 1 5 1 5 2 5 1<br />
* Elevenplus, 3 3 3 3 3 3 1 2 3 3 3 3 (approximated from [[22edo]])<br />
* Flattertone[7], 5 5 4 5 5 5 4 (diatonic)<br />
** Fun 5-tone subset of Flattertone[7], 9 5 5 4 10<br />
* Flattertone[12], 4 1 4 1 4 1 4 4 1 4 1 4 (chromatic)<br />
* Flattertone[19], 3 1 3 1 1 3 1 1 3 1 3 1 1 3 1 1 3 1 1 (enharmonic)<br />
* Iranian Calendar, 5 4 4 4 4 4 4 4<br />
* Semiquartal, 5 5 2 5 2 5 2 5 2<br />
* Semiquartal[14], 3 2 3 2 2 3 2 2 3 2 2<br />
* Blended slurpee{{idio}}, 3 1 2 2 3 3 5 3 3 2 2 4 ([[modmos]] of slurpee[12])<br />
{{Todo|expand scales list}}<br />
<br />
== Delta-rational harmony ==<br />
The tables below show chords that approximate 3-integer-limit [[delta-rational]] chords with least-squares error less than 0.001.<br />
<br />
=== Fully delta-rational triads ===<br />
{| class="mw-collapsible mw-collapsed class="wikitable sortable"<br />
|-<br />
! Steps<br />
! Delta signature<br />
! Least-squares error<br />
|-<br />
| 0,1,2<br />
| +1+1<br />
| 0.00021<br />
|-<br />
| 0,1,3<br />
| +1+2<br />
| 0.00048<br />
|-<br />
| 0,1,4<br />
| +1+3<br />
| 0.00078<br />
|-<br />
| 0,2,3<br />
| +2+1<br />
| 0.00039<br />
|-<br />
| 0,2,4<br />
| +1+1<br />
| 0.00087<br />
|-<br />
| 0,3,4<br />
| +3+1<br />
| 0.00056<br />
|-<br />
| 0,3,11<br />
| +1+3<br />
| 0.00007<br />
|-<br />
| 0,5,8<br />
| +3+2<br />
| 0.00084<br />
|-<br />
| 0,8,18<br />
| +2+3<br />
| 0.00082<br />
|-<br />
| 0,9,20<br />
| +2+3<br />
| 0.00076<br />
|-<br />
| 0,12,17<br />
| +2+1<br />
| 0.00048<br />
|-<br />
| 0,13,20<br />
| +3+2<br />
| 0.00063<br />
|-<br />
| 0,15,21<br />
| +2+1<br />
| 0.00063<br />
|-<br />
| 0,16,28<br />
| +1+1<br />
| 0.00082<br />
|-<br />
| 0,18,25<br />
| +2+1<br />
| 0.00081<br />
|-<br />
| 0,18,31<br />
| +1+1<br />
| 0.00058<br />
|-<br />
| 0,19,24<br />
| +3+1<br />
| 0.00095<br />
|}<br />
<br />
=== Partially delta-rational tetrads ===<br />
{| class="mw-collapsible mw-collapsed class="wikitable sortable"<br />
|-<br />
! Steps<br />
! Delta signature<br />
! Least-squares error<br />
|-<br />
| 0,1,2,3<br />
| +1+?+1<br />
| 0.00053<br />
|-<br />
| 0,1,2,4<br />
| +1+?+2<br />
| 0.00094<br />
|-<br />
| 0,1,3,4<br />
| +1+?+1<br />
| 0.00080<br />
|-<br />
| 0,1,17,18<br />
| +2+?+3<br />
| 0.00073<br />
|-<br />
| 0,1,17,19<br />
| +1+?+3<br />
| 0.00071<br />
|-<br />
| 0,1,18,19<br />
| +2+?+3<br />
| 0.00042<br />
|-<br />
| 0,1,18,20<br />
| +1+?+3<br />
| 0.00032<br />
|-<br />
| 0,1,19,20<br />
| +2+?+3<br />
| 0.00010<br />
|-<br />
| 0,1,19,21<br />
| +1+?+3<br />
| 0.00008<br />
|-<br />
| 0,1,20,21<br />
| +2+?+3<br />
| 0.00023<br />
|-<br />
| 0,1,20,22<br />
| +1+?+3<br />
| 0.00049<br />
|-<br />
| 0,1,21,22<br />
| +2+?+3<br />
| 0.00056<br />
|-<br />
| 0,1,21,23<br />
| +1+?+3<br />
| 0.00091<br />
|-<br />
| 0,1,22,23<br />
| +2+?+3<br />
| 0.00090<br />
|-<br />
| 0,1,31,32<br />
| +1+?+2<br />
| 0.00071<br />
|-<br />
| 0,2,3,4<br />
| +2+?+1<br />
| 0.00077<br />
|-<br />
| 0,2,6,11<br />
| +1+?+3<br />
| 0.00094<br />
|-<br />
| 0,2,7,12<br />
| +1+?+3<br />
| 0.00013<br />
|-<br />
| 0,2,8,13<br />
| +1+?+3<br />
| 0.00069<br />
|-<br />
| 0,2,12,13<br />
| +3+?+2<br />
| 0.00083<br />
|-<br />
| 0,2,12,15<br />
| +1+?+2<br />
| 0.00087<br />
|-<br />
| 0,2,13,14<br />
| +3+?+2<br />
| 0.00045<br />
|-<br />
| 0,2,13,16<br />
| +1+?+2<br />
| 0.00014<br />
|-<br />
| 0,2,14,15<br />
| +3+?+2<br />
| 0.00008<br />
|-<br />
| 0,2,14,17<br />
| +1+?+2<br />
| 0.00060<br />
|-<br />
| 0,2,15,16<br />
| +3+?+2<br />
| 0.00031<br />
|-<br />
| 0,2,16,17<br />
| +3+?+2<br />
| 0.00071<br />
|-<br />
| 0,2,18,20<br />
| +2+?+3<br />
| 0.00084<br />
|-<br />
| 0,2,18,22<br />
| +1+?+3<br />
| 0.00024<br />
|-<br />
| 0,2,19,21<br />
| +2+?+3<br />
| 0.00020<br />
|-<br />
| 0,2,19,23<br />
| +1+?+3<br />
| 0.00058<br />
|-<br />
| 0,2,20,22<br />
| +2+?+3<br />
| 0.00046<br />
|-<br />
| 0,3,4,5<br />
| +3+?+1<br />
| 0.00097<br />
|-<br />
| 0,3,5,9<br />
| +2+?+3<br />
| 0.00010<br />
|-<br />
| 0,3,6,10<br />
| +2+?+3<br />
| 0.00090<br />
|-<br />
| 0,3,7,12<br />
| +1+?+2<br />
| 0.00074<br />
|-<br />
| 0,3,8,13<br />
| +1+?+2<br />
| 0.00037<br />
|-<br />
| 0,3,10,17<br />
| +1+?+3<br />
| 0.00009<br />
|-<br />
| 0,3,17,23<br />
| +1+?+3<br />
| 0.00096<br />
|-<br />
| 0,3,18,22<br />
| +1+?+2<br />
| 0.00088<br />
|-<br />
| 0,3,18,24<br />
| +1+?+3<br />
| 0.00027<br />
|-<br />
| 0,3,19,20<br />
| +2+?+1<br />
| 0.00059<br />
|-<br />
| 0,3,19,21<br />
| +1+?+1<br />
| 0.00063<br />
|-<br />
| 0,3,19,22<br />
| +2+?+3<br />
| 0.00030<br />
|-<br />
| 0,3,19,23<br />
| +1+?+2<br />
| 0.00023<br />
|-<br />
| 0,3,20,21<br />
| +2+?+1<br />
| 0.00014<br />
|-<br />
| 0,3,20,22<br />
| +1+?+1<br />
| 0.00015<br />
|-<br />
| 0,3,20,23<br />
| +2+?+3<br />
| 0.00070<br />
|-<br />
| 0,3,21,22<br />
| +2+?+1<br />
| 0.00032<br />
|-<br />
| 0,3,21,23<br />
| +1+?+1<br />
| 0.00095<br />
|-<br />
| 0,3,22,23<br />
| +2+?+1<br />
| 0.00078<br />
|-<br />
| 0,3,27,32<br />
| +1+?+3<br />
| 0.00004<br />
|-<br />
| 0,4,5,12<br />
| +1+?+2<br />
| 0.00026<br />
|-<br />
| 0,4,6,16<br />
| +1+?+3<br />
| 0.00066<br />
|-<br />
| 0,4,8,13<br />
| +2+?+3<br />
| 0.00023<br />
|-<br />
| 0,4,11,20<br />
| +1+?+3<br />
| 0.00023<br />
|-<br />
| 0,4,13,14<br />
| +3+?+1<br />
| 0.00091<br />
|-<br />
| 0,4,13,19<br />
| +1+?+2<br />
| 0.00048<br />
|-<br />
| 0,4,14,15<br />
| +3+?+1<br />
| 0.00050<br />
|-<br />
| 0,4,14,16<br />
| +3+?+2<br />
| 0.00055<br />
|-<br />
| 0,4,14,17<br />
| +1+?+1<br />
| 0.00021<br />
|-<br />
| 0,4,15,16<br />
| +3+?+1<br />
| 0.00009<br />
|-<br />
| 0,4,15,17<br />
| +3+?+2<br />
| 0.00023<br />
|-<br />
| 0,4,15,18<br />
| +1+?+1<br />
| 0.00085<br />
|-<br />
| 0,4,16,17<br />
| +3+?+1<br />
| 0.00034<br />
|-<br />
| 0,4,17,18<br />
| +3+?+1<br />
| 0.00077<br />
|-<br />
| 0,4,17,25<br />
| +1+?+3<br />
| 0.00043<br />
|-<br />
| 0,4,19,23<br />
| +2+?+3<br />
| 0.00041<br />
|-<br />
| 0,4,20,24<br />
| +2+?+3<br />
| 0.00094<br />
|-<br />
| 0,4,22,27<br />
| +1+?+2<br />
| 0.00020<br />
|-<br />
| 0,4,24,31<br />
| +1+?+3<br />
| 0.00022<br />
|-<br />
| 0,5,6,9<br />
| +3+?+2<br />
| 0.00003<br />
|-<br />
| 0,5,7,10<br />
| +3+?+2<br />
| 0.00097<br />
|-<br />
| 0,5,7,19<br />
| +1+?+3<br />
| 0.00004<br />
|-<br />
| 0,5,9,17<br />
| +1+?+2<br />
| 0.00017<br />
|-<br />
| 0,5,10,16<br />
| +2+?+3<br />
| 0.00019<br />
|-<br />
| 0,5,11,13<br />
| +2+?+1<br />
| 0.00087<br />
|-<br />
| 0,5,11,15<br />
| +1+?+1<br />
| 0.00018<br />
|-<br />
| 0,5,12,14<br />
| +2+?+1<br />
| 0.00011<br />
|-<br />
| 0,5,12,23<br />
| +1+?+3<br />
| 0.00067<br />
|-<br />
| 0,5,13,15<br />
| +2+?+1<br />
| 0.00067<br />
|-<br />
| 0,5,16,23<br />
| +1+?+2<br />
| 0.00008<br />
|-<br />
| 0,5,17,27<br />
| +1+?+3<br />
| 0.00055<br />
|-<br />
| 0,5,19,24<br />
| +2+?+3<br />
| 0.00051<br />
|-<br />
| 0,5,22,31<br />
| +1+?+3<br />
| 0.00057<br />
|-<br />
| 0,5,24,30<br />
| +1+?+2<br />
| 0.00036<br />
|-<br />
| 0,5,25,26<br />
| +3+?+1<br />
| 0.00071<br />
|-<br />
| 0,5,25,27<br />
| +3+?+2<br />
| 0.00082<br />
|-<br />
| 0,5,25,28<br />
| +1+?+1<br />
| 0.00045<br />
|-<br />
| 0,5,26,27<br />
| +3+?+1<br />
| 0.00018<br />
|-<br />
| 0,5,26,28<br />
| +3+?+2<br />
| 0.00016<br />
|-<br />
| 0,5,26,29<br />
| +1+?+1<br />
| 0.00090<br />
|-<br />
| 0,5,27,28<br />
| +3+?+1<br />
| 0.00035<br />
|-<br />
| 0,5,28,29<br />
| +3+?+1<br />
| 0.00090<br />
|-<br />
| 0,6,7,17<br />
| +1+?+2<br />
| 0.00087<br />
|-<br />
| 0,6,8,22<br />
| +1+?+3<br />
| 0.00045<br />
|-<br />
| 0,6,9,14<br />
| +1+?+1<br />
| 0.00031<br />
|-<br />
| 0,6,11,18<br />
| +2+?+3<br />
| 0.00093<br />
|-<br />
| 0,6,12,21<br />
| +1+?+2<br />
| 0.00036<br />
|-<br />
| 0,6,12,25<br />
| +1+?+3<br />
| 0.00032<br />
|-<br />
| 0,6,15,18<br />
| +3+?+2<br />
| 0.00026<br />
|-<br />
| 0,6,16,19<br />
| +3+?+2<br />
| 0.00095<br />
|-<br />
| 0,6,16,28<br />
| +1+?+3<br />
| 0.00053<br />
|-<br />
| 0,6,18,26<br />
| +1+?+2<br />
| 0.00064<br />
|-<br />
| 0,6,19,25<br />
| +2+?+3<br />
| 0.00062<br />
|-<br />
| 0,6,20,24<br />
| +1+?+1<br />
| 0.00052<br />
|-<br />
| 0,6,21,23<br />
| +2+?+1<br />
| 0.00031<br />
|-<br />
| 0,6,21,32<br />
| +1+?+3<br />
| 0.00033<br />
|-<br />
| 0,6,22,24<br />
| +2+?+1<br />
| 0.00063<br />
|-<br />
| 0,6,25,32<br />
| +1+?+2<br />
| 0.00034<br />
|-<br />
| 0,7,8,14<br />
| +1+?+1<br />
| 0.00029<br />
|-<br />
| 0,7,8,24<br />
| +1+?+3<br />
| 0.00080<br />
|-<br />
| 0,7,9,11<br />
| +3+?+1<br />
| 0.00066<br />
|-<br />
| 0,7,9,12<br />
| +2+?+1<br />
| 0.00041<br />
|-<br />
| 0,7,9,13<br />
| +3+?+2<br />
| 0.00019<br />
|-<br />
| 0,7,10,12<br />
| +3+?+1<br />
| 0.00009<br />
|-<br />
| 0,7,10,13<br />
| +2+?+1<br />
| 0.00070<br />
|-<br />
| 0,7,11,13<br />
| +3+?+1<br />
| 0.00087<br />
|-<br />
| 0,7,12,27<br />
| +1+?+3<br />
| 0.00041<br />
|-<br />
| 0,7,16,30<br />
| +1+?+3<br />
| 0.00098<br />
|-<br />
| 0,7,17,22<br />
| +1+?+1<br />
| 0.00008<br />
|-<br />
| 0,7,19,26<br />
| +2+?+3<br />
| 0.00073<br />
|-<br />
| 0,7,20,29<br />
| +1+?+2<br />
| 0.00002<br />
|-<br />
| 0,7,23,26<br />
| +3+?+2<br />
| 0.00010<br />
|-<br />
| 0,7,28,32<br />
| +1+?+1<br />
| 0.00033<br />
|-<br />
| 0,7,29,31<br />
| +2+?+1<br />
| 0.00020<br />
|-<br />
| 0,7,30,32<br />
| +2+?+1<br />
| 0.00091<br />
|-<br />
| 0,8,12,29<br />
| +1+?+3<br />
| 0.00097<br />
|-<br />
| 0,8,13,22<br />
| +2+?+3<br />
| 0.00051<br />
|-<br />
| 0,8,15,21<br />
| +1+?+1<br />
| 0.00062<br />
|-<br />
| 0,8,15,31<br />
| +1+?+3<br />
| 0.00047<br />
|-<br />
| 0,8,16,18<br />
| +3+?+1<br />
| 0.00066<br />
|-<br />
| 0,8,16,19<br />
| +2+?+1<br />
| 0.00031<br />
|-<br />
| 0,8,16,20<br />
| +3+?+2<br />
| 0.00043<br />
|-<br />
| 0,8,16,27<br />
| +1+?+2<br />
| 0.00090<br />
|-<br />
| 0,8,17,19<br />
| +3+?+1<br />
| 0.00022<br />
|-<br />
| 0,8,17,20<br />
| +2+?+1<br />
| 0.00098<br />
|-<br />
| 0,8,19,27<br />
| +2+?+3<br />
| 0.00085<br />
|-<br />
| 0,8,24,29<br />
| +1+?+1<br />
| 0.00020<br />
|-<br />
| 0,9,11,16<br />
| +3+?+2<br />
| 0.00051<br />
|-<br />
| 0,9,13,20<br />
| +1+?+1<br />
| 0.00002<br />
|-<br />
| 0,9,14,24<br />
| +2+?+3<br />
| 0.00073<br />
|-<br />
| 0,9,18,30<br />
| +1+?+2<br />
| 0.00090<br />
|-<br />
| 0,9,19,28<br />
| +2+?+3<br />
| 0.00096<br />
|-<br />
| 0,9,21,27<br />
| +1+?+1<br />
| 0.00040<br />
|-<br />
| 0,9,22,24<br />
| +3+?+1<br />
| 0.00087<br />
|-<br />
| 0,9,22,25<br />
| +2+?+1<br />
| 0.00053<br />
|-<br />
| 0,9,22,26<br />
| +3+?+2<br />
| 0.00026<br />
|-<br />
| 0,9,23,25<br />
| +3+?+1<br />
| 0.00013<br />
|-<br />
| 0,9,23,26<br />
| +2+?+1<br />
| 0.00093<br />
|-<br />
| 0,10,11,26<br />
| +1+?+2<br />
| 0.00035<br />
|-<br />
| 0,10,11,32<br />
| +1+?+3<br />
| 0.00081<br />
|-<br />
| 0,10,12,20<br />
| +1+?+1<br />
| 0.00098<br />
|-<br />
| 0,10,14,18<br />
| +2+?+1<br />
| 0.00050<br />
|-<br />
| 0,10,14,25<br />
| +2+?+3<br />
| 0.00088<br />
|-<br />
| 0,10,15,29<br />
| +1+?+2<br />
| 0.00041<br />
|-<br />
| 0,10,16,21<br />
| +3+?+2<br />
| 0.00055<br />
|-<br />
| 0,10,19,32<br />
| +1+?+2<br />
| 0.00021<br />
|-<br />
| 0,10,27,31<br />
| +3+?+2<br />
| 0.00082<br />
|-<br />
| 0,10,28,30<br />
| +3+?+1<br />
| 0.00045<br />
|-<br />
| 0,10,28,31<br />
| +2+?+1<br />
| 0.00016<br />
|-<br />
| 0,10,29,31<br />
| +3+?+1<br />
| 0.00068<br />
|-<br />
| 0,11,12,18<br />
| +3+?+2<br />
| 0.00030<br />
|-<br />
| 0,11,13,16<br />
| +3+?+1<br />
| 0.00081<br />
|-<br />
| 0,11,14,17<br />
| +3+?+1<br />
| 0.00044<br />
|-<br />
| 0,11,16,31<br />
| +1+?+2<br />
| 0.00064<br />
|-<br />
| 0,11,17,25<br />
| +1+?+1<br />
| 0.00091<br />
|-<br />
| 0,11,19,23<br />
| +2+?+1<br />
| 0.00045<br />
|-<br />
| 0,11,21,26<br />
| +3+?+2<br />
| 0.00074<br />
|-<br />
| 0,12,15,24<br />
| +1+?+1<br />
| 0.00087<br />
|-<br />
| 0,12,15,28<br />
| +2+?+3<br />
| 0.00013<br />
|-<br />
| 0,12,17,23<br />
| +3+?+2<br />
| 0.00054<br />
|-<br />
| 0,12,18,21<br />
| +3+?+1<br />
| 0.00043<br />
|-<br />
| 0,12,19,22<br />
| +3+?+1<br />
| 0.00095<br />
|-<br />
| 0,12,23,27<br />
| +2+?+1<br />
| 0.00083<br />
|-<br />
| 0,12,26,31<br />
| +3+?+2<br />
| 0.00005<br />
|-<br />
| 0,13,14,24<br />
| +1+?+1<br />
| 0.00019<br />
|-<br />
| 0,13,17,22<br />
| +2+?+1<br />
| 0.00085<br />
|-<br />
| 0,13,21,27<br />
| +3+?+2<br />
| 0.00035<br />
|-<br />
| 0,13,22,25<br />
| +3+?+1<br />
| 0.00097<br />
|-<br />
| 0,13,23,26<br />
| +3+?+1<br />
| 0.00054<br />
|-<br />
| 0,13,28,32<br />
| +2+?+1<br />
| 0.00055<br />
|-<br />
| 0,14,17,24<br />
| +3+?+2<br />
| 0.00099<br />
|-<br />
| 0,14,18,28<br />
| +1+?+1<br />
| 0.00043<br />
|-<br />
| 0,14,21,26<br />
| +2+?+1<br />
| 0.00080<br />
|-<br />
| 0,14,25,31<br />
| +3+?+2<br />
| 0.00054<br />
|-<br />
| 0,14,27,30<br />
| +3+?+1<br />
| 0.00050<br />
|-<br />
| 0,15,16,20<br />
| +3+?+1<br />
| 0.00055<br />
|-<br />
| 0,15,17,28<br />
| +1+?+1<br />
| 0.00064<br />
|-<br />
| 0,15,21,28<br />
| +3+?+2<br />
| 0.00045<br />
|-<br />
| 0,15,22,32<br />
| +1+?+1<br />
| 0.00039<br />
|-<br />
| 0,16,18,26<br />
| +3+?+2<br />
| 0.00049<br />
|-<br />
| 0,16,19,25<br />
| +2+?+1<br />
| 0.00031<br />
|-<br />
| 0,16,20,24<br />
| +3+?+1<br />
| 0.00018<br />
|-<br />
| 0,16,25,32<br />
| +3+?+2<br />
| 0.00095<br />
|-<br />
| 0,17,22,28<br />
| +2+?+1<br />
| 0.00091<br />
|-<br />
| 0,17,23,27<br />
| +3+?+1<br />
| 0.00066<br />
|-<br />
| 0,18,27,31<br />
| +3+?+1<br />
| 0.00095<br />
|-<br />
| 0,19,21,28<br />
| +2+?+1<br />
| 0.00065<br />
|-<br />
| 0,20,24,31<br />
| +2+?+1<br />
| 0.00078<br />
|-<br />
| 0,21,22,32<br />
| +3+?+2<br />
| 0.00091<br />
|-<br />
| 0,22,27,32<br />
| +3+?+1<br />
| 0.00038<br />
|}<br />
<br />
== Instruments ==<br />
[[Lumatone mapping for 33edo]]<br />
<br />
== Music ==<br />
=== Modern renderings ===<br />
; {{W|Johann Sebastian Bach}}<br />
* [https://www.youtube.com/watch?v=IhR9oFt5zx4 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)<br />
* [https://www.youtube.com/watch?v=ynPQPm_ekos "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)<br />
<br />
=== 21st century ===<br />
; [[Bryan Deister]]<br />
* [https://www.youtube.com/watch?v=swyP6tB78k0 ''groove 33edo''] (2023)<br />
* [https://www.youtube.com/watch?v=GypR6x_Ih1I ''33edo jam''] (2025)<br />
* [https://www.youtube.com/shorts/mkaaAJEyGFU ''33edo riff''] (2025)<br />
* [https://www.youtube.com/shorts/Lf0CCX88w_w ''33edo improv''] (2025)<br />
<br />
; [[Peter Kosmorsky]]<br />
* [https://www.youtube.com/watch?v=SXgUFxyuLZo ''Deluge'']<br />
<br />
; [[Budjarn Lambeth]]<br />
* [https://youtu.be/scCuGXnj5IY ''Enchanted Shopping Mall''] (2024)<br />
<br />
; [[Claudi Meneghin]]<br />
* [https://www.youtube.com/watch?v=REkrbdesbLo ''Rising Canon on a Ground'', for Baroque Oboe, Bassoon, Violone] (2024) – ([https://www.youtube.com/watch?v=4fhcNPjFv14 for Organ])<br />
* [https://www.youtube.com/watch?v=pkYN8SX6luY ''Lytel Twyelyghte Musicke (Little Twilight Music)'', for Brass and Timpani] (2024)<br />
<br />
; [[Relyt R]]<br />
* from ''Xuixo'' (2023)<br />
** "Nongenerate" [https://relytr.bandcamp.com/track/nondegenerate-33-edo Bandcamp] | [https://open.spotify.com/track/3e2WbgFlAYC4BccPGOWHMo Spotify]<br />
** "Kolmekymmentäkolme" [https://relytr.bandcamp.com/track/kolme-kymment-kolme-33-edo Bandcamp] | [https://open.spotify.com/track/4fx1yQ1RQtEu8EYhNUtN79 Spotify]<br />
<br />
; [[Chris Vaisvil]]<br />
* [http://chrisvaisvil.com/5-5-1-mode-of-33-equal-with-video/ 5 5 1 mode of 33 equal (with video)] [http://micro.soonlabel.com/33edo/20130827_551of33.mp3 play]<br />
<br />
; [[Xeno*n*]]<br />
* [https://www.youtube.com/watch?v=EPB1Rzjwguk ''Mysteries of Thirty-Three''] (2024)<br />
<br />
[[Category:Listen]]<br />
[[Category:Meantone]]<br />
[[Category:Subgroup temperaments]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Template:Uniform_map&diff=224397Template:Uniform map2026-02-20T16:49:26Z<p>ArrowHead294: </p>
<hr />
<div><includeonly>{{#invoke: Uniform_map | print_table<br />
| {{{1|5}}}<br />
| {{{2|11.5}}}<br />
| {{{3|12.5}}}<br />
| prec = {{{prec|4}}}<br />
| edo = {{{edo|{{#rmatch: {{ed title}}|/(\d+)/|\1}}}}}<br />
| debug = {{lc: {{{debug|}}}}}<br />
}}</includeonly><noinclude><br />
{{documentation}}<br />
<br />
[[Category:Templates]]<br />
</noinclude></div>ArrowHead294https://en.xen.wiki/index.php?title=Template:Uniform_map&diff=224395Template:Uniform map2026-02-20T16:47:48Z<p>ArrowHead294: </p>
<hr />
<div><includeonly>{{#invoke: Uniform_map | print_table<br />
| {{{1|5}}}<br />
| {{{2|11.5}}}<br />
| {{{3|12.5}}}<br />
| prec = {{{prec|4}}}<br />
| edo = {{{edo|{{ed title}}}}}<br />
| debug = {{lc: {{{debug|}}}}}<br />
}}</includeonly><noinclude><br />
{{documentation}}<br />
<br />
[[Category:Templates]]<br />
</noinclude></div>ArrowHead294https://en.xen.wiki/index.php?title=Template:Interval_table&diff=224394Template:Interval table2026-02-20T16:40:52Z<p>ArrowHead294: </p>
<hr />
<div><includeonly>{{safesubst:#invoke: Interval_table | interval_table<br />
| tuning={{{1|{{safesubst:PAGENAME}}}}}<br />
| additional={{{additional|}}}<br />
| max_error={{{max_error|35}}}<br />
| debug={{safesubst:lc: {{{debug|}}}}}<br />
}}{{safesubst:#if: {{{debug|}}}||{{Todo|replace auto-generated table of intervals with manually curated table}}}}</includeonly><noinclude><br />
{{Documentation}}<br />
<br />
[[Category:Interval list templates]]<br />
</noinclude></div>ArrowHead294https://en.xen.wiki/index.php?title=S-expression&diff=224390S-expression2026-02-20T11:48:37Z<p>ArrowHead294: /* Glossary */</p>
<hr />
<div>An '''S-expression''' is any product, or ratio of products, of the '''square superparticulars''' S''k'', which are defined as the fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. Commas defined by S-expressions turn out to represent intuitive and wide-reaching families of tempered equivalences, and therefore present a very useful framework to learn for a good understanding of the [[commas]] that appear frequently in xen.<br />
<br />
== Quick rules of S-expressions ==<br />
As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply when they are [[tempering out|tempered out]]. The linked sections provide deeper information into each comma family.<br />
<br />
* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. <br>Tempering out S''k'' equates {{sfrac|''k'' + 1|''k''}} with {{sfrac|''k''|''k'' − 1}} and splits {{sfrac|''k'' + 1|''k'' − 1}} in two.<br />
* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: {{nowrap|'''S''k'' * S(''k'' + 1)'''}}, superparticular fractions of the form {{sfrac|''k''(''k'' + 1)/2|(''k'' − 1)(''k'' + 2)/2}}. <br>Tempering out {{nowrap|S''k'' * S(''k'' + 1)}} equates {{sfrac|''k'' + 2|''k'' + 1}} with {{sfrac|''k''|''k'' − 1}}, and {{sfrac|''k'' + 2|''k''}} with {{sfrac|''k'' + 1|''k'' − 1}}.<br />
* [[#Sk2 * S(k + 1) and S(k − 1) * Sk2 (lopsided commas)|Lopsided commas]]: {{nowrap|'''(S''k'')<sup>2</sup> * S(''k'' + 1)'''}} and {{nowrap|'''(S''k'')<sup>2</sup> * S(''k'' − 1)'''}}. <br>Tempering out the former equates {{sfrac|''k'' + 2|''k''}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>2</sup> and {{sfrac|''k'' + 2|''k'' − 1}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>3</sup>, and tempering out the latter equates {{sfrac|''k''|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>2</sup> and {{sfrac|''k'' + 1|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.<br />
* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out splits {{sfrac|''k'' + 2|''k'' − 1}} into {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.<br />
* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out splits {{sfrac|''k'' + 3|''k'' − 1}} into {{pars|{{sfrac|''k'' + 2|''k''}}}}<sup>2</sup>.<br />
<br />
== S''k'' (square-particulars) ==<br />
A '''square superparticular''', or ''square-particular'' for short, is a [[superparticular]] [[interval]] whose numerator is a square number, which is to say, a superparticular of the form<br />
<br />
<math>\displaystyle \frac {k^2}{k^2 - 1} = \frac {k/(k - 1)}{(k + 1)/k} </math><br />
<br />
which is square-(super)particular ''k'' for a given integer {{nowrap|''k'' &gt; 1}}. A suggested shorthand for this interval is '''S''k''''' for the ''k''-th square superparticular, where the ''S'' stands for "(Shorthand for) Second-order/Square Superparticular". This will be used later in this article as the notation will prove powerful in understanding the commas and implied tempered structures of [[regular temperament]]s. Note that this means {{nowrap|S2 {{=}} [[4/3]]}} is the first musically meaningful square-particular, as {{nowrap|S1 {{=}} 1/0}}.<br />
<br />
Also note that we use the notation S''k''<sup>''p''</sup> to mean (S''k'')<sup>''p''</sup> rather than S(''k''<sup>''p''</sup>) for convenience in the practical analysis of regular temperaments using [[S-expression]]s.<br />
<br />
=== Significance/motivation ===<br />
Square-particulars are important structurally because they are the intervals between consecutive [[superparticular]] [[interval]]s while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the ({{nowrap|''k'' + 1}})th harmonic, as well as the potential representational sacrifices that must be made from that point onward. In other words, understanding the mappings of S''k'' in a given temperament is equivalent to understanding the spacing of consecutive superparticular intervals, and thereby to understanding the way it represents (or tries to represent) the harmonic series.<br />
<br />
=== Table of square-particulars ===<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 31-limit square-particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2<br />
| ([[2/1]])/([[3/2]])<br />
| [[4/3]]<br />
| 3<br />
|-<br />
| S3<br />
| ([[3/2]])/([[4/3]])<br />
| [[9/8]]<br />
| 3<br />
|-<br />
| S4<br />
| ([[4/3]])/([[5/4]])<br />
| [[16/15]]<br />
| 5<br />
|-<br />
| S5<br />
| ([[5/4]])/([[6/5]])<br />
| [[25/24]]<br />
| 5<br />
|-<br />
| S6 = S8*S9<br />
| ([[6/5]])/([[7/6]])<br />
| [[36/35]]<br />
| 7<br />
|-<br />
| S7<br />
| ([[7/6]])/([[8/7]])<br />
| [[49/48]]<br />
| 7<br />
|-<br />
| S8<br />
| ([[8/7]])/([[9/8]])<br />
| [[64/63]]<br />
| 7<br />
|-<br />
| S9 = S6/S8<br />
| ([[9/8]])/([[10/9]])<br />
| [[81/80]]<br />
| 5<br />
|-<br />
| S10<br />
| ([[10/9]])/([[11/10]])<br />
| [[100/99]]<br />
| 11<br />
|-<br />
| S11<br />
| ([[11/10]])/([[12/11]])<br />
| [[121/120]]<br />
| 11<br />
|-<br />
| S12<br />
| ([[12/11]])/([[13/12]])<br />
| [[144/143]]<br />
| 13<br />
|-<br />
| S13<br />
| ([[13/12]])/([[14/13]])<br />
| [[169/168]]<br />
| 13<br />
|-<br />
| S14<br />
| ([[14/13]])/([[15/14]])<br />
| [[196/195]]<br />
| 13<br />
|-<br />
| S15<br />
| ([[15/14]])/([[16/15]])<br />
| [[225/224]]<br />
| 7<br />
|-<br />
| S16<br />
| ([[16/15]])/([[17/16]])<br />
| [[256/255]]<br />
| 17<br />
|-<br />
| S17<br />
| ([[17/16]])/([[18/17]])<br />
| [[289/288]]<br />
| 17<br />
|-<br />
| S18<br />
| ([[18/17]])/([[19/18]])<br />
| [[324/323]]<br />
| 19<br />
|-<br />
| S19<br />
| ([[19/18]])/([[20/19]])<br />
| [[361/360]]<br />
| 19<br />
|-<br />
| S20<br />
| ([[20/19]])/([[21/20]])<br />
| [[400/399]]<br />
| 19<br />
|-<br />
| S21<br />
| ([[21/20]])/([[22/21]])<br />
| [[441/440]]<br />
| 11<br />
|-<br />
| S22<br />
| ([[22/21]])/([[23/22]])<br />
| [[484/483]]<br />
| 23<br />
|-<br />
| S23<br />
| ([[23/22]])/([[24/23]])<br />
| [[529/528]]<br />
| 23<br />
|-<br />
| S24<br />
| ([[24/23]])/([[25/24]])<br />
| [[576/575]]<br />
| 23<br />
|-<br />
| S25<br />
| ([[25/24]])/([[26/25]])<br />
| [[625/624]]<br />
| 13<br />
|-<br />
| S26 = S13/S15<br />
| ([[26/25]])/([[27/26]])<br />
| [[676/675]]<br />
| 13<br />
|-<br />
| S27<br />
| ([[27/26]])/([[28/27]])<br />
| [[729/728]]<br />
| 13<br />
|-<br />
| S28<br />
| ([[28/27]])/([[29/28]])<br />
| [[784/783]]<br />
| 29<br />
|-<br />
| S29<br />
| ([[29/28]])/([[30/29]])<br />
| [[841/840]]<br />
| 29<br />
|-<br />
| S30<br />
| ([[30/29]])/([[31/30]])<br />
| [[900/899]]<br />
| 31<br />
|-<br />
| S31<br />
| ([[31/30]])/([[32/31]])<br />
| [[961/960]]<br />
| 31<br />
|-<br />
| S32<br />
| ([[32/31]])/([[33/32]])<br />
| [[1024/1023]]<br />
| 31<br />
|-<br />
| S33<br />
| ([[33/32]])/([[34/33]])<br />
| [[1089/1088]]<br />
| 17<br />
|-<br />
| S34<br />
| ([[34/33]])/([[35/34]])<br />
| [[1156/1155]]<br />
| 17<br />
|-<br />
| S35 = S49*S50<br />
| ([[35/34]])/([[36/35]])<br />
| [[1225/1224]]<br />
| 17<br />
|-<br />
| S39<br />
| ([[39/38]])/([[40/39]])<br />
| [[1521/1520]]<br />
| 19<br />
|-<br />
| S45<br />
| ([[45/44]])/([[46/45]])<br />
| [[2025/2024]]<br />
| 23<br />
|-<br />
| S49<br />
| ([[49/48]])/([[50/49]])<br />
| [[2401/2400]]<br />
| 7<br />
|-<br />
| S50<br />
| ([[50/49]])/([[51/50]])<br />
| [[2500/2499]]<br />
| 17<br />
|-<br />
| S51<br />
| ([[51/50]])/([[52/51]])<br />
| [[2601/2600]]<br />
| 17<br />
|-<br />
| S55 = S22/S24<br />
| ([[55/54]])/([[56/55]])<br />
| [[3025/3024]]<br />
| 11<br />
|-<br />
| S56<br />
| ([[56/55]])/([[57/56]])<br />
| [[3136/3135]]<br />
| 19<br />
|-<br />
| S57<br />
| ([[57/56]])/([[58/57]])<br />
| [[3249/3248]]<br />
| 29<br />
|-<br />
| S63<br />
| ([[63/62]])/([[64/63]])<br />
| [[3969/3968]]<br />
| 31<br />
|-<br />
| S64<br />
| ([[64/63]])/([[65/64]])<br />
| [[4096/4095]]<br />
| 13<br />
|-<br />
| S65<br />
| ([[65/64]])/([[66/65]])<br />
| [[4225/4224]]<br />
| 13<br />
|-<br />
| S69<br />
| ([[69/68]])/([[70/69]])<br />
| [[4761/4760]]<br />
| 23<br />
|-<br />
| S76<br />
| ([[76/75]])/([[77/76]])<br />
| [[5776/5775]]<br />
| 19<br />
|-<br />
| S77<br />
| ([[77/76]])/([[78/77]])<br />
| [[5929/5928]]<br />
| 19<br />
|-<br />
| S91<br />
| ([[91/90]])/([[92/91]])<br />
| [[8281/8280]]<br />
| 23<br />
|-<br />
| S92<br />
| ([[92/91]])/([[93/92]])<br />
| [[8464/8463]]<br />
| 31<br />
|-<br />
| S99 = S33/S35<br />
| ([[99/98]])/([[100/99]])<br />
| [[9801/9800]]<br />
| 11<br />
|-<br />
| S115<br />
| ([[115/114]])/([[116/115]])<br />
| [[13225/13224]]<br />
| 29<br />
|-<br />
| S116<br />
| ([[116/115]])/([[117/116]])<br />
| [[13456/13455]]<br />
| 29<br />
|-<br />
| S120<br />
| ([[120/119]])/([[121/120]])<br />
| [[14400/14399]]<br />
| 17<br />
|-<br />
| S125<br />
| ([[125/124]])/([[126/125]])<br />
| [[15625/15624]]<br />
| 31<br />
|-<br />
| S144<br />
| ([[144/143]])/([[145/144]])<br />
| [[20736/20735]]<br />
| 29<br />
|-<br />
| S153<br />
| ([[153/152]])/([[154/153]])<br />
| [[23409/23408]]<br />
| 19<br />
|-<br />
| S154<br />
| ([[154/153]])/([[155/154]])<br />
| [[23716/23715]]<br />
| 31<br />
|-<br />
| S155<br />
| ([[155/154]])/([[156/155]])<br />
| [[24025/24024]]<br />
| 31<br />
|-<br />
| S161 = S46/S48<br />
| ([[161/160]])/([[162/161]])<br />
| [[25921/25920]]<br />
| 23<br />
|-<br />
| S169<br />
| ([[169/168]])/([[170/169]])<br />
| [[28561/28560]]<br />
| 17<br />
|-<br />
| S170<br />
| ([[170/169]])/([[171/170]])<br />
| [[28900/28899]]<br />
| 19<br />
|-<br />
| S175<br />
| ([[175/174]])/([[176/175]])<br />
| [[30625/30624]]<br />
| 29<br />
|-<br />
| S208<br />
| ([[208/207]])/([[209/208]])<br />
| [[43264/43263]]<br />
| 23<br />
|-<br />
| S209<br />
| ([[209/208]])/([[210/209]])<br />
| [[43681/43680]]<br />
| 19<br />
|-<br />
| S231<br />
| ([[231/230]])/([[232/231]])<br />
| [[53361/53360]]<br />
| 29<br />
|-<br />
| S289<br />
| ([[289/288]])/([[290/289]])<br />
| [[83521/83520]]<br />
| 29<br />
|-<br />
| S323<br />
| ([[323/322]])/([[324/323]])<br />
| [[104329/104328]]<br />
| 23<br />
|-<br />
| S324<br />
| ([[324/323]])/([[325/324]])<br />
| [[104976/104975]]<br />
| 19<br />
|-<br />
| S341<br />
| ([[341/340]])/([[342/341]])<br />
| [[116281/116280]]<br />
| 31<br />
|-<br />
| S342<br />
| ([[342/341]])/([[343/342]])<br />
| [[116964/116963]]<br />
| 31<br />
|-<br />
| S351 = S78/S80<br />
| ([[351/350]])/([[352/351]])<br />
| [[123201/123200]]<br />
| 13<br />
|-<br />
| S391<br />
| ([[391/390]])/([[392/391]])<br />
| [[152881/152880]]<br />
| 23<br />
|-<br />
| S441<br />
| ([[441/440]])/([[442/441]])<br />
| [[194481/194480]]<br />
| 17<br />
|-<br />
| S494<br />
| ([[494/493]])/([[495/494]])<br />
| [[244036/244035]]<br />
| 29<br />
|-<br />
| S495<br />
| ([[495/494]])/([[496/495]])<br />
| [[245025/245024]]<br />
| 31<br />
|-<br />
| S528<br />
| ([[528/527]])/([[529/528]])<br />
| [[278784/278783]]<br />
| 31<br />
|-<br />
| S551<br />
| ([[551/550]])/([[552/551]])<br />
| [[303601/303600]]<br />
| 29<br />
|-<br />
| S714<br />
| ([[714/713]])/([[715/714]])<br />
| [[509796/509795]]<br />
| 31<br />
|-<br />
| S783<br />
| ([[783/782]])/([[784/783]])<br />
| [[613089/613088]]<br />
| 29<br />
|-<br />
| S1275<br />
| ([[1275/1274]])/([[1276/1275]])<br />
| [[1625625/1625624]]<br />
| 29<br />
|-<br />
| S1519<br />
| ([[1519/1518]])/([[1520/1519]])<br />
| [[2307361/2307360]]<br />
| 31<br />
|-<br />
| S1520<br />
| ([[1520/1519]])/([[1521/1520]])<br />
| [[2310400/2310399]]<br />
| 31<br />
|-<br />
| S2001<br />
| ([[2001/2000]])/([[2002/2001]])<br />
| [[4004001/4004000]]<br />
| 29<br />
|-<br />
| S2024<br />
| ([[2024/2023]])/([[2025/2024]])<br />
| [[4096576/4096575]]<br />
| 23<br />
|-<br />
| S2431<br />
| ([[2431/2430]])/([[2432/2431]])<br />
| [[5909761/5909760]]<br />
| 19<br />
|-<br />
| S3249<br />
| ([[3249/3248]])/([[3250/3249]])<br />
| <font style="font-size:0.88em">[[10556001/10556000]]</font><br />
| 29<br />
|-<br />
| S9801<br />
| ([[9801/9800]])/([[9802/9801]])<br />
| <font style="font-size:0.88em">[[96059601/96059600]]</font><br />
| 29<br />
|-<br />
| S13311<br />
| <font style="font-size:0.86em">([[13311/13310]])/([[13312/13311]])</font><br />
| <font style="font-size:0.79em">[[177182721/177182720]]</font><br />
| 29<br />
|-<br />
| S13455<br />
| <font style="font-size:0.86em">([[13455/13454]])/([[13456/13455]])</font><br />
| <font style="font-size:0.79em">[[181037025/181037024]]</font><br />
| 31<br />
|}<br />
<br />
=== Alternatives to tempering square-particulars ===<br />
It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas {{nowrap|U''k'' {{=}} {{sfrac|S''k''|S(''k'' + 1)}}}}, we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]<ref group="note">In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as U''k'' = S(''k'' - 1)/S''k'' and U''k'' = S(''k'' + 1)/S(''k'' + 2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.</ref>. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because {{sfrac|''k'' + 2|''k'' + 1}} and {{sfrac|''k''|''k'' − 1}} are equidistant from {{sfrac|''k'' + 1|''k''}} (because of tempering {{sfrac|S''k''|S(''k'' + 1)}}), this means that another expression for {{sfrac|S''k''|S(''k'' + 1)}} is the following:<br />
<br />
<math>\displaystyle {\rm S}k / {\rm S} (k + 1) = \frac{(k + 2) / (k - 1)}{((k + 1)/k)^3} </math><br />
<br />
This means you can read the ''k'' and {{nowrap|''k'' + 1}} from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in [[#Sk/S(k + 1) (ultraparticulars)|the table of ultraparticulars]]).<br />
<br />
Furthermore, defining another sequence of commas with [[semiparticular|formula {{sfrac|S''k''|S(''k'' + 2)}} leads to semiparticulars]] which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering consecutive ultraparticulars.<br />
<br />
== {{nowrap|S''k''*S(''k'' + 1)}} (triangle-particulars) ==<br />
=== Significance ===<br />
1. Every triangle-particular is superparticular, so these are efficient commas. (See also the [[#Short proof of the superparticularity of triangle-particulars]].)<br />
<br />
2. Often each individual triangle-particular, taken as a comma, implies other useful equivalences not necessarily corresponding to the general form, speaking of which …<br />
<br />
3. Every triangle-particular is the difference between two nearly-adjacent superparticular intervals {{nowrap|''k'' + 2|''k'' + 1}} and {{nowrap|''k''|''k'' − 1}}.<br />
<br />
4. Tempering any two consecutive square-particulars S''k'' and {{nowrap|S(''k'' + 1)}} implies tempering a triangle-particular, so these are common commas. (See also: [[lopsided comma]]s.)<br />
<br />
5. If we temper {{nowrap|S''k'' * S(''k'' + 1)}} but not S''k'' or {{nowrap|S(''k'' + 1)}}, then one or more intervals of {{sfrac|''k''|''k'' − 1}}, {{sfrac|''k'' + 1|''k''}}, and {{sfrac|''k'' + 2|''k'' + 1}} ''must'' be mapped inconsistently, because:<br />
: If {{nowrap|''k'' + 1|''k''}} is mapped above {{nowrap|{{sfrac|''k'' + 2|''k'' + 1}} ~ {{sfrac|''k''|''k'' − 1}}}} we have {{nowrap|{{sfrac|''k'' + 1|''k''}} &gt; {{sfrac|''k''|''k'' − 1}}}} and if it is mapped below we have {{nowrap|{{sfrac|''k'' + 1|''k''}} &lt; {{sfrac|''k'' + 2|''k'' + 1}}}}.<br />
: (Generalisations of this and their implications for consistency are discussed in [[#Sk*S(k + 1)*…*S(k + n − 1) (1/n-square-particulars)|the section covering 1/''n''-square-particulars]].)<br />
<br />
=== Meaning ===<br />
Notice that if we equate {{sfrac|''k'' + 2|''k'' + 1}} with {{sfrac|''k''|''k'' − 1}} (by [[tempering out]] their difference), then multiply both sides by {{sfrac|''k'' + 1|''k''}}, we have:<br />
<br />
<math>\displaystyle \left(\frac{k + 2}{k + 1}\right)\left(\frac{k + 1}{k}\right) = \left(\frac{k + 1}{k}\right)\left(\frac{k}{k - 1}\right) </math><br />
<br />
which simplifies to:<br />
<br />
<math>\displaystyle \frac{k + 2}{k} = \frac{k + 1}{k - 1} </math><br />
<br />
This means that if we temper: <math>{\rm S}k \cdot {\rm S}(k+1) = \frac{k/(k-1)}{(k+1)/k} \cdot \frac{(k+1)/k}{(k+2)/(k+1)} = \frac{k/(k-1)}{(k+2)/(k+1)}</math><br />
<br />
… then this equivalence is achieved. Note that there is little to no reason to not also temper S''k'' and {{nowrap|S(''k'' + 1)}} individually unless other considerations seem to force your hand.<br />
<br />
=== Short proof of the superparticularity of triangle-particulars ===<br />
<math>\displaystyle S(k)*S(k + 1) = \frac{\frac{k}{k - 1}}{\frac{k + 2}{k + 1}} = \frac{k(k + 1)}{(k - 1)(k + 2)} = \frac{k^2 + k}{k^2 + k - 2} </math><br />
<br />
Then notice that {{nowrap|''k''<sup>2</sup> + ''k''}} is always a multiple of 2, therefore the above always simplifies to a superparticular. Half of this superparticular is halfway between the corresponding square-particulars, and because of its composition it could therefore be reasoned that it'd likely be half as accurate as tempering either of the square-particulars individually, so these are "1/2-square-particulars" in a sense, and half of a square is a triangle, which is not a coincidence here because the numerators of all of these superparticular intervals/commas are [[triangular number]]s! (Hence the alternative name "[[triangle-particular]]".)<br />
<br />
=== Table of triangle-particulars ===<br />
For completeness, all the intervals of this form are included, because of their structural importance for JI, and for the possibility of (in)consistency of mappings when tempered for the above reason.<br />
<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 31-limit triangle-particulars<ref group="note">After 75, 76, 77, 78, streaks of four consecutive harmonics in the 23-limit become very sparse. The last few streaks are deeply related to the consistency and structure of [[311edo]], as [[311edo]] can be described as the unique 23-limit temperament that tempers all triangle-particulars from [[595/594]] up to [[21736/21735]]. It also tempers all the square-particulars composing those triangle-particulars with the exception of S169 and S170. It also maps the corresponding intervals of the 77-odd-limit consistently. 170/169 is the only place where the logic seems to "break" as it is mapped to 2 steps instead of 3 meaning the mapping of that superparticular is inconsistent.</ref><br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3<br />
| ([[3/1]])/([[2/1]])<br />
| [[3/2]]<br />
| 3<br />
|-<br />
| S3*S4<br />
| ([[3/2]])/([[5/4]])<br />
| [[6/5]]<br />
| 5<br />
|-<br />
| S4*S5<br />
| ([[4/3]])/([[6/5]])<br />
| [[10/9]]<br />
| 5<br />
|-<br />
| S5*S6<br />
| ([[5/4]])/([[7/6]])<br />
| [[15/14]]<br />
| 7<br />
|-<br />
| S6*S7<br />
| ([[6/5]])/([[8/7]])<br />
| [[21/20]]<br />
| 7<br />
|-<br />
| S7*S8 = S4/S6<br />
| ([[7/6]])([[9/8]])<br />
| [[28/27]]<br />
| 7<br />
|-<br />
| S8*S9 = S6<br />
| ([[8/7]])/([[10/9]])<br />
| [[36/35]]<br />
| 7<br />
|-<br />
| S9*S10<br />
| ([[9/8]])/([[11/10]])<br />
| [[45/44]]<br />
| 11<br />
|-<br />
| S10*S11<br />
| ([[10/9]])/([[12/11]])<br />
| [[55/54]]<br />
| 11<br />
|-<br />
| S11*S12<br />
| ([[11/10]])/([[13/12]])<br />
| [[66/65]]<br />
| 13<br />
|-<br />
| S12*S13<br />
| ([[12/11]])/([[14/13]])<br />
| [[78/77]]<br />
| 13<br />
|-<br />
| S13*S14<br />
| ([[13/12]])/([[15/14]])<br />
| [[91/90]]<br />
| 13<br />
|-<br />
| S14*S15<br />
| ([[14/13]])/([[16/15]])<br />
| [[105/104]]<br />
| 13<br />
|-<br />
| S15*S16<br />
| ([[15/14]])/([[17/16]])<br />
| [[120/119]]<br />
| 17<br />
|-<br />
| S16*S17<br />
| ([[16/15]])/([[18/17]])<br />
| [[136/135]]<br />
| 17<br />
|-<br />
| S17*S18<br />
| ([[17/16]])/([[19/18]])<br />
| [[153/152]]<br />
| 19<br />
|-<br />
| S18*S19<br />
| ([[18/17]])/([[20/19]])<br />
| [[171/170]]<br />
| 19<br />
|-<br />
| S19*S20<br />
| ([[19/18]])/([[21/20]])<br />
| [[190/189]]<br />
| 19<br />
|-<br />
| S20*S21<br />
| ([[20/19]])/([[22/21]])<br />
| [[210/209]]<br />
| 19<br />
|-<br />
| S21*S22<br />
| ([[21/20]])/([[23/22]])<br />
| [[231/230]]<br />
| 23<br />
|-<br />
| S22*S23<br />
| ([[22/21]])/([[24/23]])<br />
| [[253/252]]<br />
| 23<br />
|-<br />
| S23*S24<br />
| ([[23/22]])/([[25/24]])<br />
| [[276/275]]<br />
| 23<br />
|-<br />
| S24*S25<br />
| ([[24/23]])/([[26/25]])<br />
| [[300/299]]<br />
| 23<br />
|-<br />
| S25*S26 = S10/S12<br />
| ([[25/24]])/([[27/26]])<br />
| [[325/324]]<br />
| 13<br />
|-<br />
| S26*S27<br />
| ([[26/25]])/([[28/27]])<br />
| [[351/350]]<br />
| 13<br />
|-<br />
| S27*S28<br />
| ([[27/26]])/([[29/28]])<br />
| [[378/377]]<br />
| 29<br />
|-<br />
| S28*S29<br />
| ([[28/27]])/([[30/29]])<br />
| [[406/405]]<br />
| 29<br />
|-<br />
| S29*S30<br />
| ([[29/28]])/([[31/30]])<br />
| [[435/434]]<br />
| 31<br />
|-<br />
| S30*S31<br />
| ([[30/29]])/([[32/31]])<br />
| [[465/464]]<br />
| 31<br />
|-<br />
| S31*S32<br />
| ([[31/30]])/([[33/32]])<br />
| [[496/495]]<br />
| 31<br />
|-<br />
| S32*S33<br />
| ([[32/31]])/([[34/33]])<br />
| [[528/527]]<br />
| 31<br />
|-<br />
| S33*S34<br />
| ([[33/32]])/([[35/34]])<br />
| [[561/560]]<br />
| 17<br />
|-<br />
| S34*S35<br />
| ([[34/33]])/([[36/35]])<br />
| [[595/594]]<br />
| 17<br />
|-<br />
| S49*S50 = S35<br />
| ([[49/48]])/([[51/50]])<br />
| [[1225/1224]]<br />
| 17<br />
|-<br />
| S50*S51<br />
| ([[50/49]])/([[52/51]])<br />
| [[1275/1274]]<br />
| 17<br />
|-<br />
| S55*S56<br />
| ([[55/54]])/([[57/56]])<br />
| [[1540/1539]]<br />
| 19<br />
|-<br />
| S63*S64<br />
| ([[63/62]])/([[65/64]])<br />
| [[2016/2015]]<br />
| 31<br />
|-<br />
| S64*S65<br />
| ([[64/63]])/([[66/65]])<br />
| [[2080/2079]]<br />
| 13<br />
|-<br />
| S76*S77<br />
| ([[76/75]])/([[78/77]])<br />
| [[2926/2925]]<br />
| 19<br />
|-<br />
| S91*S92<br />
| ([[91/90]])/([[93/92]])<br />
| [[4186/4185]]<br />
| 31<br />
|-<br />
| S115*S116<br />
| ([[115/114]])/([[117/116]])<br />
| [[6670/6669]]<br />
| 29<br />
|-<br />
| S153*S154<br />
| ([[153/152]])/([[155/154]])<br />
| [[11781/11780]]<br />
| 31<br />
|-<br />
| S154*S155<br />
| ([[154/153]])/([[156/155]])<br />
| [[11935/11934]]<br />
| 31<br />
|-<br />
| S169*S170<br />
| ([[169/168]])/([[171/170]])<br />
| [[14365/14364]]<br />
| 19<br />
|-<br />
| S208*S209<br />
| ([[208/207]])/([[210/209]])<br />
| [[21736/21735]]<br />
| 19<br />
|-<br />
| S323*S324<br />
| ([[323/322]])/([[325/324]])<br />
| [[52326/52325]]<br />
| 23<br />
|-<br />
| S341*S342<br />
| ([[341/340]])/([[343/342]])<br />
| [[58311/58310]]<br />
| 31<br />
|-<br />
| S494*S495<br />
| ([[494/493]])/([[496/495]])<br />
| [[122265/122264]]<br />
| 31<br />
|-<br />
| S1519*S1520<br />
| ([[1519/1518]])/([[1521/1520]])<br />
| [[1154440/1154439]]<br />
| 31<br />
|}<br />
<br />
== {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} (1/''n''-square-particulars) ==<br />
=== Motivation ===<br />
1/''n''-square-particulars are a generalization of square- and 1/2-square-particulars to a comma/interval whose S-expression is can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}.<br />
<br />
In other words, each and every S-expression of a comma as a 1/''n''-square-particular corresponds exactly to expressing it as the ratio between two [[superparticular]] intervals, with ''n'' distance between them, where, for example, 10/9 and 11/10 are considered as having 1 distance between them, corresponding to (1/1-)square-particulars (in this case [[100/99|S10]]).<br />
<br />
These commas are important in a few ways:<br />
1. As a generalization of important special cases {{nowrap|''n'' {{=}} 0|''n'' {{=}} 1}}, and {{nowrap|''n'' {{=}} 2}}, (which are almost all superparticular; the only case where they aren't is that {{nowrap|''n'' {{=}} 3}} (1/3-square-particulars) are throdd-particular one third of the time, so this suggests these are efficient commas. A cursory look will show that many 1/n-square-particulars for small n are superparticular, and many more are the next best things (odd-particular, throdd-particular, quodd-particular, etc.) so this confirms them being a family of efficient commas.<br />
<br />
2. Because of being the ratio of two superparticular intervals, in higher-complexity cases they often correspond to small commas between large commas which we don't want to temper, for example {{nowrap|{{sfrac|[[81/80]]|[[91/90]]}} {{=}} S81 * S82 * … * S90}} {{nowrap|{{=}} [[729/728]]}} {{nowrap|{{=}} S27}}. They also often simplify in cases like these; note that a suggested shorthand is S81..90 for {{nowrap|S81 * S82 * … * S90}} and thus more generally S''a''..''b'' for {{nowrap|S''a'' * S(''a'' + 1) * … * S''b''}}.<br />
<br />
3. They often correspond to "nontrivial" equivalences that need to be dug up which are not obvious from their expression as a ratio of two superparticular intervals, for example, [[385/384|S33*S34*S35]], suggesting they are a goldmine for valuable tempering opportunities. <br />
<br />
4. Their expressions naturally make them implied by tempering consecutive square-particulars, so if you notice them present and that the individual square-particulars aren't tempered, if you want to extend your temperament and/or reduce its rank (tempering it down) and/or hope to make your temperament more efficient, you can try tempering the untempered square-particulars that a tempered 1/''n''-square-particular is composed of (although this is not always possible). There is also good theoretical motivation for wanting to do this, as the next section will discuss.<br />
<br />
5. They're relevant to understanding how much damage is present in a temperament's harmonic series representation, because they show how many superparticular intervals are either not distinguished or worse mapped inconsistently, bringing us finally to …<br />
<br />
6. They're relevant to understanding limitations of consistency (or more precisely, monotonicity) of any given temperament, as the next section will discuss.<br />
<br />
=== Significance/implications for consistency ===<br />
1/n-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} have implications for the [[consistency]] of the ({{nowrap|''k'' + ''n''}})-[[odd-limit]] when tempered. Specifically:<br />
<br />
If a temperament tempers a 1/''n''-square-particular of the form {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, it must temper all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', {{nowrap|S(''k'' + 1)}}, …, {{nowrap|S(''k'' + ''n'' − 1)}}. If it does not, it is ''necessarily'' inconsistent (more formally and weakly, not monotonic) in the ({{nowrap|''k'' + ''n''}})-odd-limit.<ref group="note">Note that this statement is a slight inaccuracy, because technically the tuning of the higher rank temperament corresponding to the lower rank temperament that tempers all of these commas is the unique ''and only'' (continuum of) tuning(s) for which this statement is false, but it's reasonable to simplify this technicality as this (continuum of) tuning(s) corresponds exactly and uniquely to tempering all the square-particulars we said were not tempered.</ref> A proof is as follows:<br />
<br />
Consider the following sequence of superparticular intervals, all of which in the ({{nowrap|''k'' + ''n''}})-odd-limit:<br />
<br />
<math>\displaystyle\frac{k + n}{k + n - 1}, \frac{k + n - 1}{k + n - 2}, …, \frac{k + 1}{k}, \frac{k}{k - 1}</math><br />
<br />
Because of tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, we require that {{nowrap|{{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} {{=}} {{sfrac|''k''|''k'' − 1}}}} consistently. Therefore, if any superparticular {{sfrac|''x''|''x'' − 1}} imbetween (meaning {{nowrap|''k'' + ''n'' &gt; ''x'' &gt; ''k''}}) is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true:<br />
<br />
<math>\displaystyle\begin{align}<br />
\operatorname{mapping}\left(\frac{k + n}{k + n - 1}\right) &> \operatorname{mapping}\left(\frac{x}{x - 1}\right) \\<br />
\operatorname{mapping}\left(\frac{k}{k - 1}\right) &< \operatorname{mapping}\left(\frac{x}{x - 1}\right)<br />
\end{align}</math><br />
<br />
Therefore any superparticular interval {{sfrac|''x''|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap|''k'' − 1}})-odd-limit.<br />
<br />
<br />
=== {{nowrap|S(''k'' − 1)*S''k''*S(''k'' + 1)}} (1/3-square-particulars) ===<br />
This section concerns commas of the form {{nowrap|S(''k'' − 1) * S''k'' * S(''k'' + 1) {{=}} {{sfrac|&nbsp;{{sfrac|''k'' − 1|''k'' − 2}}&nbsp;|&nbsp;{{sfrac|''k'' + 2|''k'' + 1}}&nbsp;}}}} which therefore do not (directly) involve the ''k''th harmonic. These are a special case of 1/''n''-square-particulars.<br />
<br />
==== Significance ====<br />
1. Two-thirds of all {{frac|1|3}}-square-particulars are superparticular and the other third are [[#Glossary|throdd-particular]], so these are efficient commas. (See also the [[#Proof of simplification of 1/3-square-particulars]].)<br />
<br />
2. They are often implied in a variety of ways by combinations of other commas discussed on this page.<br />
<br />
3. Their omission of direct relation to the ''k''th harmonic make them theoretically interesting and potentially useful. (The other type of comma on this page that does this is [[semiparticular]]s.)<br />
<br />
4. Square-particulars, {{frac|1|2}}-square-particulars (a.k.a. [[triangle-particular]]s), and {{frac|1|3}}-square-particulars are part of a more general sequence with interesting properties: [[1/n-square-particular|1/''n''-square-particular]]s.<br />
<br />
==== Proof of simplification of 1/3-square-particulars ====<br />
We can check the general algebraic expression of any 1/3-square-particular for any potential simplifications:<br />
<br />
<math>\displaystyle\begin{align}<br />
S(k-1) * S(k) * S(k+1) &= \left(\frac{\frac{k-1}{k-2}}{\frac{k}{k-1}}\right)\left(\frac{\frac{k}{k-1}}{\frac{k+1}{k}}\right)\left(\frac{\frac{k+1}{k}}{\frac{k+2}{k+1}}\right) \\<br />
&= \frac{\frac{k-1}{k-2}}{\frac{k+2}{k+1}} \\<br />
&= \frac{(k-1)(k+1)}{(k-2)(k+2)} \\<br />
&= \frac{k^2 - 1}{k^2 - 4}<br />
\end{align}</math><br />
<br />
If {{nowrap|''k'' {{=}} 3''n'' + 1}} then:<br />
<br />
<math>\displaystyle S(k-1) * Sk * S(k+1) = \frac{9n^2 + 6n}{9n^2 + 6n - 3} = \frac{3n^2 + 2n}{3n^2 + 2n - 1}</math><br />
<br />
if {{nowrap|''k'' {{=}} 3''n'' + 2}} then:<br />
<br />
<math>\displaystyle S(k-1) * Sk * S(k+1) = \frac{9n^2 + 12n + 3}{9n^2 + 12n} = \frac{3n^2 + 4n + 1}{3n^2 + 4n}</math><br />
<br />
if {{nowrap|''k'' {{=}} 3''n''}} then:<br />
<br />
<math>\displaystyle S(k-1) * Sk * S(k+1) = \frac{9n^2 - 1}{9n^2 - 4}</math><br />
<br />
In other words, what this shows is all {{frac|1|3}}-square-particulars of the form S(''k'' − 1) * S''k'' * S(''k'' + 1) are superparticular iff ''k'' is throdd (not a multiple of 3), and all {{frac|1|3}}-square-particulars of the form {{nowrap|S(3''k'' − 1) * S(3''k'') * S(3''k'' + 1)}} are throdd-particular with the numerator and denominator always being one less than a multiple of 3 (which is to say, commas of this form are throdd-particular iff ''k'' is threven and superparticular iff ''k'' is throdd).<br />
<br />
=== Tables of 1/n-square-particulars ===<br />
<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 41-limit {{frac|1|3}}-square-particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3*S4<br />
| ([[2/1]])/([[5/4]])<br />
| [[8/5]]<br />
| 5<br />
|-<br />
| S3*S4*S5<br />
| ([[3/2]])/([[6/5]])<br />
| [[5/4]]<br />
| 5<br />
|-<br />
| S4*S5*S6<br />
| ([[4/3]])/([[7/6]])<br />
| [[8/7]]<br />
| 7<br />
|-<br />
| S5*S6*S7<br />
| ([[5/4]])/([[8/7]])<br />
| [[35/32]]<br />
| 7<br />
|-<br />
| S6*S7*S8<br />
| ([[6/5]])/([[9/8]])<br />
| [[16/15]]<br />
| 5<br />
|-<br />
| S7*S8*S9<br />
| ([[7/6]])/([[10/9]])<br />
| [[21/20]]<br />
| 7<br />
|-<br />
| S8*S9*S10<br />
| ([[8/7]])/([[11/10]])<br />
| [[80/77]]<br />
| 11<br />
|-<br />
| S9*S10*S11<br />
| ([[9/8]])/([[12/11]])<br />
| [[33/32]]<br />
| 11<br />
|-<br />
| S10*S11*S12<br />
| ([[10/9]])/([[13/12]])<br />
| [[40/39]]<br />
| 13<br />
|-<br />
| S11*S12*S13<br />
| ([[11/10]])/([[14/13]])<br />
| [[143/140]]<br />
| 13<br />
|-<br />
| S12*S13*S14<br />
| ([[12/11]])/([[15/14]])<br />
| [[56/55]]<br />
| 11<br />
|-<br />
| S13*S14*S15<br />
| ([[13/12]])/([[16/15]])<br />
| [[65/64]]<br />
| 13<br />
|-<br />
| S14*S15*S16<br />
| ([[14/13]])/([[17/16]])<br />
| [[224/221]]<br />
| 17<br />
|-<br />
| S15*S16*S17<br />
| ([[15/14]])/([[18/17]])<br />
| [[85/84]]<br />
| 17<br />
|-<br />
| S16*S17*S18<br />
| ([[16/15]])/([[19/18]])<br />
| [[96/95]]<br />
| 19<br />
|-<br />
| S17*S18*S19<br />
| ([[17/16]])/([[20/19]])<br />
| [[323/320]]<br />
| 19<br />
|-<br />
| S18*S19*S20<br />
| ([[18/17]])/([[21/20]])<br />
| [[120/119]]<br />
| 17<br />
|-<br />
| S19*S20*S21<br />
| ([[19/18]])/([[22/21]])<br />
| [[133/132]]<br />
| 19<br />
|-<br />
| S20*S21*S22<br />
| ([[20/19]])/([[23/22]])<br />
| [[440/437]]<br />
| 23<br />
|-<br />
| S21*S22*S23<br />
| ([[21/20]])/([[24/23]])<br />
| [[161/160]]<br />
| 23<br />
|-<br />
| S22*S23*S24<br />
| ([[22/21]])/([[25/24]])<br />
| [[176/175]]<br />
| 11<br />
|-<br />
| S23*S24*S25<br />
| ([[23/22]])/([[26/25]])<br />
| [[575/572]]<br />
| 23<br />
|-<br />
| S24*S25*S26<br />
| ([[24/23]])/([[27/26]])<br />
| [[208/207]]<br />
| 23<br />
|-<br />
| S25*S26*S27<br />
| ([[25/24]])/([[28/27]])<br />
| [[225/224]]<br />
| 7<br />
|-<br />
| S26*S27*S28<br />
| ([[26/25]])/([[29/28]])<br />
| [[728/725]]<br />
| 29<br />
|-<br />
| S27*S28*S29<br />
| ([[27/26]])/([[30/29]])<br />
| [[261/260]]<br />
| 29<br />
|-<br />
| S28*S29*S30<br />
| ([[28/27]])/([[31/30]])<br />
| [[280/279]]<br />
| 31<br />
|-<br />
| S29*S30*S31<br />
| ([[29/28]])/([[32/31]])<br />
| [[899/896]]<br />
| 31<br />
|-<br />
| S30*S31*S32<br />
| ([[30/29]])/([[33/32]])<br />
| [[320/319]]<br />
| 29<br />
|-<br />
| S31*S32*S33<br />
| ([[31/30]])/([[34/33]])<br />
| [[341/340]]<br />
| 31<br />
|-<br />
| S32*S33*S34<br />
| ([[32/31]])/([[35/34]])<br />
| [[1088/1085]]<br />
| 31<br />
|-<br />
| S33*S34*S35<br />
| ([[33/32]])/([[36/35]])<br />
| [[385/384]]<br />
| 11<br />
|-<br />
| S34*S35*S36<br />
| ([[34/33]])/([[37/36]])<br />
| [[408/407]]<br />
| 37<br />
|-<br />
| S35*S36*S37<br />
| ([[35/34]])/([[38/37]])<br />
| [[1295/1292]]<br />
| 37<br />
|-<br />
| S36*S37*S38<br />
| ([[36/35]])/([[39/38]])<br />
| [[456/455]]<br />
| 19<br />
|-<br />
| S37*S38*S39<br />
| ([[37/36]])/([[40/39]])<br />
| [[481/480]]<br />
| 37<br />
|-<br />
| S38*S39*S40<br />
| ([[38/37]])/([[41/40]])<br />
| [[1520/1517]]<br />
| 41<br />
|-<br />
| S39*S40*S41<br />
| ([[39/38]])/([[42/41]])<br />
| [[533/532]]<br />
| 41<br />
|-<br />
| S42*S43*S44<br />
| ([[42/41]])/([[45/44]])<br />
| [[616/615]]<br />
| 41<br />
|-<br />
| S46*S47*S48<br />
| ([[46/45]])/([[49/48]])<br />
| [[736/735]]<br />
| 23<br />
|-<br />
| S49*S50*S51<br />
| ([[49/48]])/([[52/51]])<br />
| [[833/832]]<br />
| 17<br />
|-<br />
| S52*S53*S54<br />
| ([[52/51]])/([[55/54]])<br />
| [[936/935]]<br />
| 17<br />
|-<br />
| S55*S56*S57<br />
| ([[55/54]])/([[58/57]])<br />
| [[1045/1044]]<br />
| 29<br />
|-<br />
| S63*S64*S65<br />
| ([[63/62]])/([[66/65]])<br />
| [[1365/1364]]<br />
| 31<br />
|-<br />
| S66*S67*S68<br />
| ([[66/65]])/([[69/68]])<br />
| [[1496/1495]]<br />
| 23<br />
|-<br />
| S75*S76*S77<br />
| ([[75/74]])/([[78/77]])<br />
| [[1925/1924]]<br />
| 37<br />
|-<br />
| S78*S79*S80<br />
| ([[78/77]])/([[81/80]])<br />
| [[2080/2079]]<br />
| 13<br />
|-<br />
| S82*S83*S84<br />
| ([[82/81]])/([[85/84]])<br />
| [[2296/2295]]<br />
| 41<br />
|-<br />
| S85*S86*S87<br />
| ([[85/84]])/([[88/87]])<br />
| [[2465/2464]]<br />
| 29<br />
|-<br />
| S88*S89*S90<br />
| ([[88/87]])/([[91/90]])<br />
| [[2640/2639]]<br />
| 29<br />
|-<br />
| S93*S94*S95<br />
| ([[93/92]])/([[96/95]])<br />
| [[2945/2944]]<br />
| 31<br />
|-<br />
| S96*S97*S98<br />
| ([[96/95]])/([[99/98]])<br />
| [[3136/3135]]<br />
| 19<br />
|-<br />
| S112*S113*S114<br />
| ([[112/111]])/([[115/114]])<br />
| [[4256/4255]]<br />
| 37<br />
|-<br />
| S117*S118*S119<br />
| ([[117/116]])/([[120/119]])<br />
| [[4641/4640]]<br />
| 29<br />
|-<br />
| S121*S122*S123<br />
| ([[121/120]])/([[124/123]])<br />
| [[4961/4960]]<br />
| 41<br />
|-<br />
| S133*S134*S135<br />
| ([[133/132]])/([[136/135]])<br />
| [[5985/5984]]<br />
| 19<br />
|-<br />
| S145*S146*S147<br />
| ([[145/144]])/([[148/147]])<br />
| [[7105/7104]]<br />
| 37<br />
|-<br />
| S153*S154*S155<br />
| ([[153/152]])/([[156/155]])<br />
| [[7905/7904]]<br />
| 31<br />
|-<br />
| S162*S163*S164<br />
| ([[162/161]])/([[165/164]])<br />
| [[8856/8855]]<br />
| 41<br />
|-<br />
| S187*S188*S189<br />
| ([[187/186]])/([[190/189]])<br />
| [[11781/11780]]<br />
| 31<br />
|-<br />
| S205*S206*S207<br />
| ([[205/204]])/([[208/207]])<br />
| [[14145/14144]]<br />
| 41<br />
|-<br />
| S222*S223*S224<br />
| ([[222/221]])/([[225/224]])<br />
| [[16576/16575]]<br />
| 37<br />
|-<br />
| S243*S244*S245<br />
| ([[243/242]])/([[246/245]])<br />
| [[19845/19844]]<br />
| 41<br />
|-<br />
| S253*S254*S255<br />
| ([[253/252]])/([[256/255]])<br />
| [[21505/21504]]<br />
| 23<br />
|-<br />
| S273*S274*S275<br />
| ([[273/272]])/([[276/275]])<br />
| [[25025/25024]]<br />
| 23<br />
|-<br />
| S286*S287*S288<br />
| ([[286/285]])/([[289/288]])<br />
| [[27456/27455]]<br />
| 19<br />
|-<br />
| S287*S288*S289<br />
| ([[287/286]])/([[290/289]])<br />
| [[82943/82940]]<br />
| 41<br />
|-<br />
| S297*S298*S299<br />
| ([[297/296]])/([[300/299]])<br />
| [[29601/29600]]<br />
| 37<br />
|-<br />
| S320*S321*S322<br />
| ([[320/319]])/([[323/322]])<br />
| [[103040/103037]]<br />
| 29<br />
|-<br />
| S361*S362*S363<br />
| ([[361/360]])/([[364/363]])<br />
| [[43681/43680]]<br />
| 19<br />
|-<br />
| S375*S376*S377<br />
| ([[375/374]])/([[378/377]])<br />
| [[47125/47124]]<br />
| 29<br />
|}<br />
<br />
{| class="wikitable center-all\<br />
|+ style="font-size: 105%;" | 23-limit {{frac|1|4}}-square particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3*S4*S5<br />
| ([[2/1]])/([[6/5]])<br />
| [[5/3]]<br />
| 5<br />
|-<br />
| S3*S4*S5*S6<br />
| ([[3/2]])/([[7/6]])<br />
| [[9/7]]<br />
| 7<br />
|-<br />
| S4*S5*S6*S7<br />
| ([[4/3]])/([[8/7]])<br />
| [[7/6]]<br />
| 7<br />
|-<br />
| S5*S6*S7*S8<br />
| ([[5/4]])/([[9/8]])<br />
| [[10/9]]<br />
| 5<br />
|-<br />
| S6*S7*S8*S9<br />
| ([[6/5]])/([[10/9]])<br />
| [[27/25]]<br />
| 5<br />
|-<br />
| S7*S8*S9*S10<br />
| ([[7/6]])/([[11/10]])<br />
| [[35/33]]<br />
| 11<br />
|-<br />
| S8*S9*S10*S11<br />
| ([[8/7]])/([[12/11]])<br />
| [[22/21]]<br />
| 11<br />
|-<br />
| S9*S10*S11*S12<br />
| ([[9/8]])/([[13/12]])<br />
| [[27/26]]<br />
| 13<br />
|-<br />
| S10*S11*S12*S13<br />
| ([[10/9]])/([[14/13]])<br />
| [[65/63]]<br />
| 13<br />
|-<br />
| S11*S12*S13*S14<br />
| ([[11/10]])/([[15/14]])<br />
| [[77/75]]<br />
| 11<br />
|-<br />
| S12*S13*S14*S15<br />
| ([[12/11]])/([[16/15]])<br />
| [[45/44]]<br />
| 11<br />
|-<br />
| S13*S14*S15*S16<br />
| ([[13/12]])/([[17/16]])<br />
| [[52/51]]<br />
| 17<br />
|-<br />
| S14*S15*S16*S17<br />
| ([[14/13]])/([[18/17]])<br />
| [[119/117]]<br />
| 17<br />
|-<br />
| S15*S16*S17*S18<br />
| ([[15/14]])/([[19/18]])<br />
| [[135/133]]<br />
| 19<br />
|-<br />
| S16*S17*S18*S19<br />
| ([[16/15]])/([[20/19]])<br />
| [[76/75]]<br />
| 19<br />
|-<br />
| S17*S18*S19*S20<br />
| ([[17/16]])/([[21/20]])<br />
| [[85/84]]<br />
| 17<br />
|-<br />
| S18*S19*S20*S21<br />
| ([[18/17]])/([[22/21]])<br />
| [[189/187]]<br />
| 17<br />
|-<br />
| S19*S20*S21*S22<br />
| ([[19/18]])/([[23/22]])<br />
| [[209/207]]<br />
| 23<br />
|-<br />
| S20*S21*S22*S23<br />
| ([[20/19]])/([[24/23]])<br />
| [[115/114]]<br />
| 23<br />
|-<br />
| S21*S22*S23*S24<br />
| ([[21/20]])/([[25/24]])<br />
| [[126/125]]<br />
| 7<br />
|-<br />
| S22*S23*S24*S25<br />
| ([[22/21]])/([[26/25]])<br />
| [[275/273]]<br />
| 13<br />
|-<br />
| S23*S24*S25*S26<br />
| ([[23/22]])/([[27/26]])<br />
| [[299/297]]<br />
| 23<br />
|-<br />
| S24*S25*S26*S27<br />
| ([[24/23]])/([[28/27]])<br />
| [[162/161]]<br />
| 23<br />
|-<br />
| S35*S36*S37*S38<br />
| ([[35/34]])/([[39/38]])<br />
| [[665/663]]<br />
| 19<br />
|-<br />
| S36*S37*S38*S39<br />
| ([[36/35]])/([[40/39]])<br />
| [[351/350]]<br />
| 13<br />
|-<br />
| S45*S46*S47*S48<br />
| ([[45/44]])/([[49/48]])<br />
| [[540/539]]<br />
| 11<br />
|-<br />
| S46*S47*S48*S49<br />
| ([[46/45]])/([[50/49]])<br />
| [[1127/1125]]<br />
| 23<br />
|-<br />
| S51*S52*S53*S54<br />
| ([[51/50]])/([[55/54]])<br />
| [[1377/1375]]<br />
| 17<br />
|-<br />
| S52*S53*S54*S55<br />
| ([[52/51]])/([[56/55]])<br />
| [[715/714]]<br />
| 17<br />
|-<br />
| S65*S66*S67*S68<br />
| ([[65/64]])/([[69/68]])<br />
| [[1105/1104]]<br />
| 23<br />
|-<br />
| S66*S67*S68*S69<br />
| ([[66/65]])/([[70/69]])<br />
| [[2277/2275]]<br />
| 23<br />
|-<br />
| S77*S78*S79*S80<br />
| ([[77/76]])/([[81/80]])<br />
| [[1540/1539]]<br />
| 19<br />
|-<br />
| S81*S82*S83*S84<br />
| ([[81/80]])/([[85/84]])<br />
| [[1701/1700]]<br />
| 17<br />
|-<br />
| S92*S93*S94*S95<br />
| ([[92/91]])/([[96/95]])<br />
| [[2185/2184]]<br />
| 23<br />
|-<br />
| S96*S97*S98*S99<br />
| <span style="font-size:0.94em">([[96/95]])/([[100/99]])</span><br />
| [[2376/2375]]<br />
| 19<br />
|-<br />
| <span style="font-size: 0.79em;">S221*S222*S223*S224</span><br />
| <span style="font-size: 0.79em;">([[221/220]])/([[225/224]])</span><br />
| [[12376/12375]]<br />
| 17<br />
|}<br />
<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 23-limit {{frac|1|5}}-square particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3*S4*S5*S6<br />
| ([[2/1]])/([[7/6]])<br />
| [[12/7]]<br />
| 7<br />
|-<br />
| S3*S4*S5*S6*S7<br />
| ([[3/2]])/([[8/7]])<br />
| [[21/16]]<br />
| 7<br />
|-<br />
| S4*S5*S6*S7*S8<br />
| ([[4/3]])/([[9/8]])<br />
| [[32/27]]<br />
| 3<br />
|-<br />
| S5*S6*S7*S8*S9<br />
| ([[5/4]])/([[10/9]])<br />
| [[9/8]]<br />
| 3<br />
|-<br />
| S6*S7*S8*S9*S10<br />
| ([[6/5]])/([[11/10]])<br />
| [[12/11]]<br />
| 11<br />
|-<br />
| S7*S8*S9*S10*S11<br />
| ([[7/6]])/([[12/11]])<br />
| [[77/72]]<br />
| 11<br />
|-<br />
| S8*S9*S10*S11*S12<br />
| ([[8/7]])/([[13/12]])<br />
| [[96/91]]<br />
| 13<br />
|-<br />
| S9*S10*S11*S12*S13<br />
| ([[9/8]])/([[14/13]])<br />
| [[117/112]]<br />
| 13<br />
|-<br />
| S10*S11*S12*S13*S14<br />
| ([[10/9]])/([[15/14]])<br />
| [[28/27]]<br />
| 7<br />
|-<br />
| S11*S12*S13*S14*S15<br />
| ([[11/10]])/([[16/15]])<br />
| [[33/32]]<br />
| 11<br />
|-<br />
| S12*S13*S14*S15*S16<br />
| ([[12/11]])/([[17/16]])<br />
| [[192/187]]<br />
| 17<br />
|-<br />
| S13*S14*S15*S16*S17<br />
| ([[13/12]])/([[18/17]])<br />
| [[221/216]]<br />
| 17<br />
|-<br />
| S14*S15*S16*S17*S18<br />
| ([[14/13]])/([[19/18]])<br />
| [[252/247]]<br />
| 19<br />
|-<br />
| S15*S16*S17*S18*S19<br />
| ([[15/14]])/([[20/19]])<br />
| [[57/56]]<br />
| 19<br />
|-<br />
| S16*S17*S18*S19*S20<br />
| ([[16/15]])/([[21/20]])<br />
| [[64/63]]<br />
| 7<br />
|-<br />
| S17*S18*S19*S20*S21<br />
| ([[17/16]])/([[22/21]])<br />
| [[357/352]]<br />
| 17<br />
|-<br />
| S18*S19*S20*S21*S22<br />
| ([[18/17]])/([[23/22]])<br />
| [[396/391]]<br />
| 23<br />
|-<br />
| S19*S20*S21*S22*S23<br />
| ([[19/18]])/([[24/23]])<br />
| [[437/432]]<br />
| 23<br />
|-<br />
| S20*S21*S22*S23*S24<br />
| ([[20/19]])/([[25/24]])<br />
| [[96/95]]<br />
| 19<br />
|-<br />
| S21*S22*S23*S24*S25<br />
| ([[21/20]])/([[26/25]])<br />
| [[105/104]]<br />
| 13<br />
|-<br />
| S22*S23*S24*S25*S26<br />
| ([[22/21]])/([[27/26]])<br />
| [[572/567]]<br />
| 13<br />
|-<br />
| S23*S24*S25*S26*S27<br />
| ([[23/22]])/([[28/27]])<br />
| [[621/616]]<br />
| 23<br />
|-<br />
| S28*S29*S30*S31*S32<br />
| ([[28/27]])/([[33/32]])<br />
| [[896/891]]<br />
| 11<br />
|-<br />
| S34*S35*S36*S37*S38<br />
| ([[34/33]])/([[39/38]])<br />
| [[1292/1287]]<br />
| 19<br />
|-<br />
| S35*S36*S37*S38*S39<br />
| ([[35/34]])/([[40/39]])<br />
| [[273/272]]<br />
| 17<br />
|-<br />
| S40*S41*S42*S43*S44<br />
| ([[40/39]])/([[45/44]])<br />
| [[352/351]]<br />
| 13<br />
|-<br />
| S45*S46*S47*S48*S49<br />
| ([[45/44]])/([[50/49]])<br />
| [[441/440]]<br />
| 11<br />
|-<br />
| S46*S47*S48*S49*S50<br />
| ([[46/45]])/([[51/50]])<br />
| [[460/459]]<br />
| 23<br />
|-<br />
| S50*S51*S52*S53*S54<br />
| ([[50/49]])/([[55/54]])<br />
| [[540/539]]<br />
| 11<br />
|-<br />
| S51*S52*S53*S54*S55<br />
| ([[51/50]])/([[56/55]])<br />
| [[561/560]]<br />
| 17<br />
|-<br />
| S52*S53*S54*S55*S56<br />
| ([[52/51]])/([[57/56]])<br />
| [[2912/2907]]<br />
| 19<br />
|-<br />
| S64*S65*S66*S67*S68<br />
| ([[64/63]])/([[69/68]])<br />
| [[4352/4347]]<br />
| 23<br />
|-<br />
| S65*S66*S67*S68*S69<br />
| ([[65/64]])/([[70/69]])<br />
| [[897/896]]<br />
| 23<br />
|-<br />
| S76*S77*S78*S79*S80<br />
| ([[76/75]])/([[81/80]])<br />
| [[1216/1215]]<br />
| 19<br />
|-<br />
| S91*S92*S93*S94*S95<br />
| ([[91/90]])/([[96/95]])<br />
| [[1729/1728]]<br />
| 19<br />
|-<br />
| <span style="font-size: 0.79em;">S100*S101*S102*S103*S104</span><br />
| <span style="font-size: 0.83em;">([[100/99]])/([[105/104]])</span><br />
| [[2080/2079]]<br />
| 13<br />
|-<br />
| <span style="font-size: 0.79em;">S115*S116*S117*S118*S119</span><br />
| <span style="font-size: 0.79em;">([[115/114]])/([[120/119]])</span><br />
| [[2737/2736]]<br />
| 23<br />
|-<br />
| <span style="font-size: 0.79em;">S121*S122*S123*S124*S125</span><br />
| <span style="font-size: 0.79em;">([[121/120]])/([[126/125]])</span><br />
| [[3025/3024]]<br />
| 11<br />
|-<br />
| <span style="font-size: 0.79em;">S171*S172*S173*S174*S175</span><br />
| <span style="font-size: 0.79em;">([[171/170]])/([[176/175]])</span><br />
| [[5985/5984]]<br />
| 19<br />
|}<br />
<br />
== {{nowrap|S''k''/S(''k'' + 1)}} (ultraparticulars) ==<br />
=== Motivational example ===<br />
Often it is desirable to make consecutive [[superparticular]] intervals equidistant. This has a number of nice consequences, many of which not explained here—see the motivation section for each infinite family of commas defined on this page.<br />
<br />
For example, if you want 6/5 equidistant from 5/4 and 7/6, you must equate {{nowrap|{{sfrac|[[5/4]]|[[6/5]]}} {{=}} [[25/24]]}} {{nowrap|{{=}} S5}} with {{nowrap|{{sfrac|[[6/5]]|[[7/6]]}} {{=}} [[36/35]]}} {{nowrap|{{=}} S6}}, hence tempering {{nowrap|{{sfrac|S5|S6}} {{=}} {{sfrac|25/24|36/35}}}} {{nowrap|{{=}} [[875/864]]}}, but it's actually often not necessary to know the specific numbers, often familiarizing yourself with and understanding the "S''k''" notation will give you a lot of insight, as we'll see.<br />
<br />
Back to our example: we know that {{nowrap|S5 ~ S6}} (because we're tempering S5/S6); from this we can deduce that the intervals must be arranged like this: {{nowrap|7/6 &larr; S5~S6 &rarr; 6/5 &larr; S5~S6 &rarr; 5/4}}.<br />
<br />
From this you can deduce that {{nowrap|([[6/5]])<sup>3</sup> &rarr; [[7/4]]}}, because you can lower one of the 6/5's to [[7/6]] (lowering it by S6) and raise another of the 6/5's to [[5/4]] (raising it by S5). Then because we've tempered S5 and S6 together, we've lowered and raised by the same amount, so the result of {{nowrap|7/6 * 6/5 * 5/4 {{=}} 7/4}} must be the same as the result of {{nowrap|6/5 * 6/5 * 6/5}} in this temperament.<br />
<br />
Familiarize yourself with the structure of this argument, as [[S-expression/Advanced results#Mathematical derivations|it generalizes to arbitrary S''k'']]; the algebraic proof is tedious, but the intuition is the same:<br />
<br />
<math>\displaystyle \frac{k+2}{k+1} \leftarrow S(k+1)~Sk \rightarrow \frac{k+1}{k} \leftarrow S(k+1)~Sk \rightarrow \frac{k}{k-1}</math><br />
<br />
… implies that three {{sfrac|''k'' + 1|''k''}} give {{sfrac|''k'' + 2|''k'' − 1}} iff we temper {{sfrac|S''k''|S(''k'' + 1)}}.&nbsp;{{qed}}<br />
<br />
=== Significance ===<br />
1. Tempering any two consecutive square-particulars S''k'' and S({{nowrap|''k'' + 1}}) will naturally imply tempering the ultraparticular between them, {{sfrac|S''k''|S(''k'' + 1)}}, meaning they are very common implicit commas.<br />
<br />
2. Tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness.<br />
<br />
3. Tempering the ultraparticular S''k''/S({{nowrap|''k'' + 1}}) along with either the corresponding 1/2-square-particular {{nowrap|S''k'' * S(''k'' + 1)}} or one of the two corresponding lopsided commas {{nowrap|S''k''<sup>2</sup> * S(''k'' + 1)}} or {{nowrap|S''k'' * S(''k'' + 1)<sup>2</sup>}} implies tempering both of S''k'' and S({{nowrap|''k'' + 1}}) individually, and vice versa, so that there is a total of ''five'' equivalences—corresponding to ''five'' infinite families of commas—for every such S''k'' and S({{nowrap|''k''+1}}). This only gets better if you temper a third consecutive square-particular. This is an abundance of "at a glance" essential tempering information that is fully general so only needs to be learned once, and is the motivation of the use of '''S-expressions'''. (For example, {{nowrap|{S16, S17} &rarr; {{(}}S16 * S17, S16/S17, S16<sup>2</sup> * S17, S16 * S17<sup>2</sup>{{)}} }}, and any of the two commas in the latter set imply all the other commas too.)<br />
<br />
=== Table of ultraparticulars ===<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Cube Relation<br />
! Comma<br />
! Cents<br />
|-<br />
| S2/S3 = ([[4/3]])/([[9/8]])<br />
| ([[4/1]])/([[3/2]])<sup>3</sup><br />
| [[32/27]]<br />
| 294.135<br />
|-<br />
| S3/S4 = ([[9/8]])/([[16/15]])<br />
| ([[5/2]])/([[4/3]])<sup>3</sup><br />
| [[135/128]]<br />
| 92.179<br />
|-<br />
| S4/S5 = ([[16/15]])/([[25/24]])<br />
| ([[2/1]])/([[5/4]])<sup>3</sup><br />
| [[128/125]]<br />
| 41.059<br />
|-<br />
| S5/S6 = ([[25/24]])/([[36/35]])<br />
| ([[7/4]])/([[6/5]])<sup>3</sup><br />
| [[875/864]]<br />
| 21.902<br />
|-<br />
| S6/S7 = ([[36/35]])/([[49/48]])<br />
| ([[8/5]])/([[7/6]])<sup>3</sup><br />
| [[1728/1715]]<br />
| 13.074<br />
|-<br />
| S7/S8 = ([[49/48]])/([[64/63]])<br />
| ([[3/2]])/([[8/7]])<sup>3</sup><br />
| [[1029/1024]]<br />
| 8.433<br />
|-<br />
| S8/S9 = ([[64/63]])/([[81/80]])<br />
| ([[10/7]])/([[9/8]])<sup>3</sup><br />
| [[5120/5103]]<br />
| 5.758<br />
|-<br />
| S9/S10 = ([[81/80]])/([[100/99]])<br />
| ([[11/8]])/([[10/9]])<sup>3</sup><br />
| [[8019/8000]]<br />
| 4.107<br />
|-<br />
| S10/S11 = ([[100/99]])/([[121/120]])<br />
| ([[4/3]])/([[11/10]])<sup>3</sup><br />
| [[4000/3993]]<br />
| 3.032<br />
|-<br />
| S11/S12 = ([[121/120]])/([[144/143]])<br />
| ([[13/10]])/([[12/11]])<sup>3</sup><br />
| [[17303/17280]]<br />
| 2.303<br />
|-<br />
| S12/S13 = ([[144/143]])/([[169/168]])<br />
| ([[14/11]])/([[13/12]])<sup>3</sup><br />
| [[24192/24167]]<br />
| 1.79<br />
|-<br />
| S13/S14 = ([[169/168]])/([[196/195]])<br />
| ([[5/4]])/([[14/13]])<sup>3</sup><br />
| [[10985/10976]]<br />
| 1.419<br />
|-<br />
| S14/S15 = ([[196/195]])/([[225/224]])<br />
| ([[16/13]])/([[15/14]])<sup>3</sup><br />
| [[43904/43875]]<br />
| 1.144<br />
|-<br />
| S15/S16 = ([[225/224]])/([[256/255]])<br />
| ([[17/14]])/([[16/15]])<sup>3</sup><br />
| [[57375/57344]]<br />
| 0.936<br />
|-<br />
| S16/S17 = ([[256/255]])/([[289/288]])<br />
| ([[6/5]])/([[17/16]])<sup>3</sup><br />
| [[24576/24565]]<br />
| 0.775<br />
|-<br />
| S17/S18 = ([[289/288]])/([[324/323]])<br />
| ([[19/16]])/([[18/17]])<sup>3</sup><br />
| [[93347/93312]]<br />
| 0.649<br />
|-<br />
| S18/S19 = ([[324/323]])/([[361/360]])<br />
| ([[20/17]])/([[19/18]])<sup>3</sup><br />
| [[116640/116603]]<br />
| 0.549<br />
|-<br />
| S19/S20 = ([[361/360]])/([[400/399]])<br />
| ([[7/6]])/([[20/19]])<sup>3</sup><br />
| [[48013/48000]]<br />
| 0.469<br />
|-<br />
| S20/S21 = ([[400/399]])/([[441/440]])<br />
| ([[22/19]])/([[21/20]])<sup>3</sup><br />
| [[176000/175959]]<br />
| 0.403<br />
|-<br />
| S21/S22 = ([[441/440]])/([[484/483]])<br />
| ([[23/20]])/([[22/21]])<sup>3</sup><br />
| [[213003/212960]]<br />
| 0.35<br />
|-<br />
| S22/S23 = ([[484/483]])/([[529/528]])<br />
| ([[8/7]])/([[23/22]])<sup>3</sup><br />
| [[85184/85169]]<br />
| 0.305<br />
|-<br />
| S23/S24 = ([[529/528]])/([[576/575]])<br />
| ([[25/22]])/([[24/23]])<sup>3</sup><br />
| [[304175/304128]]<br />
| 0.268<br />
|-<br />
| S24/S25 = ([[576/575]])/([[625/624]])<br />
| ([[26/23]])/([[25/24]])<sup>3</sup><br />
| [[359424/359375]]<br />
| 0.236<br />
|-<br />
| S25/S26 = ([[625/624]])/([[676/675]])<br />
| ([[9/8]])/([[26/25]])<sup>3</sup><br />
| [[140625/140608]]<br />
| 0.209<br />
|-<br />
| S26/S27 = ([[676/675]])/([[729/728]])<br />
| ([[28/25]])/([[27/26]])<sup>3</sup><br />
| [[492128/492075]]<br />
| 0.186<br />
|-<br />
| S27/S28 = ([[729/728]])/([[784/783]])<br />
| ([[29/26]])/([[28/27]])<sup>3</sup><br />
| [[570807/570752]]<br />
| 0.167<br />
|-<br />
| S28/S29 = ([[784/783]])/([[841/840]])<br />
| ([[10/9]])/([[29/28]])<sup>3</sup><br />
| [[219520/219501]]<br />
| 0.15<br />
|-<br />
| S31/S32 = ([[961/960]])/([[1024/1023]])<br />
| ([[11/10]])/([[32/31]])<sup>3</sup><br />
| [[327701/327680]]<br />
| 0.111<br />
|-<br />
| S33/S34 = ([[1089/1088]])/([[1156/1155]])<br />
| ([[35/32]])/([[34/33]])<sup>3</sup><br />
| [[1257795/1257728]]<br />
| 0.092<br />
|-<br />
| S34/S35 = ([[1156/1155]])/([[1225/1224]])<br />
| ([[12/11]])/([[35/34]])<sup>3</sup><br />
| [[471648/471625]]<br />
| 0.084<br />
|-<br />
| S37/S38 = ([[1369/1368]])/([[1444/1443]])<br />
| ([[13/12]])/([[38/37]])<sup>3</sup><br />
| [[658489/658464]]<br />
| 0.066<br />
|-<br />
| S40/S41 = ([[1600/1599]])/([[1681/1680]])<br />
| ([[14/13]])/([[41/40]])<sup>3</sup><br />
| [[896000/895973]]<br />
| 0.052<br />
|-<br />
| S43/S44 = ([[1849/1848]])/([[1936/1935]])<br />
| ([[15/14]])/([[44/43]])<sup>3</sup><br />
| [[1192605/1192576]]<br />
| 0.042<br />
|-<br />
| S46/S47 = ([[2116/2115]])/([[2209/2208]])<br />
| ([[16/15]])/([[47/46]])<sup>3</sup><br />
| [[1557376/1557345]]<br />
| 0.034<br />
|-<br />
| S49/S50 = ([[2401/2400]])/([[2500/2499]])<br />
| ([[17/16]])/([[50/49]])<sup>3</sup><br />
| [[2000033/2000000]]<br />
| 0.029<br />
|-<br />
| S50/S51 = ([[2500/2499]])/([[2601/2600]])<br />
| ([[52/49]])/([[51/50]])<sup>3</sup><br />
| [[6500000/6499899]]<br />
| 0.027<br />
|-<br />
| S55/S56 = ([[3025/3024]])/([[3136/3135]])<br />
| ([[19/18]])/([[56/55]])<sup>3</sup><br />
| [[3161125/3161088]]<br />
| 0.02<br />
|-<br />
| S64/S65 = ([[4096/4095]])/([[4225/4224]])<br />
| ([[22/21]])/([[65/64]])<sup>3</sup><br />
| [[5767168/5767125]]<br />
| 0.013<br />
|}<br />
<br />
The above table is a list of all [[23-limit]] ultraparticulars corresponding to S''k'' with ''k'' < 77, plus ultraparticulars corresponding to dividing a [[superparticular interval]] into three equal parts up to [[17/16]] (or up to [[19/18]] but excluding 18/17 because of it requiring a large prime, 53), plus S27/S28 so that we have all ultraparticulars up to S28/S29 listed rather than up to S26/S27.<br />
<br />
This table has been expanded following every ultraparticular from S2/S3 to S16/S17 having its own page. Note that ultraparticulars are, in general, extremely precise commas so that usually one wouldn't consider tempering them directly rather than through tempering the square-particulars S''k'' which they are composed of. As an example of this, notice that [[4000/3993|S10/S11]] is the largest ultraparticular categorised as an [[unnoticeable comma]], which means not unnoticeable in the absolute sense but rather in the sense of being smaller than the melodic just-noticeable difference, despite only dividing a superparticular as simple and unremarkable as [[4/3]]. For this reason, a [[cent]]s column has been included to aid an appreciation of their precision. The cent value of a [[semiparticular]] is roughly double that of any of the two ultraparticulars it is composed of; this becomes more true the higher you go.<br />
<br />
Note also from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably [[consistent]]) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3''k'' + 1)/S(3''k''+ 2) for a positive integer ''k'', because then the superparticular can be expressed as:<br />
<br />
<math>\frac{(3k + 3)/3k}{((3k + 2)(3k + 1))^3} = \frac{(k + 1)/k}{((3k + 2)(3k + 1))^3}</math><br />
<br />
Also note that if you temper multiple adjacent ultraparticulars, you sometimes are not required to use those ultraparticulars in the comma list as description of (the bulk of) the tempering may be possible through [[#Sk/S(k + 2) (semiparticulars)|semiparticulars]], discussed next.<br />
<br />
== {{nowrap|S''k''/S(''k'' + 2)}} (semiparticulars) ==<br />
=== Motivational examples ===<br />
If we want to halve one JI interval into two of another JI interval, there is a powerful and elegant pattern for doing so:<br />
* [[4/3]] is approximately half of [[9/5]]<br />
* [[9/7]] is approximately half of [[5/3]] (=&nbsp;10/6)<br />
* [[5/4]] is approximately half of [[11/7]]<br />
* [[11/9]] is approximately half of [[3/2]] (=&nbsp;12/8)<br />
* [[6/5]] is approximately half of [[13/9]]<br />
* [[13/11]] is approximately half of [[7/5]] (=&nbsp;14/10)<br />
* [[7/6]] is approximately half of [[15/11]]<br />
* [[15/13]] is approximately half of [[4/3]] (=&nbsp;16/12)<br />
* [[8/7]] is approximately half of [[17/13]]<br />
* [[17/15]] is approximately half of [[9/7]] (=&nbsp;18/14)<br />
* [[9/8]] is approximately half of [[19/15]]<br />
* [[19/17]] is approximately half of [[5/4]] (=&nbsp;20/16)<br />
<br />
These properties show a pattern: take some arbitrary [[#Glossary|quodd-particular]] (''k'' + 4)/''k''; observe that we can split it into (''k'' + 4)/(''k'' + 2) * (''k'' + 2)/''k''.<br />
<br />
Now observe that (''k'' + 2)/''k'' > (''k'' + 3)/(''k'' + 1) > (''k'' + 4)/(''k'' + 2); in fact, it can be shown fairly easily that (''k'' + 3)/(''k'' + 1) is the [[mediant]] of (''k'' + 4)/(''k'' + 2) and (''k'' + 2)/''k''.<br />
<br />
It turns out that making this mediant — (''k'' + 3)/(''k'' + 1) — equal to half of (''k'' + 4)/''k'' is equivalent to tempering S(''k'' + 1)/S(''k'' + 3).<br />
<br />
=== Significance ===<br />
1. For differences between square-particulars of the form S''k''/S(''k'' + 2), the resulting comma is either [[superparticular]] or [[#Glossary|odd-particular]], so these are efficient commas. (This terminology also suggests [[#Glossary|throdd-particular]] and [[#Glossary|quodd-particular]] as generalizations.)<br />
<br />
2. Tempering any two consecutive [[ultraparticular]]s implies tempering a semiparticular, so from two adjacent "thirding" equivalences you get a "halving" equivalence for free!<br />
<br />
3. Tempering any two nearly-consecutive square-particulars (S''k'' and S(''k'' + 2)) implies tempering a semiparticular; this is generally much more ideal than tempering two consecutive S''k'' because it is a lot lower damage (see [[lopsided comma]]s for (relatively) large commas implied by this higher-damage strategy).<br />
<br />
4. On top of the halving equivalence, there is a number of subtler structural implications, [[discussed below, that may be desirable to the temperament designer.<br />
<br />
=== Meaning ===<br />
: '''Reader notes:''' In the below, we use S(''k'' - 1)/S(''k'' + 1) for symmetry around ''k'' to make the math visually simpler, but keep in mind it's equivalent to using an offset ''k''.<br />
: '''Also:''' keep in mind that ''k'' - ''a'' (for positive ''a'') is smaller than ''k'', so that ''k''/(''k'' - ''a'') > (''k'' + ''a'')/''k'' (because the former appears earlier in the harmonic series & is thus larger); this is an important and useful intuition to learn.<br />
<br />
Tempering S(''k'' - 1)/S(''k'' + 1) implies that (''k'' + 2)/(''k'' - 2) is divisible exactly into two halves of (''k'' + 1)/(''k'' - 1). It also implies that the intervals (''k'' + 2)/''k'' (=s) and ''k''/(''k'' - 2) (=L) are equidistant from (''k'' + 1)/(''k'' - 1) (=M) because to make them equidistant we need to temper:<br />
<br />
<math>\displaystyle \frac{L/M}{M/s} = \frac{ \left(\frac{k}{k-2}\right)/\left(\frac{k+1}{k-1}\right) }{ \left(\frac{k+1}{k-1}\right)/\left(\frac{k+2}{k}\right) } = \frac{Ls}{M^2} = \frac{\frac{k+2}{k-2}}{\left(\frac{k+1}{k-1}\right)^2} </math><br />
<br />
… and notice that the latter expression is the one we've [[S-expression/Advanced results#Mathematical derivations|shown is equal to S(''k''-1)/S(''k''+1)]] (up to an offset ''k''). In other words, you could interpret that a reason that tempering S(''k''-1)/S(''k''+1) results in (''k''+1)/(''k''-1) being half of (''k''+2)/(''k''-2) is because it makes the following three intervals equidistant:<br />
(''k''+2)/''k'', (''k''+1)/(''k''-1), ''k''/(''k''-2)<br />
<br />
Also note that in the above, (''k'' + 1)/(''k'' - 1) is the [[mediant]] of the adjacent two intervals, meaning that division of an interval into two via tempering a semiparticular is in some sense 'optimal' relative to the complexity. This also means that if ''k'' is a multiple of 2, this corresponds to a natural way to split the square superparticular S(''k''/2) into two parts. For example, if ''k'' = 10 then we have (10+2)/10, (10+1)/(10-1), 10/(10-2) as equidistant, which simplified is 6/5, 11/9, 5/4, with 11/9 being the mediant of 6/5 and 5/4, and therefore the corresponding superparticular S5 = (5/4)/(6/5) is split into two parts which are tempered together: (5/4)/(11/9) = 45/44 and (11/9)/(6/5) = 55/54. The semiparticular is therefore S(10-1)/S(10+1) = S9/S11 = 243/242 = (45/44)/(55/54) = ((10+2)/(10-2))/((10+1)/(10-1))<sup>2</sup>.<br />
<br />
This form of comma has been named "semiparticular", because most of the time it is superparticular but less often it is odd-particular, and because when tempered out they all cause an interval to be divided into two equal parts where each part is a (tempered version of a) superparticular or odd-particular, and the interval being divided in half is sometimes quodd-particular, sometimes odd-particular and sometimes superparticular. Specifically:<br />
<br />
* To find out what a superparticular (''a''+1)/''a'' is approximately half of, temper the semiparticular S(2''a'')/S(2''a''+2) and you can observe that (2''a''+3)/(2''a''-1) is the interval it is approximately half of.<br />
<br />
* To find out what an odd-particular (2''a''+1)/(2''a''-1) is approximately half of, temper the semiparticular S(2''a''-1)/S(2''a''+1) and you can observe that (2''a''+2)/(2''a''-2) = (''a''+1)/(''a''-1), a superparticular or odd-particular, is the interval it is approximately half of.<br />
<br />
* To find out what splits a superparticular (''a''+1)/''a'' in half, temper the semiparticular S(4''a''+1)/S(4''a''+3) and you can observe that (4''a''+3)/(4''a''+1), an odd-particular, is the interval that is approximately half of it.<br />
<br />
* To find out what splits an odd-particular (2''a''+1)/(2''a''-1) in half, temper the semiparticular S(4''a''-2)/S(4''a''+2) and you can observe that (4''a''-1)/(4''a''+1), an odd-particular, is the interval that is approximately half of it.<br />
<br />
Also, the interval in the denominator of an expression of a semiparticular of the form (a/b)/(c/d)<sup>2</sup> is significant in that it has a special relationship: specifically, consider tempering (a/b)/(c/d)<sup>2</sup> so therefore the interval c/d is equal to the interval (a/b)/(c/d). This is significant because it allows the intuitive replacement of two consecutive superparticulars (whose product is a superparticular or odd-particular) with the two superparticulars directly adjacent to them.<br />
<br />
For example, as 9/8 = 18/17 * 17/16 we can replace 18/17 with 19/18 and 17/16 with 16/15 by tempering S16/S18 = (19/15)/(9/8)<sup>2</sup> because we can multiply 9/8 by the tempered comma (19/15)/(9/8)<sup>2</sup> to get (19/15)/(9/8) = (19/18)(16/15) (because 9/8 = 18/16), or as 13/11 = 13/12 * 12/11 we can replace 13/12 with 14/13 and 12/11 with 11/10 by tempering S11/S13 = (7/5)/(13/11)<sup>2</sup> because we can multiply 13/11 by the tempered comma (7/5)/(13/11)<sup>2</sup> to get (7/5)/(13/11) = (14/13)(11/10) (because 7/5 = 14/10). Note we have to replace ''both'' intervals ''simultaneously'' as this is lower error, and note that if we want to be able to replace them individually we must pick the higher error route of tempering S16 and S18 or S11 and S13 individually (for which tempering the semiparticular is then an implied consequence). (The broader lesson is that you can rewrite exact JI equivalences with the commas you are tempering to find new interesting consequences of those commas.)<br />
<br />
=== Table of semiparticulars ===<br />
Here follows a table of [[23-limit]] semiparticulars corresponding to square-particulars S''k'' for ''k'' < 96, plus all semiparticulars up to [[9801/9800|S33/S35 = S99]], an exceptional [[11-limit]] comma, plus all semiparticulars dividing [[superparticular interval]]s up to [[13/12]] (corresponding to the [[17-limit]] semiparticular [[31213/31212|S49/S51]]) for completeness. This table also shows all semiparticulars corresponding to splitting an [[#Glossary|odd-particular]] in two up to [[17/15]] (although a common strategy is to temper the square-particular that is the difference between the two superparticular intervals the odd-particular is composed of instead). The bound ''k'' < 96 was chosen as it corresponds to another remarkable semiparticular [[123201/123200|S78/S80 = S351]]. Perhaps many of the patterns will become clearer if you examine this table:<br />
<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Square Relation<br />
! Ratio<br />
|-<br />
| S2/S4 = ([[4/3]])/([[16/15]])<br />
| ([[5/1]])/([[2/1]])<sup>2</sup><br />
| [[5/4]]<br />
|-<br />
| S3/S5 = ([[9/8]])/([[25/24]])<br />
| ([[3/1]])/([[5/3]])<sup>2</sup><br />
| [[27/25]]<br />
|-<br />
| S4/S6 = ([[16/15]])/([[36/35]])<br />
| ([[7/3]])/([[3/2]])<sup>2</sup><br />
| [[28/27]]<br />
|-<br />
| S5/S7 = ([[25/24]])/([[49/48]])<br />
| ([[2/1]])/([[7/5]])<sup>2</sup><br />
| [[50/49]]<br />
|-<br />
| S6/S8 = ([[36/35]])/([[64/63]])<br />
| ([[9/5]])/([[4/3]])<sup>2</sup><br />
| [[81/80]]<br />
|-<br />
| S7/S9 = ([[49/48]])/([[81/80]])<br />
| ([[5/3]])/([[9/7]])<sup>2</sup><br />
| [[245/243]]<br />
|-<br />
| S8/S10 = ([[64/63]])/([[100/99]])<br />
| ([[11/7]])/([[5/4]])<sup>2</sup><br />
| [[176/175]]<br />
|-<br />
| S9/S11 = ([[81/80]])/([[121/120]])<br />
| ([[3/2]])/([[11/9]])<sup>2</sup><br />
| [[243/242]]<br />
|-<br />
| S10/S12 = ([[100/99]])/([[144/143]])<br />
| ([[13/9]])/([[6/5]])<sup>2</sup><br />
| [[325/324]]<br />
|-<br />
| S11/S13 = ([[121/120]])/([[169/168]])<br />
| ([[7/5]])/([[13/11]])<sup>2</sup><br />
| [[847/845]]<br />
|-<br />
| S12/S14 = ([[144/143]])/([[196/195]])<br />
| ([[15/11]])/([[7/6]])<sup>2</sup><br />
| [[540/539]]<br />
|-<br />
| S13/S15 = ([[169/168]])/([[225/224]])<br />
| ([[4/3]])/([[15/13]])<sup>2</sup><br />
| [[676/675]]<br />
|-<br />
| S14/S16 = ([[196/195]])/([[256/255]])<br />
| ([[17/13]])/([[8/7]])<sup>2</sup><br />
| [[833/832]]<br />
|-<br />
| S15/S17 = ([[225/224]])/([[289/288]])<br />
| ([[9/7]])/([[17/15]])<sup>2</sup><br />
| [[2025/2023]]<br />
|-<br />
| S16/S18 = ([[256/255]])/([[324/323]])<br />
| ([[19/15]])/([[9/8]])<sup>2</sup><br />
| [[1216/1215]]<br />
|-<br />
| S17/S19 = ([[289/288]])/([[361/360]])<br />
| ([[5/4]])/([[19/17]])<sup>2</sup><br />
| [[1445/1444]]<br />
|-<br />
| S18/S20 = ([[324/323]])/([[400/399]])<br />
| ([[21/17]])/([[10/9]])<sup>2</sup><br />
| [[1701/1700]]<br />
|-<br />
| S19/S21 = ([[361/360]])/([[441/440]])<br />
| ([[11/9]])/([[21/19]])<sup>2</sup><br />
| [[3971/3969]]<br />
|-<br />
| S20/S22 = ([[400/399]])/([[484/483]])<br />
| ([[23/19]])/([[11/10]])<sup>2</sup><br />
| [[2300/2299]]<br />
|-<br />
| S21/S23 = ([[441/440]])/([[529/528]])<br />
| ([[6/5]])/([[23/21]])<sup>2</sup><br />
| [[2646/2645]]<br />
|-<br />
| S22/S24 = ([[484/483]])/([[576/575]])<br />
| ([[25/21]])/([[12/11]])<sup>2</sup><br />
| [[3025/3024]]<br />
|-<br />
| S23/S25 = ([[529/528]])/([[625/624]])<br />
| ([[13/11]])/([[25/23]])<sup>2</sup><br />
| [[6877/6875]]<br />
|-<br />
| S24/S26 = ([[576/575]])/([[676/675]])<br />
| ([[27/23]])/([[13/12]])<sup>2</sup><br />
| [[3888/3887]]<br />
|-<br />
| S25/S27 = ([[625/624]])/([[729/728]])<br />
| ([[7/6]])/([[27/25]])<sup>2</sup><br />
| [[4375/4374]]<br />
|-<br />
| S26/S28 = ([[676/675]])/([[784/783]])<br />
| ([[29/25]])/([[14/13]])<sup>2</sup><br />
| [[4901/4900]]<br />
|-<br />
| S27/S29 = ([[729/728]])/([[841/840]])<br />
| ([[15/13]])/([[29/27]])<sup>2</sup><br />
| [[10935/10933]]<br />
|-<br />
| S28/S30 = ([[784/783]])/([[900/899]])<br />
| ([[31/27]])/([[15/14]])<sup>2</sup><br />
| [[6076/6075]]<br />
|-<br />
| S29/S31 = ([[841/840]])/([[961/960]])<br />
| ([[8/7]])/([[31/29]])<sup>2</sup><br />
| [[6728/6727]]<br />
|-<br />
| S30/S32 = ([[900/899]])/([[1024/1023]])<br />
| ([[33/29]])/([[16/15]])<sup>2</sup><br />
| [[7425/7424]]<br />
|-<br />
| S31/S33 = ([[961/960]])/([[1089/1088]])<br />
| ([[17/15]])/([[33/31]])<sup>2</sup><br />
| [[16337/16335]]<br />
|-<br />
| S32/S34 = ([[1024/1023]])/([[1156/1155]])<br />
| ([[35/31]])/([[17/16]])<sup>2</sup><br />
| [[8960/8959]]<br />
|-<br />
| S33/S35 = ([[1089/1088]])/([[1225/1224]])<br />
| ([[9/8]])/([[35/33]])<sup>2</sup><br />
| [[9801/9800]]<br />
|-<br />
| S36/S38 = ([[1296/1295]])/([[1444/1443]])<br />
| ([[39/35]])/([[19/18]])<sup>2</sup><br />
| [[12636/12635]]<br />
|-<br />
| S37/S39 = ([[1369/1368]])/([[1521/1520]])<br />
| ([[10/9]])/([[39/37]])<sup>2</sup><br />
| [[13690/13689]]<br />
|-<br />
| S41/S43 = ([[1681/1680]])/([[1849/1848]])<br />
| ([[11/10]])/([[43/41]])<sup>2</sup><br />
| [[18491/18490]]<br />
|-<br />
| S45/S47 = ([[2025/2024]])/([[2209/2208]])<br />
| ([[12/11]])/([[47/45]])<sup>2</sup><br />
| [[24300/24299]]<br />
|-<br />
| S46/S48 = ([[2116/2115]])/([[2304/2303]])<br />
| ([[49/45]])/([[24/23]])<sup>2</sup><br />
| [[25921/25920]]<br />
|-<br />
| S49/S51 = ([[2401/2400]])/([[2601/2600]])<br />
| ([[13/12]])/([[51/49]])<sup>2</sup><br />
| [[31213/31212]]<br />
|-<br />
| S52/S54 = ([[2704/2703]])/([[2916/2915]])<br />
| ([[55/51]])/([[27/26]])<sup>2</sup><br />
| [[37180/37179]]<br />
|-<br />
| S66/S68 = ([[4356/4355]])/([[4624/4623]])<br />
| ([[69/65]])/([[34/33]])<sup>2</sup><br />
| [[75141/75140]]<br />
|-<br />
| S78/S80 = ([[6084/6083]])/([[6400/6399]])<br />
| ([[81/77]])/([[40/39]])<sup>2</sup><br />
| [[123201/123200]]<br />
|}<br />
<br />
(Note that while a lot of these have pages, not all of them do, although that doesn't mean they shouldn't. A noticeable streak of commas currently without pages correspond to when dividing a superparticular interval implicates intervals from a higher [[prime limit]], as a surprising amount of 23-limit semiparticulars shown here already have pages.)<br />
<br />
== {{nowrap|S''k''² * S(''k'' + 1)}} and {{nowrap|S(''k'' − 1) * S''k''²}} (lopsided commas) ==<br />
=== Significance ===<br />
1. Tempering any two consecutive square-particulars, S''k'' and S(''k'' + 1), implies tempering the two associated lopsided commas as well as the associated [[triangle-particular]] and [[ultraparticular]], so the lopsided commas represent the general form of the highest-damage relations/consequences of doing so.<br />
<br />
2. If a comma (such as the diaschisma, [[2048/2025]]), admits an expression as a lopsided comma, it means that one is likely missing out on tempering opportunities by not also tempering the square-particulars composing it (such as [[256/255|S16]] and [[289/288|S17]] in the case of the diaschisma), often involving expanding the subgroup and adding a number of new equivalence relations (as previously explained) while simultaneously making the temperament more efficient and more precise.<br />
<br />
3. It is surprising that there are fairly simple general equivalence relations for these S-expressions, essentially being "free" to read off of an S-expression-based comma list, once you know the general form.<br />
<br />
=== Derivation of equivalence relation ===<br />
Using the clarity of [[S-expression/Advanced results#Using S-factorizations to understand the significance of S-expressions|S-factorizations]], we can show the interval relations implicated by these two new "lopsided" forms, which will make clear the reason for their name:<br />
<br />
S''k''<sup>2</sup> * S(''k''+1) = [''k''-1, ''k'', ''k''+1, ''k''+2]^(2[-1, 2, -1, 0] + [0, -1, 2, -1] = [-2, 4, -2, 0] + [0, -1, 2, -1] = [-2, 3, 0, -1]) implies:<br />
<br />
S''k''<sup>2</sup> * S(''k''+1) = (''k''/(''k''-1))<sup>2</sup> / ((''k''+2)/''k'') through [-2, 3, 0, -1] = [-2, 2, 0, 0] - [0, -1, 0, 1].<br />
<br />
S(''k''-1) * S''k''<sup>2</sup> = [''k''-2, ''k''-1, ''k'', ''k''+1]^([-1, 2, -1, 0] + 2[0, -1, 2, -1] = [-1, 2, -1, 0] + [0, -2, 4, -2] = [-1, 0, 3, -2]) implies:<br />
<br />
S(''k''-1) * S''k''<sup>2</sup> = (''k''/(''k''-2)) / ((''k''+1)/''k'')<sup>2</sup> through [-1, 0, 3, -2] = [-1, 0, 1, 0] - [0, 0, -2, 2].<br />
<br />
=== Tables ===<br />
Below are two tables of [[43-limit]] lopsided commas. First, the "top heavy" lopsided commas, where the squared interval is in the numerator, then the "bottom heavy" lopsided commas, where the squared interval is in the denominator. These tables are so big because these commas are quite large so the more interesting commas appear later. For this reason and for completeness, the tables show up to until a little past the largest known lopsided commas that have their own page: the [[olympia]] and the [[phaotic comma]].<br />
<br />
==== Top-heavy lopsided commas ====<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Square Relation<br />
! Ratio<br />
|-<br />
| S2<sup>2</sup>*S3 = [[3/2]] * [[4/3]]<br />
| ([[2/1]])<sup>2</sup> / ([[2/1]])<br />
| [[2/1]]<br />
|-<br />
| S3<sup>2</sup>*S4 = [[6/5]] * [[9/8]]<br />
| ([[3/2]])<sup>2</sup> / ([[5/3]])<br />
| [[27/20]]<br />
|-<br />
| S4<sup>2</sup>*S5 = [[10/9]] * [[16/15]]<br />
| ([[4/3]])<sup>2</sup> / ([[3/2]])<br />
| [[32/27]]<br />
|-<br />
| S5<sup>2</sup>*S6 = [[15/14]] * [[25/24]]<br />
| ([[5/4]])<sup>2</sup> / ([[7/5]])<br />
| [[125/112]]<br />
|-<br />
| S6<sup>2</sup>*S7 = [[21/20]] * [[36/35]]<br />
| ([[6/5]])<sup>2</sup> / ([[4/3]])<br />
| [[27/25]]<br />
|-<br />
| S7<sup>2</sup>*S8 = [[28/27]] * [[49/48]]<br />
| ([[7/6]])<sup>2</sup> / ([[9/7]])<br />
| [[343/324]]<br />
|-<br />
| S8<sup>2</sup>*S9 = [[36/35]] * [[64/63]]<br />
| ([[8/7]])<sup>2</sup> / ([[5/4]])<br />
| [[256/245]]<br />
|-<br />
| S9<sup>2</sup>*S10 = [[45/44]] * [[81/80]]<br />
| ([[9/8]])<sup>2</sup> / ([[11/9]])<br />
| [[729/704]]<br />
|-<br />
| S10<sup>2</sup>*S11 = [[55/54]] * [[100/99]]<br />
| ([[10/9]])<sup>2</sup> / ([[6/5]])<br />
| [[250/243]]<br />
|-<br />
| S11<sup>2</sup>*S12 = [[66/65]] * [[121/120]]<br />
| ([[11/10]])<sup>2</sup> / ([[13/11]])<br />
| [[1331/1300]]<br />
|-<br />
| S12<sup>2</sup>*S13 = [[78/77]] * [[144/143]]<br />
| ([[12/11]])<sup>2</sup> / ([[7/6]])<br />
| [[864/847]]<br />
|-<br />
| S13<sup>2</sup>*S14 = [[91/90]] * [[169/168]]<br />
| ([[13/12]])<sup>2</sup> / ([[15/13]])<br />
| [[2197/2160]]<br />
|-<br />
| S14<sup>2</sup>*S15 = [[105/104]] * [[196/195]]<br />
| ([[14/13]])<sup>2</sup> / ([[8/7]])<br />
| [[343/338]]<br />
|-<br />
| S15<sup>2</sup>*S16 = [[120/119]] * [[225/224]]<br />
| ([[15/14]])<sup>2</sup> / ([[17/15]])<br />
| [[3375/3332]]<br />
|-<br />
| S16<sup>2</sup>*S17 = [[136/135]] * [[256/255]]<br />
| ([[16/15]])<sup>2</sup> / ([[9/8]])<br />
| [[2048/2025]]<br />
|-<br />
| S17<sup>2</sup>*S18 = [[153/152]] * [[289/288]]<br />
| ([[17/16]])<sup>2</sup> / ([[19/17]])<br />
| [[4913/4864]]<br />
|-<br />
| S18<sup>2</sup>*S19 = [[171/170]] * [[324/323]]<br />
| ([[18/17]])<sup>2</sup> / ([[10/9]])<br />
| [[1458/1445]]<br />
|-<br />
| S19<sup>2</sup>*S20 = [[190/189]] * [[361/360]]<br />
| ([[19/18]])<sup>2</sup> / ([[21/19]])<br />
| [[6859/6804]]<br />
|-<br />
| S20<sup>2</sup>*S21 = [[210/209]] * [[400/399]]<br />
| ([[20/19]])<sup>2</sup> / ([[11/10]])<br />
| [[4000/3971]]<br />
|-<br />
| S21<sup>2</sup>*S22 = [[231/230]] * [[441/440]]<br />
| ([[21/20]])<sup>2</sup> / ([[23/21]])<br />
| [[9261/9200]]<br />
|-<br />
| S22<sup>2</sup>*S23 = [[253/252]] * [[484/483]]<br />
| ([[22/21]])<sup>2</sup> / ([[12/11]])<br />
| [[1331/1323]]<br />
|-<br />
| S23<sup>2</sup>*S24 = [[276/275]] * [[529/528]]<br />
| ([[23/22]])<sup>2</sup> / ([[25/23]])<br />
| [[12167/12100]]<br />
|-<br />
| S24<sup>2</sup>*S25 = [[300/299]] * [[576/575]]<br />
| ([[24/23]])<sup>2</sup> / ([[13/12]])<br />
| [[6912/6877]]<br />
|-<br />
| S25<sup>2</sup>*S26 = [[325/324]] * [[625/624]]<br />
| ([[25/24]])<sup>2</sup> / ([[27/25]])<br />
| [[15625/15552]]<br />
|-<br />
| S26<sup>2</sup>*S27 = [[351/350]] * [[676/675]]<br />
| ([[26/25]])<sup>2</sup> / ([[14/13]])<br />
| [[4394/4375]]<br />
|-<br />
| S27<sup>2</sup>*S28 = [[378/377]] * [[729/728]]<br />
| ([[27/26]])<sup>2</sup> / ([[29/27]])<br />
| [[19683/19604]]<br />
|-<br />
| S28<sup>2</sup>*S29 = [[406/405]] * [[784/783]]<br />
| ([[28/27]])<sup>2</sup> / ([[15/14]])<br />
| [[10976/10935]]<br />
|-<br />
| S29<sup>2</sup>*S30 = [[435/434]] * [[841/840]]<br />
| ([[29/28]])<sup>2</sup> / ([[31/29]])<br />
| [[24389/24304]]<br />
|-<br />
| S30<sup>2</sup>*S31 = [[465/464]] * [[900/899]]<br />
| ([[30/29]])<sup>2</sup> / ([[16/15]])<br />
| [[3375/3364]]<br />
|-<br />
| S31<sup>2</sup>*S32 = [[496/495]] * [[961/960]]<br />
| ([[31/30]])<sup>2</sup> / ([[33/31]])<br />
| [[29791/29700]]<br />
|-<br />
| S32<sup>2</sup>*S33 = [[528/527]] * [[1024/1023]]<br />
| ([[32/31]])<sup>2</sup> / ([[17/16]])<br />
| [[16384/16337]]<br />
|-<br />
| S33<sup>2</sup>*S34 = [[561/560]] * [[1089/1088]]<br />
| ([[33/32]])<sup>2</sup> / ([[35/33]])<br />
| [[35937/35840]]<br />
|-<br />
| S34<sup>2</sup>*S35 = [[595/594]] * [[1156/1155]]<br />
| ([[34/33]])<sup>2</sup> / ([[18/17]])<br />
| [[9826/9801]]<br />
|-<br />
| S35<sup>2</sup>*S36 = [[630/629]] * [[1225/1224]]<br />
| ([[35/34]])<sup>2</sup> / ([[37/35]])<br />
| [[42875/42772]]<br />
|-<br />
| S36<sup>2</sup>*S37 = [[666/665]] * [[1296/1295]]<br />
| ([[36/35]])<sup>2</sup> / ([[19/18]])<br />
| [[23328/23275]]<br />
|-<br />
| S37<sup>2</sup>*S38 = [[703/702]] * [[1369/1368]]<br />
| ([[37/36]])<sup>2</sup> / ([[39/37]])<br />
| [[50653/50544]]<br />
|-<br />
| S38<sup>2</sup>*S39 = [[741/740]] * [[1444/1443]]<br />
| ([[38/37]])<sup>2</sup> / ([[20/19]])<br />
| [[6859/6845]]<br />
|-<br />
| S39<sup>2</sup>*S40 = [[780/779]] * [[1521/1520]]<br />
| ([[39/38]])<sup>2</sup> / ([[41/39]])<br />
| [[59319/59204]]<br />
|-<br />
| S40<sup>2</sup>*S41 = [[820/819]] * [[1600/1599]]<br />
| ([[40/39]])<sup>2</sup> / ([[21/20]])<br />
| [[32000/31941]]<br />
|-<br />
| S41<sup>2</sup>*S42 = [[861/860]] * [[1681/1680]]<br />
| ([[41/40]])<sup>2</sup> / ([[43/41]])<br />
| [[68921/68800]]<br />
|-<br />
| S42<sup>2</sup>*S43 = [[903/902]] * [[1764/1763]]<br />
| ([[42/41]])<sup>2</sup> / ([[22/21]])<br />
| [[18522/18491]]<br />
|-<br />
| S43<sup>2</sup>*S44 = [[946/945]] * [[1849/1848]]<br />
| ([[43/42]])<sup>2</sup> / ([[45/43]])<br />
| [[79507/79380]]<br />
|-<br />
| S44<sup>2</sup>*S45 = [[990/989]] * [[1936/1935]]<br />
| ([[44/43]])<sup>2</sup> / ([[23/22]])<br />
| [[42592/42527]]<br />
|-<br />
| S46<sup>2</sup>*S47 = [[1081/1080]] * [[2116/2115]]<br />
| ([[46/45]])<sup>2</sup> / ([[24/23]])<br />
| [[12167/12150]]<br />
|-<br />
| S49<sup>2</sup>*S50 = [[1225/1224]] * [[2401/2400]]<br />
| ([[49/48]])<sup>2</sup> / ([[51/49]])<br />
| [[117649/117504]]<br />
|-<br />
| S50<sup>2</sup>*S51 = [[1275/1274]] * [[2500/2499]]<br />
| ([[50/49]])<sup>2</sup> / ([[26/25]])<br />
| [[31250/31213]]<br />
|-<br />
| S52<sup>2</sup>*S53 = [[1378/1377]] * [[2704/2703]]<br />
| ([[52/51]])<sup>2</sup> / ([[27/26]])<br />
| [[70304/70227]]<br />
|-<br />
| S55<sup>2</sup>*S56 = [[1540/1539]] * [[3025/3024]]<br />
| ([[55/54]])<sup>2</sup> / ([[57/55]])<br />
| [[166375/166212]]<br />
|-<br />
| S56<sup>2</sup>*S57 = [[1596/1595]] * [[3136/3135]]<br />
| ([[56/55]])<sup>2</sup> / ([[29/28]])<br />
| [[87808/87725]]<br />
|-<br />
| S58<sup>2</sup>*S59 = [[1711/1710]] * [[3364/3363]]<br />
| ([[58/57]])<sup>2</sup> / ([[30/29]])<br />
| [[48778/48735]]<br />
|-<br />
| S63<sup>2</sup>*S64 = [[2016/2015]] * [[3969/3968]]<br />
| ([[63/62]])<sup>2</sup> / ([[65/63]])<br />
| [[250047/249860]]<br />
|-<br />
| S64<sup>2</sup>*S65 = [[2080/2079]] * [[4096/4095]]<br />
| ([[64/63]])<sup>2</sup> / ([[33/32]])<br />
| [[131072/130977]]<br />
|-<br />
| S66<sup>2</sup>*S67 = [[2211/2210]] * [[4356/4355]]<br />
| ([[66/65]])<sup>2</sup> / ([[34/33]])<br />
| [[71874/71825]]<br />
|-<br />
| S70<sup>2</sup>*S71 = [[2485/2484]] * [[4900/4899]]<br />
| ([[70/69]])<sup>2</sup> / ([[36/35]])<br />
| [[42875/42849]]<br />
|-<br />
| S75<sup>2</sup>*S76 = [[2850/2849]] * [[5625/5624]]<br />
| ([[75/74]])<sup>2</sup> / ([[77/75]])<br />
| [[421875/421652]]<br />
|-<br />
| S76<sup>2</sup>*S77 = [[2926/2925]] * [[5776/5775]]<br />
| ([[76/75]])<sup>2</sup> / ([[39/38]])<br />
| [[219488/219375]]<br />
|-<br />
| S78<sup>2</sup>*S79 = [[3081/3080]] * [[6084/6083]]<br />
| ([[78/77]])<sup>2</sup> / ([[40/39]])<br />
| [[59319/59290]]<br />
|}<br />
<br />
==== Bottom-heavy lopsided commas ====<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Square Relation<br />
! Ratio<br />
|-<br />
| S3<sup>2</sup>*S2 = [[3/2]] * [[9/8]]<br />
| ([[3/1]]) / ([[4/3]])<sup>2</sup><br />
| [[27/16]]<br />
|-<br />
| S4<sup>2</sup>*S3 = [[6/5]] * [[16/15]]<br />
| ([[2/1]]) / ([[5/4]])<sup>2</sup><br />
| [[32/25]]<br />
|-<br />
| S5<sup>2</sup>*S4 = [[10/9]] * [[25/24]]<br />
| ([[5/3]]) / ([[6/5]])<sup>2</sup><br />
| [[125/108]]<br />
|-<br />
| S6<sup>2</sup>*S5 = [[15/14]] * [[36/35]]<br />
| ([[3/2]]) / ([[7/6]])<sup>2</sup><br />
| [[54/49]]<br />
|-<br />
| S7<sup>2</sup>*S6 = [[21/20]] * [[49/48]]<br />
| ([[7/5]]) / ([[8/7]])<sup>2</sup><br />
| [[343/320]]<br />
|-<br />
| S8<sup>2</sup>*S7 = [[28/27]] * [[64/63]]<br />
| ([[4/3]]) / ([[9/8]])<sup>2</sup><br />
| [[256/243]]<br />
|-<br />
| S9<sup>2</sup>*S8 = [[36/35]] * [[81/80]]<br />
| ([[9/7]]) / ([[10/9]])<sup>2</sup><br />
| [[729/700]]<br />
|-<br />
| S10<sup>2</sup>*S9 = [[45/44]] * [[100/99]]<br />
| ([[5/4]]) / ([[11/10]])<sup>2</sup><br />
| [[125/121]]<br />
|-<br />
| S11<sup>2</sup>*S10 = [[55/54]] * [[121/120]]<br />
| ([[11/9]]) / ([[12/11]])<sup>2</sup><br />
| [[1331/1296]]<br />
|-<br />
| S12<sup>2</sup>*S11 = [[66/65]] * [[144/143]]<br />
| ([[6/5]]) / ([[13/12]])<sup>2</sup><br />
| [[864/845]]<br />
|-<br />
| S13<sup>2</sup>*S12 = [[78/77]] * [[169/168]]<br />
| ([[13/11]]) / ([[14/13]])<sup>2</sup><br />
| [[2197/2156]]<br />
|-<br />
| S14<sup>2</sup>*S13 = [[91/90]] * [[196/195]]<br />
| ([[7/6]]) / ([[15/14]])<sup>2</sup><br />
| [[686/675]]<br />
|-<br />
| S15<sup>2</sup>*S14 = [[105/104]] * [[225/224]]<br />
| ([[15/13]]) / ([[16/15]])<sup>2</sup><br />
| [[3375/3328]]<br />
|-<br />
| S16<sup>2</sup>*S15 = [[120/119]] * [[256/255]]<br />
| ([[8/7]]) / ([[17/16]])<sup>2</sup><br />
| [[2048/2023]]<br />
|-<br />
| S17<sup>2</sup>*S16 = [[136/135]] * [[289/288]]<br />
| ([[17/15]]) / ([[18/17]])<sup>2</sup><br />
| [[4913/4860]]<br />
|-<br />
| S18<sup>2</sup>*S17 = [[153/152]] * [[324/323]]<br />
| ([[9/8]]) / ([[19/18]])<sup>2</sup><br />
| [[729/722]]<br />
|-<br />
| S19<sup>2</sup>*S18 = [[171/170]] * [[361/360]]<br />
| ([[19/17]]) / ([[20/19]])<sup>2</sup><br />
| [[6859/6800]]<br />
|-<br />
| S20<sup>2</sup>*S19 = [[190/189]] * [[400/399]]<br />
| ([[10/9]]) / ([[21/20]])<sup>2</sup><br />
| [[4000/3969]]<br />
|-<br />
| S21<sup>2</sup>*S20 = [[210/209]] * [[441/440]]<br />
| ([[21/19]]) / ([[22/21]])<sup>2</sup><br />
| [[9261/9196]]<br />
|-<br />
| S22<sup>2</sup>*S21 = [[231/230]] * [[484/483]]<br />
| ([[11/10]]) / ([[23/22]])<sup>2</sup><br />
| [[2662/2645]]<br />
|-<br />
| S23<sup>2</sup>*S22 = [[253/252]] * [[529/528]]<br />
| ([[23/21]]) / ([[24/23]])<sup>2</sup><br />
| [[12167/12096]]<br />
|-<br />
| S24<sup>2</sup>*S23 = [[276/275]] * [[576/575]]<br />
| ([[12/11]]) / ([[25/24]])<sup>2</sup><br />
| [[6912/6875]]<br />
|-<br />
| S25<sup>2</sup>*S24 = [[300/299]] * [[625/624]]<br />
| ([[25/23]]) / ([[26/25]])<sup>2</sup><br />
| [[15625/15548]]<br />
|-<br />
| S26<sup>2</sup>*S25 = [[325/324]] * [[676/675]]<br />
| ([[13/12]]) / ([[27/26]])<sup>2</sup><br />
| [[2197/2187]]<br />
|-<br />
| S27<sup>2</sup>*S26 = [[351/350]] * [[729/728]]<br />
| ([[27/25]]) / ([[28/27]])<sup>2</sup><br />
| [[19683/19600]]<br />
|-<br />
| S28<sup>2</sup>*S27 = [[378/377]] * [[784/783]]<br />
| ([[14/13]]) / ([[29/28]])<sup>2</sup><br />
| [[10976/10933]]<br />
|-<br />
| S29<sup>2</sup>*S28 = [[406/405]] * [[841/840]]<br />
| ([[29/27]]) / ([[30/29]])<sup>2</sup><br />
| [[24389/24300]]<br />
|-<br />
| S30<sup>2</sup>*S29 = [[435/434]] * [[900/899]]<br />
| ([[15/14]]) / ([[31/30]])<sup>2</sup><br />
| [[6750/6727]]<br />
|-<br />
| S31<sup>2</sup>*S30 = [[465/464]] * [[961/960]]<br />
| ([[31/29]]) / ([[32/31]])<sup>2</sup><br />
| [[29791/29696]]<br />
|-<br />
| S32<sup>2</sup>*S31 = [[496/495]] * [[1024/1023]]<br />
| ([[16/15]]) / ([[33/32]])<sup>2</sup><br />
| [[16384/16335]]<br />
|-<br />
| S33<sup>2</sup>*S32 = [[528/527]] * [[1089/1088]]<br />
| ([[33/31]]) / ([[34/33]])<sup>2</sup><br />
| [[35937/35836]]<br />
|-<br />
| S34<sup>2</sup>*S33 = [[561/560]] * [[1156/1155]]<br />
| ([[17/16]]) / ([[35/34]])<sup>2</sup><br />
| [[4913/4900]]<br />
|-<br />
| S35<sup>2</sup>*S34 = [[595/594]] * [[1225/1224]]<br />
| ([[35/33]]) / ([[36/35]])<sup>2</sup><br />
| [[42875/42768]]<br />
|-<br />
| S36<sup>2</sup>*S35 = [[630/629]] * [[1296/1295]]<br />
| ([[18/17]]) / ([[37/36]])<sup>2</sup><br />
| [[23328/23273]]<br />
|-<br />
| S37<sup>2</sup>*S36 = [[666/665]] * [[1369/1368]]<br />
| ([[37/35]]) / ([[38/37]])<sup>2</sup><br />
| [[50653/50540]]<br />
|-<br />
| S38<sup>2</sup>*S37 = [[703/702]] * [[1444/1443]]<br />
| ([[19/18]]) / ([[39/38]])<sup>2</sup><br />
| [[13718/13689]]<br />
|-<br />
| S39<sup>2</sup>*S38 = [[741/740]] * [[1521/1520]]<br />
| ([[39/37]]) / ([[40/39]])<sup>2</sup><br />
| [[59319/59200]]<br />
|-<br />
| S40<sup>2</sup>*S39 = [[780/779]] * [[1600/1599]]<br />
| ([[20/19]]) / ([[41/40]])<sup>2</sup><br />
| [[32000/31939]]<br />
|-<br />
| S41<sup>2</sup>*S40 = [[820/819]] * [[1681/1680]]<br />
| ([[41/39]]) / ([[42/41]])<sup>2</sup><br />
| [[68921/68796]]<br />
|-<br />
| S42<sup>2</sup>*S41 = [[861/860]] * [[1764/1763]]<br />
| ([[21/20]]) / ([[43/42]])<sup>2</sup><br />
| [[9261/9245]]<br />
|-<br />
| S43<sup>2</sup>*S42 = [[903/902]] * [[1849/1848]]<br />
| ([[43/41]]) / ([[44/43]])<sup>2</sup><br />
| [[79507/79376]]<br />
|-<br />
| S44<sup>2</sup>*S43 = [[946/945]] * [[1936/1935]]<br />
| ([[22/21]]) / ([[45/44]])<sup>2</sup><br />
| [[42592/42525]]<br />
|-<br />
| S45<sup>2</sup>*S44 = [[990/989]] * [[2025/2024]]<br />
| ([[45/43]]) / ([[46/45]])<sup>2</sup><br />
| [[91125/90988]]<br />
|-<br />
| S48<sup>2</sup>*S47 = [[1128/1127]] * [[2304/2303]]<br />
| ([[24/23]]) / ([[49/48]])<sup>2</sup><br />
| [[55296/55223]]<br />
|-<br />
| S50<sup>2</sup>*S49 = [[1225/1224]] * [[2500/2499]]<br />
| ([[25/24]]) / ([[51/50]])<sup>2</sup><br />
| [[15625/15606]]<br />
|-<br />
| S51<sup>2</sup>*S50 = [[1275/1274]] * [[2601/2600]]<br />
| ([[51/49]]) / ([[52/51]])<sup>2</sup><br />
| [[132651/132496]]<br />
|-<br />
| S54<sup>2</sup>*S53 = [[1431/1430]] * [[2916/2915]]<br />
| ([[27/26]]) / ([[55/54]])<sup>2</sup><br />
| [[39366/39325]]<br />
|-<br />
| S56<sup>2</sup>*S55 = [[1540/1539]] * [[3136/3135]]<br />
| ([[28/27]]) / ([[57/56]])<sup>2</sup><br />
| [[87808/87723]]<br />
|-<br />
| S57<sup>2</sup>*S56 = [[1596/1595]] * [[3249/3248]]<br />
| ([[57/55]]) / ([[58/57]])<sup>2</sup><br />
| [[185193/185020]]<br />
|-<br />
| S62<sup>2</sup>*S61 = [[1891/1890]] * [[3844/3843]]<br />
| ([[31/30]]) / ([[63/62]])<sup>2</sup><br />
| [[59582/59535]]<br />
|-<br />
| S64<sup>2</sup>*S63 = [[2016/2015]] * [[4096/4095]]<br />
| ([[32/31]]) / ([[65/64]])<sup>2</sup><br />
| [[131072/130975]]<br />
|-<br />
| S65<sup>2</sup>*S64 = [[2080/2079]] * [[4225/4224]]<br />
| ([[65/63]]) / ([[66/65]])<sup>2</sup><br />
| [[274625/274428]]<br />
|-<br />
| S68<sup>2</sup>*S67 = [[2278/2277]] * [[4624/4623]]<br />
| ([[34/33]]) / ([[69/68]])<sup>2</sup><br />
| [[157216/157113]]<br />
|-<br />
| S74<sup>2</sup>*S73 = [[2701/2700]] * [[5476/5475]]<br />
| ([[37/36]]) / ([[75/74]])<sup>2</sup><br />
| [[50653/50625]]<br />
|-<br />
| S76<sup>2</sup>*S75 = [[2850/2849]] * [[5776/5775]]<br />
| ([[38/37]]) / ([[77/76]])<sup>2</sup><br />
| [[219488/219373]]<br />
|-<br />
| S77<sup>2</sup>*S76 = [[2926/2925]] * [[5929/5928]]<br />
| ([[77/75]]) / ([[78/77]])<sup>2</sup><br />
| [[456533/456300]]<br />
|-<br />
| S80<sup>2</sup>*S79 = [[3160/3159]] * [[6400/6399]]<br />
| ([[40/39]]) / ([[81/80]])<sup>2</sup><br />
| [[256000/255879]]<br />
|}<br />
<br />
== Equivalent S-expressions ==<br />
=== Significance and meaning ===<br />
All S-expressions have other equivalent S-expressions, however when the equivalence makes one comma a member of two of the infinite families discussed on this page, or otherwise makes it equal to a product or ratio between two such commas, this often means exceptional and nontrivial ("deep") tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending how you temper further. Generally we exclude 1/n-square-particulars, only noting up to 1/3-square-particulars, because equivalent 1/n-square-particular expressions become very common as you allow higher n, but are still quite rare for small n.<br />
<br />
=== A useful general rule ===<br />
While there are likely arbitrarily many ways of rewriting S-expressions due to the redundancy in representation, the following equivalence is, due to its simplicity and elegance, arguably most likely to be useful:<br />
<br />
<math><br />
\large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)<br />
</math><br />
<br />
This is important to note because using this simple rule we can derive an infinite amount of trivially and obviously equivalent S-expressions that should be discarded from [[#Examples]]. See [[S-expression/Advanced results]] for mathematical details.<br />
<br />
=== Examples ===<br />
Here is an incomplete list of examples (feel free to expand with any equivalences you find that you think are valuable).<br />
<br />
'''Importantly:''' examples that can ''easily'' (with one or two algebraic rewriting steps) be shown to result from the aforementioned [[#A useful general rule|useful general rule]] are considered invalid/trivial.<br />
<br />
{| class="wikitable center-all<br />
|-<br />
! Comma<br />
! S-expressions<br />
|-<br />
| [[64/63]]<br />
| (S4*S5*S6)/S3 = S4/(S6*S7) = S8<br />
|-<br />
| [[81/80]]<br />
| S6/S8 = S9<br />
|-<br />
| [[176/175]]<br />
| S8/S10 = S22*S23*S24<br />
|-<br />
| [[243/242]]<br />
| S9/S11 = S15/([[3025/3024|S22/S24 = S55]])<br />
|-<br />
| [[325/324]]<br />
| S10/S12 = S25*S26<br />
|-<br />
| [[540/539]]<br />
| S12/S14 = (S9*S10)/S7 = (S6/S7)/(S8/S10)<br />
|-<br />
| [[676/675]]<br />
| S13/S15 = S26<br />
|-<br />
| [[1225/1224]]<br />
| S35 = S49*S50<br />
|-<br />
| [[3025/3024]]<br />
| S22/S24 = S55 = S25/S27 * S99<br />
|-<br />
| [[2601/2600]]<br />
| S17/(S25*S26) = S51<br />
|-<br />
| [[9801/9800]]<br />
| S99 = S33/S35<br />
|-<br />
| [[25921/25920]]<br />
| S161 = S46/S48<br />
|-<br />
| [[123201/123200]]<br />
| S351 = S78/S80<br />
|}<br />
<br />
Note: Where a comma written in the form a/b is used in an S-expression, this means to replace that comma with any equivalent S-expression. This is done in the case of [[3025/3024]] as there are many S-expressions for it so restating them each time it appears seems inconvenient.<br />
<br />
A proof that every positive rational number (and thus every JI interval) can be written as an S-expression follows.<br />
<br />
It suffices to show every superparticular number including 2/1 has an expression using square-particulars:<br />
<br />
<math>\displaystyle<br />
\begin{align}<br />
& 2/1 = S_2 \cdot S_2 \cdot S_3\ ,\\<br />
& 3/2 = S_2 \cdot S_3\ ,\\<br />
& 4/3 = S_2\ ,\\<br />
& \frac{a/(a - 1)}{(b + 1)/b} = \prod_{k=a}^b \left( S_k = \frac{k/(k - 1)}{(k + 1)/k} \right) \\<br />
& \ \ \ = \frac{a/(a - 1)}{(a + 1)/a} \cdot \frac{(a + 1)/a}{(a + 2)/(a + 1)} \cdot \frac{(a + 2)/(a + 1)}{(a + 3)/(a + 2)} \cdot\ \ldots \cdot \frac{b/(b - 1)}{(b + 1)/b} = \frac{a/(a - 1)}{(b + 1)/b} \\<br />
& \implies \frac{a/(a - 1)}{(b + 1)/b} = S_a \cdot S_{a + 1} \cdot S_{a + 2} \cdot\ \ldots \cdot S_b \\<br />
& \implies \frac{S_2 \cdot S_2 \cdot S_3}{\prod_{a = 2}^k S_a} = 2 \cdot \left( \frac{2/(2 - 1)}{(k + 1)/k} \right)^{-1} = 2 \cdot \left( \frac{(k + 1)/k}{2} \right) = (k + 1)/k<br />
\end{align}<br />
</math><br />
<br />
From here it should not be hard to see how to make any positive rational number. For 11/6, for example, we can do (11/10)(10/9)(9/8)…(2/1) = 11 and then divide that by (6/5)(5/4)(4/3)(3/2)(2/1), meaning 11/6 = (11/10)(10/9)(9/8)(8/7)(7/6) because of the cancellations, then each of those superparticulars we replace with the corresponding S-expression to get the final S-expression. This final S-expression is likely to be far from the most efficient or interesting expression; the redundancy in S-expressions is a strength and feature, as it tells us that there are more than the trivial connections between commas and intervals and that S-expressions can be wielded as a mathematical tool/language to investigate and identify them.<br />
<br />
== Glossary ==<br />
<br />
; Superparticular<br />
: The interval/comma between two consecutive harmonics. See [[superparticular]].<br />
: These are of the form {{sfrac|''k'' + 1|''k''}}.<br />
<br />
; Square-particular<br />
: A superparticular interval/comma whose numerator is a square number. A shorthand (nick)name for square superparticular.<br />
: These are of the form {{nowrap|{{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}} {{=}} S''k''}}.<br />
<br />
; Triangle-particular<br />
: A superparticular interval/comma whose numerator is a [[triangular number]]. A shorthand (nick)name for triangular superparticular. An alternative name for 1/2-square-particular.<br />
: These are of the form {{sfrac|''k''<sup>2</sup> + ''k''|''k''<sup>2</sup> + ''k'' − 2}}. (This always simplifies to a superparticular.)<br />
<br />
; 1/''n''-square-particular<br />
: A comma which is the product of ''n'' consecutive square-particulars and which can therefore be expressed as the ratio between two superparticulars.<br />
: These are of the form {{nowrap|S''a'' * S(''a'' + 1) * … * S''b'' {{=}} {{sfrac|{{sfrac|''a''|''a'' − 1}}|{{sfrac|''b'' + 1|''b''}}}}}} {{nowrap|{{=}} {{sfrac|''ab''|(''a'' − 1)(''b'' + 1)}}}}.<br />
: Replacing/substituting ''a'' with ''k'' and ''b'' with ''k'' + ''n'' - 1 gives us an equivalent expression that includes the number of square-particulars ''n'':<br />
: {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1) {{=}} {{sfrac|{{sfrac|''k''|''k'' − 1}}|{{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}}}}} {{nowrap|{{=}} {{sfrac|''k''(''k'' + ''n'' − 1)|(''k'' − 1)(''k'' + ''n''}}}}<br />
: For {{nowrap|''b'' {{=}} ''a'' + 1}} these can also be called triangle-particulars, in which case they are always superparticular.<br />
: These have implications for whether consistency in the {{nowrap|(''n'' + ''k'') {{=}} (''b'' + 1)}}-[[odd-limit]] is ''potentially'' possible in a given temperament; see the [[#Sk*S(k + 1)*…*S(k + n - 1) (1/n-square-particulars)|section on 1/''n''-square-particulars]].<br />
<br />
; Odd-particular<br />
: An interval/comma between two consecutive odd harmonics. The odd analogue of superparticular.<br />
: These are of the form {{sfrac|2''k'' + 1|2''k'' − 1}}.<br />
<br />
; Throdd-particular<br />
: An interval/comma between two harmonics 3 apart which is not superparticular.<br />
: These are of the form {{sfrac|3''k'' + 1|3''k'' − 2}} or {{sfrac|3''k'' + 2|3''k'' − 1}}.<br />
<br />
; Quodd-particular<br />
: An interval/comma between two harmonics 4 apart which is not superparticular or odd-particular.<br />
: These are of the form {{sfrac|4''k'' + 1|4''k'' − 3}} or {{sfrac|4''k'' + 3|4''k'' − 1}}.<br />
<br />
; ''n''-odd-particular<br />
: An interval/comma between two coprime harmonics ''n'' apart (also called as [[Delta-N ratio|delta-''n'' ratio]]). It is the generalization of superparticular, odd-particular, throdd-particular, and quodd-particular.<br />
: If ''n'' is a prime, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is not superparticular. For example, 5-odd-particular intervals are of the form {{sfrac|5''k'' + 1|5''k'' − 4}}, {{sfrac|5''k'' + 2|5''k'' − 3}}, {{sfrac|5''k'' + 3|5''k'' − 2}}, or {{sfrac|5''k'' + 4|5''k'' − 1}}.<br />
: If ''n'' is a composite, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is neither superparticular nor of ''m''-odd-particular intervals where ''m'' is any other divisor of ''n''. For example, 6-odd-particular intervals are of the form {{sfrac|6''k'' + 1|6''k'' − 5}} or {{sfrac|6''k'' + 5|6''k'' − 1}}.<br />
<br />
; Ultraparticular<br />
: An interval/comma which is the ratio of two consecutive square-particulars.<br />
: These are of the form {{sfrac|S''k''|S(''k'' + 1)}}.<br />
<br />
; Semiparticular<br />
: A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:<br />
: These are of the form {{sfrac|S''k''|S(''k'' + 2)}}.<br />
<br />
; S-expression<br />
: An expression using the S''k'' shorthand notation corresponding strictly to multiplying and dividing only (arbitrary) square-particulars. S-expressions include singular square superparticulars and expressions for other superparticulars in terms of square superparticulars.<br />
<br />
; S-factorization<br />
: An expression that takes a list of consecutive integer harmonics including the ''k''th harmonic and raises them to integer powers, similar to a [[smonzo]] but uniquely suited to analysing S-expressions.<br />
: For example: {{nowrap|S''k'' {{=}} [''k'' − 1, ''k'', ''k'' + 1]<sup>[−1, 2, −1]</sup>}} because {{nowrap|S''k'' {{=}} (''k'' − 1)<sup>−1</sup>''k''<sup>2</sup>(''k'' + 1)<sup>−1</sup>}}.<br />
<br />
; S-comma<br />
: Any comma within one of the infinite families of commas discussed here, excluding 1/n-square-particulars for n>5 (so square-particulars, triangle-particulars, 1/3-square-particulars, 1/4-square-particulars and 1/5-square-particulars are included, but not anything beyond; this bound is used for exclusion (rather than n>3) to allow the utility of 1/5-square-particulars in avoiding twin primes by equating superparticular intervals on either side of the twin primes).<br />
<br />
; Indirect S-comma<br />
: Any comma that is the product or ratio of two S-commas. These appear frequently as S-expressions for commas that are more challenging/nontrivial to represent from the perspective of S-expressions; for example, the [[schisma]] admits at least three such representations.<br />
<br />
== See further ==<br />
* [[S-expression/Advanced results|Advanced results]] – for the harder-to-reach algebra<br />
<br />
== Notes ==<br />
<references group="note" /><br />
<br />
[[Category:Elementary math]]<br />
[[Category:Pages with proofs]]<br />
[[Category:Ratio]]<br />
[[Category:Terms]]</div>ArrowHead294https://en.xen.wiki/index.php?title=S-expression&diff=224389S-expression2026-02-20T11:47:56Z<p>ArrowHead294: /* Glossary */</p>
<hr />
<div>An '''S-expression''' is any product, or ratio of products, of the '''square superparticulars''' S''k'', which are defined as the fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. Commas defined by S-expressions turn out to represent intuitive and wide-reaching families of tempered equivalences, and therefore present a very useful framework to learn for a good understanding of the [[commas]] that appear frequently in xen.<br />
<br />
== Quick rules of S-expressions ==<br />
As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply when they are [[tempering out|tempered out]]. The linked sections provide deeper information into each comma family.<br />
<br />
* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. <br>Tempering out S''k'' equates {{sfrac|''k'' + 1|''k''}} with {{sfrac|''k''|''k'' − 1}} and splits {{sfrac|''k'' + 1|''k'' − 1}} in two.<br />
* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: {{nowrap|'''S''k'' * S(''k'' + 1)'''}}, superparticular fractions of the form {{sfrac|''k''(''k'' + 1)/2|(''k'' − 1)(''k'' + 2)/2}}. <br>Tempering out {{nowrap|S''k'' * S(''k'' + 1)}} equates {{sfrac|''k'' + 2|''k'' + 1}} with {{sfrac|''k''|''k'' − 1}}, and {{sfrac|''k'' + 2|''k''}} with {{sfrac|''k'' + 1|''k'' − 1}}.<br />
* [[#Sk2 * S(k + 1) and S(k − 1) * Sk2 (lopsided commas)|Lopsided commas]]: {{nowrap|'''(S''k'')<sup>2</sup> * S(''k'' + 1)'''}} and {{nowrap|'''(S''k'')<sup>2</sup> * S(''k'' − 1)'''}}. <br>Tempering out the former equates {{sfrac|''k'' + 2|''k''}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>2</sup> and {{sfrac|''k'' + 2|''k'' − 1}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>3</sup>, and tempering out the latter equates {{sfrac|''k''|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>2</sup> and {{sfrac|''k'' + 1|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.<br />
* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out splits {{sfrac|''k'' + 2|''k'' − 1}} into {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.<br />
* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out splits {{sfrac|''k'' + 3|''k'' − 1}} into {{pars|{{sfrac|''k'' + 2|''k''}}}}<sup>2</sup>.<br />
<br />
== S''k'' (square-particulars) ==<br />
A '''square superparticular''', or ''square-particular'' for short, is a [[superparticular]] [[interval]] whose numerator is a square number, which is to say, a superparticular of the form<br />
<br />
<math>\displaystyle \frac {k^2}{k^2 - 1} = \frac {k/(k - 1)}{(k + 1)/k} </math><br />
<br />
which is square-(super)particular ''k'' for a given integer {{nowrap|''k'' &gt; 1}}. A suggested shorthand for this interval is '''S''k''''' for the ''k''-th square superparticular, where the ''S'' stands for "(Shorthand for) Second-order/Square Superparticular". This will be used later in this article as the notation will prove powerful in understanding the commas and implied tempered structures of [[regular temperament]]s. Note that this means {{nowrap|S2 {{=}} [[4/3]]}} is the first musically meaningful square-particular, as {{nowrap|S1 {{=}} 1/0}}.<br />
<br />
Also note that we use the notation S''k''<sup>''p''</sup> to mean (S''k'')<sup>''p''</sup> rather than S(''k''<sup>''p''</sup>) for convenience in the practical analysis of regular temperaments using [[S-expression]]s.<br />
<br />
=== Significance/motivation ===<br />
Square-particulars are important structurally because they are the intervals between consecutive [[superparticular]] [[interval]]s while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the ({{nowrap|''k'' + 1}})th harmonic, as well as the potential representational sacrifices that must be made from that point onward. In other words, understanding the mappings of S''k'' in a given temperament is equivalent to understanding the spacing of consecutive superparticular intervals, and thereby to understanding the way it represents (or tries to represent) the harmonic series.<br />
<br />
=== Table of square-particulars ===<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 31-limit square-particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2<br />
| ([[2/1]])/([[3/2]])<br />
| [[4/3]]<br />
| 3<br />
|-<br />
| S3<br />
| ([[3/2]])/([[4/3]])<br />
| [[9/8]]<br />
| 3<br />
|-<br />
| S4<br />
| ([[4/3]])/([[5/4]])<br />
| [[16/15]]<br />
| 5<br />
|-<br />
| S5<br />
| ([[5/4]])/([[6/5]])<br />
| [[25/24]]<br />
| 5<br />
|-<br />
| S6 = S8*S9<br />
| ([[6/5]])/([[7/6]])<br />
| [[36/35]]<br />
| 7<br />
|-<br />
| S7<br />
| ([[7/6]])/([[8/7]])<br />
| [[49/48]]<br />
| 7<br />
|-<br />
| S8<br />
| ([[8/7]])/([[9/8]])<br />
| [[64/63]]<br />
| 7<br />
|-<br />
| S9 = S6/S8<br />
| ([[9/8]])/([[10/9]])<br />
| [[81/80]]<br />
| 5<br />
|-<br />
| S10<br />
| ([[10/9]])/([[11/10]])<br />
| [[100/99]]<br />
| 11<br />
|-<br />
| S11<br />
| ([[11/10]])/([[12/11]])<br />
| [[121/120]]<br />
| 11<br />
|-<br />
| S12<br />
| ([[12/11]])/([[13/12]])<br />
| [[144/143]]<br />
| 13<br />
|-<br />
| S13<br />
| ([[13/12]])/([[14/13]])<br />
| [[169/168]]<br />
| 13<br />
|-<br />
| S14<br />
| ([[14/13]])/([[15/14]])<br />
| [[196/195]]<br />
| 13<br />
|-<br />
| S15<br />
| ([[15/14]])/([[16/15]])<br />
| [[225/224]]<br />
| 7<br />
|-<br />
| S16<br />
| ([[16/15]])/([[17/16]])<br />
| [[256/255]]<br />
| 17<br />
|-<br />
| S17<br />
| ([[17/16]])/([[18/17]])<br />
| [[289/288]]<br />
| 17<br />
|-<br />
| S18<br />
| ([[18/17]])/([[19/18]])<br />
| [[324/323]]<br />
| 19<br />
|-<br />
| S19<br />
| ([[19/18]])/([[20/19]])<br />
| [[361/360]]<br />
| 19<br />
|-<br />
| S20<br />
| ([[20/19]])/([[21/20]])<br />
| [[400/399]]<br />
| 19<br />
|-<br />
| S21<br />
| ([[21/20]])/([[22/21]])<br />
| [[441/440]]<br />
| 11<br />
|-<br />
| S22<br />
| ([[22/21]])/([[23/22]])<br />
| [[484/483]]<br />
| 23<br />
|-<br />
| S23<br />
| ([[23/22]])/([[24/23]])<br />
| [[529/528]]<br />
| 23<br />
|-<br />
| S24<br />
| ([[24/23]])/([[25/24]])<br />
| [[576/575]]<br />
| 23<br />
|-<br />
| S25<br />
| ([[25/24]])/([[26/25]])<br />
| [[625/624]]<br />
| 13<br />
|-<br />
| S26 = S13/S15<br />
| ([[26/25]])/([[27/26]])<br />
| [[676/675]]<br />
| 13<br />
|-<br />
| S27<br />
| ([[27/26]])/([[28/27]])<br />
| [[729/728]]<br />
| 13<br />
|-<br />
| S28<br />
| ([[28/27]])/([[29/28]])<br />
| [[784/783]]<br />
| 29<br />
|-<br />
| S29<br />
| ([[29/28]])/([[30/29]])<br />
| [[841/840]]<br />
| 29<br />
|-<br />
| S30<br />
| ([[30/29]])/([[31/30]])<br />
| [[900/899]]<br />
| 31<br />
|-<br />
| S31<br />
| ([[31/30]])/([[32/31]])<br />
| [[961/960]]<br />
| 31<br />
|-<br />
| S32<br />
| ([[32/31]])/([[33/32]])<br />
| [[1024/1023]]<br />
| 31<br />
|-<br />
| S33<br />
| ([[33/32]])/([[34/33]])<br />
| [[1089/1088]]<br />
| 17<br />
|-<br />
| S34<br />
| ([[34/33]])/([[35/34]])<br />
| [[1156/1155]]<br />
| 17<br />
|-<br />
| S35 = S49*S50<br />
| ([[35/34]])/([[36/35]])<br />
| [[1225/1224]]<br />
| 17<br />
|-<br />
| S39<br />
| ([[39/38]])/([[40/39]])<br />
| [[1521/1520]]<br />
| 19<br />
|-<br />
| S45<br />
| ([[45/44]])/([[46/45]])<br />
| [[2025/2024]]<br />
| 23<br />
|-<br />
| S49<br />
| ([[49/48]])/([[50/49]])<br />
| [[2401/2400]]<br />
| 7<br />
|-<br />
| S50<br />
| ([[50/49]])/([[51/50]])<br />
| [[2500/2499]]<br />
| 17<br />
|-<br />
| S51<br />
| ([[51/50]])/([[52/51]])<br />
| [[2601/2600]]<br />
| 17<br />
|-<br />
| S55 = S22/S24<br />
| ([[55/54]])/([[56/55]])<br />
| [[3025/3024]]<br />
| 11<br />
|-<br />
| S56<br />
| ([[56/55]])/([[57/56]])<br />
| [[3136/3135]]<br />
| 19<br />
|-<br />
| S57<br />
| ([[57/56]])/([[58/57]])<br />
| [[3249/3248]]<br />
| 29<br />
|-<br />
| S63<br />
| ([[63/62]])/([[64/63]])<br />
| [[3969/3968]]<br />
| 31<br />
|-<br />
| S64<br />
| ([[64/63]])/([[65/64]])<br />
| [[4096/4095]]<br />
| 13<br />
|-<br />
| S65<br />
| ([[65/64]])/([[66/65]])<br />
| [[4225/4224]]<br />
| 13<br />
|-<br />
| S69<br />
| ([[69/68]])/([[70/69]])<br />
| [[4761/4760]]<br />
| 23<br />
|-<br />
| S76<br />
| ([[76/75]])/([[77/76]])<br />
| [[5776/5775]]<br />
| 19<br />
|-<br />
| S77<br />
| ([[77/76]])/([[78/77]])<br />
| [[5929/5928]]<br />
| 19<br />
|-<br />
| S91<br />
| ([[91/90]])/([[92/91]])<br />
| [[8281/8280]]<br />
| 23<br />
|-<br />
| S92<br />
| ([[92/91]])/([[93/92]])<br />
| [[8464/8463]]<br />
| 31<br />
|-<br />
| S99 = S33/S35<br />
| ([[99/98]])/([[100/99]])<br />
| [[9801/9800]]<br />
| 11<br />
|-<br />
| S115<br />
| ([[115/114]])/([[116/115]])<br />
| [[13225/13224]]<br />
| 29<br />
|-<br />
| S116<br />
| ([[116/115]])/([[117/116]])<br />
| [[13456/13455]]<br />
| 29<br />
|-<br />
| S120<br />
| ([[120/119]])/([[121/120]])<br />
| [[14400/14399]]<br />
| 17<br />
|-<br />
| S125<br />
| ([[125/124]])/([[126/125]])<br />
| [[15625/15624]]<br />
| 31<br />
|-<br />
| S144<br />
| ([[144/143]])/([[145/144]])<br />
| [[20736/20735]]<br />
| 29<br />
|-<br />
| S153<br />
| ([[153/152]])/([[154/153]])<br />
| [[23409/23408]]<br />
| 19<br />
|-<br />
| S154<br />
| ([[154/153]])/([[155/154]])<br />
| [[23716/23715]]<br />
| 31<br />
|-<br />
| S155<br />
| ([[155/154]])/([[156/155]])<br />
| [[24025/24024]]<br />
| 31<br />
|-<br />
| S161 = S46/S48<br />
| ([[161/160]])/([[162/161]])<br />
| [[25921/25920]]<br />
| 23<br />
|-<br />
| S169<br />
| ([[169/168]])/([[170/169]])<br />
| [[28561/28560]]<br />
| 17<br />
|-<br />
| S170<br />
| ([[170/169]])/([[171/170]])<br />
| [[28900/28899]]<br />
| 19<br />
|-<br />
| S175<br />
| ([[175/174]])/([[176/175]])<br />
| [[30625/30624]]<br />
| 29<br />
|-<br />
| S208<br />
| ([[208/207]])/([[209/208]])<br />
| [[43264/43263]]<br />
| 23<br />
|-<br />
| S209<br />
| ([[209/208]])/([[210/209]])<br />
| [[43681/43680]]<br />
| 19<br />
|-<br />
| S231<br />
| ([[231/230]])/([[232/231]])<br />
| [[53361/53360]]<br />
| 29<br />
|-<br />
| S289<br />
| ([[289/288]])/([[290/289]])<br />
| [[83521/83520]]<br />
| 29<br />
|-<br />
| S323<br />
| ([[323/322]])/([[324/323]])<br />
| [[104329/104328]]<br />
| 23<br />
|-<br />
| S324<br />
| ([[324/323]])/([[325/324]])<br />
| [[104976/104975]]<br />
| 19<br />
|-<br />
| S341<br />
| ([[341/340]])/([[342/341]])<br />
| [[116281/116280]]<br />
| 31<br />
|-<br />
| S342<br />
| ([[342/341]])/([[343/342]])<br />
| [[116964/116963]]<br />
| 31<br />
|-<br />
| S351 = S78/S80<br />
| ([[351/350]])/([[352/351]])<br />
| [[123201/123200]]<br />
| 13<br />
|-<br />
| S391<br />
| ([[391/390]])/([[392/391]])<br />
| [[152881/152880]]<br />
| 23<br />
|-<br />
| S441<br />
| ([[441/440]])/([[442/441]])<br />
| [[194481/194480]]<br />
| 17<br />
|-<br />
| S494<br />
| ([[494/493]])/([[495/494]])<br />
| [[244036/244035]]<br />
| 29<br />
|-<br />
| S495<br />
| ([[495/494]])/([[496/495]])<br />
| [[245025/245024]]<br />
| 31<br />
|-<br />
| S528<br />
| ([[528/527]])/([[529/528]])<br />
| [[278784/278783]]<br />
| 31<br />
|-<br />
| S551<br />
| ([[551/550]])/([[552/551]])<br />
| [[303601/303600]]<br />
| 29<br />
|-<br />
| S714<br />
| ([[714/713]])/([[715/714]])<br />
| [[509796/509795]]<br />
| 31<br />
|-<br />
| S783<br />
| ([[783/782]])/([[784/783]])<br />
| [[613089/613088]]<br />
| 29<br />
|-<br />
| S1275<br />
| ([[1275/1274]])/([[1276/1275]])<br />
| [[1625625/1625624]]<br />
| 29<br />
|-<br />
| S1519<br />
| ([[1519/1518]])/([[1520/1519]])<br />
| [[2307361/2307360]]<br />
| 31<br />
|-<br />
| S1520<br />
| ([[1520/1519]])/([[1521/1520]])<br />
| [[2310400/2310399]]<br />
| 31<br />
|-<br />
| S2001<br />
| ([[2001/2000]])/([[2002/2001]])<br />
| [[4004001/4004000]]<br />
| 29<br />
|-<br />
| S2024<br />
| ([[2024/2023]])/([[2025/2024]])<br />
| [[4096576/4096575]]<br />
| 23<br />
|-<br />
| S2431<br />
| ([[2431/2430]])/([[2432/2431]])<br />
| [[5909761/5909760]]<br />
| 19<br />
|-<br />
| S3249<br />
| ([[3249/3248]])/([[3250/3249]])<br />
| <font style="font-size:0.88em">[[10556001/10556000]]</font><br />
| 29<br />
|-<br />
| S9801<br />
| ([[9801/9800]])/([[9802/9801]])<br />
| <font style="font-size:0.88em">[[96059601/96059600]]</font><br />
| 29<br />
|-<br />
| S13311<br />
| <font style="font-size:0.86em">([[13311/13310]])/([[13312/13311]])</font><br />
| <font style="font-size:0.79em">[[177182721/177182720]]</font><br />
| 29<br />
|-<br />
| S13455<br />
| <font style="font-size:0.86em">([[13455/13454]])/([[13456/13455]])</font><br />
| <font style="font-size:0.79em">[[181037025/181037024]]</font><br />
| 31<br />
|}<br />
<br />
=== Alternatives to tempering square-particulars ===<br />
It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas {{nowrap|U''k'' {{=}} {{sfrac|S''k''|S(''k'' + 1)}}}}, we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]<ref group="note">In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as U''k'' = S(''k'' - 1)/S''k'' and U''k'' = S(''k'' + 1)/S(''k'' + 2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.</ref>. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because {{sfrac|''k'' + 2|''k'' + 1}} and {{sfrac|''k''|''k'' − 1}} are equidistant from {{sfrac|''k'' + 1|''k''}} (because of tempering {{sfrac|S''k''|S(''k'' + 1)}}), this means that another expression for {{sfrac|S''k''|S(''k'' + 1)}} is the following:<br />
<br />
<math>\displaystyle {\rm S}k / {\rm S} (k + 1) = \frac{(k + 2) / (k - 1)}{((k + 1)/k)^3} </math><br />
<br />
This means you can read the ''k'' and {{nowrap|''k'' + 1}} from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in [[#Sk/S(k + 1) (ultraparticulars)|the table of ultraparticulars]]).<br />
<br />
Furthermore, defining another sequence of commas with [[semiparticular|formula {{sfrac|S''k''|S(''k'' + 2)}} leads to semiparticulars]] which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering consecutive ultraparticulars.<br />
<br />
== {{nowrap|S''k''*S(''k'' + 1)}} (triangle-particulars) ==<br />
=== Significance ===<br />
1. Every triangle-particular is superparticular, so these are efficient commas. (See also the [[#Short proof of the superparticularity of triangle-particulars]].)<br />
<br />
2. Often each individual triangle-particular, taken as a comma, implies other useful equivalences not necessarily corresponding to the general form, speaking of which …<br />
<br />
3. Every triangle-particular is the difference between two nearly-adjacent superparticular intervals {{nowrap|''k'' + 2|''k'' + 1}} and {{nowrap|''k''|''k'' − 1}}.<br />
<br />
4. Tempering any two consecutive square-particulars S''k'' and {{nowrap|S(''k'' + 1)}} implies tempering a triangle-particular, so these are common commas. (See also: [[lopsided comma]]s.)<br />
<br />
5. If we temper {{nowrap|S''k'' * S(''k'' + 1)}} but not S''k'' or {{nowrap|S(''k'' + 1)}}, then one or more intervals of {{sfrac|''k''|''k'' − 1}}, {{sfrac|''k'' + 1|''k''}}, and {{sfrac|''k'' + 2|''k'' + 1}} ''must'' be mapped inconsistently, because:<br />
: If {{nowrap|''k'' + 1|''k''}} is mapped above {{nowrap|{{sfrac|''k'' + 2|''k'' + 1}} ~ {{sfrac|''k''|''k'' − 1}}}} we have {{nowrap|{{sfrac|''k'' + 1|''k''}} &gt; {{sfrac|''k''|''k'' − 1}}}} and if it is mapped below we have {{nowrap|{{sfrac|''k'' + 1|''k''}} &lt; {{sfrac|''k'' + 2|''k'' + 1}}}}.<br />
: (Generalisations of this and their implications for consistency are discussed in [[#Sk*S(k + 1)*…*S(k + n − 1) (1/n-square-particulars)|the section covering 1/''n''-square-particulars]].)<br />
<br />
=== Meaning ===<br />
Notice that if we equate {{sfrac|''k'' + 2|''k'' + 1}} with {{sfrac|''k''|''k'' − 1}} (by [[tempering out]] their difference), then multiply both sides by {{sfrac|''k'' + 1|''k''}}, we have:<br />
<br />
<math>\displaystyle \left(\frac{k + 2}{k + 1}\right)\left(\frac{k + 1}{k}\right) = \left(\frac{k + 1}{k}\right)\left(\frac{k}{k - 1}\right) </math><br />
<br />
which simplifies to:<br />
<br />
<math>\displaystyle \frac{k + 2}{k} = \frac{k + 1}{k - 1} </math><br />
<br />
This means that if we temper: <math>{\rm S}k \cdot {\rm S}(k+1) = \frac{k/(k-1)}{(k+1)/k} \cdot \frac{(k+1)/k}{(k+2)/(k+1)} = \frac{k/(k-1)}{(k+2)/(k+1)}</math><br />
<br />
… then this equivalence is achieved. Note that there is little to no reason to not also temper S''k'' and {{nowrap|S(''k'' + 1)}} individually unless other considerations seem to force your hand.<br />
<br />
=== Short proof of the superparticularity of triangle-particulars ===<br />
<math>\displaystyle S(k)*S(k + 1) = \frac{\frac{k}{k - 1}}{\frac{k + 2}{k + 1}} = \frac{k(k + 1)}{(k - 1)(k + 2)} = \frac{k^2 + k}{k^2 + k - 2} </math><br />
<br />
Then notice that {{nowrap|''k''<sup>2</sup> + ''k''}} is always a multiple of 2, therefore the above always simplifies to a superparticular. Half of this superparticular is halfway between the corresponding square-particulars, and because of its composition it could therefore be reasoned that it'd likely be half as accurate as tempering either of the square-particulars individually, so these are "1/2-square-particulars" in a sense, and half of a square is a triangle, which is not a coincidence here because the numerators of all of these superparticular intervals/commas are [[triangular number]]s! (Hence the alternative name "[[triangle-particular]]".)<br />
<br />
=== Table of triangle-particulars ===<br />
For completeness, all the intervals of this form are included, because of their structural importance for JI, and for the possibility of (in)consistency of mappings when tempered for the above reason.<br />
<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 31-limit triangle-particulars<ref group="note">After 75, 76, 77, 78, streaks of four consecutive harmonics in the 23-limit become very sparse. The last few streaks are deeply related to the consistency and structure of [[311edo]], as [[311edo]] can be described as the unique 23-limit temperament that tempers all triangle-particulars from [[595/594]] up to [[21736/21735]]. It also tempers all the square-particulars composing those triangle-particulars with the exception of S169 and S170. It also maps the corresponding intervals of the 77-odd-limit consistently. 170/169 is the only place where the logic seems to "break" as it is mapped to 2 steps instead of 3 meaning the mapping of that superparticular is inconsistent.</ref><br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3<br />
| ([[3/1]])/([[2/1]])<br />
| [[3/2]]<br />
| 3<br />
|-<br />
| S3*S4<br />
| ([[3/2]])/([[5/4]])<br />
| [[6/5]]<br />
| 5<br />
|-<br />
| S4*S5<br />
| ([[4/3]])/([[6/5]])<br />
| [[10/9]]<br />
| 5<br />
|-<br />
| S5*S6<br />
| ([[5/4]])/([[7/6]])<br />
| [[15/14]]<br />
| 7<br />
|-<br />
| S6*S7<br />
| ([[6/5]])/([[8/7]])<br />
| [[21/20]]<br />
| 7<br />
|-<br />
| S7*S8 = S4/S6<br />
| ([[7/6]])([[9/8]])<br />
| [[28/27]]<br />
| 7<br />
|-<br />
| S8*S9 = S6<br />
| ([[8/7]])/([[10/9]])<br />
| [[36/35]]<br />
| 7<br />
|-<br />
| S9*S10<br />
| ([[9/8]])/([[11/10]])<br />
| [[45/44]]<br />
| 11<br />
|-<br />
| S10*S11<br />
| ([[10/9]])/([[12/11]])<br />
| [[55/54]]<br />
| 11<br />
|-<br />
| S11*S12<br />
| ([[11/10]])/([[13/12]])<br />
| [[66/65]]<br />
| 13<br />
|-<br />
| S12*S13<br />
| ([[12/11]])/([[14/13]])<br />
| [[78/77]]<br />
| 13<br />
|-<br />
| S13*S14<br />
| ([[13/12]])/([[15/14]])<br />
| [[91/90]]<br />
| 13<br />
|-<br />
| S14*S15<br />
| ([[14/13]])/([[16/15]])<br />
| [[105/104]]<br />
| 13<br />
|-<br />
| S15*S16<br />
| ([[15/14]])/([[17/16]])<br />
| [[120/119]]<br />
| 17<br />
|-<br />
| S16*S17<br />
| ([[16/15]])/([[18/17]])<br />
| [[136/135]]<br />
| 17<br />
|-<br />
| S17*S18<br />
| ([[17/16]])/([[19/18]])<br />
| [[153/152]]<br />
| 19<br />
|-<br />
| S18*S19<br />
| ([[18/17]])/([[20/19]])<br />
| [[171/170]]<br />
| 19<br />
|-<br />
| S19*S20<br />
| ([[19/18]])/([[21/20]])<br />
| [[190/189]]<br />
| 19<br />
|-<br />
| S20*S21<br />
| ([[20/19]])/([[22/21]])<br />
| [[210/209]]<br />
| 19<br />
|-<br />
| S21*S22<br />
| ([[21/20]])/([[23/22]])<br />
| [[231/230]]<br />
| 23<br />
|-<br />
| S22*S23<br />
| ([[22/21]])/([[24/23]])<br />
| [[253/252]]<br />
| 23<br />
|-<br />
| S23*S24<br />
| ([[23/22]])/([[25/24]])<br />
| [[276/275]]<br />
| 23<br />
|-<br />
| S24*S25<br />
| ([[24/23]])/([[26/25]])<br />
| [[300/299]]<br />
| 23<br />
|-<br />
| S25*S26 = S10/S12<br />
| ([[25/24]])/([[27/26]])<br />
| [[325/324]]<br />
| 13<br />
|-<br />
| S26*S27<br />
| ([[26/25]])/([[28/27]])<br />
| [[351/350]]<br />
| 13<br />
|-<br />
| S27*S28<br />
| ([[27/26]])/([[29/28]])<br />
| [[378/377]]<br />
| 29<br />
|-<br />
| S28*S29<br />
| ([[28/27]])/([[30/29]])<br />
| [[406/405]]<br />
| 29<br />
|-<br />
| S29*S30<br />
| ([[29/28]])/([[31/30]])<br />
| [[435/434]]<br />
| 31<br />
|-<br />
| S30*S31<br />
| ([[30/29]])/([[32/31]])<br />
| [[465/464]]<br />
| 31<br />
|-<br />
| S31*S32<br />
| ([[31/30]])/([[33/32]])<br />
| [[496/495]]<br />
| 31<br />
|-<br />
| S32*S33<br />
| ([[32/31]])/([[34/33]])<br />
| [[528/527]]<br />
| 31<br />
|-<br />
| S33*S34<br />
| ([[33/32]])/([[35/34]])<br />
| [[561/560]]<br />
| 17<br />
|-<br />
| S34*S35<br />
| ([[34/33]])/([[36/35]])<br />
| [[595/594]]<br />
| 17<br />
|-<br />
| S49*S50 = S35<br />
| ([[49/48]])/([[51/50]])<br />
| [[1225/1224]]<br />
| 17<br />
|-<br />
| S50*S51<br />
| ([[50/49]])/([[52/51]])<br />
| [[1275/1274]]<br />
| 17<br />
|-<br />
| S55*S56<br />
| ([[55/54]])/([[57/56]])<br />
| [[1540/1539]]<br />
| 19<br />
|-<br />
| S63*S64<br />
| ([[63/62]])/([[65/64]])<br />
| [[2016/2015]]<br />
| 31<br />
|-<br />
| S64*S65<br />
| ([[64/63]])/([[66/65]])<br />
| [[2080/2079]]<br />
| 13<br />
|-<br />
| S76*S77<br />
| ([[76/75]])/([[78/77]])<br />
| [[2926/2925]]<br />
| 19<br />
|-<br />
| S91*S92<br />
| ([[91/90]])/([[93/92]])<br />
| [[4186/4185]]<br />
| 31<br />
|-<br />
| S115*S116<br />
| ([[115/114]])/([[117/116]])<br />
| [[6670/6669]]<br />
| 29<br />
|-<br />
| S153*S154<br />
| ([[153/152]])/([[155/154]])<br />
| [[11781/11780]]<br />
| 31<br />
|-<br />
| S154*S155<br />
| ([[154/153]])/([[156/155]])<br />
| [[11935/11934]]<br />
| 31<br />
|-<br />
| S169*S170<br />
| ([[169/168]])/([[171/170]])<br />
| [[14365/14364]]<br />
| 19<br />
|-<br />
| S208*S209<br />
| ([[208/207]])/([[210/209]])<br />
| [[21736/21735]]<br />
| 19<br />
|-<br />
| S323*S324<br />
| ([[323/322]])/([[325/324]])<br />
| [[52326/52325]]<br />
| 23<br />
|-<br />
| S341*S342<br />
| ([[341/340]])/([[343/342]])<br />
| [[58311/58310]]<br />
| 31<br />
|-<br />
| S494*S495<br />
| ([[494/493]])/([[496/495]])<br />
| [[122265/122264]]<br />
| 31<br />
|-<br />
| S1519*S1520<br />
| ([[1519/1518]])/([[1521/1520]])<br />
| [[1154440/1154439]]<br />
| 31<br />
|}<br />
<br />
== {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} (1/''n''-square-particulars) ==<br />
=== Motivation ===<br />
1/''n''-square-particulars are a generalization of square- and 1/2-square-particulars to a comma/interval whose S-expression is can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}.<br />
<br />
In other words, each and every S-expression of a comma as a 1/''n''-square-particular corresponds exactly to expressing it as the ratio between two [[superparticular]] intervals, with ''n'' distance between them, where, for example, 10/9 and 11/10 are considered as having 1 distance between them, corresponding to (1/1-)square-particulars (in this case [[100/99|S10]]).<br />
<br />
These commas are important in a few ways:<br />
1. As a generalization of important special cases {{nowrap|''n'' {{=}} 0|''n'' {{=}} 1}}, and {{nowrap|''n'' {{=}} 2}}, (which are almost all superparticular; the only case where they aren't is that {{nowrap|''n'' {{=}} 3}} (1/3-square-particulars) are throdd-particular one third of the time, so this suggests these are efficient commas. A cursory look will show that many 1/n-square-particulars for small n are superparticular, and many more are the next best things (odd-particular, throdd-particular, quodd-particular, etc.) so this confirms them being a family of efficient commas.<br />
<br />
2. Because of being the ratio of two superparticular intervals, in higher-complexity cases they often correspond to small commas between large commas which we don't want to temper, for example {{nowrap|{{sfrac|[[81/80]]|[[91/90]]}} {{=}} S81 * S82 * … * S90}} {{nowrap|{{=}} [[729/728]]}} {{nowrap|{{=}} S27}}. They also often simplify in cases like these; note that a suggested shorthand is S81..90 for {{nowrap|S81 * S82 * … * S90}} and thus more generally S''a''..''b'' for {{nowrap|S''a'' * S(''a'' + 1) * … * S''b''}}.<br />
<br />
3. They often correspond to "nontrivial" equivalences that need to be dug up which are not obvious from their expression as a ratio of two superparticular intervals, for example, [[385/384|S33*S34*S35]], suggesting they are a goldmine for valuable tempering opportunities. <br />
<br />
4. Their expressions naturally make them implied by tempering consecutive square-particulars, so if you notice them present and that the individual square-particulars aren't tempered, if you want to extend your temperament and/or reduce its rank (tempering it down) and/or hope to make your temperament more efficient, you can try tempering the untempered square-particulars that a tempered 1/''n''-square-particular is composed of (although this is not always possible). There is also good theoretical motivation for wanting to do this, as the next section will discuss.<br />
<br />
5. They're relevant to understanding how much damage is present in a temperament's harmonic series representation, because they show how many superparticular intervals are either not distinguished or worse mapped inconsistently, bringing us finally to …<br />
<br />
6. They're relevant to understanding limitations of consistency (or more precisely, monotonicity) of any given temperament, as the next section will discuss.<br />
<br />
=== Significance/implications for consistency ===<br />
1/n-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} have implications for the [[consistency]] of the ({{nowrap|''k'' + ''n''}})-[[odd-limit]] when tempered. Specifically:<br />
<br />
If a temperament tempers a 1/''n''-square-particular of the form {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, it must temper all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', {{nowrap|S(''k'' + 1)}}, …, {{nowrap|S(''k'' + ''n'' − 1)}}. If it does not, it is ''necessarily'' inconsistent (more formally and weakly, not monotonic) in the ({{nowrap|''k'' + ''n''}})-odd-limit.<ref group="note">Note that this statement is a slight inaccuracy, because technically the tuning of the higher rank temperament corresponding to the lower rank temperament that tempers all of these commas is the unique ''and only'' (continuum of) tuning(s) for which this statement is false, but it's reasonable to simplify this technicality as this (continuum of) tuning(s) corresponds exactly and uniquely to tempering all the square-particulars we said were not tempered.</ref> A proof is as follows:<br />
<br />
Consider the following sequence of superparticular intervals, all of which in the ({{nowrap|''k'' + ''n''}})-odd-limit:<br />
<br />
<math>\displaystyle\frac{k + n}{k + n - 1}, \frac{k + n - 1}{k + n - 2}, …, \frac{k + 1}{k}, \frac{k}{k - 1}</math><br />
<br />
Because of tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, we require that {{nowrap|{{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} {{=}} {{sfrac|''k''|''k'' − 1}}}} consistently. Therefore, if any superparticular {{sfrac|''x''|''x'' − 1}} imbetween (meaning {{nowrap|''k'' + ''n'' &gt; ''x'' &gt; ''k''}}) is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true:<br />
<br />
<math>\displaystyle\begin{align}<br />
\operatorname{mapping}\left(\frac{k + n}{k + n - 1}\right) &> \operatorname{mapping}\left(\frac{x}{x - 1}\right) \\<br />
\operatorname{mapping}\left(\frac{k}{k - 1}\right) &< \operatorname{mapping}\left(\frac{x}{x - 1}\right)<br />
\end{align}</math><br />
<br />
Therefore any superparticular interval {{sfrac|''x''|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap|''k'' − 1}})-odd-limit.<br />
<br />
<br />
=== {{nowrap|S(''k'' − 1)*S''k''*S(''k'' + 1)}} (1/3-square-particulars) ===<br />
This section concerns commas of the form {{nowrap|S(''k'' − 1) * S''k'' * S(''k'' + 1) {{=}} {{sfrac|&nbsp;{{sfrac|''k'' − 1|''k'' − 2}}&nbsp;|&nbsp;{{sfrac|''k'' + 2|''k'' + 1}}&nbsp;}}}} which therefore do not (directly) involve the ''k''th harmonic. These are a special case of 1/''n''-square-particulars.<br />
<br />
==== Significance ====<br />
1. Two-thirds of all {{frac|1|3}}-square-particulars are superparticular and the other third are [[#Glossary|throdd-particular]], so these are efficient commas. (See also the [[#Proof of simplification of 1/3-square-particulars]].)<br />
<br />
2. They are often implied in a variety of ways by combinations of other commas discussed on this page.<br />
<br />
3. Their omission of direct relation to the ''k''th harmonic make them theoretically interesting and potentially useful. (The other type of comma on this page that does this is [[semiparticular]]s.)<br />
<br />
4. Square-particulars, {{frac|1|2}}-square-particulars (a.k.a. [[triangle-particular]]s), and {{frac|1|3}}-square-particulars are part of a more general sequence with interesting properties: [[1/n-square-particular|1/''n''-square-particular]]s.<br />
<br />
==== Proof of simplification of 1/3-square-particulars ====<br />
We can check the general algebraic expression of any 1/3-square-particular for any potential simplifications:<br />
<br />
<math>\displaystyle\begin{align}<br />
S(k-1) * S(k) * S(k+1) &= \left(\frac{\frac{k-1}{k-2}}{\frac{k}{k-1}}\right)\left(\frac{\frac{k}{k-1}}{\frac{k+1}{k}}\right)\left(\frac{\frac{k+1}{k}}{\frac{k+2}{k+1}}\right) \\<br />
&= \frac{\frac{k-1}{k-2}}{\frac{k+2}{k+1}} \\<br />
&= \frac{(k-1)(k+1)}{(k-2)(k+2)} \\<br />
&= \frac{k^2 - 1}{k^2 - 4}<br />
\end{align}</math><br />
<br />
If {{nowrap|''k'' {{=}} 3''n'' + 1}} then:<br />
<br />
<math>\displaystyle S(k-1) * Sk * S(k+1) = \frac{9n^2 + 6n}{9n^2 + 6n - 3} = \frac{3n^2 + 2n}{3n^2 + 2n - 1}</math><br />
<br />
if {{nowrap|''k'' {{=}} 3''n'' + 2}} then:<br />
<br />
<math>\displaystyle S(k-1) * Sk * S(k+1) = \frac{9n^2 + 12n + 3}{9n^2 + 12n} = \frac{3n^2 + 4n + 1}{3n^2 + 4n}</math><br />
<br />
if {{nowrap|''k'' {{=}} 3''n''}} then:<br />
<br />
<math>\displaystyle S(k-1) * Sk * S(k+1) = \frac{9n^2 - 1}{9n^2 - 4}</math><br />
<br />
In other words, what this shows is all {{frac|1|3}}-square-particulars of the form S(''k'' − 1) * S''k'' * S(''k'' + 1) are superparticular iff ''k'' is throdd (not a multiple of 3), and all {{frac|1|3}}-square-particulars of the form {{nowrap|S(3''k'' − 1) * S(3''k'') * S(3''k'' + 1)}} are throdd-particular with the numerator and denominator always being one less than a multiple of 3 (which is to say, commas of this form are throdd-particular iff ''k'' is threven and superparticular iff ''k'' is throdd).<br />
<br />
=== Tables of 1/n-square-particulars ===<br />
<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 41-limit {{frac|1|3}}-square-particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3*S4<br />
| ([[2/1]])/([[5/4]])<br />
| [[8/5]]<br />
| 5<br />
|-<br />
| S3*S4*S5<br />
| ([[3/2]])/([[6/5]])<br />
| [[5/4]]<br />
| 5<br />
|-<br />
| S4*S5*S6<br />
| ([[4/3]])/([[7/6]])<br />
| [[8/7]]<br />
| 7<br />
|-<br />
| S5*S6*S7<br />
| ([[5/4]])/([[8/7]])<br />
| [[35/32]]<br />
| 7<br />
|-<br />
| S6*S7*S8<br />
| ([[6/5]])/([[9/8]])<br />
| [[16/15]]<br />
| 5<br />
|-<br />
| S7*S8*S9<br />
| ([[7/6]])/([[10/9]])<br />
| [[21/20]]<br />
| 7<br />
|-<br />
| S8*S9*S10<br />
| ([[8/7]])/([[11/10]])<br />
| [[80/77]]<br />
| 11<br />
|-<br />
| S9*S10*S11<br />
| ([[9/8]])/([[12/11]])<br />
| [[33/32]]<br />
| 11<br />
|-<br />
| S10*S11*S12<br />
| ([[10/9]])/([[13/12]])<br />
| [[40/39]]<br />
| 13<br />
|-<br />
| S11*S12*S13<br />
| ([[11/10]])/([[14/13]])<br />
| [[143/140]]<br />
| 13<br />
|-<br />
| S12*S13*S14<br />
| ([[12/11]])/([[15/14]])<br />
| [[56/55]]<br />
| 11<br />
|-<br />
| S13*S14*S15<br />
| ([[13/12]])/([[16/15]])<br />
| [[65/64]]<br />
| 13<br />
|-<br />
| S14*S15*S16<br />
| ([[14/13]])/([[17/16]])<br />
| [[224/221]]<br />
| 17<br />
|-<br />
| S15*S16*S17<br />
| ([[15/14]])/([[18/17]])<br />
| [[85/84]]<br />
| 17<br />
|-<br />
| S16*S17*S18<br />
| ([[16/15]])/([[19/18]])<br />
| [[96/95]]<br />
| 19<br />
|-<br />
| S17*S18*S19<br />
| ([[17/16]])/([[20/19]])<br />
| [[323/320]]<br />
| 19<br />
|-<br />
| S18*S19*S20<br />
| ([[18/17]])/([[21/20]])<br />
| [[120/119]]<br />
| 17<br />
|-<br />
| S19*S20*S21<br />
| ([[19/18]])/([[22/21]])<br />
| [[133/132]]<br />
| 19<br />
|-<br />
| S20*S21*S22<br />
| ([[20/19]])/([[23/22]])<br />
| [[440/437]]<br />
| 23<br />
|-<br />
| S21*S22*S23<br />
| ([[21/20]])/([[24/23]])<br />
| [[161/160]]<br />
| 23<br />
|-<br />
| S22*S23*S24<br />
| ([[22/21]])/([[25/24]])<br />
| [[176/175]]<br />
| 11<br />
|-<br />
| S23*S24*S25<br />
| ([[23/22]])/([[26/25]])<br />
| [[575/572]]<br />
| 23<br />
|-<br />
| S24*S25*S26<br />
| ([[24/23]])/([[27/26]])<br />
| [[208/207]]<br />
| 23<br />
|-<br />
| S25*S26*S27<br />
| ([[25/24]])/([[28/27]])<br />
| [[225/224]]<br />
| 7<br />
|-<br />
| S26*S27*S28<br />
| ([[26/25]])/([[29/28]])<br />
| [[728/725]]<br />
| 29<br />
|-<br />
| S27*S28*S29<br />
| ([[27/26]])/([[30/29]])<br />
| [[261/260]]<br />
| 29<br />
|-<br />
| S28*S29*S30<br />
| ([[28/27]])/([[31/30]])<br />
| [[280/279]]<br />
| 31<br />
|-<br />
| S29*S30*S31<br />
| ([[29/28]])/([[32/31]])<br />
| [[899/896]]<br />
| 31<br />
|-<br />
| S30*S31*S32<br />
| ([[30/29]])/([[33/32]])<br />
| [[320/319]]<br />
| 29<br />
|-<br />
| S31*S32*S33<br />
| ([[31/30]])/([[34/33]])<br />
| [[341/340]]<br />
| 31<br />
|-<br />
| S32*S33*S34<br />
| ([[32/31]])/([[35/34]])<br />
| [[1088/1085]]<br />
| 31<br />
|-<br />
| S33*S34*S35<br />
| ([[33/32]])/([[36/35]])<br />
| [[385/384]]<br />
| 11<br />
|-<br />
| S34*S35*S36<br />
| ([[34/33]])/([[37/36]])<br />
| [[408/407]]<br />
| 37<br />
|-<br />
| S35*S36*S37<br />
| ([[35/34]])/([[38/37]])<br />
| [[1295/1292]]<br />
| 37<br />
|-<br />
| S36*S37*S38<br />
| ([[36/35]])/([[39/38]])<br />
| [[456/455]]<br />
| 19<br />
|-<br />
| S37*S38*S39<br />
| ([[37/36]])/([[40/39]])<br />
| [[481/480]]<br />
| 37<br />
|-<br />
| S38*S39*S40<br />
| ([[38/37]])/([[41/40]])<br />
| [[1520/1517]]<br />
| 41<br />
|-<br />
| S39*S40*S41<br />
| ([[39/38]])/([[42/41]])<br />
| [[533/532]]<br />
| 41<br />
|-<br />
| S42*S43*S44<br />
| ([[42/41]])/([[45/44]])<br />
| [[616/615]]<br />
| 41<br />
|-<br />
| S46*S47*S48<br />
| ([[46/45]])/([[49/48]])<br />
| [[736/735]]<br />
| 23<br />
|-<br />
| S49*S50*S51<br />
| ([[49/48]])/([[52/51]])<br />
| [[833/832]]<br />
| 17<br />
|-<br />
| S52*S53*S54<br />
| ([[52/51]])/([[55/54]])<br />
| [[936/935]]<br />
| 17<br />
|-<br />
| S55*S56*S57<br />
| ([[55/54]])/([[58/57]])<br />
| [[1045/1044]]<br />
| 29<br />
|-<br />
| S63*S64*S65<br />
| ([[63/62]])/([[66/65]])<br />
| [[1365/1364]]<br />
| 31<br />
|-<br />
| S66*S67*S68<br />
| ([[66/65]])/([[69/68]])<br />
| [[1496/1495]]<br />
| 23<br />
|-<br />
| S75*S76*S77<br />
| ([[75/74]])/([[78/77]])<br />
| [[1925/1924]]<br />
| 37<br />
|-<br />
| S78*S79*S80<br />
| ([[78/77]])/([[81/80]])<br />
| [[2080/2079]]<br />
| 13<br />
|-<br />
| S82*S83*S84<br />
| ([[82/81]])/([[85/84]])<br />
| [[2296/2295]]<br />
| 41<br />
|-<br />
| S85*S86*S87<br />
| ([[85/84]])/([[88/87]])<br />
| [[2465/2464]]<br />
| 29<br />
|-<br />
| S88*S89*S90<br />
| ([[88/87]])/([[91/90]])<br />
| [[2640/2639]]<br />
| 29<br />
|-<br />
| S93*S94*S95<br />
| ([[93/92]])/([[96/95]])<br />
| [[2945/2944]]<br />
| 31<br />
|-<br />
| S96*S97*S98<br />
| ([[96/95]])/([[99/98]])<br />
| [[3136/3135]]<br />
| 19<br />
|-<br />
| S112*S113*S114<br />
| ([[112/111]])/([[115/114]])<br />
| [[4256/4255]]<br />
| 37<br />
|-<br />
| S117*S118*S119<br />
| ([[117/116]])/([[120/119]])<br />
| [[4641/4640]]<br />
| 29<br />
|-<br />
| S121*S122*S123<br />
| ([[121/120]])/([[124/123]])<br />
| [[4961/4960]]<br />
| 41<br />
|-<br />
| S133*S134*S135<br />
| ([[133/132]])/([[136/135]])<br />
| [[5985/5984]]<br />
| 19<br />
|-<br />
| S145*S146*S147<br />
| ([[145/144]])/([[148/147]])<br />
| [[7105/7104]]<br />
| 37<br />
|-<br />
| S153*S154*S155<br />
| ([[153/152]])/([[156/155]])<br />
| [[7905/7904]]<br />
| 31<br />
|-<br />
| S162*S163*S164<br />
| ([[162/161]])/([[165/164]])<br />
| [[8856/8855]]<br />
| 41<br />
|-<br />
| S187*S188*S189<br />
| ([[187/186]])/([[190/189]])<br />
| [[11781/11780]]<br />
| 31<br />
|-<br />
| S205*S206*S207<br />
| ([[205/204]])/([[208/207]])<br />
| [[14145/14144]]<br />
| 41<br />
|-<br />
| S222*S223*S224<br />
| ([[222/221]])/([[225/224]])<br />
| [[16576/16575]]<br />
| 37<br />
|-<br />
| S243*S244*S245<br />
| ([[243/242]])/([[246/245]])<br />
| [[19845/19844]]<br />
| 41<br />
|-<br />
| S253*S254*S255<br />
| ([[253/252]])/([[256/255]])<br />
| [[21505/21504]]<br />
| 23<br />
|-<br />
| S273*S274*S275<br />
| ([[273/272]])/([[276/275]])<br />
| [[25025/25024]]<br />
| 23<br />
|-<br />
| S286*S287*S288<br />
| ([[286/285]])/([[289/288]])<br />
| [[27456/27455]]<br />
| 19<br />
|-<br />
| S287*S288*S289<br />
| ([[287/286]])/([[290/289]])<br />
| [[82943/82940]]<br />
| 41<br />
|-<br />
| S297*S298*S299<br />
| ([[297/296]])/([[300/299]])<br />
| [[29601/29600]]<br />
| 37<br />
|-<br />
| S320*S321*S322<br />
| ([[320/319]])/([[323/322]])<br />
| [[103040/103037]]<br />
| 29<br />
|-<br />
| S361*S362*S363<br />
| ([[361/360]])/([[364/363]])<br />
| [[43681/43680]]<br />
| 19<br />
|-<br />
| S375*S376*S377<br />
| ([[375/374]])/([[378/377]])<br />
| [[47125/47124]]<br />
| 29<br />
|}<br />
<br />
{| class="wikitable center-all\<br />
|+ style="font-size: 105%;" | 23-limit {{frac|1|4}}-square particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3*S4*S5<br />
| ([[2/1]])/([[6/5]])<br />
| [[5/3]]<br />
| 5<br />
|-<br />
| S3*S4*S5*S6<br />
| ([[3/2]])/([[7/6]])<br />
| [[9/7]]<br />
| 7<br />
|-<br />
| S4*S5*S6*S7<br />
| ([[4/3]])/([[8/7]])<br />
| [[7/6]]<br />
| 7<br />
|-<br />
| S5*S6*S7*S8<br />
| ([[5/4]])/([[9/8]])<br />
| [[10/9]]<br />
| 5<br />
|-<br />
| S6*S7*S8*S9<br />
| ([[6/5]])/([[10/9]])<br />
| [[27/25]]<br />
| 5<br />
|-<br />
| S7*S8*S9*S10<br />
| ([[7/6]])/([[11/10]])<br />
| [[35/33]]<br />
| 11<br />
|-<br />
| S8*S9*S10*S11<br />
| ([[8/7]])/([[12/11]])<br />
| [[22/21]]<br />
| 11<br />
|-<br />
| S9*S10*S11*S12<br />
| ([[9/8]])/([[13/12]])<br />
| [[27/26]]<br />
| 13<br />
|-<br />
| S10*S11*S12*S13<br />
| ([[10/9]])/([[14/13]])<br />
| [[65/63]]<br />
| 13<br />
|-<br />
| S11*S12*S13*S14<br />
| ([[11/10]])/([[15/14]])<br />
| [[77/75]]<br />
| 11<br />
|-<br />
| S12*S13*S14*S15<br />
| ([[12/11]])/([[16/15]])<br />
| [[45/44]]<br />
| 11<br />
|-<br />
| S13*S14*S15*S16<br />
| ([[13/12]])/([[17/16]])<br />
| [[52/51]]<br />
| 17<br />
|-<br />
| S14*S15*S16*S17<br />
| ([[14/13]])/([[18/17]])<br />
| [[119/117]]<br />
| 17<br />
|-<br />
| S15*S16*S17*S18<br />
| ([[15/14]])/([[19/18]])<br />
| [[135/133]]<br />
| 19<br />
|-<br />
| S16*S17*S18*S19<br />
| ([[16/15]])/([[20/19]])<br />
| [[76/75]]<br />
| 19<br />
|-<br />
| S17*S18*S19*S20<br />
| ([[17/16]])/([[21/20]])<br />
| [[85/84]]<br />
| 17<br />
|-<br />
| S18*S19*S20*S21<br />
| ([[18/17]])/([[22/21]])<br />
| [[189/187]]<br />
| 17<br />
|-<br />
| S19*S20*S21*S22<br />
| ([[19/18]])/([[23/22]])<br />
| [[209/207]]<br />
| 23<br />
|-<br />
| S20*S21*S22*S23<br />
| ([[20/19]])/([[24/23]])<br />
| [[115/114]]<br />
| 23<br />
|-<br />
| S21*S22*S23*S24<br />
| ([[21/20]])/([[25/24]])<br />
| [[126/125]]<br />
| 7<br />
|-<br />
| S22*S23*S24*S25<br />
| ([[22/21]])/([[26/25]])<br />
| [[275/273]]<br />
| 13<br />
|-<br />
| S23*S24*S25*S26<br />
| ([[23/22]])/([[27/26]])<br />
| [[299/297]]<br />
| 23<br />
|-<br />
| S24*S25*S26*S27<br />
| ([[24/23]])/([[28/27]])<br />
| [[162/161]]<br />
| 23<br />
|-<br />
| S35*S36*S37*S38<br />
| ([[35/34]])/([[39/38]])<br />
| [[665/663]]<br />
| 19<br />
|-<br />
| S36*S37*S38*S39<br />
| ([[36/35]])/([[40/39]])<br />
| [[351/350]]<br />
| 13<br />
|-<br />
| S45*S46*S47*S48<br />
| ([[45/44]])/([[49/48]])<br />
| [[540/539]]<br />
| 11<br />
|-<br />
| S46*S47*S48*S49<br />
| ([[46/45]])/([[50/49]])<br />
| [[1127/1125]]<br />
| 23<br />
|-<br />
| S51*S52*S53*S54<br />
| ([[51/50]])/([[55/54]])<br />
| [[1377/1375]]<br />
| 17<br />
|-<br />
| S52*S53*S54*S55<br />
| ([[52/51]])/([[56/55]])<br />
| [[715/714]]<br />
| 17<br />
|-<br />
| S65*S66*S67*S68<br />
| ([[65/64]])/([[69/68]])<br />
| [[1105/1104]]<br />
| 23<br />
|-<br />
| S66*S67*S68*S69<br />
| ([[66/65]])/([[70/69]])<br />
| [[2277/2275]]<br />
| 23<br />
|-<br />
| S77*S78*S79*S80<br />
| ([[77/76]])/([[81/80]])<br />
| [[1540/1539]]<br />
| 19<br />
|-<br />
| S81*S82*S83*S84<br />
| ([[81/80]])/([[85/84]])<br />
| [[1701/1700]]<br />
| 17<br />
|-<br />
| S92*S93*S94*S95<br />
| ([[92/91]])/([[96/95]])<br />
| [[2185/2184]]<br />
| 23<br />
|-<br />
| S96*S97*S98*S99<br />
| <span style="font-size:0.94em">([[96/95]])/([[100/99]])</span><br />
| [[2376/2375]]<br />
| 19<br />
|-<br />
| <span style="font-size: 0.79em;">S221*S222*S223*S224</span><br />
| <span style="font-size: 0.79em;">([[221/220]])/([[225/224]])</span><br />
| [[12376/12375]]<br />
| 17<br />
|}<br />
<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 23-limit {{frac|1|5}}-square particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3*S4*S5*S6<br />
| ([[2/1]])/([[7/6]])<br />
| [[12/7]]<br />
| 7<br />
|-<br />
| S3*S4*S5*S6*S7<br />
| ([[3/2]])/([[8/7]])<br />
| [[21/16]]<br />
| 7<br />
|-<br />
| S4*S5*S6*S7*S8<br />
| ([[4/3]])/([[9/8]])<br />
| [[32/27]]<br />
| 3<br />
|-<br />
| S5*S6*S7*S8*S9<br />
| ([[5/4]])/([[10/9]])<br />
| [[9/8]]<br />
| 3<br />
|-<br />
| S6*S7*S8*S9*S10<br />
| ([[6/5]])/([[11/10]])<br />
| [[12/11]]<br />
| 11<br />
|-<br />
| S7*S8*S9*S10*S11<br />
| ([[7/6]])/([[12/11]])<br />
| [[77/72]]<br />
| 11<br />
|-<br />
| S8*S9*S10*S11*S12<br />
| ([[8/7]])/([[13/12]])<br />
| [[96/91]]<br />
| 13<br />
|-<br />
| S9*S10*S11*S12*S13<br />
| ([[9/8]])/([[14/13]])<br />
| [[117/112]]<br />
| 13<br />
|-<br />
| S10*S11*S12*S13*S14<br />
| ([[10/9]])/([[15/14]])<br />
| [[28/27]]<br />
| 7<br />
|-<br />
| S11*S12*S13*S14*S15<br />
| ([[11/10]])/([[16/15]])<br />
| [[33/32]]<br />
| 11<br />
|-<br />
| S12*S13*S14*S15*S16<br />
| ([[12/11]])/([[17/16]])<br />
| [[192/187]]<br />
| 17<br />
|-<br />
| S13*S14*S15*S16*S17<br />
| ([[13/12]])/([[18/17]])<br />
| [[221/216]]<br />
| 17<br />
|-<br />
| S14*S15*S16*S17*S18<br />
| ([[14/13]])/([[19/18]])<br />
| [[252/247]]<br />
| 19<br />
|-<br />
| S15*S16*S17*S18*S19<br />
| ([[15/14]])/([[20/19]])<br />
| [[57/56]]<br />
| 19<br />
|-<br />
| S16*S17*S18*S19*S20<br />
| ([[16/15]])/([[21/20]])<br />
| [[64/63]]<br />
| 7<br />
|-<br />
| S17*S18*S19*S20*S21<br />
| ([[17/16]])/([[22/21]])<br />
| [[357/352]]<br />
| 17<br />
|-<br />
| S18*S19*S20*S21*S22<br />
| ([[18/17]])/([[23/22]])<br />
| [[396/391]]<br />
| 23<br />
|-<br />
| S19*S20*S21*S22*S23<br />
| ([[19/18]])/([[24/23]])<br />
| [[437/432]]<br />
| 23<br />
|-<br />
| S20*S21*S22*S23*S24<br />
| ([[20/19]])/([[25/24]])<br />
| [[96/95]]<br />
| 19<br />
|-<br />
| S21*S22*S23*S24*S25<br />
| ([[21/20]])/([[26/25]])<br />
| [[105/104]]<br />
| 13<br />
|-<br />
| S22*S23*S24*S25*S26<br />
| ([[22/21]])/([[27/26]])<br />
| [[572/567]]<br />
| 13<br />
|-<br />
| S23*S24*S25*S26*S27<br />
| ([[23/22]])/([[28/27]])<br />
| [[621/616]]<br />
| 23<br />
|-<br />
| S28*S29*S30*S31*S32<br />
| ([[28/27]])/([[33/32]])<br />
| [[896/891]]<br />
| 11<br />
|-<br />
| S34*S35*S36*S37*S38<br />
| ([[34/33]])/([[39/38]])<br />
| [[1292/1287]]<br />
| 19<br />
|-<br />
| S35*S36*S37*S38*S39<br />
| ([[35/34]])/([[40/39]])<br />
| [[273/272]]<br />
| 17<br />
|-<br />
| S40*S41*S42*S43*S44<br />
| ([[40/39]])/([[45/44]])<br />
| [[352/351]]<br />
| 13<br />
|-<br />
| S45*S46*S47*S48*S49<br />
| ([[45/44]])/([[50/49]])<br />
| [[441/440]]<br />
| 11<br />
|-<br />
| S46*S47*S48*S49*S50<br />
| ([[46/45]])/([[51/50]])<br />
| [[460/459]]<br />
| 23<br />
|-<br />
| S50*S51*S52*S53*S54<br />
| ([[50/49]])/([[55/54]])<br />
| [[540/539]]<br />
| 11<br />
|-<br />
| S51*S52*S53*S54*S55<br />
| ([[51/50]])/([[56/55]])<br />
| [[561/560]]<br />
| 17<br />
|-<br />
| S52*S53*S54*S55*S56<br />
| ([[52/51]])/([[57/56]])<br />
| [[2912/2907]]<br />
| 19<br />
|-<br />
| S64*S65*S66*S67*S68<br />
| ([[64/63]])/([[69/68]])<br />
| [[4352/4347]]<br />
| 23<br />
|-<br />
| S65*S66*S67*S68*S69<br />
| ([[65/64]])/([[70/69]])<br />
| [[897/896]]<br />
| 23<br />
|-<br />
| S76*S77*S78*S79*S80<br />
| ([[76/75]])/([[81/80]])<br />
| [[1216/1215]]<br />
| 19<br />
|-<br />
| S91*S92*S93*S94*S95<br />
| ([[91/90]])/([[96/95]])<br />
| [[1729/1728]]<br />
| 19<br />
|-<br />
| <span style="font-size: 0.79em;">S100*S101*S102*S103*S104</span><br />
| <span style="font-size: 0.83em;">([[100/99]])/([[105/104]])</span><br />
| [[2080/2079]]<br />
| 13<br />
|-<br />
| <span style="font-size: 0.79em;">S115*S116*S117*S118*S119</span><br />
| <span style="font-size: 0.79em;">([[115/114]])/([[120/119]])</span><br />
| [[2737/2736]]<br />
| 23<br />
|-<br />
| <span style="font-size: 0.79em;">S121*S122*S123*S124*S125</span><br />
| <span style="font-size: 0.79em;">([[121/120]])/([[126/125]])</span><br />
| [[3025/3024]]<br />
| 11<br />
|-<br />
| <span style="font-size: 0.79em;">S171*S172*S173*S174*S175</span><br />
| <span style="font-size: 0.79em;">([[171/170]])/([[176/175]])</span><br />
| [[5985/5984]]<br />
| 19<br />
|}<br />
<br />
== {{nowrap|S''k''/S(''k'' + 1)}} (ultraparticulars) ==<br />
=== Motivational example ===<br />
Often it is desirable to make consecutive [[superparticular]] intervals equidistant. This has a number of nice consequences, many of which not explained here—see the motivation section for each infinite family of commas defined on this page.<br />
<br />
For example, if you want 6/5 equidistant from 5/4 and 7/6, you must equate {{nowrap|{{sfrac|[[5/4]]|[[6/5]]}} {{=}} [[25/24]]}} {{nowrap|{{=}} S5}} with {{nowrap|{{sfrac|[[6/5]]|[[7/6]]}} {{=}} [[36/35]]}} {{nowrap|{{=}} S6}}, hence tempering {{nowrap|{{sfrac|S5|S6}} {{=}} {{sfrac|25/24|36/35}}}} {{nowrap|{{=}} [[875/864]]}}, but it's actually often not necessary to know the specific numbers, often familiarizing yourself with and understanding the "S''k''" notation will give you a lot of insight, as we'll see.<br />
<br />
Back to our example: we know that {{nowrap|S5 ~ S6}} (because we're tempering S5/S6); from this we can deduce that the intervals must be arranged like this: {{nowrap|7/6 &larr; S5~S6 &rarr; 6/5 &larr; S5~S6 &rarr; 5/4}}.<br />
<br />
From this you can deduce that {{nowrap|([[6/5]])<sup>3</sup> &rarr; [[7/4]]}}, because you can lower one of the 6/5's to [[7/6]] (lowering it by S6) and raise another of the 6/5's to [[5/4]] (raising it by S5). Then because we've tempered S5 and S6 together, we've lowered and raised by the same amount, so the result of {{nowrap|7/6 * 6/5 * 5/4 {{=}} 7/4}} must be the same as the result of {{nowrap|6/5 * 6/5 * 6/5}} in this temperament.<br />
<br />
Familiarize yourself with the structure of this argument, as [[S-expression/Advanced results#Mathematical derivations|it generalizes to arbitrary S''k'']]; the algebraic proof is tedious, but the intuition is the same:<br />
<br />
<math>\displaystyle \frac{k+2}{k+1} \leftarrow S(k+1)~Sk \rightarrow \frac{k+1}{k} \leftarrow S(k+1)~Sk \rightarrow \frac{k}{k-1}</math><br />
<br />
… implies that three {{sfrac|''k'' + 1|''k''}} give {{sfrac|''k'' + 2|''k'' − 1}} iff we temper {{sfrac|S''k''|S(''k'' + 1)}}.&nbsp;{{qed}}<br />
<br />
=== Significance ===<br />
1. Tempering any two consecutive square-particulars S''k'' and S({{nowrap|''k'' + 1}}) will naturally imply tempering the ultraparticular between them, {{sfrac|S''k''|S(''k'' + 1)}}, meaning they are very common implicit commas.<br />
<br />
2. Tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness.<br />
<br />
3. Tempering the ultraparticular S''k''/S({{nowrap|''k'' + 1}}) along with either the corresponding 1/2-square-particular {{nowrap|S''k'' * S(''k'' + 1)}} or one of the two corresponding lopsided commas {{nowrap|S''k''<sup>2</sup> * S(''k'' + 1)}} or {{nowrap|S''k'' * S(''k'' + 1)<sup>2</sup>}} implies tempering both of S''k'' and S({{nowrap|''k'' + 1}}) individually, and vice versa, so that there is a total of ''five'' equivalences—corresponding to ''five'' infinite families of commas—for every such S''k'' and S({{nowrap|''k''+1}}). This only gets better if you temper a third consecutive square-particular. This is an abundance of "at a glance" essential tempering information that is fully general so only needs to be learned once, and is the motivation of the use of '''S-expressions'''. (For example, {{nowrap|{S16, S17} &rarr; {{(}}S16 * S17, S16/S17, S16<sup>2</sup> * S17, S16 * S17<sup>2</sup>{{)}} }}, and any of the two commas in the latter set imply all the other commas too.)<br />
<br />
=== Table of ultraparticulars ===<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Cube Relation<br />
! Comma<br />
! Cents<br />
|-<br />
| S2/S3 = ([[4/3]])/([[9/8]])<br />
| ([[4/1]])/([[3/2]])<sup>3</sup><br />
| [[32/27]]<br />
| 294.135<br />
|-<br />
| S3/S4 = ([[9/8]])/([[16/15]])<br />
| ([[5/2]])/([[4/3]])<sup>3</sup><br />
| [[135/128]]<br />
| 92.179<br />
|-<br />
| S4/S5 = ([[16/15]])/([[25/24]])<br />
| ([[2/1]])/([[5/4]])<sup>3</sup><br />
| [[128/125]]<br />
| 41.059<br />
|-<br />
| S5/S6 = ([[25/24]])/([[36/35]])<br />
| ([[7/4]])/([[6/5]])<sup>3</sup><br />
| [[875/864]]<br />
| 21.902<br />
|-<br />
| S6/S7 = ([[36/35]])/([[49/48]])<br />
| ([[8/5]])/([[7/6]])<sup>3</sup><br />
| [[1728/1715]]<br />
| 13.074<br />
|-<br />
| S7/S8 = ([[49/48]])/([[64/63]])<br />
| ([[3/2]])/([[8/7]])<sup>3</sup><br />
| [[1029/1024]]<br />
| 8.433<br />
|-<br />
| S8/S9 = ([[64/63]])/([[81/80]])<br />
| ([[10/7]])/([[9/8]])<sup>3</sup><br />
| [[5120/5103]]<br />
| 5.758<br />
|-<br />
| S9/S10 = ([[81/80]])/([[100/99]])<br />
| ([[11/8]])/([[10/9]])<sup>3</sup><br />
| [[8019/8000]]<br />
| 4.107<br />
|-<br />
| S10/S11 = ([[100/99]])/([[121/120]])<br />
| ([[4/3]])/([[11/10]])<sup>3</sup><br />
| [[4000/3993]]<br />
| 3.032<br />
|-<br />
| S11/S12 = ([[121/120]])/([[144/143]])<br />
| ([[13/10]])/([[12/11]])<sup>3</sup><br />
| [[17303/17280]]<br />
| 2.303<br />
|-<br />
| S12/S13 = ([[144/143]])/([[169/168]])<br />
| ([[14/11]])/([[13/12]])<sup>3</sup><br />
| [[24192/24167]]<br />
| 1.79<br />
|-<br />
| S13/S14 = ([[169/168]])/([[196/195]])<br />
| ([[5/4]])/([[14/13]])<sup>3</sup><br />
| [[10985/10976]]<br />
| 1.419<br />
|-<br />
| S14/S15 = ([[196/195]])/([[225/224]])<br />
| ([[16/13]])/([[15/14]])<sup>3</sup><br />
| [[43904/43875]]<br />
| 1.144<br />
|-<br />
| S15/S16 = ([[225/224]])/([[256/255]])<br />
| ([[17/14]])/([[16/15]])<sup>3</sup><br />
| [[57375/57344]]<br />
| 0.936<br />
|-<br />
| S16/S17 = ([[256/255]])/([[289/288]])<br />
| ([[6/5]])/([[17/16]])<sup>3</sup><br />
| [[24576/24565]]<br />
| 0.775<br />
|-<br />
| S17/S18 = ([[289/288]])/([[324/323]])<br />
| ([[19/16]])/([[18/17]])<sup>3</sup><br />
| [[93347/93312]]<br />
| 0.649<br />
|-<br />
| S18/S19 = ([[324/323]])/([[361/360]])<br />
| ([[20/17]])/([[19/18]])<sup>3</sup><br />
| [[116640/116603]]<br />
| 0.549<br />
|-<br />
| S19/S20 = ([[361/360]])/([[400/399]])<br />
| ([[7/6]])/([[20/19]])<sup>3</sup><br />
| [[48013/48000]]<br />
| 0.469<br />
|-<br />
| S20/S21 = ([[400/399]])/([[441/440]])<br />
| ([[22/19]])/([[21/20]])<sup>3</sup><br />
| [[176000/175959]]<br />
| 0.403<br />
|-<br />
| S21/S22 = ([[441/440]])/([[484/483]])<br />
| ([[23/20]])/([[22/21]])<sup>3</sup><br />
| [[213003/212960]]<br />
| 0.35<br />
|-<br />
| S22/S23 = ([[484/483]])/([[529/528]])<br />
| ([[8/7]])/([[23/22]])<sup>3</sup><br />
| [[85184/85169]]<br />
| 0.305<br />
|-<br />
| S23/S24 = ([[529/528]])/([[576/575]])<br />
| ([[25/22]])/([[24/23]])<sup>3</sup><br />
| [[304175/304128]]<br />
| 0.268<br />
|-<br />
| S24/S25 = ([[576/575]])/([[625/624]])<br />
| ([[26/23]])/([[25/24]])<sup>3</sup><br />
| [[359424/359375]]<br />
| 0.236<br />
|-<br />
| S25/S26 = ([[625/624]])/([[676/675]])<br />
| ([[9/8]])/([[26/25]])<sup>3</sup><br />
| [[140625/140608]]<br />
| 0.209<br />
|-<br />
| S26/S27 = ([[676/675]])/([[729/728]])<br />
| ([[28/25]])/([[27/26]])<sup>3</sup><br />
| [[492128/492075]]<br />
| 0.186<br />
|-<br />
| S27/S28 = ([[729/728]])/([[784/783]])<br />
| ([[29/26]])/([[28/27]])<sup>3</sup><br />
| [[570807/570752]]<br />
| 0.167<br />
|-<br />
| S28/S29 = ([[784/783]])/([[841/840]])<br />
| ([[10/9]])/([[29/28]])<sup>3</sup><br />
| [[219520/219501]]<br />
| 0.15<br />
|-<br />
| S31/S32 = ([[961/960]])/([[1024/1023]])<br />
| ([[11/10]])/([[32/31]])<sup>3</sup><br />
| [[327701/327680]]<br />
| 0.111<br />
|-<br />
| S33/S34 = ([[1089/1088]])/([[1156/1155]])<br />
| ([[35/32]])/([[34/33]])<sup>3</sup><br />
| [[1257795/1257728]]<br />
| 0.092<br />
|-<br />
| S34/S35 = ([[1156/1155]])/([[1225/1224]])<br />
| ([[12/11]])/([[35/34]])<sup>3</sup><br />
| [[471648/471625]]<br />
| 0.084<br />
|-<br />
| S37/S38 = ([[1369/1368]])/([[1444/1443]])<br />
| ([[13/12]])/([[38/37]])<sup>3</sup><br />
| [[658489/658464]]<br />
| 0.066<br />
|-<br />
| S40/S41 = ([[1600/1599]])/([[1681/1680]])<br />
| ([[14/13]])/([[41/40]])<sup>3</sup><br />
| [[896000/895973]]<br />
| 0.052<br />
|-<br />
| S43/S44 = ([[1849/1848]])/([[1936/1935]])<br />
| ([[15/14]])/([[44/43]])<sup>3</sup><br />
| [[1192605/1192576]]<br />
| 0.042<br />
|-<br />
| S46/S47 = ([[2116/2115]])/([[2209/2208]])<br />
| ([[16/15]])/([[47/46]])<sup>3</sup><br />
| [[1557376/1557345]]<br />
| 0.034<br />
|-<br />
| S49/S50 = ([[2401/2400]])/([[2500/2499]])<br />
| ([[17/16]])/([[50/49]])<sup>3</sup><br />
| [[2000033/2000000]]<br />
| 0.029<br />
|-<br />
| S50/S51 = ([[2500/2499]])/([[2601/2600]])<br />
| ([[52/49]])/([[51/50]])<sup>3</sup><br />
| [[6500000/6499899]]<br />
| 0.027<br />
|-<br />
| S55/S56 = ([[3025/3024]])/([[3136/3135]])<br />
| ([[19/18]])/([[56/55]])<sup>3</sup><br />
| [[3161125/3161088]]<br />
| 0.02<br />
|-<br />
| S64/S65 = ([[4096/4095]])/([[4225/4224]])<br />
| ([[22/21]])/([[65/64]])<sup>3</sup><br />
| [[5767168/5767125]]<br />
| 0.013<br />
|}<br />
<br />
The above table is a list of all [[23-limit]] ultraparticulars corresponding to S''k'' with ''k'' < 77, plus ultraparticulars corresponding to dividing a [[superparticular interval]] into three equal parts up to [[17/16]] (or up to [[19/18]] but excluding 18/17 because of it requiring a large prime, 53), plus S27/S28 so that we have all ultraparticulars up to S28/S29 listed rather than up to S26/S27.<br />
<br />
This table has been expanded following every ultraparticular from S2/S3 to S16/S17 having its own page. Note that ultraparticulars are, in general, extremely precise commas so that usually one wouldn't consider tempering them directly rather than through tempering the square-particulars S''k'' which they are composed of. As an example of this, notice that [[4000/3993|S10/S11]] is the largest ultraparticular categorised as an [[unnoticeable comma]], which means not unnoticeable in the absolute sense but rather in the sense of being smaller than the melodic just-noticeable difference, despite only dividing a superparticular as simple and unremarkable as [[4/3]]. For this reason, a [[cent]]s column has been included to aid an appreciation of their precision. The cent value of a [[semiparticular]] is roughly double that of any of the two ultraparticulars it is composed of; this becomes more true the higher you go.<br />
<br />
Note also from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably [[consistent]]) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3''k'' + 1)/S(3''k''+ 2) for a positive integer ''k'', because then the superparticular can be expressed as:<br />
<br />
<math>\frac{(3k + 3)/3k}{((3k + 2)(3k + 1))^3} = \frac{(k + 1)/k}{((3k + 2)(3k + 1))^3}</math><br />
<br />
Also note that if you temper multiple adjacent ultraparticulars, you sometimes are not required to use those ultraparticulars in the comma list as description of (the bulk of) the tempering may be possible through [[#Sk/S(k + 2) (semiparticulars)|semiparticulars]], discussed next.<br />
<br />
== {{nowrap|S''k''/S(''k'' + 2)}} (semiparticulars) ==<br />
=== Motivational examples ===<br />
If we want to halve one JI interval into two of another JI interval, there is a powerful and elegant pattern for doing so:<br />
* [[4/3]] is approximately half of [[9/5]]<br />
* [[9/7]] is approximately half of [[5/3]] (=&nbsp;10/6)<br />
* [[5/4]] is approximately half of [[11/7]]<br />
* [[11/9]] is approximately half of [[3/2]] (=&nbsp;12/8)<br />
* [[6/5]] is approximately half of [[13/9]]<br />
* [[13/11]] is approximately half of [[7/5]] (=&nbsp;14/10)<br />
* [[7/6]] is approximately half of [[15/11]]<br />
* [[15/13]] is approximately half of [[4/3]] (=&nbsp;16/12)<br />
* [[8/7]] is approximately half of [[17/13]]<br />
* [[17/15]] is approximately half of [[9/7]] (=&nbsp;18/14)<br />
* [[9/8]] is approximately half of [[19/15]]<br />
* [[19/17]] is approximately half of [[5/4]] (=&nbsp;20/16)<br />
<br />
These properties show a pattern: take some arbitrary [[#Glossary|quodd-particular]] (''k'' + 4)/''k''; observe that we can split it into (''k'' + 4)/(''k'' + 2) * (''k'' + 2)/''k''.<br />
<br />
Now observe that (''k'' + 2)/''k'' > (''k'' + 3)/(''k'' + 1) > (''k'' + 4)/(''k'' + 2); in fact, it can be shown fairly easily that (''k'' + 3)/(''k'' + 1) is the [[mediant]] of (''k'' + 4)/(''k'' + 2) and (''k'' + 2)/''k''.<br />
<br />
It turns out that making this mediant — (''k'' + 3)/(''k'' + 1) — equal to half of (''k'' + 4)/''k'' is equivalent to tempering S(''k'' + 1)/S(''k'' + 3).<br />
<br />
=== Significance ===<br />
1. For differences between square-particulars of the form S''k''/S(''k'' + 2), the resulting comma is either [[superparticular]] or [[#Glossary|odd-particular]], so these are efficient commas. (This terminology also suggests [[#Glossary|throdd-particular]] and [[#Glossary|quodd-particular]] as generalizations.)<br />
<br />
2. Tempering any two consecutive [[ultraparticular]]s implies tempering a semiparticular, so from two adjacent "thirding" equivalences you get a "halving" equivalence for free!<br />
<br />
3. Tempering any two nearly-consecutive square-particulars (S''k'' and S(''k'' + 2)) implies tempering a semiparticular; this is generally much more ideal than tempering two consecutive S''k'' because it is a lot lower damage (see [[lopsided comma]]s for (relatively) large commas implied by this higher-damage strategy).<br />
<br />
4. On top of the halving equivalence, there is a number of subtler structural implications, [[discussed below, that may be desirable to the temperament designer.<br />
<br />
=== Meaning ===<br />
: '''Reader notes:''' In the below, we use S(''k'' - 1)/S(''k'' + 1) for symmetry around ''k'' to make the math visually simpler, but keep in mind it's equivalent to using an offset ''k''.<br />
: '''Also:''' keep in mind that ''k'' - ''a'' (for positive ''a'') is smaller than ''k'', so that ''k''/(''k'' - ''a'') > (''k'' + ''a'')/''k'' (because the former appears earlier in the harmonic series & is thus larger); this is an important and useful intuition to learn.<br />
<br />
Tempering S(''k'' - 1)/S(''k'' + 1) implies that (''k'' + 2)/(''k'' - 2) is divisible exactly into two halves of (''k'' + 1)/(''k'' - 1). It also implies that the intervals (''k'' + 2)/''k'' (=s) and ''k''/(''k'' - 2) (=L) are equidistant from (''k'' + 1)/(''k'' - 1) (=M) because to make them equidistant we need to temper:<br />
<br />
<math>\displaystyle \frac{L/M}{M/s} = \frac{ \left(\frac{k}{k-2}\right)/\left(\frac{k+1}{k-1}\right) }{ \left(\frac{k+1}{k-1}\right)/\left(\frac{k+2}{k}\right) } = \frac{Ls}{M^2} = \frac{\frac{k+2}{k-2}}{\left(\frac{k+1}{k-1}\right)^2} </math><br />
<br />
… and notice that the latter expression is the one we've [[S-expression/Advanced results#Mathematical derivations|shown is equal to S(''k''-1)/S(''k''+1)]] (up to an offset ''k''). In other words, you could interpret that a reason that tempering S(''k''-1)/S(''k''+1) results in (''k''+1)/(''k''-1) being half of (''k''+2)/(''k''-2) is because it makes the following three intervals equidistant:<br />
(''k''+2)/''k'', (''k''+1)/(''k''-1), ''k''/(''k''-2)<br />
<br />
Also note that in the above, (''k'' + 1)/(''k'' - 1) is the [[mediant]] of the adjacent two intervals, meaning that division of an interval into two via tempering a semiparticular is in some sense 'optimal' relative to the complexity. This also means that if ''k'' is a multiple of 2, this corresponds to a natural way to split the square superparticular S(''k''/2) into two parts. For example, if ''k'' = 10 then we have (10+2)/10, (10+1)/(10-1), 10/(10-2) as equidistant, which simplified is 6/5, 11/9, 5/4, with 11/9 being the mediant of 6/5 and 5/4, and therefore the corresponding superparticular S5 = (5/4)/(6/5) is split into two parts which are tempered together: (5/4)/(11/9) = 45/44 and (11/9)/(6/5) = 55/54. The semiparticular is therefore S(10-1)/S(10+1) = S9/S11 = 243/242 = (45/44)/(55/54) = ((10+2)/(10-2))/((10+1)/(10-1))<sup>2</sup>.<br />
<br />
This form of comma has been named "semiparticular", because most of the time it is superparticular but less often it is odd-particular, and because when tempered out they all cause an interval to be divided into two equal parts where each part is a (tempered version of a) superparticular or odd-particular, and the interval being divided in half is sometimes quodd-particular, sometimes odd-particular and sometimes superparticular. Specifically:<br />
<br />
* To find out what a superparticular (''a''+1)/''a'' is approximately half of, temper the semiparticular S(2''a'')/S(2''a''+2) and you can observe that (2''a''+3)/(2''a''-1) is the interval it is approximately half of.<br />
<br />
* To find out what an odd-particular (2''a''+1)/(2''a''-1) is approximately half of, temper the semiparticular S(2''a''-1)/S(2''a''+1) and you can observe that (2''a''+2)/(2''a''-2) = (''a''+1)/(''a''-1), a superparticular or odd-particular, is the interval it is approximately half of.<br />
<br />
* To find out what splits a superparticular (''a''+1)/''a'' in half, temper the semiparticular S(4''a''+1)/S(4''a''+3) and you can observe that (4''a''+3)/(4''a''+1), an odd-particular, is the interval that is approximately half of it.<br />
<br />
* To find out what splits an odd-particular (2''a''+1)/(2''a''-1) in half, temper the semiparticular S(4''a''-2)/S(4''a''+2) and you can observe that (4''a''-1)/(4''a''+1), an odd-particular, is the interval that is approximately half of it.<br />
<br />
Also, the interval in the denominator of an expression of a semiparticular of the form (a/b)/(c/d)<sup>2</sup> is significant in that it has a special relationship: specifically, consider tempering (a/b)/(c/d)<sup>2</sup> so therefore the interval c/d is equal to the interval (a/b)/(c/d). This is significant because it allows the intuitive replacement of two consecutive superparticulars (whose product is a superparticular or odd-particular) with the two superparticulars directly adjacent to them.<br />
<br />
For example, as 9/8 = 18/17 * 17/16 we can replace 18/17 with 19/18 and 17/16 with 16/15 by tempering S16/S18 = (19/15)/(9/8)<sup>2</sup> because we can multiply 9/8 by the tempered comma (19/15)/(9/8)<sup>2</sup> to get (19/15)/(9/8) = (19/18)(16/15) (because 9/8 = 18/16), or as 13/11 = 13/12 * 12/11 we can replace 13/12 with 14/13 and 12/11 with 11/10 by tempering S11/S13 = (7/5)/(13/11)<sup>2</sup> because we can multiply 13/11 by the tempered comma (7/5)/(13/11)<sup>2</sup> to get (7/5)/(13/11) = (14/13)(11/10) (because 7/5 = 14/10). Note we have to replace ''both'' intervals ''simultaneously'' as this is lower error, and note that if we want to be able to replace them individually we must pick the higher error route of tempering S16 and S18 or S11 and S13 individually (for which tempering the semiparticular is then an implied consequence). (The broader lesson is that you can rewrite exact JI equivalences with the commas you are tempering to find new interesting consequences of those commas.)<br />
<br />
=== Table of semiparticulars ===<br />
Here follows a table of [[23-limit]] semiparticulars corresponding to square-particulars S''k'' for ''k'' < 96, plus all semiparticulars up to [[9801/9800|S33/S35 = S99]], an exceptional [[11-limit]] comma, plus all semiparticulars dividing [[superparticular interval]]s up to [[13/12]] (corresponding to the [[17-limit]] semiparticular [[31213/31212|S49/S51]]) for completeness. This table also shows all semiparticulars corresponding to splitting an [[#Glossary|odd-particular]] in two up to [[17/15]] (although a common strategy is to temper the square-particular that is the difference between the two superparticular intervals the odd-particular is composed of instead). The bound ''k'' < 96 was chosen as it corresponds to another remarkable semiparticular [[123201/123200|S78/S80 = S351]]. Perhaps many of the patterns will become clearer if you examine this table:<br />
<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Square Relation<br />
! Ratio<br />
|-<br />
| S2/S4 = ([[4/3]])/([[16/15]])<br />
| ([[5/1]])/([[2/1]])<sup>2</sup><br />
| [[5/4]]<br />
|-<br />
| S3/S5 = ([[9/8]])/([[25/24]])<br />
| ([[3/1]])/([[5/3]])<sup>2</sup><br />
| [[27/25]]<br />
|-<br />
| S4/S6 = ([[16/15]])/([[36/35]])<br />
| ([[7/3]])/([[3/2]])<sup>2</sup><br />
| [[28/27]]<br />
|-<br />
| S5/S7 = ([[25/24]])/([[49/48]])<br />
| ([[2/1]])/([[7/5]])<sup>2</sup><br />
| [[50/49]]<br />
|-<br />
| S6/S8 = ([[36/35]])/([[64/63]])<br />
| ([[9/5]])/([[4/3]])<sup>2</sup><br />
| [[81/80]]<br />
|-<br />
| S7/S9 = ([[49/48]])/([[81/80]])<br />
| ([[5/3]])/([[9/7]])<sup>2</sup><br />
| [[245/243]]<br />
|-<br />
| S8/S10 = ([[64/63]])/([[100/99]])<br />
| ([[11/7]])/([[5/4]])<sup>2</sup><br />
| [[176/175]]<br />
|-<br />
| S9/S11 = ([[81/80]])/([[121/120]])<br />
| ([[3/2]])/([[11/9]])<sup>2</sup><br />
| [[243/242]]<br />
|-<br />
| S10/S12 = ([[100/99]])/([[144/143]])<br />
| ([[13/9]])/([[6/5]])<sup>2</sup><br />
| [[325/324]]<br />
|-<br />
| S11/S13 = ([[121/120]])/([[169/168]])<br />
| ([[7/5]])/([[13/11]])<sup>2</sup><br />
| [[847/845]]<br />
|-<br />
| S12/S14 = ([[144/143]])/([[196/195]])<br />
| ([[15/11]])/([[7/6]])<sup>2</sup><br />
| [[540/539]]<br />
|-<br />
| S13/S15 = ([[169/168]])/([[225/224]])<br />
| ([[4/3]])/([[15/13]])<sup>2</sup><br />
| [[676/675]]<br />
|-<br />
| S14/S16 = ([[196/195]])/([[256/255]])<br />
| ([[17/13]])/([[8/7]])<sup>2</sup><br />
| [[833/832]]<br />
|-<br />
| S15/S17 = ([[225/224]])/([[289/288]])<br />
| ([[9/7]])/([[17/15]])<sup>2</sup><br />
| [[2025/2023]]<br />
|-<br />
| S16/S18 = ([[256/255]])/([[324/323]])<br />
| ([[19/15]])/([[9/8]])<sup>2</sup><br />
| [[1216/1215]]<br />
|-<br />
| S17/S19 = ([[289/288]])/([[361/360]])<br />
| ([[5/4]])/([[19/17]])<sup>2</sup><br />
| [[1445/1444]]<br />
|-<br />
| S18/S20 = ([[324/323]])/([[400/399]])<br />
| ([[21/17]])/([[10/9]])<sup>2</sup><br />
| [[1701/1700]]<br />
|-<br />
| S19/S21 = ([[361/360]])/([[441/440]])<br />
| ([[11/9]])/([[21/19]])<sup>2</sup><br />
| [[3971/3969]]<br />
|-<br />
| S20/S22 = ([[400/399]])/([[484/483]])<br />
| ([[23/19]])/([[11/10]])<sup>2</sup><br />
| [[2300/2299]]<br />
|-<br />
| S21/S23 = ([[441/440]])/([[529/528]])<br />
| ([[6/5]])/([[23/21]])<sup>2</sup><br />
| [[2646/2645]]<br />
|-<br />
| S22/S24 = ([[484/483]])/([[576/575]])<br />
| ([[25/21]])/([[12/11]])<sup>2</sup><br />
| [[3025/3024]]<br />
|-<br />
| S23/S25 = ([[529/528]])/([[625/624]])<br />
| ([[13/11]])/([[25/23]])<sup>2</sup><br />
| [[6877/6875]]<br />
|-<br />
| S24/S26 = ([[576/575]])/([[676/675]])<br />
| ([[27/23]])/([[13/12]])<sup>2</sup><br />
| [[3888/3887]]<br />
|-<br />
| S25/S27 = ([[625/624]])/([[729/728]])<br />
| ([[7/6]])/([[27/25]])<sup>2</sup><br />
| [[4375/4374]]<br />
|-<br />
| S26/S28 = ([[676/675]])/([[784/783]])<br />
| ([[29/25]])/([[14/13]])<sup>2</sup><br />
| [[4901/4900]]<br />
|-<br />
| S27/S29 = ([[729/728]])/([[841/840]])<br />
| ([[15/13]])/([[29/27]])<sup>2</sup><br />
| [[10935/10933]]<br />
|-<br />
| S28/S30 = ([[784/783]])/([[900/899]])<br />
| ([[31/27]])/([[15/14]])<sup>2</sup><br />
| [[6076/6075]]<br />
|-<br />
| S29/S31 = ([[841/840]])/([[961/960]])<br />
| ([[8/7]])/([[31/29]])<sup>2</sup><br />
| [[6728/6727]]<br />
|-<br />
| S30/S32 = ([[900/899]])/([[1024/1023]])<br />
| ([[33/29]])/([[16/15]])<sup>2</sup><br />
| [[7425/7424]]<br />
|-<br />
| S31/S33 = ([[961/960]])/([[1089/1088]])<br />
| ([[17/15]])/([[33/31]])<sup>2</sup><br />
| [[16337/16335]]<br />
|-<br />
| S32/S34 = ([[1024/1023]])/([[1156/1155]])<br />
| ([[35/31]])/([[17/16]])<sup>2</sup><br />
| [[8960/8959]]<br />
|-<br />
| S33/S35 = ([[1089/1088]])/([[1225/1224]])<br />
| ([[9/8]])/([[35/33]])<sup>2</sup><br />
| [[9801/9800]]<br />
|-<br />
| S36/S38 = ([[1296/1295]])/([[1444/1443]])<br />
| ([[39/35]])/([[19/18]])<sup>2</sup><br />
| [[12636/12635]]<br />
|-<br />
| S37/S39 = ([[1369/1368]])/([[1521/1520]])<br />
| ([[10/9]])/([[39/37]])<sup>2</sup><br />
| [[13690/13689]]<br />
|-<br />
| S41/S43 = ([[1681/1680]])/([[1849/1848]])<br />
| ([[11/10]])/([[43/41]])<sup>2</sup><br />
| [[18491/18490]]<br />
|-<br />
| S45/S47 = ([[2025/2024]])/([[2209/2208]])<br />
| ([[12/11]])/([[47/45]])<sup>2</sup><br />
| [[24300/24299]]<br />
|-<br />
| S46/S48 = ([[2116/2115]])/([[2304/2303]])<br />
| ([[49/45]])/([[24/23]])<sup>2</sup><br />
| [[25921/25920]]<br />
|-<br />
| S49/S51 = ([[2401/2400]])/([[2601/2600]])<br />
| ([[13/12]])/([[51/49]])<sup>2</sup><br />
| [[31213/31212]]<br />
|-<br />
| S52/S54 = ([[2704/2703]])/([[2916/2915]])<br />
| ([[55/51]])/([[27/26]])<sup>2</sup><br />
| [[37180/37179]]<br />
|-<br />
| S66/S68 = ([[4356/4355]])/([[4624/4623]])<br />
| ([[69/65]])/([[34/33]])<sup>2</sup><br />
| [[75141/75140]]<br />
|-<br />
| S78/S80 = ([[6084/6083]])/([[6400/6399]])<br />
| ([[81/77]])/([[40/39]])<sup>2</sup><br />
| [[123201/123200]]<br />
|}<br />
<br />
(Note that while a lot of these have pages, not all of them do, although that doesn't mean they shouldn't. A noticeable streak of commas currently without pages correspond to when dividing a superparticular interval implicates intervals from a higher [[prime limit]], as a surprising amount of 23-limit semiparticulars shown here already have pages.)<br />
<br />
== {{nowrap|S''k''² * S(''k'' + 1)}} and {{nowrap|S(''k'' − 1) * S''k''²}} (lopsided commas) ==<br />
=== Significance ===<br />
1. Tempering any two consecutive square-particulars, S''k'' and S(''k'' + 1), implies tempering the two associated lopsided commas as well as the associated [[triangle-particular]] and [[ultraparticular]], so the lopsided commas represent the general form of the highest-damage relations/consequences of doing so.<br />
<br />
2. If a comma (such as the diaschisma, [[2048/2025]]), admits an expression as a lopsided comma, it means that one is likely missing out on tempering opportunities by not also tempering the square-particulars composing it (such as [[256/255|S16]] and [[289/288|S17]] in the case of the diaschisma), often involving expanding the subgroup and adding a number of new equivalence relations (as previously explained) while simultaneously making the temperament more efficient and more precise.<br />
<br />
3. It is surprising that there are fairly simple general equivalence relations for these S-expressions, essentially being "free" to read off of an S-expression-based comma list, once you know the general form.<br />
<br />
=== Derivation of equivalence relation ===<br />
Using the clarity of [[S-expression/Advanced results#Using S-factorizations to understand the significance of S-expressions|S-factorizations]], we can show the interval relations implicated by these two new "lopsided" forms, which will make clear the reason for their name:<br />
<br />
S''k''<sup>2</sup> * S(''k''+1) = [''k''-1, ''k'', ''k''+1, ''k''+2]^(2[-1, 2, -1, 0] + [0, -1, 2, -1] = [-2, 4, -2, 0] + [0, -1, 2, -1] = [-2, 3, 0, -1]) implies:<br />
<br />
S''k''<sup>2</sup> * S(''k''+1) = (''k''/(''k''-1))<sup>2</sup> / ((''k''+2)/''k'') through [-2, 3, 0, -1] = [-2, 2, 0, 0] - [0, -1, 0, 1].<br />
<br />
S(''k''-1) * S''k''<sup>2</sup> = [''k''-2, ''k''-1, ''k'', ''k''+1]^([-1, 2, -1, 0] + 2[0, -1, 2, -1] = [-1, 2, -1, 0] + [0, -2, 4, -2] = [-1, 0, 3, -2]) implies:<br />
<br />
S(''k''-1) * S''k''<sup>2</sup> = (''k''/(''k''-2)) / ((''k''+1)/''k'')<sup>2</sup> through [-1, 0, 3, -2] = [-1, 0, 1, 0] - [0, 0, -2, 2].<br />
<br />
=== Tables ===<br />
Below are two tables of [[43-limit]] lopsided commas. First, the "top heavy" lopsided commas, where the squared interval is in the numerator, then the "bottom heavy" lopsided commas, where the squared interval is in the denominator. These tables are so big because these commas are quite large so the more interesting commas appear later. For this reason and for completeness, the tables show up to until a little past the largest known lopsided commas that have their own page: the [[olympia]] and the [[phaotic comma]].<br />
<br />
==== Top-heavy lopsided commas ====<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Square Relation<br />
! Ratio<br />
|-<br />
| S2<sup>2</sup>*S3 = [[3/2]] * [[4/3]]<br />
| ([[2/1]])<sup>2</sup> / ([[2/1]])<br />
| [[2/1]]<br />
|-<br />
| S3<sup>2</sup>*S4 = [[6/5]] * [[9/8]]<br />
| ([[3/2]])<sup>2</sup> / ([[5/3]])<br />
| [[27/20]]<br />
|-<br />
| S4<sup>2</sup>*S5 = [[10/9]] * [[16/15]]<br />
| ([[4/3]])<sup>2</sup> / ([[3/2]])<br />
| [[32/27]]<br />
|-<br />
| S5<sup>2</sup>*S6 = [[15/14]] * [[25/24]]<br />
| ([[5/4]])<sup>2</sup> / ([[7/5]])<br />
| [[125/112]]<br />
|-<br />
| S6<sup>2</sup>*S7 = [[21/20]] * [[36/35]]<br />
| ([[6/5]])<sup>2</sup> / ([[4/3]])<br />
| [[27/25]]<br />
|-<br />
| S7<sup>2</sup>*S8 = [[28/27]] * [[49/48]]<br />
| ([[7/6]])<sup>2</sup> / ([[9/7]])<br />
| [[343/324]]<br />
|-<br />
| S8<sup>2</sup>*S9 = [[36/35]] * [[64/63]]<br />
| ([[8/7]])<sup>2</sup> / ([[5/4]])<br />
| [[256/245]]<br />
|-<br />
| S9<sup>2</sup>*S10 = [[45/44]] * [[81/80]]<br />
| ([[9/8]])<sup>2</sup> / ([[11/9]])<br />
| [[729/704]]<br />
|-<br />
| S10<sup>2</sup>*S11 = [[55/54]] * [[100/99]]<br />
| ([[10/9]])<sup>2</sup> / ([[6/5]])<br />
| [[250/243]]<br />
|-<br />
| S11<sup>2</sup>*S12 = [[66/65]] * [[121/120]]<br />
| ([[11/10]])<sup>2</sup> / ([[13/11]])<br />
| [[1331/1300]]<br />
|-<br />
| S12<sup>2</sup>*S13 = [[78/77]] * [[144/143]]<br />
| ([[12/11]])<sup>2</sup> / ([[7/6]])<br />
| [[864/847]]<br />
|-<br />
| S13<sup>2</sup>*S14 = [[91/90]] * [[169/168]]<br />
| ([[13/12]])<sup>2</sup> / ([[15/13]])<br />
| [[2197/2160]]<br />
|-<br />
| S14<sup>2</sup>*S15 = [[105/104]] * [[196/195]]<br />
| ([[14/13]])<sup>2</sup> / ([[8/7]])<br />
| [[343/338]]<br />
|-<br />
| S15<sup>2</sup>*S16 = [[120/119]] * [[225/224]]<br />
| ([[15/14]])<sup>2</sup> / ([[17/15]])<br />
| [[3375/3332]]<br />
|-<br />
| S16<sup>2</sup>*S17 = [[136/135]] * [[256/255]]<br />
| ([[16/15]])<sup>2</sup> / ([[9/8]])<br />
| [[2048/2025]]<br />
|-<br />
| S17<sup>2</sup>*S18 = [[153/152]] * [[289/288]]<br />
| ([[17/16]])<sup>2</sup> / ([[19/17]])<br />
| [[4913/4864]]<br />
|-<br />
| S18<sup>2</sup>*S19 = [[171/170]] * [[324/323]]<br />
| ([[18/17]])<sup>2</sup> / ([[10/9]])<br />
| [[1458/1445]]<br />
|-<br />
| S19<sup>2</sup>*S20 = [[190/189]] * [[361/360]]<br />
| ([[19/18]])<sup>2</sup> / ([[21/19]])<br />
| [[6859/6804]]<br />
|-<br />
| S20<sup>2</sup>*S21 = [[210/209]] * [[400/399]]<br />
| ([[20/19]])<sup>2</sup> / ([[11/10]])<br />
| [[4000/3971]]<br />
|-<br />
| S21<sup>2</sup>*S22 = [[231/230]] * [[441/440]]<br />
| ([[21/20]])<sup>2</sup> / ([[23/21]])<br />
| [[9261/9200]]<br />
|-<br />
| S22<sup>2</sup>*S23 = [[253/252]] * [[484/483]]<br />
| ([[22/21]])<sup>2</sup> / ([[12/11]])<br />
| [[1331/1323]]<br />
|-<br />
| S23<sup>2</sup>*S24 = [[276/275]] * [[529/528]]<br />
| ([[23/22]])<sup>2</sup> / ([[25/23]])<br />
| [[12167/12100]]<br />
|-<br />
| S24<sup>2</sup>*S25 = [[300/299]] * [[576/575]]<br />
| ([[24/23]])<sup>2</sup> / ([[13/12]])<br />
| [[6912/6877]]<br />
|-<br />
| S25<sup>2</sup>*S26 = [[325/324]] * [[625/624]]<br />
| ([[25/24]])<sup>2</sup> / ([[27/25]])<br />
| [[15625/15552]]<br />
|-<br />
| S26<sup>2</sup>*S27 = [[351/350]] * [[676/675]]<br />
| ([[26/25]])<sup>2</sup> / ([[14/13]])<br />
| [[4394/4375]]<br />
|-<br />
| S27<sup>2</sup>*S28 = [[378/377]] * [[729/728]]<br />
| ([[27/26]])<sup>2</sup> / ([[29/27]])<br />
| [[19683/19604]]<br />
|-<br />
| S28<sup>2</sup>*S29 = [[406/405]] * [[784/783]]<br />
| ([[28/27]])<sup>2</sup> / ([[15/14]])<br />
| [[10976/10935]]<br />
|-<br />
| S29<sup>2</sup>*S30 = [[435/434]] * [[841/840]]<br />
| ([[29/28]])<sup>2</sup> / ([[31/29]])<br />
| [[24389/24304]]<br />
|-<br />
| S30<sup>2</sup>*S31 = [[465/464]] * [[900/899]]<br />
| ([[30/29]])<sup>2</sup> / ([[16/15]])<br />
| [[3375/3364]]<br />
|-<br />
| S31<sup>2</sup>*S32 = [[496/495]] * [[961/960]]<br />
| ([[31/30]])<sup>2</sup> / ([[33/31]])<br />
| [[29791/29700]]<br />
|-<br />
| S32<sup>2</sup>*S33 = [[528/527]] * [[1024/1023]]<br />
| ([[32/31]])<sup>2</sup> / ([[17/16]])<br />
| [[16384/16337]]<br />
|-<br />
| S33<sup>2</sup>*S34 = [[561/560]] * [[1089/1088]]<br />
| ([[33/32]])<sup>2</sup> / ([[35/33]])<br />
| [[35937/35840]]<br />
|-<br />
| S34<sup>2</sup>*S35 = [[595/594]] * [[1156/1155]]<br />
| ([[34/33]])<sup>2</sup> / ([[18/17]])<br />
| [[9826/9801]]<br />
|-<br />
| S35<sup>2</sup>*S36 = [[630/629]] * [[1225/1224]]<br />
| ([[35/34]])<sup>2</sup> / ([[37/35]])<br />
| [[42875/42772]]<br />
|-<br />
| S36<sup>2</sup>*S37 = [[666/665]] * [[1296/1295]]<br />
| ([[36/35]])<sup>2</sup> / ([[19/18]])<br />
| [[23328/23275]]<br />
|-<br />
| S37<sup>2</sup>*S38 = [[703/702]] * [[1369/1368]]<br />
| ([[37/36]])<sup>2</sup> / ([[39/37]])<br />
| [[50653/50544]]<br />
|-<br />
| S38<sup>2</sup>*S39 = [[741/740]] * [[1444/1443]]<br />
| ([[38/37]])<sup>2</sup> / ([[20/19]])<br />
| [[6859/6845]]<br />
|-<br />
| S39<sup>2</sup>*S40 = [[780/779]] * [[1521/1520]]<br />
| ([[39/38]])<sup>2</sup> / ([[41/39]])<br />
| [[59319/59204]]<br />
|-<br />
| S40<sup>2</sup>*S41 = [[820/819]] * [[1600/1599]]<br />
| ([[40/39]])<sup>2</sup> / ([[21/20]])<br />
| [[32000/31941]]<br />
|-<br />
| S41<sup>2</sup>*S42 = [[861/860]] * [[1681/1680]]<br />
| ([[41/40]])<sup>2</sup> / ([[43/41]])<br />
| [[68921/68800]]<br />
|-<br />
| S42<sup>2</sup>*S43 = [[903/902]] * [[1764/1763]]<br />
| ([[42/41]])<sup>2</sup> / ([[22/21]])<br />
| [[18522/18491]]<br />
|-<br />
| S43<sup>2</sup>*S44 = [[946/945]] * [[1849/1848]]<br />
| ([[43/42]])<sup>2</sup> / ([[45/43]])<br />
| [[79507/79380]]<br />
|-<br />
| S44<sup>2</sup>*S45 = [[990/989]] * [[1936/1935]]<br />
| ([[44/43]])<sup>2</sup> / ([[23/22]])<br />
| [[42592/42527]]<br />
|-<br />
| S46<sup>2</sup>*S47 = [[1081/1080]] * [[2116/2115]]<br />
| ([[46/45]])<sup>2</sup> / ([[24/23]])<br />
| [[12167/12150]]<br />
|-<br />
| S49<sup>2</sup>*S50 = [[1225/1224]] * [[2401/2400]]<br />
| ([[49/48]])<sup>2</sup> / ([[51/49]])<br />
| [[117649/117504]]<br />
|-<br />
| S50<sup>2</sup>*S51 = [[1275/1274]] * [[2500/2499]]<br />
| ([[50/49]])<sup>2</sup> / ([[26/25]])<br />
| [[31250/31213]]<br />
|-<br />
| S52<sup>2</sup>*S53 = [[1378/1377]] * [[2704/2703]]<br />
| ([[52/51]])<sup>2</sup> / ([[27/26]])<br />
| [[70304/70227]]<br />
|-<br />
| S55<sup>2</sup>*S56 = [[1540/1539]] * [[3025/3024]]<br />
| ([[55/54]])<sup>2</sup> / ([[57/55]])<br />
| [[166375/166212]]<br />
|-<br />
| S56<sup>2</sup>*S57 = [[1596/1595]] * [[3136/3135]]<br />
| ([[56/55]])<sup>2</sup> / ([[29/28]])<br />
| [[87808/87725]]<br />
|-<br />
| S58<sup>2</sup>*S59 = [[1711/1710]] * [[3364/3363]]<br />
| ([[58/57]])<sup>2</sup> / ([[30/29]])<br />
| [[48778/48735]]<br />
|-<br />
| S63<sup>2</sup>*S64 = [[2016/2015]] * [[3969/3968]]<br />
| ([[63/62]])<sup>2</sup> / ([[65/63]])<br />
| [[250047/249860]]<br />
|-<br />
| S64<sup>2</sup>*S65 = [[2080/2079]] * [[4096/4095]]<br />
| ([[64/63]])<sup>2</sup> / ([[33/32]])<br />
| [[131072/130977]]<br />
|-<br />
| S66<sup>2</sup>*S67 = [[2211/2210]] * [[4356/4355]]<br />
| ([[66/65]])<sup>2</sup> / ([[34/33]])<br />
| [[71874/71825]]<br />
|-<br />
| S70<sup>2</sup>*S71 = [[2485/2484]] * [[4900/4899]]<br />
| ([[70/69]])<sup>2</sup> / ([[36/35]])<br />
| [[42875/42849]]<br />
|-<br />
| S75<sup>2</sup>*S76 = [[2850/2849]] * [[5625/5624]]<br />
| ([[75/74]])<sup>2</sup> / ([[77/75]])<br />
| [[421875/421652]]<br />
|-<br />
| S76<sup>2</sup>*S77 = [[2926/2925]] * [[5776/5775]]<br />
| ([[76/75]])<sup>2</sup> / ([[39/38]])<br />
| [[219488/219375]]<br />
|-<br />
| S78<sup>2</sup>*S79 = [[3081/3080]] * [[6084/6083]]<br />
| ([[78/77]])<sup>2</sup> / ([[40/39]])<br />
| [[59319/59290]]<br />
|}<br />
<br />
==== Bottom-heavy lopsided commas ====<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Square Relation<br />
! Ratio<br />
|-<br />
| S3<sup>2</sup>*S2 = [[3/2]] * [[9/8]]<br />
| ([[3/1]]) / ([[4/3]])<sup>2</sup><br />
| [[27/16]]<br />
|-<br />
| S4<sup>2</sup>*S3 = [[6/5]] * [[16/15]]<br />
| ([[2/1]]) / ([[5/4]])<sup>2</sup><br />
| [[32/25]]<br />
|-<br />
| S5<sup>2</sup>*S4 = [[10/9]] * [[25/24]]<br />
| ([[5/3]]) / ([[6/5]])<sup>2</sup><br />
| [[125/108]]<br />
|-<br />
| S6<sup>2</sup>*S5 = [[15/14]] * [[36/35]]<br />
| ([[3/2]]) / ([[7/6]])<sup>2</sup><br />
| [[54/49]]<br />
|-<br />
| S7<sup>2</sup>*S6 = [[21/20]] * [[49/48]]<br />
| ([[7/5]]) / ([[8/7]])<sup>2</sup><br />
| [[343/320]]<br />
|-<br />
| S8<sup>2</sup>*S7 = [[28/27]] * [[64/63]]<br />
| ([[4/3]]) / ([[9/8]])<sup>2</sup><br />
| [[256/243]]<br />
|-<br />
| S9<sup>2</sup>*S8 = [[36/35]] * [[81/80]]<br />
| ([[9/7]]) / ([[10/9]])<sup>2</sup><br />
| [[729/700]]<br />
|-<br />
| S10<sup>2</sup>*S9 = [[45/44]] * [[100/99]]<br />
| ([[5/4]]) / ([[11/10]])<sup>2</sup><br />
| [[125/121]]<br />
|-<br />
| S11<sup>2</sup>*S10 = [[55/54]] * [[121/120]]<br />
| ([[11/9]]) / ([[12/11]])<sup>2</sup><br />
| [[1331/1296]]<br />
|-<br />
| S12<sup>2</sup>*S11 = [[66/65]] * [[144/143]]<br />
| ([[6/5]]) / ([[13/12]])<sup>2</sup><br />
| [[864/845]]<br />
|-<br />
| S13<sup>2</sup>*S12 = [[78/77]] * [[169/168]]<br />
| ([[13/11]]) / ([[14/13]])<sup>2</sup><br />
| [[2197/2156]]<br />
|-<br />
| S14<sup>2</sup>*S13 = [[91/90]] * [[196/195]]<br />
| ([[7/6]]) / ([[15/14]])<sup>2</sup><br />
| [[686/675]]<br />
|-<br />
| S15<sup>2</sup>*S14 = [[105/104]] * [[225/224]]<br />
| ([[15/13]]) / ([[16/15]])<sup>2</sup><br />
| [[3375/3328]]<br />
|-<br />
| S16<sup>2</sup>*S15 = [[120/119]] * [[256/255]]<br />
| ([[8/7]]) / ([[17/16]])<sup>2</sup><br />
| [[2048/2023]]<br />
|-<br />
| S17<sup>2</sup>*S16 = [[136/135]] * [[289/288]]<br />
| ([[17/15]]) / ([[18/17]])<sup>2</sup><br />
| [[4913/4860]]<br />
|-<br />
| S18<sup>2</sup>*S17 = [[153/152]] * [[324/323]]<br />
| ([[9/8]]) / ([[19/18]])<sup>2</sup><br />
| [[729/722]]<br />
|-<br />
| S19<sup>2</sup>*S18 = [[171/170]] * [[361/360]]<br />
| ([[19/17]]) / ([[20/19]])<sup>2</sup><br />
| [[6859/6800]]<br />
|-<br />
| S20<sup>2</sup>*S19 = [[190/189]] * [[400/399]]<br />
| ([[10/9]]) / ([[21/20]])<sup>2</sup><br />
| [[4000/3969]]<br />
|-<br />
| S21<sup>2</sup>*S20 = [[210/209]] * [[441/440]]<br />
| ([[21/19]]) / ([[22/21]])<sup>2</sup><br />
| [[9261/9196]]<br />
|-<br />
| S22<sup>2</sup>*S21 = [[231/230]] * [[484/483]]<br />
| ([[11/10]]) / ([[23/22]])<sup>2</sup><br />
| [[2662/2645]]<br />
|-<br />
| S23<sup>2</sup>*S22 = [[253/252]] * [[529/528]]<br />
| ([[23/21]]) / ([[24/23]])<sup>2</sup><br />
| [[12167/12096]]<br />
|-<br />
| S24<sup>2</sup>*S23 = [[276/275]] * [[576/575]]<br />
| ([[12/11]]) / ([[25/24]])<sup>2</sup><br />
| [[6912/6875]]<br />
|-<br />
| S25<sup>2</sup>*S24 = [[300/299]] * [[625/624]]<br />
| ([[25/23]]) / ([[26/25]])<sup>2</sup><br />
| [[15625/15548]]<br />
|-<br />
| S26<sup>2</sup>*S25 = [[325/324]] * [[676/675]]<br />
| ([[13/12]]) / ([[27/26]])<sup>2</sup><br />
| [[2197/2187]]<br />
|-<br />
| S27<sup>2</sup>*S26 = [[351/350]] * [[729/728]]<br />
| ([[27/25]]) / ([[28/27]])<sup>2</sup><br />
| [[19683/19600]]<br />
|-<br />
| S28<sup>2</sup>*S27 = [[378/377]] * [[784/783]]<br />
| ([[14/13]]) / ([[29/28]])<sup>2</sup><br />
| [[10976/10933]]<br />
|-<br />
| S29<sup>2</sup>*S28 = [[406/405]] * [[841/840]]<br />
| ([[29/27]]) / ([[30/29]])<sup>2</sup><br />
| [[24389/24300]]<br />
|-<br />
| S30<sup>2</sup>*S29 = [[435/434]] * [[900/899]]<br />
| ([[15/14]]) / ([[31/30]])<sup>2</sup><br />
| [[6750/6727]]<br />
|-<br />
| S31<sup>2</sup>*S30 = [[465/464]] * [[961/960]]<br />
| ([[31/29]]) / ([[32/31]])<sup>2</sup><br />
| [[29791/29696]]<br />
|-<br />
| S32<sup>2</sup>*S31 = [[496/495]] * [[1024/1023]]<br />
| ([[16/15]]) / ([[33/32]])<sup>2</sup><br />
| [[16384/16335]]<br />
|-<br />
| S33<sup>2</sup>*S32 = [[528/527]] * [[1089/1088]]<br />
| ([[33/31]]) / ([[34/33]])<sup>2</sup><br />
| [[35937/35836]]<br />
|-<br />
| S34<sup>2</sup>*S33 = [[561/560]] * [[1156/1155]]<br />
| ([[17/16]]) / ([[35/34]])<sup>2</sup><br />
| [[4913/4900]]<br />
|-<br />
| S35<sup>2</sup>*S34 = [[595/594]] * [[1225/1224]]<br />
| ([[35/33]]) / ([[36/35]])<sup>2</sup><br />
| [[42875/42768]]<br />
|-<br />
| S36<sup>2</sup>*S35 = [[630/629]] * [[1296/1295]]<br />
| ([[18/17]]) / ([[37/36]])<sup>2</sup><br />
| [[23328/23273]]<br />
|-<br />
| S37<sup>2</sup>*S36 = [[666/665]] * [[1369/1368]]<br />
| ([[37/35]]) / ([[38/37]])<sup>2</sup><br />
| [[50653/50540]]<br />
|-<br />
| S38<sup>2</sup>*S37 = [[703/702]] * [[1444/1443]]<br />
| ([[19/18]]) / ([[39/38]])<sup>2</sup><br />
| [[13718/13689]]<br />
|-<br />
| S39<sup>2</sup>*S38 = [[741/740]] * [[1521/1520]]<br />
| ([[39/37]]) / ([[40/39]])<sup>2</sup><br />
| [[59319/59200]]<br />
|-<br />
| S40<sup>2</sup>*S39 = [[780/779]] * [[1600/1599]]<br />
| ([[20/19]]) / ([[41/40]])<sup>2</sup><br />
| [[32000/31939]]<br />
|-<br />
| S41<sup>2</sup>*S40 = [[820/819]] * [[1681/1680]]<br />
| ([[41/39]]) / ([[42/41]])<sup>2</sup><br />
| [[68921/68796]]<br />
|-<br />
| S42<sup>2</sup>*S41 = [[861/860]] * [[1764/1763]]<br />
| ([[21/20]]) / ([[43/42]])<sup>2</sup><br />
| [[9261/9245]]<br />
|-<br />
| S43<sup>2</sup>*S42 = [[903/902]] * [[1849/1848]]<br />
| ([[43/41]]) / ([[44/43]])<sup>2</sup><br />
| [[79507/79376]]<br />
|-<br />
| S44<sup>2</sup>*S43 = [[946/945]] * [[1936/1935]]<br />
| ([[22/21]]) / ([[45/44]])<sup>2</sup><br />
| [[42592/42525]]<br />
|-<br />
| S45<sup>2</sup>*S44 = [[990/989]] * [[2025/2024]]<br />
| ([[45/43]]) / ([[46/45]])<sup>2</sup><br />
| [[91125/90988]]<br />
|-<br />
| S48<sup>2</sup>*S47 = [[1128/1127]] * [[2304/2303]]<br />
| ([[24/23]]) / ([[49/48]])<sup>2</sup><br />
| [[55296/55223]]<br />
|-<br />
| S50<sup>2</sup>*S49 = [[1225/1224]] * [[2500/2499]]<br />
| ([[25/24]]) / ([[51/50]])<sup>2</sup><br />
| [[15625/15606]]<br />
|-<br />
| S51<sup>2</sup>*S50 = [[1275/1274]] * [[2601/2600]]<br />
| ([[51/49]]) / ([[52/51]])<sup>2</sup><br />
| [[132651/132496]]<br />
|-<br />
| S54<sup>2</sup>*S53 = [[1431/1430]] * [[2916/2915]]<br />
| ([[27/26]]) / ([[55/54]])<sup>2</sup><br />
| [[39366/39325]]<br />
|-<br />
| S56<sup>2</sup>*S55 = [[1540/1539]] * [[3136/3135]]<br />
| ([[28/27]]) / ([[57/56]])<sup>2</sup><br />
| [[87808/87723]]<br />
|-<br />
| S57<sup>2</sup>*S56 = [[1596/1595]] * [[3249/3248]]<br />
| ([[57/55]]) / ([[58/57]])<sup>2</sup><br />
| [[185193/185020]]<br />
|-<br />
| S62<sup>2</sup>*S61 = [[1891/1890]] * [[3844/3843]]<br />
| ([[31/30]]) / ([[63/62]])<sup>2</sup><br />
| [[59582/59535]]<br />
|-<br />
| S64<sup>2</sup>*S63 = [[2016/2015]] * [[4096/4095]]<br />
| ([[32/31]]) / ([[65/64]])<sup>2</sup><br />
| [[131072/130975]]<br />
|-<br />
| S65<sup>2</sup>*S64 = [[2080/2079]] * [[4225/4224]]<br />
| ([[65/63]]) / ([[66/65]])<sup>2</sup><br />
| [[274625/274428]]<br />
|-<br />
| S68<sup>2</sup>*S67 = [[2278/2277]] * [[4624/4623]]<br />
| ([[34/33]]) / ([[69/68]])<sup>2</sup><br />
| [[157216/157113]]<br />
|-<br />
| S74<sup>2</sup>*S73 = [[2701/2700]] * [[5476/5475]]<br />
| ([[37/36]]) / ([[75/74]])<sup>2</sup><br />
| [[50653/50625]]<br />
|-<br />
| S76<sup>2</sup>*S75 = [[2850/2849]] * [[5776/5775]]<br />
| ([[38/37]]) / ([[77/76]])<sup>2</sup><br />
| [[219488/219373]]<br />
|-<br />
| S77<sup>2</sup>*S76 = [[2926/2925]] * [[5929/5928]]<br />
| ([[77/75]]) / ([[78/77]])<sup>2</sup><br />
| [[456533/456300]]<br />
|-<br />
| S80<sup>2</sup>*S79 = [[3160/3159]] * [[6400/6399]]<br />
| ([[40/39]]) / ([[81/80]])<sup>2</sup><br />
| [[256000/255879]]<br />
|}<br />
<br />
== Equivalent S-expressions ==<br />
=== Significance and meaning ===<br />
All S-expressions have other equivalent S-expressions, however when the equivalence makes one comma a member of two of the infinite families discussed on this page, or otherwise makes it equal to a product or ratio between two such commas, this often means exceptional and nontrivial ("deep") tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending how you temper further. Generally we exclude 1/n-square-particulars, only noting up to 1/3-square-particulars, because equivalent 1/n-square-particular expressions become very common as you allow higher n, but are still quite rare for small n.<br />
<br />
=== A useful general rule ===<br />
While there are likely arbitrarily many ways of rewriting S-expressions due to the redundancy in representation, the following equivalence is, due to its simplicity and elegance, arguably most likely to be useful:<br />
<br />
<math><br />
\large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)<br />
</math><br />
<br />
This is important to note because using this simple rule we can derive an infinite amount of trivially and obviously equivalent S-expressions that should be discarded from [[#Examples]]. See [[S-expression/Advanced results]] for mathematical details.<br />
<br />
=== Examples ===<br />
Here is an incomplete list of examples (feel free to expand with any equivalences you find that you think are valuable).<br />
<br />
'''Importantly:''' examples that can ''easily'' (with one or two algebraic rewriting steps) be shown to result from the aforementioned [[#A useful general rule|useful general rule]] are considered invalid/trivial.<br />
<br />
{| class="wikitable center-all<br />
|-<br />
! Comma<br />
! S-expressions<br />
|-<br />
| [[64/63]]<br />
| (S4*S5*S6)/S3 = S4/(S6*S7) = S8<br />
|-<br />
| [[81/80]]<br />
| S6/S8 = S9<br />
|-<br />
| [[176/175]]<br />
| S8/S10 = S22*S23*S24<br />
|-<br />
| [[243/242]]<br />
| S9/S11 = S15/([[3025/3024|S22/S24 = S55]])<br />
|-<br />
| [[325/324]]<br />
| S10/S12 = S25*S26<br />
|-<br />
| [[540/539]]<br />
| S12/S14 = (S9*S10)/S7 = (S6/S7)/(S8/S10)<br />
|-<br />
| [[676/675]]<br />
| S13/S15 = S26<br />
|-<br />
| [[1225/1224]]<br />
| S35 = S49*S50<br />
|-<br />
| [[3025/3024]]<br />
| S22/S24 = S55 = S25/S27 * S99<br />
|-<br />
| [[2601/2600]]<br />
| S17/(S25*S26) = S51<br />
|-<br />
| [[9801/9800]]<br />
| S99 = S33/S35<br />
|-<br />
| [[25921/25920]]<br />
| S161 = S46/S48<br />
|-<br />
| [[123201/123200]]<br />
| S351 = S78/S80<br />
|}<br />
<br />
Note: Where a comma written in the form a/b is used in an S-expression, this means to replace that comma with any equivalent S-expression. This is done in the case of [[3025/3024]] as there are many S-expressions for it so restating them each time it appears seems inconvenient.<br />
<br />
A proof that every positive rational number (and thus every JI interval) can be written as an S-expression follows.<br />
<br />
It suffices to show every superparticular number including 2/1 has an expression using square-particulars:<br />
<br />
<math>\displaystyle<br />
\begin{align}<br />
& 2/1 = S_2 \cdot S_2 \cdot S_3\ ,\\<br />
& 3/2 = S_2 \cdot S_3\ ,\\<br />
& 4/3 = S_2\ ,\\<br />
& \frac{a/(a - 1)}{(b + 1)/b} = \prod_{k=a}^b \left( S_k = \frac{k/(k - 1)}{(k + 1)/k} \right) \\<br />
& \ \ \ = \frac{a/(a - 1)}{(a + 1)/a} \cdot \frac{(a + 1)/a}{(a + 2)/(a + 1)} \cdot \frac{(a + 2)/(a + 1)}{(a + 3)/(a + 2)} \cdot\ \ldots \cdot \frac{b/(b - 1)}{(b + 1)/b} = \frac{a/(a - 1)}{(b + 1)/b} \\<br />
& \implies \frac{a/(a - 1)}{(b + 1)/b} = S_a \cdot S_{a + 1} \cdot S_{a + 2} \cdot\ \ldots \cdot S_b \\<br />
& \implies \frac{S_2 \cdot S_2 \cdot S_3}{\prod_{a = 2}^k S_a} = 2 \cdot \left( \frac{2/(2 - 1)}{(k + 1)/k} \right)^{-1} = 2 \cdot \left( \frac{(k + 1)/k}{2} \right) = (k + 1)/k<br />
\end{align}<br />
</math><br />
<br />
From here it should not be hard to see how to make any positive rational number. For 11/6, for example, we can do (11/10)(10/9)(9/8)…(2/1) = 11 and then divide that by (6/5)(5/4)(4/3)(3/2)(2/1), meaning 11/6 = (11/10)(10/9)(9/8)(8/7)(7/6) because of the cancellations, then each of those superparticulars we replace with the corresponding S-expression to get the final S-expression. This final S-expression is likely to be far from the most efficient or interesting expression; the redundancy in S-expressions is a strength and feature, as it tells us that there are more than the trivial connections between commas and intervals and that S-expressions can be wielded as a mathematical tool/language to investigate and identify them.<br />
<br />
== Glossary ==<br />
<br />
; Superparticular<br />
: The interval/comma between two consecutive harmonics. See [[superparticular]].<br />
: These are of the form {{sfrac|''k'' + 1|''k''}}.<br />
<br />
; Square-particular<br />
: A superparticular interval/comma whose numerator is a square number. A shorthand (nick)name for square superparticular.<br />
: These are of the form {{nowrap|{{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}} {{=}} S''k''}}.<br />
<br />
; Triangle-particular<br />
: A superparticular interval/comma whose numerator is a [[triangular number]]. A shorthand (nick)name for triangular superparticular. An alternative name for 1/2-square-particular.<br />
: These are of the form {{sfrac|''k''<sup>2</sup> + ''k''|''k''<sup>2</sup> + ''k'' − 2}}. (This always simplifies to a superparticular.)<br />
<br />
; 1/''n''-square-particular<br />
: A comma which is the product of ''n'' consecutive square-particulars and which can therefore be expressed as the ratio between two superparticulars.<br />
: These are of the form {{nowrap|S''a'' * S(''a'' + 1) * … * S''b'' {{=}} {{sfrac|{{sfrac|''a''|''a'' − 1}}|{{sfrac|''b'' + 1|''b''}}}}}} {{nowrap|{{=}} {{sfrac|''ab''|(''a'' − 1)(''b'' + 1)}}}}.<br />
: Replacing/substituting ''a'' with ''k'' and ''b'' with ''k'' + ''n'' - 1 gives us an equivalent expression that includes the number of square-particulars ''n'':<br />
: {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1) {{=}} {{sfrac|{{sfrac|''k''|''k'' − 1}}|{{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}}}}} {{nowrap|{{=}} {{sfrac|''k''(''k'' + ''n'' − 1)|(''k'' − 1)(''k'' + ''n''}}}}<br />
: For {{nowrap|''b'' {{=}} ''a'' + 1}} these can also be called triangle-particulars, in which case they are always superparticular.<br />
: These have implications for whether consistency in the {{nowrap|(''n'' + ''k'') {{=}} (''b'' + 1)}}-[[odd-limit]] is ''potentially'' possible in a given temperament; see the [[#Sk*S(k + 1)*…*S(k + n - 1) (1/n-square-particulars)|section on 1/''n''-square-particulars]].<br />
<br />
; Odd-particular<br />
: An interval/comma between two consecutive odd harmonics. The odd analogue of superparticular.<br />
: These are of the form {{sfrac|2''k'' + 1|2''k'' − 1}}.<br />
<br />
; Throdd-particular<br />
: An interval/comma between two harmonics 3 apart which is not superparticular.<br />
: These are of the form {{sfrac|3''k'' + 1|3''k'' − 2}} or {{sfrac|3''k'' + 2|3''k'' − 1}}.<br />
<br />
; Quodd-particular<br />
: An interval/comma between two harmonics 4 apart which is not superparticular or odd-particular.<br />
: These are of the form {{sfrac|4''k'' + 1|4''k'' − 3}} or {{sfrac|4''k'' + 3|4''k'' − 1}}.<br />
<br />
; ''n''-odd-particular<br />
: An interval/comma between two coprime harmonics ''n'' apart (also called as [[Delta-N ratio|delta-''n'' ratio]]). It is the generalization of superparticular, odd-particular, throdd-particular, and quodd-particular.<br />
: If ''n'' is a prime, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is not superparticular. For example, 5-odd-particular intervals are of the form {{sfrac|5''k'' + 1|5''k'' − 4}}, {{sfrac|5''k'' + 2|5''k'' − 3}}, {{sfrac|5''k'' + 3|5''k'' − 2}}, or {{sfrac|5''k'' + 4|5''k'' − 1}}.<br />
: If ''n'' is a composite, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is neither superparticular nor of ''m''-odd-particular intervals where ''m'' is any other divisor of ''n''. For example, 6-odd-particular intervals are of the form {{sfrac|6''k'' + 1|6''k'' − 5}} or {{sfrac|6''k'' + 5|6''k'' − 1}}.<br />
<br />
; Ultraparticular<br />
: An interval/comma which is the ratio of two consecutive square-particulars.<br />
: These are of the form {{sfrac|S''k''|S(''k'' + 1)}}.<br />
<br />
; Semiparticular<br />
: A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:<br />
: These are of the form {{sfrac|S''k''|S(''k'' + 2)}}.<br />
<br />
; S-expression<br />
: An expression using the S''k'' shorthand notation corresponding strictly to multiplying and dividing only (arbitrary) square-particulars. S-expressions include singular square superparticulars and expressions for other superparticulars in terms of square superparticulars.<br />
<br />
; S-factorization<br />
: An expression that takes a list of consecutive integer harmonics including the ''k''th harmonic and raises them to integer powers, similar to a [[smonzo]] but uniquely suited to analysing S-expressions.<br />
: For example: {{sfrac|S''k'' {{=}} [''k'' − 1, ''k'', ''k'' + 1]<sup>[−1, 2, −1]</sup>}} because {{nowrap|S''k'' {{=}} (''k'' − 1)<sup>−1</sup>''k''<sup>2</sup>(''k'' + 1)<sup>−1</sup>}}.<br />
<br />
; S-comma<br />
: Any comma within one of the infinite families of commas discussed here, excluding 1/n-square-particulars for n>5 (so square-particulars, triangle-particulars, 1/3-square-particulars, 1/4-square-particulars and 1/5-square-particulars are included, but not anything beyond; this bound is used for exclusion (rather than n>3) to allow the utility of 1/5-square-particulars in avoiding twin primes by equating superparticular intervals on either side of the twin primes).<br />
<br />
; Indirect S-comma<br />
: Any comma that is the product or ratio of two S-commas. These appear frequently as S-expressions for commas that are more challenging/nontrivial to represent from the perspective of S-expressions; for example, the [[schisma]] admits at least three such representations.<br />
<br />
== See further ==<br />
* [[S-expression/Advanced results|Advanced results]] – for the harder-to-reach algebra<br />
<br />
== Notes ==<br />
<references group="note" /><br />
<br />
[[Category:Elementary math]]<br />
[[Category:Pages with proofs]]<br />
[[Category:Ratio]]<br />
[[Category:Terms]]</div>ArrowHead294https://en.xen.wiki/index.php?title=S-expression&diff=224388S-expression2026-02-20T11:36:46Z<p>ArrowHead294: /* Quick rules of S-expressions */</p>
<hr />
<div>An '''S-expression''' is any product, or ratio of products, of the '''square superparticulars''' S''k'', which are defined as the fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. Commas defined by S-expressions turn out to represent intuitive and wide-reaching families of tempered equivalences, and therefore present a very useful framework to learn for a good understanding of the [[commas]] that appear frequently in xen.<br />
<br />
== Quick rules of S-expressions ==<br />
As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply when they are [[tempering out|tempered out]]. The linked sections provide deeper information into each comma family.<br />
<br />
* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. <br>Tempering out S''k'' equates {{sfrac|''k'' + 1|''k''}} with {{sfrac|''k''|''k'' − 1}} and splits {{sfrac|''k'' + 1|''k'' − 1}} in two.<br />
* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: {{nowrap|'''S''k'' * S(''k'' + 1)'''}}, superparticular fractions of the form {{sfrac|''k''(''k'' + 1)/2|(''k'' − 1)(''k'' + 2)/2}}. <br>Tempering out {{nowrap|S''k'' * S(''k'' + 1)}} equates {{sfrac|''k'' + 2|''k'' + 1}} with {{sfrac|''k''|''k'' − 1}}, and {{sfrac|''k'' + 2|''k''}} with {{sfrac|''k'' + 1|''k'' − 1}}.<br />
* [[#Sk2 * S(k + 1) and S(k − 1) * Sk2 (lopsided commas)|Lopsided commas]]: {{nowrap|'''(S''k'')<sup>2</sup> * S(''k'' + 1)'''}} and {{nowrap|'''(S''k'')<sup>2</sup> * S(''k'' − 1)'''}}. <br>Tempering out the former equates {{sfrac|''k'' + 2|''k''}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>2</sup> and {{sfrac|''k'' + 2|''k'' − 1}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>3</sup>, and tempering out the latter equates {{sfrac|''k''|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>2</sup> and {{sfrac|''k'' + 1|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.<br />
* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out splits {{sfrac|''k'' + 2|''k'' − 1}} into {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.<br />
* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out splits {{sfrac|''k'' + 3|''k'' − 1}} into {{pars|{{sfrac|''k'' + 2|''k''}}}}<sup>2</sup>.<br />
<br />
== S''k'' (square-particulars) ==<br />
A '''square superparticular''', or ''square-particular'' for short, is a [[superparticular]] [[interval]] whose numerator is a square number, which is to say, a superparticular of the form<br />
<br />
<math>\displaystyle \frac {k^2}{k^2 - 1} = \frac {k/(k - 1)}{(k + 1)/k} </math><br />
<br />
which is square-(super)particular ''k'' for a given integer {{nowrap|''k'' &gt; 1}}. A suggested shorthand for this interval is '''S''k''''' for the ''k''-th square superparticular, where the ''S'' stands for "(Shorthand for) Second-order/Square Superparticular". This will be used later in this article as the notation will prove powerful in understanding the commas and implied tempered structures of [[regular temperament]]s. Note that this means {{nowrap|S2 {{=}} [[4/3]]}} is the first musically meaningful square-particular, as {{nowrap|S1 {{=}} 1/0}}.<br />
<br />
Also note that we use the notation S''k''<sup>''p''</sup> to mean (S''k'')<sup>''p''</sup> rather than S(''k''<sup>''p''</sup>) for convenience in the practical analysis of regular temperaments using [[S-expression]]s.<br />
<br />
=== Significance/motivation ===<br />
Square-particulars are important structurally because they are the intervals between consecutive [[superparticular]] [[interval]]s while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the ({{nowrap|''k'' + 1}})th harmonic, as well as the potential representational sacrifices that must be made from that point onward. In other words, understanding the mappings of S''k'' in a given temperament is equivalent to understanding the spacing of consecutive superparticular intervals, and thereby to understanding the way it represents (or tries to represent) the harmonic series.<br />
<br />
=== Table of square-particulars ===<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 31-limit square-particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2<br />
| ([[2/1]])/([[3/2]])<br />
| [[4/3]]<br />
| 3<br />
|-<br />
| S3<br />
| ([[3/2]])/([[4/3]])<br />
| [[9/8]]<br />
| 3<br />
|-<br />
| S4<br />
| ([[4/3]])/([[5/4]])<br />
| [[16/15]]<br />
| 5<br />
|-<br />
| S5<br />
| ([[5/4]])/([[6/5]])<br />
| [[25/24]]<br />
| 5<br />
|-<br />
| S6 = S8*S9<br />
| ([[6/5]])/([[7/6]])<br />
| [[36/35]]<br />
| 7<br />
|-<br />
| S7<br />
| ([[7/6]])/([[8/7]])<br />
| [[49/48]]<br />
| 7<br />
|-<br />
| S8<br />
| ([[8/7]])/([[9/8]])<br />
| [[64/63]]<br />
| 7<br />
|-<br />
| S9 = S6/S8<br />
| ([[9/8]])/([[10/9]])<br />
| [[81/80]]<br />
| 5<br />
|-<br />
| S10<br />
| ([[10/9]])/([[11/10]])<br />
| [[100/99]]<br />
| 11<br />
|-<br />
| S11<br />
| ([[11/10]])/([[12/11]])<br />
| [[121/120]]<br />
| 11<br />
|-<br />
| S12<br />
| ([[12/11]])/([[13/12]])<br />
| [[144/143]]<br />
| 13<br />
|-<br />
| S13<br />
| ([[13/12]])/([[14/13]])<br />
| [[169/168]]<br />
| 13<br />
|-<br />
| S14<br />
| ([[14/13]])/([[15/14]])<br />
| [[196/195]]<br />
| 13<br />
|-<br />
| S15<br />
| ([[15/14]])/([[16/15]])<br />
| [[225/224]]<br />
| 7<br />
|-<br />
| S16<br />
| ([[16/15]])/([[17/16]])<br />
| [[256/255]]<br />
| 17<br />
|-<br />
| S17<br />
| ([[17/16]])/([[18/17]])<br />
| [[289/288]]<br />
| 17<br />
|-<br />
| S18<br />
| ([[18/17]])/([[19/18]])<br />
| [[324/323]]<br />
| 19<br />
|-<br />
| S19<br />
| ([[19/18]])/([[20/19]])<br />
| [[361/360]]<br />
| 19<br />
|-<br />
| S20<br />
| ([[20/19]])/([[21/20]])<br />
| [[400/399]]<br />
| 19<br />
|-<br />
| S21<br />
| ([[21/20]])/([[22/21]])<br />
| [[441/440]]<br />
| 11<br />
|-<br />
| S22<br />
| ([[22/21]])/([[23/22]])<br />
| [[484/483]]<br />
| 23<br />
|-<br />
| S23<br />
| ([[23/22]])/([[24/23]])<br />
| [[529/528]]<br />
| 23<br />
|-<br />
| S24<br />
| ([[24/23]])/([[25/24]])<br />
| [[576/575]]<br />
| 23<br />
|-<br />
| S25<br />
| ([[25/24]])/([[26/25]])<br />
| [[625/624]]<br />
| 13<br />
|-<br />
| S26 = S13/S15<br />
| ([[26/25]])/([[27/26]])<br />
| [[676/675]]<br />
| 13<br />
|-<br />
| S27<br />
| ([[27/26]])/([[28/27]])<br />
| [[729/728]]<br />
| 13<br />
|-<br />
| S28<br />
| ([[28/27]])/([[29/28]])<br />
| [[784/783]]<br />
| 29<br />
|-<br />
| S29<br />
| ([[29/28]])/([[30/29]])<br />
| [[841/840]]<br />
| 29<br />
|-<br />
| S30<br />
| ([[30/29]])/([[31/30]])<br />
| [[900/899]]<br />
| 31<br />
|-<br />
| S31<br />
| ([[31/30]])/([[32/31]])<br />
| [[961/960]]<br />
| 31<br />
|-<br />
| S32<br />
| ([[32/31]])/([[33/32]])<br />
| [[1024/1023]]<br />
| 31<br />
|-<br />
| S33<br />
| ([[33/32]])/([[34/33]])<br />
| [[1089/1088]]<br />
| 17<br />
|-<br />
| S34<br />
| ([[34/33]])/([[35/34]])<br />
| [[1156/1155]]<br />
| 17<br />
|-<br />
| S35 = S49*S50<br />
| ([[35/34]])/([[36/35]])<br />
| [[1225/1224]]<br />
| 17<br />
|-<br />
| S39<br />
| ([[39/38]])/([[40/39]])<br />
| [[1521/1520]]<br />
| 19<br />
|-<br />
| S45<br />
| ([[45/44]])/([[46/45]])<br />
| [[2025/2024]]<br />
| 23<br />
|-<br />
| S49<br />
| ([[49/48]])/([[50/49]])<br />
| [[2401/2400]]<br />
| 7<br />
|-<br />
| S50<br />
| ([[50/49]])/([[51/50]])<br />
| [[2500/2499]]<br />
| 17<br />
|-<br />
| S51<br />
| ([[51/50]])/([[52/51]])<br />
| [[2601/2600]]<br />
| 17<br />
|-<br />
| S55 = S22/S24<br />
| ([[55/54]])/([[56/55]])<br />
| [[3025/3024]]<br />
| 11<br />
|-<br />
| S56<br />
| ([[56/55]])/([[57/56]])<br />
| [[3136/3135]]<br />
| 19<br />
|-<br />
| S57<br />
| ([[57/56]])/([[58/57]])<br />
| [[3249/3248]]<br />
| 29<br />
|-<br />
| S63<br />
| ([[63/62]])/([[64/63]])<br />
| [[3969/3968]]<br />
| 31<br />
|-<br />
| S64<br />
| ([[64/63]])/([[65/64]])<br />
| [[4096/4095]]<br />
| 13<br />
|-<br />
| S65<br />
| ([[65/64]])/([[66/65]])<br />
| [[4225/4224]]<br />
| 13<br />
|-<br />
| S69<br />
| ([[69/68]])/([[70/69]])<br />
| [[4761/4760]]<br />
| 23<br />
|-<br />
| S76<br />
| ([[76/75]])/([[77/76]])<br />
| [[5776/5775]]<br />
| 19<br />
|-<br />
| S77<br />
| ([[77/76]])/([[78/77]])<br />
| [[5929/5928]]<br />
| 19<br />
|-<br />
| S91<br />
| ([[91/90]])/([[92/91]])<br />
| [[8281/8280]]<br />
| 23<br />
|-<br />
| S92<br />
| ([[92/91]])/([[93/92]])<br />
| [[8464/8463]]<br />
| 31<br />
|-<br />
| S99 = S33/S35<br />
| ([[99/98]])/([[100/99]])<br />
| [[9801/9800]]<br />
| 11<br />
|-<br />
| S115<br />
| ([[115/114]])/([[116/115]])<br />
| [[13225/13224]]<br />
| 29<br />
|-<br />
| S116<br />
| ([[116/115]])/([[117/116]])<br />
| [[13456/13455]]<br />
| 29<br />
|-<br />
| S120<br />
| ([[120/119]])/([[121/120]])<br />
| [[14400/14399]]<br />
| 17<br />
|-<br />
| S125<br />
| ([[125/124]])/([[126/125]])<br />
| [[15625/15624]]<br />
| 31<br />
|-<br />
| S144<br />
| ([[144/143]])/([[145/144]])<br />
| [[20736/20735]]<br />
| 29<br />
|-<br />
| S153<br />
| ([[153/152]])/([[154/153]])<br />
| [[23409/23408]]<br />
| 19<br />
|-<br />
| S154<br />
| ([[154/153]])/([[155/154]])<br />
| [[23716/23715]]<br />
| 31<br />
|-<br />
| S155<br />
| ([[155/154]])/([[156/155]])<br />
| [[24025/24024]]<br />
| 31<br />
|-<br />
| S161 = S46/S48<br />
| ([[161/160]])/([[162/161]])<br />
| [[25921/25920]]<br />
| 23<br />
|-<br />
| S169<br />
| ([[169/168]])/([[170/169]])<br />
| [[28561/28560]]<br />
| 17<br />
|-<br />
| S170<br />
| ([[170/169]])/([[171/170]])<br />
| [[28900/28899]]<br />
| 19<br />
|-<br />
| S175<br />
| ([[175/174]])/([[176/175]])<br />
| [[30625/30624]]<br />
| 29<br />
|-<br />
| S208<br />
| ([[208/207]])/([[209/208]])<br />
| [[43264/43263]]<br />
| 23<br />
|-<br />
| S209<br />
| ([[209/208]])/([[210/209]])<br />
| [[43681/43680]]<br />
| 19<br />
|-<br />
| S231<br />
| ([[231/230]])/([[232/231]])<br />
| [[53361/53360]]<br />
| 29<br />
|-<br />
| S289<br />
| ([[289/288]])/([[290/289]])<br />
| [[83521/83520]]<br />
| 29<br />
|-<br />
| S323<br />
| ([[323/322]])/([[324/323]])<br />
| [[104329/104328]]<br />
| 23<br />
|-<br />
| S324<br />
| ([[324/323]])/([[325/324]])<br />
| [[104976/104975]]<br />
| 19<br />
|-<br />
| S341<br />
| ([[341/340]])/([[342/341]])<br />
| [[116281/116280]]<br />
| 31<br />
|-<br />
| S342<br />
| ([[342/341]])/([[343/342]])<br />
| [[116964/116963]]<br />
| 31<br />
|-<br />
| S351 = S78/S80<br />
| ([[351/350]])/([[352/351]])<br />
| [[123201/123200]]<br />
| 13<br />
|-<br />
| S391<br />
| ([[391/390]])/([[392/391]])<br />
| [[152881/152880]]<br />
| 23<br />
|-<br />
| S441<br />
| ([[441/440]])/([[442/441]])<br />
| [[194481/194480]]<br />
| 17<br />
|-<br />
| S494<br />
| ([[494/493]])/([[495/494]])<br />
| [[244036/244035]]<br />
| 29<br />
|-<br />
| S495<br />
| ([[495/494]])/([[496/495]])<br />
| [[245025/245024]]<br />
| 31<br />
|-<br />
| S528<br />
| ([[528/527]])/([[529/528]])<br />
| [[278784/278783]]<br />
| 31<br />
|-<br />
| S551<br />
| ([[551/550]])/([[552/551]])<br />
| [[303601/303600]]<br />
| 29<br />
|-<br />
| S714<br />
| ([[714/713]])/([[715/714]])<br />
| [[509796/509795]]<br />
| 31<br />
|-<br />
| S783<br />
| ([[783/782]])/([[784/783]])<br />
| [[613089/613088]]<br />
| 29<br />
|-<br />
| S1275<br />
| ([[1275/1274]])/([[1276/1275]])<br />
| [[1625625/1625624]]<br />
| 29<br />
|-<br />
| S1519<br />
| ([[1519/1518]])/([[1520/1519]])<br />
| [[2307361/2307360]]<br />
| 31<br />
|-<br />
| S1520<br />
| ([[1520/1519]])/([[1521/1520]])<br />
| [[2310400/2310399]]<br />
| 31<br />
|-<br />
| S2001<br />
| ([[2001/2000]])/([[2002/2001]])<br />
| [[4004001/4004000]]<br />
| 29<br />
|-<br />
| S2024<br />
| ([[2024/2023]])/([[2025/2024]])<br />
| [[4096576/4096575]]<br />
| 23<br />
|-<br />
| S2431<br />
| ([[2431/2430]])/([[2432/2431]])<br />
| [[5909761/5909760]]<br />
| 19<br />
|-<br />
| S3249<br />
| ([[3249/3248]])/([[3250/3249]])<br />
| <font style="font-size:0.88em">[[10556001/10556000]]</font><br />
| 29<br />
|-<br />
| S9801<br />
| ([[9801/9800]])/([[9802/9801]])<br />
| <font style="font-size:0.88em">[[96059601/96059600]]</font><br />
| 29<br />
|-<br />
| S13311<br />
| <font style="font-size:0.86em">([[13311/13310]])/([[13312/13311]])</font><br />
| <font style="font-size:0.79em">[[177182721/177182720]]</font><br />
| 29<br />
|-<br />
| S13455<br />
| <font style="font-size:0.86em">([[13455/13454]])/([[13456/13455]])</font><br />
| <font style="font-size:0.79em">[[181037025/181037024]]</font><br />
| 31<br />
|}<br />
<br />
=== Alternatives to tempering square-particulars ===<br />
It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas {{nowrap|U''k'' {{=}} {{sfrac|S''k''|S(''k'' + 1)}}}}, we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]<ref group="note">In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as U''k'' = S(''k'' - 1)/S''k'' and U''k'' = S(''k'' + 1)/S(''k'' + 2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.</ref>. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because {{sfrac|''k'' + 2|''k'' + 1}} and {{sfrac|''k''|''k'' − 1}} are equidistant from {{sfrac|''k'' + 1|''k''}} (because of tempering {{sfrac|S''k''|S(''k'' + 1)}}), this means that another expression for {{sfrac|S''k''|S(''k'' + 1)}} is the following:<br />
<br />
<math>\displaystyle {\rm S}k / {\rm S} (k + 1) = \frac{(k + 2) / (k - 1)}{((k + 1)/k)^3} </math><br />
<br />
This means you can read the ''k'' and {{nowrap|''k'' + 1}} from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in [[#Sk/S(k + 1) (ultraparticulars)|the table of ultraparticulars]]).<br />
<br />
Furthermore, defining another sequence of commas with [[semiparticular|formula {{sfrac|S''k''|S(''k'' + 2)}} leads to semiparticulars]] which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering consecutive ultraparticulars.<br />
<br />
== {{nowrap|S''k''*S(''k'' + 1)}} (triangle-particulars) ==<br />
=== Significance ===<br />
1. Every triangle-particular is superparticular, so these are efficient commas. (See also the [[#Short proof of the superparticularity of triangle-particulars]].)<br />
<br />
2. Often each individual triangle-particular, taken as a comma, implies other useful equivalences not necessarily corresponding to the general form, speaking of which …<br />
<br />
3. Every triangle-particular is the difference between two nearly-adjacent superparticular intervals {{nowrap|''k'' + 2|''k'' + 1}} and {{nowrap|''k''|''k'' − 1}}.<br />
<br />
4. Tempering any two consecutive square-particulars S''k'' and {{nowrap|S(''k'' + 1)}} implies tempering a triangle-particular, so these are common commas. (See also: [[lopsided comma]]s.)<br />
<br />
5. If we temper {{nowrap|S''k'' * S(''k'' + 1)}} but not S''k'' or {{nowrap|S(''k'' + 1)}}, then one or more intervals of {{sfrac|''k''|''k'' − 1}}, {{sfrac|''k'' + 1|''k''}}, and {{sfrac|''k'' + 2|''k'' + 1}} ''must'' be mapped inconsistently, because:<br />
: If {{nowrap|''k'' + 1|''k''}} is mapped above {{nowrap|{{sfrac|''k'' + 2|''k'' + 1}} ~ {{sfrac|''k''|''k'' − 1}}}} we have {{nowrap|{{sfrac|''k'' + 1|''k''}} &gt; {{sfrac|''k''|''k'' − 1}}}} and if it is mapped below we have {{nowrap|{{sfrac|''k'' + 1|''k''}} &lt; {{sfrac|''k'' + 2|''k'' + 1}}}}.<br />
: (Generalisations of this and their implications for consistency are discussed in [[#Sk*S(k + 1)*…*S(k + n − 1) (1/n-square-particulars)|the section covering 1/''n''-square-particulars]].)<br />
<br />
=== Meaning ===<br />
Notice that if we equate {{sfrac|''k'' + 2|''k'' + 1}} with {{sfrac|''k''|''k'' − 1}} (by [[tempering out]] their difference), then multiply both sides by {{sfrac|''k'' + 1|''k''}}, we have:<br />
<br />
<math>\displaystyle \left(\frac{k + 2}{k + 1}\right)\left(\frac{k + 1}{k}\right) = \left(\frac{k + 1}{k}\right)\left(\frac{k}{k - 1}\right) </math><br />
<br />
which simplifies to:<br />
<br />
<math>\displaystyle \frac{k + 2}{k} = \frac{k + 1}{k - 1} </math><br />
<br />
This means that if we temper: <math>{\rm S}k \cdot {\rm S}(k+1) = \frac{k/(k-1)}{(k+1)/k} \cdot \frac{(k+1)/k}{(k+2)/(k+1)} = \frac{k/(k-1)}{(k+2)/(k+1)}</math><br />
<br />
… then this equivalence is achieved. Note that there is little to no reason to not also temper S''k'' and {{nowrap|S(''k'' + 1)}} individually unless other considerations seem to force your hand.<br />
<br />
=== Short proof of the superparticularity of triangle-particulars ===<br />
<math>\displaystyle S(k)*S(k + 1) = \frac{\frac{k}{k - 1}}{\frac{k + 2}{k + 1}} = \frac{k(k + 1)}{(k - 1)(k + 2)} = \frac{k^2 + k}{k^2 + k - 2} </math><br />
<br />
Then notice that {{nowrap|''k''<sup>2</sup> + ''k''}} is always a multiple of 2, therefore the above always simplifies to a superparticular. Half of this superparticular is halfway between the corresponding square-particulars, and because of its composition it could therefore be reasoned that it'd likely be half as accurate as tempering either of the square-particulars individually, so these are "1/2-square-particulars" in a sense, and half of a square is a triangle, which is not a coincidence here because the numerators of all of these superparticular intervals/commas are [[triangular number]]s! (Hence the alternative name "[[triangle-particular]]".)<br />
<br />
=== Table of triangle-particulars ===<br />
For completeness, all the intervals of this form are included, because of their structural importance for JI, and for the possibility of (in)consistency of mappings when tempered for the above reason.<br />
<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 31-limit triangle-particulars<ref group="note">After 75, 76, 77, 78, streaks of four consecutive harmonics in the 23-limit become very sparse. The last few streaks are deeply related to the consistency and structure of [[311edo]], as [[311edo]] can be described as the unique 23-limit temperament that tempers all triangle-particulars from [[595/594]] up to [[21736/21735]]. It also tempers all the square-particulars composing those triangle-particulars with the exception of S169 and S170. It also maps the corresponding intervals of the 77-odd-limit consistently. 170/169 is the only place where the logic seems to "break" as it is mapped to 2 steps instead of 3 meaning the mapping of that superparticular is inconsistent.</ref><br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3<br />
| ([[3/1]])/([[2/1]])<br />
| [[3/2]]<br />
| 3<br />
|-<br />
| S3*S4<br />
| ([[3/2]])/([[5/4]])<br />
| [[6/5]]<br />
| 5<br />
|-<br />
| S4*S5<br />
| ([[4/3]])/([[6/5]])<br />
| [[10/9]]<br />
| 5<br />
|-<br />
| S5*S6<br />
| ([[5/4]])/([[7/6]])<br />
| [[15/14]]<br />
| 7<br />
|-<br />
| S6*S7<br />
| ([[6/5]])/([[8/7]])<br />
| [[21/20]]<br />
| 7<br />
|-<br />
| S7*S8 = S4/S6<br />
| ([[7/6]])([[9/8]])<br />
| [[28/27]]<br />
| 7<br />
|-<br />
| S8*S9 = S6<br />
| ([[8/7]])/([[10/9]])<br />
| [[36/35]]<br />
| 7<br />
|-<br />
| S9*S10<br />
| ([[9/8]])/([[11/10]])<br />
| [[45/44]]<br />
| 11<br />
|-<br />
| S10*S11<br />
| ([[10/9]])/([[12/11]])<br />
| [[55/54]]<br />
| 11<br />
|-<br />
| S11*S12<br />
| ([[11/10]])/([[13/12]])<br />
| [[66/65]]<br />
| 13<br />
|-<br />
| S12*S13<br />
| ([[12/11]])/([[14/13]])<br />
| [[78/77]]<br />
| 13<br />
|-<br />
| S13*S14<br />
| ([[13/12]])/([[15/14]])<br />
| [[91/90]]<br />
| 13<br />
|-<br />
| S14*S15<br />
| ([[14/13]])/([[16/15]])<br />
| [[105/104]]<br />
| 13<br />
|-<br />
| S15*S16<br />
| ([[15/14]])/([[17/16]])<br />
| [[120/119]]<br />
| 17<br />
|-<br />
| S16*S17<br />
| ([[16/15]])/([[18/17]])<br />
| [[136/135]]<br />
| 17<br />
|-<br />
| S17*S18<br />
| ([[17/16]])/([[19/18]])<br />
| [[153/152]]<br />
| 19<br />
|-<br />
| S18*S19<br />
| ([[18/17]])/([[20/19]])<br />
| [[171/170]]<br />
| 19<br />
|-<br />
| S19*S20<br />
| ([[19/18]])/([[21/20]])<br />
| [[190/189]]<br />
| 19<br />
|-<br />
| S20*S21<br />
| ([[20/19]])/([[22/21]])<br />
| [[210/209]]<br />
| 19<br />
|-<br />
| S21*S22<br />
| ([[21/20]])/([[23/22]])<br />
| [[231/230]]<br />
| 23<br />
|-<br />
| S22*S23<br />
| ([[22/21]])/([[24/23]])<br />
| [[253/252]]<br />
| 23<br />
|-<br />
| S23*S24<br />
| ([[23/22]])/([[25/24]])<br />
| [[276/275]]<br />
| 23<br />
|-<br />
| S24*S25<br />
| ([[24/23]])/([[26/25]])<br />
| [[300/299]]<br />
| 23<br />
|-<br />
| S25*S26 = S10/S12<br />
| ([[25/24]])/([[27/26]])<br />
| [[325/324]]<br />
| 13<br />
|-<br />
| S26*S27<br />
| ([[26/25]])/([[28/27]])<br />
| [[351/350]]<br />
| 13<br />
|-<br />
| S27*S28<br />
| ([[27/26]])/([[29/28]])<br />
| [[378/377]]<br />
| 29<br />
|-<br />
| S28*S29<br />
| ([[28/27]])/([[30/29]])<br />
| [[406/405]]<br />
| 29<br />
|-<br />
| S29*S30<br />
| ([[29/28]])/([[31/30]])<br />
| [[435/434]]<br />
| 31<br />
|-<br />
| S30*S31<br />
| ([[30/29]])/([[32/31]])<br />
| [[465/464]]<br />
| 31<br />
|-<br />
| S31*S32<br />
| ([[31/30]])/([[33/32]])<br />
| [[496/495]]<br />
| 31<br />
|-<br />
| S32*S33<br />
| ([[32/31]])/([[34/33]])<br />
| [[528/527]]<br />
| 31<br />
|-<br />
| S33*S34<br />
| ([[33/32]])/([[35/34]])<br />
| [[561/560]]<br />
| 17<br />
|-<br />
| S34*S35<br />
| ([[34/33]])/([[36/35]])<br />
| [[595/594]]<br />
| 17<br />
|-<br />
| S49*S50 = S35<br />
| ([[49/48]])/([[51/50]])<br />
| [[1225/1224]]<br />
| 17<br />
|-<br />
| S50*S51<br />
| ([[50/49]])/([[52/51]])<br />
| [[1275/1274]]<br />
| 17<br />
|-<br />
| S55*S56<br />
| ([[55/54]])/([[57/56]])<br />
| [[1540/1539]]<br />
| 19<br />
|-<br />
| S63*S64<br />
| ([[63/62]])/([[65/64]])<br />
| [[2016/2015]]<br />
| 31<br />
|-<br />
| S64*S65<br />
| ([[64/63]])/([[66/65]])<br />
| [[2080/2079]]<br />
| 13<br />
|-<br />
| S76*S77<br />
| ([[76/75]])/([[78/77]])<br />
| [[2926/2925]]<br />
| 19<br />
|-<br />
| S91*S92<br />
| ([[91/90]])/([[93/92]])<br />
| [[4186/4185]]<br />
| 31<br />
|-<br />
| S115*S116<br />
| ([[115/114]])/([[117/116]])<br />
| [[6670/6669]]<br />
| 29<br />
|-<br />
| S153*S154<br />
| ([[153/152]])/([[155/154]])<br />
| [[11781/11780]]<br />
| 31<br />
|-<br />
| S154*S155<br />
| ([[154/153]])/([[156/155]])<br />
| [[11935/11934]]<br />
| 31<br />
|-<br />
| S169*S170<br />
| ([[169/168]])/([[171/170]])<br />
| [[14365/14364]]<br />
| 19<br />
|-<br />
| S208*S209<br />
| ([[208/207]])/([[210/209]])<br />
| [[21736/21735]]<br />
| 19<br />
|-<br />
| S323*S324<br />
| ([[323/322]])/([[325/324]])<br />
| [[52326/52325]]<br />
| 23<br />
|-<br />
| S341*S342<br />
| ([[341/340]])/([[343/342]])<br />
| [[58311/58310]]<br />
| 31<br />
|-<br />
| S494*S495<br />
| ([[494/493]])/([[496/495]])<br />
| [[122265/122264]]<br />
| 31<br />
|-<br />
| S1519*S1520<br />
| ([[1519/1518]])/([[1521/1520]])<br />
| [[1154440/1154439]]<br />
| 31<br />
|}<br />
<br />
== {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} (1/''n''-square-particulars) ==<br />
=== Motivation ===<br />
1/''n''-square-particulars are a generalization of square- and 1/2-square-particulars to a comma/interval whose S-expression is can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}}.<br />
<br />
In other words, each and every S-expression of a comma as a 1/''n''-square-particular corresponds exactly to expressing it as the ratio between two [[superparticular]] intervals, with ''n'' distance between them, where, for example, 10/9 and 11/10 are considered as having 1 distance between them, corresponding to (1/1-)square-particulars (in this case [[100/99|S10]]).<br />
<br />
These commas are important in a few ways:<br />
1. As a generalization of important special cases {{nowrap|''n'' {{=}} 0|''n'' {{=}} 1}}, and {{nowrap|''n'' {{=}} 2}}, (which are almost all superparticular; the only case where they aren't is that {{nowrap|''n'' {{=}} 3}} (1/3-square-particulars) are throdd-particular one third of the time, so this suggests these are efficient commas. A cursory look will show that many 1/n-square-particulars for small n are superparticular, and many more are the next best things (odd-particular, throdd-particular, quodd-particular, etc.) so this confirms them being a family of efficient commas.<br />
<br />
2. Because of being the ratio of two superparticular intervals, in higher-complexity cases they often correspond to small commas between large commas which we don't want to temper, for example {{nowrap|{{sfrac|[[81/80]]|[[91/90]]}} {{=}} S81 * S82 * … * S90}} {{nowrap|{{=}} [[729/728]]}} {{nowrap|{{=}} S27}}. They also often simplify in cases like these; note that a suggested shorthand is S81..90 for {{nowrap|S81 * S82 * … * S90}} and thus more generally S''a''..''b'' for {{nowrap|S''a'' * S(''a'' + 1) * … * S''b''}}.<br />
<br />
3. They often correspond to "nontrivial" equivalences that need to be dug up which are not obvious from their expression as a ratio of two superparticular intervals, for example, [[385/384|S33*S34*S35]], suggesting they are a goldmine for valuable tempering opportunities. <br />
<br />
4. Their expressions naturally make them implied by tempering consecutive square-particulars, so if you notice them present and that the individual square-particulars aren't tempered, if you want to extend your temperament and/or reduce its rank (tempering it down) and/or hope to make your temperament more efficient, you can try tempering the untempered square-particulars that a tempered 1/''n''-square-particular is composed of (although this is not always possible). There is also good theoretical motivation for wanting to do this, as the next section will discuss.<br />
<br />
5. They're relevant to understanding how much damage is present in a temperament's harmonic series representation, because they show how many superparticular intervals are either not distinguished or worse mapped inconsistently, bringing us finally to …<br />
<br />
6. They're relevant to understanding limitations of consistency (or more precisely, monotonicity) of any given temperament, as the next section will discuss.<br />
<br />
=== Significance/implications for consistency ===<br />
1/n-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including {{nowrap|S(''k'' + ''n'')}}) and which can therefore be written as the ratio between the two superparticulars {{sfrac|''k''|''k'' − 1}} and {{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} have implications for the [[consistency]] of the ({{nowrap|''k'' + ''n''}})-[[odd-limit]] when tempered. Specifically:<br />
<br />
If a temperament tempers a 1/''n''-square-particular of the form {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, it must temper all of the ''n'' square-particulars that compose it, which is to say it must also temper all of S''k'', {{nowrap|S(''k'' + 1)}}, …, {{nowrap|S(''k'' + ''n'' − 1)}}. If it does not, it is ''necessarily'' inconsistent (more formally and weakly, not monotonic) in the ({{nowrap|''k'' + ''n''}})-odd-limit.<ref group="note">Note that this statement is a slight inaccuracy, because technically the tuning of the higher rank temperament corresponding to the lower rank temperament that tempers all of these commas is the unique ''and only'' (continuum of) tuning(s) for which this statement is false, but it's reasonable to simplify this technicality as this (continuum of) tuning(s) corresponds exactly and uniquely to tempering all the square-particulars we said were not tempered.</ref> A proof is as follows:<br />
<br />
Consider the following sequence of superparticular intervals, all of which in the ({{nowrap|''k'' + ''n''}})-odd-limit:<br />
<br />
<math>\displaystyle\frac{k + n}{k + n - 1}, \frac{k + n - 1}{k + n - 2}, …, \frac{k + 1}{k}, \frac{k}{k - 1}</math><br />
<br />
Because of tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}}, we require that {{nowrap|{{sfrac|''k'' + ''n''|''k'' + ''n'' − 1}} {{=}} {{sfrac|''k''|''k'' − 1}}}} consistently. Therefore, if any superparticular {{sfrac|''x''|''x'' − 1}} imbetween (meaning {{nowrap|''k'' + ''n'' &gt; ''x'' &gt; ''k''}}) is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true:<br />
<br />
<math>\displaystyle\begin{align}<br />
\operatorname{mapping}\left(\frac{k + n}{k + n - 1}\right) &> \operatorname{mapping}\left(\frac{x}{x - 1}\right) \\<br />
\operatorname{mapping}\left(\frac{k}{k - 1}\right) &< \operatorname{mapping}\left(\frac{x}{x - 1}\right)<br />
\end{align}</math><br />
<br />
Therefore any superparticular interval {{sfrac|''x''|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap|S''k''*S(''k'' + 1)*…*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap|''k'' − 1}})-odd-limit.<br />
<br />
<br />
=== {{nowrap|S(''k'' − 1)*S''k''*S(''k'' + 1)}} (1/3-square-particulars) ===<br />
This section concerns commas of the form {{nowrap|S(''k'' − 1) * S''k'' * S(''k'' + 1) {{=}} {{sfrac|&nbsp;{{sfrac|''k'' − 1|''k'' − 2}}&nbsp;|&nbsp;{{sfrac|''k'' + 2|''k'' + 1}}&nbsp;}}}} which therefore do not (directly) involve the ''k''th harmonic. These are a special case of 1/''n''-square-particulars.<br />
<br />
==== Significance ====<br />
1. Two-thirds of all {{frac|1|3}}-square-particulars are superparticular and the other third are [[#Glossary|throdd-particular]], so these are efficient commas. (See also the [[#Proof of simplification of 1/3-square-particulars]].)<br />
<br />
2. They are often implied in a variety of ways by combinations of other commas discussed on this page.<br />
<br />
3. Their omission of direct relation to the ''k''th harmonic make them theoretically interesting and potentially useful. (The other type of comma on this page that does this is [[semiparticular]]s.)<br />
<br />
4. Square-particulars, {{frac|1|2}}-square-particulars (a.k.a. [[triangle-particular]]s), and {{frac|1|3}}-square-particulars are part of a more general sequence with interesting properties: [[1/n-square-particular|1/''n''-square-particular]]s.<br />
<br />
==== Proof of simplification of 1/3-square-particulars ====<br />
We can check the general algebraic expression of any 1/3-square-particular for any potential simplifications:<br />
<br />
<math>\displaystyle\begin{align}<br />
S(k-1) * S(k) * S(k+1) &= \left(\frac{\frac{k-1}{k-2}}{\frac{k}{k-1}}\right)\left(\frac{\frac{k}{k-1}}{\frac{k+1}{k}}\right)\left(\frac{\frac{k+1}{k}}{\frac{k+2}{k+1}}\right) \\<br />
&= \frac{\frac{k-1}{k-2}}{\frac{k+2}{k+1}} \\<br />
&= \frac{(k-1)(k+1)}{(k-2)(k+2)} \\<br />
&= \frac{k^2 - 1}{k^2 - 4}<br />
\end{align}</math><br />
<br />
If {{nowrap|''k'' {{=}} 3''n'' + 1}} then:<br />
<br />
<math>\displaystyle S(k-1) * Sk * S(k+1) = \frac{9n^2 + 6n}{9n^2 + 6n - 3} = \frac{3n^2 + 2n}{3n^2 + 2n - 1}</math><br />
<br />
if {{nowrap|''k'' {{=}} 3''n'' + 2}} then:<br />
<br />
<math>\displaystyle S(k-1) * Sk * S(k+1) = \frac{9n^2 + 12n + 3}{9n^2 + 12n} = \frac{3n^2 + 4n + 1}{3n^2 + 4n}</math><br />
<br />
if {{nowrap|''k'' {{=}} 3''n''}} then:<br />
<br />
<math>\displaystyle S(k-1) * Sk * S(k+1) = \frac{9n^2 - 1}{9n^2 - 4}</math><br />
<br />
In other words, what this shows is all {{frac|1|3}}-square-particulars of the form S(''k'' − 1) * S''k'' * S(''k'' + 1) are superparticular iff ''k'' is throdd (not a multiple of 3), and all {{frac|1|3}}-square-particulars of the form {{nowrap|S(3''k'' − 1) * S(3''k'') * S(3''k'' + 1)}} are throdd-particular with the numerator and denominator always being one less than a multiple of 3 (which is to say, commas of this form are throdd-particular iff ''k'' is threven and superparticular iff ''k'' is throdd).<br />
<br />
=== Tables of 1/n-square-particulars ===<br />
<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 41-limit {{frac|1|3}}-square-particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3*S4<br />
| ([[2/1]])/([[5/4]])<br />
| [[8/5]]<br />
| 5<br />
|-<br />
| S3*S4*S5<br />
| ([[3/2]])/([[6/5]])<br />
| [[5/4]]<br />
| 5<br />
|-<br />
| S4*S5*S6<br />
| ([[4/3]])/([[7/6]])<br />
| [[8/7]]<br />
| 7<br />
|-<br />
| S5*S6*S7<br />
| ([[5/4]])/([[8/7]])<br />
| [[35/32]]<br />
| 7<br />
|-<br />
| S6*S7*S8<br />
| ([[6/5]])/([[9/8]])<br />
| [[16/15]]<br />
| 5<br />
|-<br />
| S7*S8*S9<br />
| ([[7/6]])/([[10/9]])<br />
| [[21/20]]<br />
| 7<br />
|-<br />
| S8*S9*S10<br />
| ([[8/7]])/([[11/10]])<br />
| [[80/77]]<br />
| 11<br />
|-<br />
| S9*S10*S11<br />
| ([[9/8]])/([[12/11]])<br />
| [[33/32]]<br />
| 11<br />
|-<br />
| S10*S11*S12<br />
| ([[10/9]])/([[13/12]])<br />
| [[40/39]]<br />
| 13<br />
|-<br />
| S11*S12*S13<br />
| ([[11/10]])/([[14/13]])<br />
| [[143/140]]<br />
| 13<br />
|-<br />
| S12*S13*S14<br />
| ([[12/11]])/([[15/14]])<br />
| [[56/55]]<br />
| 11<br />
|-<br />
| S13*S14*S15<br />
| ([[13/12]])/([[16/15]])<br />
| [[65/64]]<br />
| 13<br />
|-<br />
| S14*S15*S16<br />
| ([[14/13]])/([[17/16]])<br />
| [[224/221]]<br />
| 17<br />
|-<br />
| S15*S16*S17<br />
| ([[15/14]])/([[18/17]])<br />
| [[85/84]]<br />
| 17<br />
|-<br />
| S16*S17*S18<br />
| ([[16/15]])/([[19/18]])<br />
| [[96/95]]<br />
| 19<br />
|-<br />
| S17*S18*S19<br />
| ([[17/16]])/([[20/19]])<br />
| [[323/320]]<br />
| 19<br />
|-<br />
| S18*S19*S20<br />
| ([[18/17]])/([[21/20]])<br />
| [[120/119]]<br />
| 17<br />
|-<br />
| S19*S20*S21<br />
| ([[19/18]])/([[22/21]])<br />
| [[133/132]]<br />
| 19<br />
|-<br />
| S20*S21*S22<br />
| ([[20/19]])/([[23/22]])<br />
| [[440/437]]<br />
| 23<br />
|-<br />
| S21*S22*S23<br />
| ([[21/20]])/([[24/23]])<br />
| [[161/160]]<br />
| 23<br />
|-<br />
| S22*S23*S24<br />
| ([[22/21]])/([[25/24]])<br />
| [[176/175]]<br />
| 11<br />
|-<br />
| S23*S24*S25<br />
| ([[23/22]])/([[26/25]])<br />
| [[575/572]]<br />
| 23<br />
|-<br />
| S24*S25*S26<br />
| ([[24/23]])/([[27/26]])<br />
| [[208/207]]<br />
| 23<br />
|-<br />
| S25*S26*S27<br />
| ([[25/24]])/([[28/27]])<br />
| [[225/224]]<br />
| 7<br />
|-<br />
| S26*S27*S28<br />
| ([[26/25]])/([[29/28]])<br />
| [[728/725]]<br />
| 29<br />
|-<br />
| S27*S28*S29<br />
| ([[27/26]])/([[30/29]])<br />
| [[261/260]]<br />
| 29<br />
|-<br />
| S28*S29*S30<br />
| ([[28/27]])/([[31/30]])<br />
| [[280/279]]<br />
| 31<br />
|-<br />
| S29*S30*S31<br />
| ([[29/28]])/([[32/31]])<br />
| [[899/896]]<br />
| 31<br />
|-<br />
| S30*S31*S32<br />
| ([[30/29]])/([[33/32]])<br />
| [[320/319]]<br />
| 29<br />
|-<br />
| S31*S32*S33<br />
| ([[31/30]])/([[34/33]])<br />
| [[341/340]]<br />
| 31<br />
|-<br />
| S32*S33*S34<br />
| ([[32/31]])/([[35/34]])<br />
| [[1088/1085]]<br />
| 31<br />
|-<br />
| S33*S34*S35<br />
| ([[33/32]])/([[36/35]])<br />
| [[385/384]]<br />
| 11<br />
|-<br />
| S34*S35*S36<br />
| ([[34/33]])/([[37/36]])<br />
| [[408/407]]<br />
| 37<br />
|-<br />
| S35*S36*S37<br />
| ([[35/34]])/([[38/37]])<br />
| [[1295/1292]]<br />
| 37<br />
|-<br />
| S36*S37*S38<br />
| ([[36/35]])/([[39/38]])<br />
| [[456/455]]<br />
| 19<br />
|-<br />
| S37*S38*S39<br />
| ([[37/36]])/([[40/39]])<br />
| [[481/480]]<br />
| 37<br />
|-<br />
| S38*S39*S40<br />
| ([[38/37]])/([[41/40]])<br />
| [[1520/1517]]<br />
| 41<br />
|-<br />
| S39*S40*S41<br />
| ([[39/38]])/([[42/41]])<br />
| [[533/532]]<br />
| 41<br />
|-<br />
| S42*S43*S44<br />
| ([[42/41]])/([[45/44]])<br />
| [[616/615]]<br />
| 41<br />
|-<br />
| S46*S47*S48<br />
| ([[46/45]])/([[49/48]])<br />
| [[736/735]]<br />
| 23<br />
|-<br />
| S49*S50*S51<br />
| ([[49/48]])/([[52/51]])<br />
| [[833/832]]<br />
| 17<br />
|-<br />
| S52*S53*S54<br />
| ([[52/51]])/([[55/54]])<br />
| [[936/935]]<br />
| 17<br />
|-<br />
| S55*S56*S57<br />
| ([[55/54]])/([[58/57]])<br />
| [[1045/1044]]<br />
| 29<br />
|-<br />
| S63*S64*S65<br />
| ([[63/62]])/([[66/65]])<br />
| [[1365/1364]]<br />
| 31<br />
|-<br />
| S66*S67*S68<br />
| ([[66/65]])/([[69/68]])<br />
| [[1496/1495]]<br />
| 23<br />
|-<br />
| S75*S76*S77<br />
| ([[75/74]])/([[78/77]])<br />
| [[1925/1924]]<br />
| 37<br />
|-<br />
| S78*S79*S80<br />
| ([[78/77]])/([[81/80]])<br />
| [[2080/2079]]<br />
| 13<br />
|-<br />
| S82*S83*S84<br />
| ([[82/81]])/([[85/84]])<br />
| [[2296/2295]]<br />
| 41<br />
|-<br />
| S85*S86*S87<br />
| ([[85/84]])/([[88/87]])<br />
| [[2465/2464]]<br />
| 29<br />
|-<br />
| S88*S89*S90<br />
| ([[88/87]])/([[91/90]])<br />
| [[2640/2639]]<br />
| 29<br />
|-<br />
| S93*S94*S95<br />
| ([[93/92]])/([[96/95]])<br />
| [[2945/2944]]<br />
| 31<br />
|-<br />
| S96*S97*S98<br />
| ([[96/95]])/([[99/98]])<br />
| [[3136/3135]]<br />
| 19<br />
|-<br />
| S112*S113*S114<br />
| ([[112/111]])/([[115/114]])<br />
| [[4256/4255]]<br />
| 37<br />
|-<br />
| S117*S118*S119<br />
| ([[117/116]])/([[120/119]])<br />
| [[4641/4640]]<br />
| 29<br />
|-<br />
| S121*S122*S123<br />
| ([[121/120]])/([[124/123]])<br />
| [[4961/4960]]<br />
| 41<br />
|-<br />
| S133*S134*S135<br />
| ([[133/132]])/([[136/135]])<br />
| [[5985/5984]]<br />
| 19<br />
|-<br />
| S145*S146*S147<br />
| ([[145/144]])/([[148/147]])<br />
| [[7105/7104]]<br />
| 37<br />
|-<br />
| S153*S154*S155<br />
| ([[153/152]])/([[156/155]])<br />
| [[7905/7904]]<br />
| 31<br />
|-<br />
| S162*S163*S164<br />
| ([[162/161]])/([[165/164]])<br />
| [[8856/8855]]<br />
| 41<br />
|-<br />
| S187*S188*S189<br />
| ([[187/186]])/([[190/189]])<br />
| [[11781/11780]]<br />
| 31<br />
|-<br />
| S205*S206*S207<br />
| ([[205/204]])/([[208/207]])<br />
| [[14145/14144]]<br />
| 41<br />
|-<br />
| S222*S223*S224<br />
| ([[222/221]])/([[225/224]])<br />
| [[16576/16575]]<br />
| 37<br />
|-<br />
| S243*S244*S245<br />
| ([[243/242]])/([[246/245]])<br />
| [[19845/19844]]<br />
| 41<br />
|-<br />
| S253*S254*S255<br />
| ([[253/252]])/([[256/255]])<br />
| [[21505/21504]]<br />
| 23<br />
|-<br />
| S273*S274*S275<br />
| ([[273/272]])/([[276/275]])<br />
| [[25025/25024]]<br />
| 23<br />
|-<br />
| S286*S287*S288<br />
| ([[286/285]])/([[289/288]])<br />
| [[27456/27455]]<br />
| 19<br />
|-<br />
| S287*S288*S289<br />
| ([[287/286]])/([[290/289]])<br />
| [[82943/82940]]<br />
| 41<br />
|-<br />
| S297*S298*S299<br />
| ([[297/296]])/([[300/299]])<br />
| [[29601/29600]]<br />
| 37<br />
|-<br />
| S320*S321*S322<br />
| ([[320/319]])/([[323/322]])<br />
| [[103040/103037]]<br />
| 29<br />
|-<br />
| S361*S362*S363<br />
| ([[361/360]])/([[364/363]])<br />
| [[43681/43680]]<br />
| 19<br />
|-<br />
| S375*S376*S377<br />
| ([[375/374]])/([[378/377]])<br />
| [[47125/47124]]<br />
| 29<br />
|}<br />
<br />
{| class="wikitable center-all\<br />
|+ style="font-size: 105%;" | 23-limit {{frac|1|4}}-square particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3*S4*S5<br />
| ([[2/1]])/([[6/5]])<br />
| [[5/3]]<br />
| 5<br />
|-<br />
| S3*S4*S5*S6<br />
| ([[3/2]])/([[7/6]])<br />
| [[9/7]]<br />
| 7<br />
|-<br />
| S4*S5*S6*S7<br />
| ([[4/3]])/([[8/7]])<br />
| [[7/6]]<br />
| 7<br />
|-<br />
| S5*S6*S7*S8<br />
| ([[5/4]])/([[9/8]])<br />
| [[10/9]]<br />
| 5<br />
|-<br />
| S6*S7*S8*S9<br />
| ([[6/5]])/([[10/9]])<br />
| [[27/25]]<br />
| 5<br />
|-<br />
| S7*S8*S9*S10<br />
| ([[7/6]])/([[11/10]])<br />
| [[35/33]]<br />
| 11<br />
|-<br />
| S8*S9*S10*S11<br />
| ([[8/7]])/([[12/11]])<br />
| [[22/21]]<br />
| 11<br />
|-<br />
| S9*S10*S11*S12<br />
| ([[9/8]])/([[13/12]])<br />
| [[27/26]]<br />
| 13<br />
|-<br />
| S10*S11*S12*S13<br />
| ([[10/9]])/([[14/13]])<br />
| [[65/63]]<br />
| 13<br />
|-<br />
| S11*S12*S13*S14<br />
| ([[11/10]])/([[15/14]])<br />
| [[77/75]]<br />
| 11<br />
|-<br />
| S12*S13*S14*S15<br />
| ([[12/11]])/([[16/15]])<br />
| [[45/44]]<br />
| 11<br />
|-<br />
| S13*S14*S15*S16<br />
| ([[13/12]])/([[17/16]])<br />
| [[52/51]]<br />
| 17<br />
|-<br />
| S14*S15*S16*S17<br />
| ([[14/13]])/([[18/17]])<br />
| [[119/117]]<br />
| 17<br />
|-<br />
| S15*S16*S17*S18<br />
| ([[15/14]])/([[19/18]])<br />
| [[135/133]]<br />
| 19<br />
|-<br />
| S16*S17*S18*S19<br />
| ([[16/15]])/([[20/19]])<br />
| [[76/75]]<br />
| 19<br />
|-<br />
| S17*S18*S19*S20<br />
| ([[17/16]])/([[21/20]])<br />
| [[85/84]]<br />
| 17<br />
|-<br />
| S18*S19*S20*S21<br />
| ([[18/17]])/([[22/21]])<br />
| [[189/187]]<br />
| 17<br />
|-<br />
| S19*S20*S21*S22<br />
| ([[19/18]])/([[23/22]])<br />
| [[209/207]]<br />
| 23<br />
|-<br />
| S20*S21*S22*S23<br />
| ([[20/19]])/([[24/23]])<br />
| [[115/114]]<br />
| 23<br />
|-<br />
| S21*S22*S23*S24<br />
| ([[21/20]])/([[25/24]])<br />
| [[126/125]]<br />
| 7<br />
|-<br />
| S22*S23*S24*S25<br />
| ([[22/21]])/([[26/25]])<br />
| [[275/273]]<br />
| 13<br />
|-<br />
| S23*S24*S25*S26<br />
| ([[23/22]])/([[27/26]])<br />
| [[299/297]]<br />
| 23<br />
|-<br />
| S24*S25*S26*S27<br />
| ([[24/23]])/([[28/27]])<br />
| [[162/161]]<br />
| 23<br />
|-<br />
| S35*S36*S37*S38<br />
| ([[35/34]])/([[39/38]])<br />
| [[665/663]]<br />
| 19<br />
|-<br />
| S36*S37*S38*S39<br />
| ([[36/35]])/([[40/39]])<br />
| [[351/350]]<br />
| 13<br />
|-<br />
| S45*S46*S47*S48<br />
| ([[45/44]])/([[49/48]])<br />
| [[540/539]]<br />
| 11<br />
|-<br />
| S46*S47*S48*S49<br />
| ([[46/45]])/([[50/49]])<br />
| [[1127/1125]]<br />
| 23<br />
|-<br />
| S51*S52*S53*S54<br />
| ([[51/50]])/([[55/54]])<br />
| [[1377/1375]]<br />
| 17<br />
|-<br />
| S52*S53*S54*S55<br />
| ([[52/51]])/([[56/55]])<br />
| [[715/714]]<br />
| 17<br />
|-<br />
| S65*S66*S67*S68<br />
| ([[65/64]])/([[69/68]])<br />
| [[1105/1104]]<br />
| 23<br />
|-<br />
| S66*S67*S68*S69<br />
| ([[66/65]])/([[70/69]])<br />
| [[2277/2275]]<br />
| 23<br />
|-<br />
| S77*S78*S79*S80<br />
| ([[77/76]])/([[81/80]])<br />
| [[1540/1539]]<br />
| 19<br />
|-<br />
| S81*S82*S83*S84<br />
| ([[81/80]])/([[85/84]])<br />
| [[1701/1700]]<br />
| 17<br />
|-<br />
| S92*S93*S94*S95<br />
| ([[92/91]])/([[96/95]])<br />
| [[2185/2184]]<br />
| 23<br />
|-<br />
| S96*S97*S98*S99<br />
| <span style="font-size:0.94em">([[96/95]])/([[100/99]])</span><br />
| [[2376/2375]]<br />
| 19<br />
|-<br />
| <span style="font-size: 0.79em;">S221*S222*S223*S224</span><br />
| <span style="font-size: 0.79em;">([[221/220]])/([[225/224]])</span><br />
| [[12376/12375]]<br />
| 17<br />
|}<br />
<br />
{| class="wikitable center-all<br />
|+ style="font-size: 105%;" | 23-limit {{frac|1|5}}-square particulars<br />
|-<br />
! S-expression<br />
! Interval relation<br />
! Ratio<br />
! Prime limit<br />
|-<br />
| S2*S3*S4*S5*S6<br />
| ([[2/1]])/([[7/6]])<br />
| [[12/7]]<br />
| 7<br />
|-<br />
| S3*S4*S5*S6*S7<br />
| ([[3/2]])/([[8/7]])<br />
| [[21/16]]<br />
| 7<br />
|-<br />
| S4*S5*S6*S7*S8<br />
| ([[4/3]])/([[9/8]])<br />
| [[32/27]]<br />
| 3<br />
|-<br />
| S5*S6*S7*S8*S9<br />
| ([[5/4]])/([[10/9]])<br />
| [[9/8]]<br />
| 3<br />
|-<br />
| S6*S7*S8*S9*S10<br />
| ([[6/5]])/([[11/10]])<br />
| [[12/11]]<br />
| 11<br />
|-<br />
| S7*S8*S9*S10*S11<br />
| ([[7/6]])/([[12/11]])<br />
| [[77/72]]<br />
| 11<br />
|-<br />
| S8*S9*S10*S11*S12<br />
| ([[8/7]])/([[13/12]])<br />
| [[96/91]]<br />
| 13<br />
|-<br />
| S9*S10*S11*S12*S13<br />
| ([[9/8]])/([[14/13]])<br />
| [[117/112]]<br />
| 13<br />
|-<br />
| S10*S11*S12*S13*S14<br />
| ([[10/9]])/([[15/14]])<br />
| [[28/27]]<br />
| 7<br />
|-<br />
| S11*S12*S13*S14*S15<br />
| ([[11/10]])/([[16/15]])<br />
| [[33/32]]<br />
| 11<br />
|-<br />
| S12*S13*S14*S15*S16<br />
| ([[12/11]])/([[17/16]])<br />
| [[192/187]]<br />
| 17<br />
|-<br />
| S13*S14*S15*S16*S17<br />
| ([[13/12]])/([[18/17]])<br />
| [[221/216]]<br />
| 17<br />
|-<br />
| S14*S15*S16*S17*S18<br />
| ([[14/13]])/([[19/18]])<br />
| [[252/247]]<br />
| 19<br />
|-<br />
| S15*S16*S17*S18*S19<br />
| ([[15/14]])/([[20/19]])<br />
| [[57/56]]<br />
| 19<br />
|-<br />
| S16*S17*S18*S19*S20<br />
| ([[16/15]])/([[21/20]])<br />
| [[64/63]]<br />
| 7<br />
|-<br />
| S17*S18*S19*S20*S21<br />
| ([[17/16]])/([[22/21]])<br />
| [[357/352]]<br />
| 17<br />
|-<br />
| S18*S19*S20*S21*S22<br />
| ([[18/17]])/([[23/22]])<br />
| [[396/391]]<br />
| 23<br />
|-<br />
| S19*S20*S21*S22*S23<br />
| ([[19/18]])/([[24/23]])<br />
| [[437/432]]<br />
| 23<br />
|-<br />
| S20*S21*S22*S23*S24<br />
| ([[20/19]])/([[25/24]])<br />
| [[96/95]]<br />
| 19<br />
|-<br />
| S21*S22*S23*S24*S25<br />
| ([[21/20]])/([[26/25]])<br />
| [[105/104]]<br />
| 13<br />
|-<br />
| S22*S23*S24*S25*S26<br />
| ([[22/21]])/([[27/26]])<br />
| [[572/567]]<br />
| 13<br />
|-<br />
| S23*S24*S25*S26*S27<br />
| ([[23/22]])/([[28/27]])<br />
| [[621/616]]<br />
| 23<br />
|-<br />
| S28*S29*S30*S31*S32<br />
| ([[28/27]])/([[33/32]])<br />
| [[896/891]]<br />
| 11<br />
|-<br />
| S34*S35*S36*S37*S38<br />
| ([[34/33]])/([[39/38]])<br />
| [[1292/1287]]<br />
| 19<br />
|-<br />
| S35*S36*S37*S38*S39<br />
| ([[35/34]])/([[40/39]])<br />
| [[273/272]]<br />
| 17<br />
|-<br />
| S40*S41*S42*S43*S44<br />
| ([[40/39]])/([[45/44]])<br />
| [[352/351]]<br />
| 13<br />
|-<br />
| S45*S46*S47*S48*S49<br />
| ([[45/44]])/([[50/49]])<br />
| [[441/440]]<br />
| 11<br />
|-<br />
| S46*S47*S48*S49*S50<br />
| ([[46/45]])/([[51/50]])<br />
| [[460/459]]<br />
| 23<br />
|-<br />
| S50*S51*S52*S53*S54<br />
| ([[50/49]])/([[55/54]])<br />
| [[540/539]]<br />
| 11<br />
|-<br />
| S51*S52*S53*S54*S55<br />
| ([[51/50]])/([[56/55]])<br />
| [[561/560]]<br />
| 17<br />
|-<br />
| S52*S53*S54*S55*S56<br />
| ([[52/51]])/([[57/56]])<br />
| [[2912/2907]]<br />
| 19<br />
|-<br />
| S64*S65*S66*S67*S68<br />
| ([[64/63]])/([[69/68]])<br />
| [[4352/4347]]<br />
| 23<br />
|-<br />
| S65*S66*S67*S68*S69<br />
| ([[65/64]])/([[70/69]])<br />
| [[897/896]]<br />
| 23<br />
|-<br />
| S76*S77*S78*S79*S80<br />
| ([[76/75]])/([[81/80]])<br />
| [[1216/1215]]<br />
| 19<br />
|-<br />
| S91*S92*S93*S94*S95<br />
| ([[91/90]])/([[96/95]])<br />
| [[1729/1728]]<br />
| 19<br />
|-<br />
| <span style="font-size: 0.79em;">S100*S101*S102*S103*S104</span><br />
| <span style="font-size: 0.83em;">([[100/99]])/([[105/104]])</span><br />
| [[2080/2079]]<br />
| 13<br />
|-<br />
| <span style="font-size: 0.79em;">S115*S116*S117*S118*S119</span><br />
| <span style="font-size: 0.79em;">([[115/114]])/([[120/119]])</span><br />
| [[2737/2736]]<br />
| 23<br />
|-<br />
| <span style="font-size: 0.79em;">S121*S122*S123*S124*S125</span><br />
| <span style="font-size: 0.79em;">([[121/120]])/([[126/125]])</span><br />
| [[3025/3024]]<br />
| 11<br />
|-<br />
| <span style="font-size: 0.79em;">S171*S172*S173*S174*S175</span><br />
| <span style="font-size: 0.79em;">([[171/170]])/([[176/175]])</span><br />
| [[5985/5984]]<br />
| 19<br />
|}<br />
<br />
== {{nowrap|S''k''/S(''k'' + 1)}} (ultraparticulars) ==<br />
=== Motivational example ===<br />
Often it is desirable to make consecutive [[superparticular]] intervals equidistant. This has a number of nice consequences, many of which not explained here—see the motivation section for each infinite family of commas defined on this page.<br />
<br />
For example, if you want 6/5 equidistant from 5/4 and 7/6, you must equate {{nowrap|{{sfrac|[[5/4]]|[[6/5]]}} {{=}} [[25/24]]}} {{nowrap|{{=}} S5}} with {{nowrap|{{sfrac|[[6/5]]|[[7/6]]}} {{=}} [[36/35]]}} {{nowrap|{{=}} S6}}, hence tempering {{nowrap|{{sfrac|S5|S6}} {{=}} {{sfrac|25/24|36/35}}}} {{nowrap|{{=}} [[875/864]]}}, but it's actually often not necessary to know the specific numbers, often familiarizing yourself with and understanding the "S''k''" notation will give you a lot of insight, as we'll see.<br />
<br />
Back to our example: we know that {{nowrap|S5 ~ S6}} (because we're tempering S5/S6); from this we can deduce that the intervals must be arranged like this: {{nowrap|7/6 &larr; S5~S6 &rarr; 6/5 &larr; S5~S6 &rarr; 5/4}}.<br />
<br />
From this you can deduce that {{nowrap|([[6/5]])<sup>3</sup> &rarr; [[7/4]]}}, because you can lower one of the 6/5's to [[7/6]] (lowering it by S6) and raise another of the 6/5's to [[5/4]] (raising it by S5). Then because we've tempered S5 and S6 together, we've lowered and raised by the same amount, so the result of {{nowrap|7/6 * 6/5 * 5/4 {{=}} 7/4}} must be the same as the result of {{nowrap|6/5 * 6/5 * 6/5}} in this temperament.<br />
<br />
Familiarize yourself with the structure of this argument, as [[S-expression/Advanced results#Mathematical derivations|it generalizes to arbitrary S''k'']]; the algebraic proof is tedious, but the intuition is the same:<br />
<br />
<math>\displaystyle \frac{k+2}{k+1} \leftarrow S(k+1)~Sk \rightarrow \frac{k+1}{k} \leftarrow S(k+1)~Sk \rightarrow \frac{k}{k-1}</math><br />
<br />
… implies that three {{sfrac|''k'' + 1|''k''}} give {{sfrac|''k'' + 2|''k'' − 1}} iff we temper {{sfrac|S''k''|S(''k'' + 1)}}.&nbsp;{{qed}}<br />
<br />
=== Significance ===<br />
1. Tempering any two consecutive square-particulars S''k'' and S({{nowrap|''k'' + 1}}) will naturally imply tempering the ultraparticular between them, {{sfrac|S''k''|S(''k'' + 1)}}, meaning they are very common implicit commas.<br />
<br />
2. Tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness.<br />
<br />
3. Tempering the ultraparticular S''k''/S({{nowrap|''k'' + 1}}) along with either the corresponding 1/2-square-particular {{nowrap|S''k'' * S(''k'' + 1)}} or one of the two corresponding lopsided commas {{nowrap|S''k''<sup>2</sup> * S(''k'' + 1)}} or {{nowrap|S''k'' * S(''k'' + 1)<sup>2</sup>}} implies tempering both of S''k'' and S({{nowrap|''k'' + 1}}) individually, and vice versa, so that there is a total of ''five'' equivalences—corresponding to ''five'' infinite families of commas—for every such S''k'' and S({{nowrap|''k''+1}}). This only gets better if you temper a third consecutive square-particular. This is an abundance of "at a glance" essential tempering information that is fully general so only needs to be learned once, and is the motivation of the use of '''S-expressions'''. (For example, {{nowrap|{S16, S17} &rarr; {{(}}S16 * S17, S16/S17, S16<sup>2</sup> * S17, S16 * S17<sup>2</sup>{{)}} }}, and any of the two commas in the latter set imply all the other commas too.)<br />
<br />
=== Table of ultraparticulars ===<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Cube Relation<br />
! Comma<br />
! Cents<br />
|-<br />
| S2/S3 = ([[4/3]])/([[9/8]])<br />
| ([[4/1]])/([[3/2]])<sup>3</sup><br />
| [[32/27]]<br />
| 294.135<br />
|-<br />
| S3/S4 = ([[9/8]])/([[16/15]])<br />
| ([[5/2]])/([[4/3]])<sup>3</sup><br />
| [[135/128]]<br />
| 92.179<br />
|-<br />
| S4/S5 = ([[16/15]])/([[25/24]])<br />
| ([[2/1]])/([[5/4]])<sup>3</sup><br />
| [[128/125]]<br />
| 41.059<br />
|-<br />
| S5/S6 = ([[25/24]])/([[36/35]])<br />
| ([[7/4]])/([[6/5]])<sup>3</sup><br />
| [[875/864]]<br />
| 21.902<br />
|-<br />
| S6/S7 = ([[36/35]])/([[49/48]])<br />
| ([[8/5]])/([[7/6]])<sup>3</sup><br />
| [[1728/1715]]<br />
| 13.074<br />
|-<br />
| S7/S8 = ([[49/48]])/([[64/63]])<br />
| ([[3/2]])/([[8/7]])<sup>3</sup><br />
| [[1029/1024]]<br />
| 8.433<br />
|-<br />
| S8/S9 = ([[64/63]])/([[81/80]])<br />
| ([[10/7]])/([[9/8]])<sup>3</sup><br />
| [[5120/5103]]<br />
| 5.758<br />
|-<br />
| S9/S10 = ([[81/80]])/([[100/99]])<br />
| ([[11/8]])/([[10/9]])<sup>3</sup><br />
| [[8019/8000]]<br />
| 4.107<br />
|-<br />
| S10/S11 = ([[100/99]])/([[121/120]])<br />
| ([[4/3]])/([[11/10]])<sup>3</sup><br />
| [[4000/3993]]<br />
| 3.032<br />
|-<br />
| S11/S12 = ([[121/120]])/([[144/143]])<br />
| ([[13/10]])/([[12/11]])<sup>3</sup><br />
| [[17303/17280]]<br />
| 2.303<br />
|-<br />
| S12/S13 = ([[144/143]])/([[169/168]])<br />
| ([[14/11]])/([[13/12]])<sup>3</sup><br />
| [[24192/24167]]<br />
| 1.79<br />
|-<br />
| S13/S14 = ([[169/168]])/([[196/195]])<br />
| ([[5/4]])/([[14/13]])<sup>3</sup><br />
| [[10985/10976]]<br />
| 1.419<br />
|-<br />
| S14/S15 = ([[196/195]])/([[225/224]])<br />
| ([[16/13]])/([[15/14]])<sup>3</sup><br />
| [[43904/43875]]<br />
| 1.144<br />
|-<br />
| S15/S16 = ([[225/224]])/([[256/255]])<br />
| ([[17/14]])/([[16/15]])<sup>3</sup><br />
| [[57375/57344]]<br />
| 0.936<br />
|-<br />
| S16/S17 = ([[256/255]])/([[289/288]])<br />
| ([[6/5]])/([[17/16]])<sup>3</sup><br />
| [[24576/24565]]<br />
| 0.775<br />
|-<br />
| S17/S18 = ([[289/288]])/([[324/323]])<br />
| ([[19/16]])/([[18/17]])<sup>3</sup><br />
| [[93347/93312]]<br />
| 0.649<br />
|-<br />
| S18/S19 = ([[324/323]])/([[361/360]])<br />
| ([[20/17]])/([[19/18]])<sup>3</sup><br />
| [[116640/116603]]<br />
| 0.549<br />
|-<br />
| S19/S20 = ([[361/360]])/([[400/399]])<br />
| ([[7/6]])/([[20/19]])<sup>3</sup><br />
| [[48013/48000]]<br />
| 0.469<br />
|-<br />
| S20/S21 = ([[400/399]])/([[441/440]])<br />
| ([[22/19]])/([[21/20]])<sup>3</sup><br />
| [[176000/175959]]<br />
| 0.403<br />
|-<br />
| S21/S22 = ([[441/440]])/([[484/483]])<br />
| ([[23/20]])/([[22/21]])<sup>3</sup><br />
| [[213003/212960]]<br />
| 0.35<br />
|-<br />
| S22/S23 = ([[484/483]])/([[529/528]])<br />
| ([[8/7]])/([[23/22]])<sup>3</sup><br />
| [[85184/85169]]<br />
| 0.305<br />
|-<br />
| S23/S24 = ([[529/528]])/([[576/575]])<br />
| ([[25/22]])/([[24/23]])<sup>3</sup><br />
| [[304175/304128]]<br />
| 0.268<br />
|-<br />
| S24/S25 = ([[576/575]])/([[625/624]])<br />
| ([[26/23]])/([[25/24]])<sup>3</sup><br />
| [[359424/359375]]<br />
| 0.236<br />
|-<br />
| S25/S26 = ([[625/624]])/([[676/675]])<br />
| ([[9/8]])/([[26/25]])<sup>3</sup><br />
| [[140625/140608]]<br />
| 0.209<br />
|-<br />
| S26/S27 = ([[676/675]])/([[729/728]])<br />
| ([[28/25]])/([[27/26]])<sup>3</sup><br />
| [[492128/492075]]<br />
| 0.186<br />
|-<br />
| S27/S28 = ([[729/728]])/([[784/783]])<br />
| ([[29/26]])/([[28/27]])<sup>3</sup><br />
| [[570807/570752]]<br />
| 0.167<br />
|-<br />
| S28/S29 = ([[784/783]])/([[841/840]])<br />
| ([[10/9]])/([[29/28]])<sup>3</sup><br />
| [[219520/219501]]<br />
| 0.15<br />
|-<br />
| S31/S32 = ([[961/960]])/([[1024/1023]])<br />
| ([[11/10]])/([[32/31]])<sup>3</sup><br />
| [[327701/327680]]<br />
| 0.111<br />
|-<br />
| S33/S34 = ([[1089/1088]])/([[1156/1155]])<br />
| ([[35/32]])/([[34/33]])<sup>3</sup><br />
| [[1257795/1257728]]<br />
| 0.092<br />
|-<br />
| S34/S35 = ([[1156/1155]])/([[1225/1224]])<br />
| ([[12/11]])/([[35/34]])<sup>3</sup><br />
| [[471648/471625]]<br />
| 0.084<br />
|-<br />
| S37/S38 = ([[1369/1368]])/([[1444/1443]])<br />
| ([[13/12]])/([[38/37]])<sup>3</sup><br />
| [[658489/658464]]<br />
| 0.066<br />
|-<br />
| S40/S41 = ([[1600/1599]])/([[1681/1680]])<br />
| ([[14/13]])/([[41/40]])<sup>3</sup><br />
| [[896000/895973]]<br />
| 0.052<br />
|-<br />
| S43/S44 = ([[1849/1848]])/([[1936/1935]])<br />
| ([[15/14]])/([[44/43]])<sup>3</sup><br />
| [[1192605/1192576]]<br />
| 0.042<br />
|-<br />
| S46/S47 = ([[2116/2115]])/([[2209/2208]])<br />
| ([[16/15]])/([[47/46]])<sup>3</sup><br />
| [[1557376/1557345]]<br />
| 0.034<br />
|-<br />
| S49/S50 = ([[2401/2400]])/([[2500/2499]])<br />
| ([[17/16]])/([[50/49]])<sup>3</sup><br />
| [[2000033/2000000]]<br />
| 0.029<br />
|-<br />
| S50/S51 = ([[2500/2499]])/([[2601/2600]])<br />
| ([[52/49]])/([[51/50]])<sup>3</sup><br />
| [[6500000/6499899]]<br />
| 0.027<br />
|-<br />
| S55/S56 = ([[3025/3024]])/([[3136/3135]])<br />
| ([[19/18]])/([[56/55]])<sup>3</sup><br />
| [[3161125/3161088]]<br />
| 0.02<br />
|-<br />
| S64/S65 = ([[4096/4095]])/([[4225/4224]])<br />
| ([[22/21]])/([[65/64]])<sup>3</sup><br />
| [[5767168/5767125]]<br />
| 0.013<br />
|}<br />
<br />
The above table is a list of all [[23-limit]] ultraparticulars corresponding to S''k'' with ''k'' < 77, plus ultraparticulars corresponding to dividing a [[superparticular interval]] into three equal parts up to [[17/16]] (or up to [[19/18]] but excluding 18/17 because of it requiring a large prime, 53), plus S27/S28 so that we have all ultraparticulars up to S28/S29 listed rather than up to S26/S27.<br />
<br />
This table has been expanded following every ultraparticular from S2/S3 to S16/S17 having its own page. Note that ultraparticulars are, in general, extremely precise commas so that usually one wouldn't consider tempering them directly rather than through tempering the square-particulars S''k'' which they are composed of. As an example of this, notice that [[4000/3993|S10/S11]] is the largest ultraparticular categorised as an [[unnoticeable comma]], which means not unnoticeable in the absolute sense but rather in the sense of being smaller than the melodic just-noticeable difference, despite only dividing a superparticular as simple and unremarkable as [[4/3]]. For this reason, a [[cent]]s column has been included to aid an appreciation of their precision. The cent value of a [[semiparticular]] is roughly double that of any of the two ultraparticulars it is composed of; this becomes more true the higher you go.<br />
<br />
Note also from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably [[consistent]]) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3''k'' + 1)/S(3''k''+ 2) for a positive integer ''k'', because then the superparticular can be expressed as:<br />
<br />
<math>\frac{(3k + 3)/3k}{((3k + 2)(3k + 1))^3} = \frac{(k + 1)/k}{((3k + 2)(3k + 1))^3}</math><br />
<br />
Also note that if you temper multiple adjacent ultraparticulars, you sometimes are not required to use those ultraparticulars in the comma list as description of (the bulk of) the tempering may be possible through [[#Sk/S(k + 2) (semiparticulars)|semiparticulars]], discussed next.<br />
<br />
== {{nowrap|S''k''/S(''k'' + 2)}} (semiparticulars) ==<br />
=== Motivational examples ===<br />
If we want to halve one JI interval into two of another JI interval, there is a powerful and elegant pattern for doing so:<br />
* [[4/3]] is approximately half of [[9/5]]<br />
* [[9/7]] is approximately half of [[5/3]] (=&nbsp;10/6)<br />
* [[5/4]] is approximately half of [[11/7]]<br />
* [[11/9]] is approximately half of [[3/2]] (=&nbsp;12/8)<br />
* [[6/5]] is approximately half of [[13/9]]<br />
* [[13/11]] is approximately half of [[7/5]] (=&nbsp;14/10)<br />
* [[7/6]] is approximately half of [[15/11]]<br />
* [[15/13]] is approximately half of [[4/3]] (=&nbsp;16/12)<br />
* [[8/7]] is approximately half of [[17/13]]<br />
* [[17/15]] is approximately half of [[9/7]] (=&nbsp;18/14)<br />
* [[9/8]] is approximately half of [[19/15]]<br />
* [[19/17]] is approximately half of [[5/4]] (=&nbsp;20/16)<br />
<br />
These properties show a pattern: take some arbitrary [[#Glossary|quodd-particular]] (''k'' + 4)/''k''; observe that we can split it into (''k'' + 4)/(''k'' + 2) * (''k'' + 2)/''k''.<br />
<br />
Now observe that (''k'' + 2)/''k'' > (''k'' + 3)/(''k'' + 1) > (''k'' + 4)/(''k'' + 2); in fact, it can be shown fairly easily that (''k'' + 3)/(''k'' + 1) is the [[mediant]] of (''k'' + 4)/(''k'' + 2) and (''k'' + 2)/''k''.<br />
<br />
It turns out that making this mediant — (''k'' + 3)/(''k'' + 1) — equal to half of (''k'' + 4)/''k'' is equivalent to tempering S(''k'' + 1)/S(''k'' + 3).<br />
<br />
=== Significance ===<br />
1. For differences between square-particulars of the form S''k''/S(''k'' + 2), the resulting comma is either [[superparticular]] or [[#Glossary|odd-particular]], so these are efficient commas. (This terminology also suggests [[#Glossary|throdd-particular]] and [[#Glossary|quodd-particular]] as generalizations.)<br />
<br />
2. Tempering any two consecutive [[ultraparticular]]s implies tempering a semiparticular, so from two adjacent "thirding" equivalences you get a "halving" equivalence for free!<br />
<br />
3. Tempering any two nearly-consecutive square-particulars (S''k'' and S(''k'' + 2)) implies tempering a semiparticular; this is generally much more ideal than tempering two consecutive S''k'' because it is a lot lower damage (see [[lopsided comma]]s for (relatively) large commas implied by this higher-damage strategy).<br />
<br />
4. On top of the halving equivalence, there is a number of subtler structural implications, [[discussed below, that may be desirable to the temperament designer.<br />
<br />
=== Meaning ===<br />
: '''Reader notes:''' In the below, we use S(''k'' - 1)/S(''k'' + 1) for symmetry around ''k'' to make the math visually simpler, but keep in mind it's equivalent to using an offset ''k''.<br />
: '''Also:''' keep in mind that ''k'' - ''a'' (for positive ''a'') is smaller than ''k'', so that ''k''/(''k'' - ''a'') > (''k'' + ''a'')/''k'' (because the former appears earlier in the harmonic series & is thus larger); this is an important and useful intuition to learn.<br />
<br />
Tempering S(''k'' - 1)/S(''k'' + 1) implies that (''k'' + 2)/(''k'' - 2) is divisible exactly into two halves of (''k'' + 1)/(''k'' - 1). It also implies that the intervals (''k'' + 2)/''k'' (=s) and ''k''/(''k'' - 2) (=L) are equidistant from (''k'' + 1)/(''k'' - 1) (=M) because to make them equidistant we need to temper:<br />
<br />
<math>\displaystyle \frac{L/M}{M/s} = \frac{ \left(\frac{k}{k-2}\right)/\left(\frac{k+1}{k-1}\right) }{ \left(\frac{k+1}{k-1}\right)/\left(\frac{k+2}{k}\right) } = \frac{Ls}{M^2} = \frac{\frac{k+2}{k-2}}{\left(\frac{k+1}{k-1}\right)^2} </math><br />
<br />
… and notice that the latter expression is the one we've [[S-expression/Advanced results#Mathematical derivations|shown is equal to S(''k''-1)/S(''k''+1)]] (up to an offset ''k''). In other words, you could interpret that a reason that tempering S(''k''-1)/S(''k''+1) results in (''k''+1)/(''k''-1) being half of (''k''+2)/(''k''-2) is because it makes the following three intervals equidistant:<br />
(''k''+2)/''k'', (''k''+1)/(''k''-1), ''k''/(''k''-2)<br />
<br />
Also note that in the above, (''k'' + 1)/(''k'' - 1) is the [[mediant]] of the adjacent two intervals, meaning that division of an interval into two via tempering a semiparticular is in some sense 'optimal' relative to the complexity. This also means that if ''k'' is a multiple of 2, this corresponds to a natural way to split the square superparticular S(''k''/2) into two parts. For example, if ''k'' = 10 then we have (10+2)/10, (10+1)/(10-1), 10/(10-2) as equidistant, which simplified is 6/5, 11/9, 5/4, with 11/9 being the mediant of 6/5 and 5/4, and therefore the corresponding superparticular S5 = (5/4)/(6/5) is split into two parts which are tempered together: (5/4)/(11/9) = 45/44 and (11/9)/(6/5) = 55/54. The semiparticular is therefore S(10-1)/S(10+1) = S9/S11 = 243/242 = (45/44)/(55/54) = ((10+2)/(10-2))/((10+1)/(10-1))<sup>2</sup>.<br />
<br />
This form of comma has been named "semiparticular", because most of the time it is superparticular but less often it is odd-particular, and because when tempered out they all cause an interval to be divided into two equal parts where each part is a (tempered version of a) superparticular or odd-particular, and the interval being divided in half is sometimes quodd-particular, sometimes odd-particular and sometimes superparticular. Specifically:<br />
<br />
* To find out what a superparticular (''a''+1)/''a'' is approximately half of, temper the semiparticular S(2''a'')/S(2''a''+2) and you can observe that (2''a''+3)/(2''a''-1) is the interval it is approximately half of.<br />
<br />
* To find out what an odd-particular (2''a''+1)/(2''a''-1) is approximately half of, temper the semiparticular S(2''a''-1)/S(2''a''+1) and you can observe that (2''a''+2)/(2''a''-2) = (''a''+1)/(''a''-1), a superparticular or odd-particular, is the interval it is approximately half of.<br />
<br />
* To find out what splits a superparticular (''a''+1)/''a'' in half, temper the semiparticular S(4''a''+1)/S(4''a''+3) and you can observe that (4''a''+3)/(4''a''+1), an odd-particular, is the interval that is approximately half of it.<br />
<br />
* To find out what splits an odd-particular (2''a''+1)/(2''a''-1) in half, temper the semiparticular S(4''a''-2)/S(4''a''+2) and you can observe that (4''a''-1)/(4''a''+1), an odd-particular, is the interval that is approximately half of it.<br />
<br />
Also, the interval in the denominator of an expression of a semiparticular of the form (a/b)/(c/d)<sup>2</sup> is significant in that it has a special relationship: specifically, consider tempering (a/b)/(c/d)<sup>2</sup> so therefore the interval c/d is equal to the interval (a/b)/(c/d). This is significant because it allows the intuitive replacement of two consecutive superparticulars (whose product is a superparticular or odd-particular) with the two superparticulars directly adjacent to them.<br />
<br />
For example, as 9/8 = 18/17 * 17/16 we can replace 18/17 with 19/18 and 17/16 with 16/15 by tempering S16/S18 = (19/15)/(9/8)<sup>2</sup> because we can multiply 9/8 by the tempered comma (19/15)/(9/8)<sup>2</sup> to get (19/15)/(9/8) = (19/18)(16/15) (because 9/8 = 18/16), or as 13/11 = 13/12 * 12/11 we can replace 13/12 with 14/13 and 12/11 with 11/10 by tempering S11/S13 = (7/5)/(13/11)<sup>2</sup> because we can multiply 13/11 by the tempered comma (7/5)/(13/11)<sup>2</sup> to get (7/5)/(13/11) = (14/13)(11/10) (because 7/5 = 14/10). Note we have to replace ''both'' intervals ''simultaneously'' as this is lower error, and note that if we want to be able to replace them individually we must pick the higher error route of tempering S16 and S18 or S11 and S13 individually (for which tempering the semiparticular is then an implied consequence). (The broader lesson is that you can rewrite exact JI equivalences with the commas you are tempering to find new interesting consequences of those commas.)<br />
<br />
=== Table of semiparticulars ===<br />
Here follows a table of [[23-limit]] semiparticulars corresponding to square-particulars S''k'' for ''k'' < 96, plus all semiparticulars up to [[9801/9800|S33/S35 = S99]], an exceptional [[11-limit]] comma, plus all semiparticulars dividing [[superparticular interval]]s up to [[13/12]] (corresponding to the [[17-limit]] semiparticular [[31213/31212|S49/S51]]) for completeness. This table also shows all semiparticulars corresponding to splitting an [[#Glossary|odd-particular]] in two up to [[17/15]] (although a common strategy is to temper the square-particular that is the difference between the two superparticular intervals the odd-particular is composed of instead). The bound ''k'' < 96 was chosen as it corresponds to another remarkable semiparticular [[123201/123200|S78/S80 = S351]]. Perhaps many of the patterns will become clearer if you examine this table:<br />
<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Square Relation<br />
! Ratio<br />
|-<br />
| S2/S4 = ([[4/3]])/([[16/15]])<br />
| ([[5/1]])/([[2/1]])<sup>2</sup><br />
| [[5/4]]<br />
|-<br />
| S3/S5 = ([[9/8]])/([[25/24]])<br />
| ([[3/1]])/([[5/3]])<sup>2</sup><br />
| [[27/25]]<br />
|-<br />
| S4/S6 = ([[16/15]])/([[36/35]])<br />
| ([[7/3]])/([[3/2]])<sup>2</sup><br />
| [[28/27]]<br />
|-<br />
| S5/S7 = ([[25/24]])/([[49/48]])<br />
| ([[2/1]])/([[7/5]])<sup>2</sup><br />
| [[50/49]]<br />
|-<br />
| S6/S8 = ([[36/35]])/([[64/63]])<br />
| ([[9/5]])/([[4/3]])<sup>2</sup><br />
| [[81/80]]<br />
|-<br />
| S7/S9 = ([[49/48]])/([[81/80]])<br />
| ([[5/3]])/([[9/7]])<sup>2</sup><br />
| [[245/243]]<br />
|-<br />
| S8/S10 = ([[64/63]])/([[100/99]])<br />
| ([[11/7]])/([[5/4]])<sup>2</sup><br />
| [[176/175]]<br />
|-<br />
| S9/S11 = ([[81/80]])/([[121/120]])<br />
| ([[3/2]])/([[11/9]])<sup>2</sup><br />
| [[243/242]]<br />
|-<br />
| S10/S12 = ([[100/99]])/([[144/143]])<br />
| ([[13/9]])/([[6/5]])<sup>2</sup><br />
| [[325/324]]<br />
|-<br />
| S11/S13 = ([[121/120]])/([[169/168]])<br />
| ([[7/5]])/([[13/11]])<sup>2</sup><br />
| [[847/845]]<br />
|-<br />
| S12/S14 = ([[144/143]])/([[196/195]])<br />
| ([[15/11]])/([[7/6]])<sup>2</sup><br />
| [[540/539]]<br />
|-<br />
| S13/S15 = ([[169/168]])/([[225/224]])<br />
| ([[4/3]])/([[15/13]])<sup>2</sup><br />
| [[676/675]]<br />
|-<br />
| S14/S16 = ([[196/195]])/([[256/255]])<br />
| ([[17/13]])/([[8/7]])<sup>2</sup><br />
| [[833/832]]<br />
|-<br />
| S15/S17 = ([[225/224]])/([[289/288]])<br />
| ([[9/7]])/([[17/15]])<sup>2</sup><br />
| [[2025/2023]]<br />
|-<br />
| S16/S18 = ([[256/255]])/([[324/323]])<br />
| ([[19/15]])/([[9/8]])<sup>2</sup><br />
| [[1216/1215]]<br />
|-<br />
| S17/S19 = ([[289/288]])/([[361/360]])<br />
| ([[5/4]])/([[19/17]])<sup>2</sup><br />
| [[1445/1444]]<br />
|-<br />
| S18/S20 = ([[324/323]])/([[400/399]])<br />
| ([[21/17]])/([[10/9]])<sup>2</sup><br />
| [[1701/1700]]<br />
|-<br />
| S19/S21 = ([[361/360]])/([[441/440]])<br />
| ([[11/9]])/([[21/19]])<sup>2</sup><br />
| [[3971/3969]]<br />
|-<br />
| S20/S22 = ([[400/399]])/([[484/483]])<br />
| ([[23/19]])/([[11/10]])<sup>2</sup><br />
| [[2300/2299]]<br />
|-<br />
| S21/S23 = ([[441/440]])/([[529/528]])<br />
| ([[6/5]])/([[23/21]])<sup>2</sup><br />
| [[2646/2645]]<br />
|-<br />
| S22/S24 = ([[484/483]])/([[576/575]])<br />
| ([[25/21]])/([[12/11]])<sup>2</sup><br />
| [[3025/3024]]<br />
|-<br />
| S23/S25 = ([[529/528]])/([[625/624]])<br />
| ([[13/11]])/([[25/23]])<sup>2</sup><br />
| [[6877/6875]]<br />
|-<br />
| S24/S26 = ([[576/575]])/([[676/675]])<br />
| ([[27/23]])/([[13/12]])<sup>2</sup><br />
| [[3888/3887]]<br />
|-<br />
| S25/S27 = ([[625/624]])/([[729/728]])<br />
| ([[7/6]])/([[27/25]])<sup>2</sup><br />
| [[4375/4374]]<br />
|-<br />
| S26/S28 = ([[676/675]])/([[784/783]])<br />
| ([[29/25]])/([[14/13]])<sup>2</sup><br />
| [[4901/4900]]<br />
|-<br />
| S27/S29 = ([[729/728]])/([[841/840]])<br />
| ([[15/13]])/([[29/27]])<sup>2</sup><br />
| [[10935/10933]]<br />
|-<br />
| S28/S30 = ([[784/783]])/([[900/899]])<br />
| ([[31/27]])/([[15/14]])<sup>2</sup><br />
| [[6076/6075]]<br />
|-<br />
| S29/S31 = ([[841/840]])/([[961/960]])<br />
| ([[8/7]])/([[31/29]])<sup>2</sup><br />
| [[6728/6727]]<br />
|-<br />
| S30/S32 = ([[900/899]])/([[1024/1023]])<br />
| ([[33/29]])/([[16/15]])<sup>2</sup><br />
| [[7425/7424]]<br />
|-<br />
| S31/S33 = ([[961/960]])/([[1089/1088]])<br />
| ([[17/15]])/([[33/31]])<sup>2</sup><br />
| [[16337/16335]]<br />
|-<br />
| S32/S34 = ([[1024/1023]])/([[1156/1155]])<br />
| ([[35/31]])/([[17/16]])<sup>2</sup><br />
| [[8960/8959]]<br />
|-<br />
| S33/S35 = ([[1089/1088]])/([[1225/1224]])<br />
| ([[9/8]])/([[35/33]])<sup>2</sup><br />
| [[9801/9800]]<br />
|-<br />
| S36/S38 = ([[1296/1295]])/([[1444/1443]])<br />
| ([[39/35]])/([[19/18]])<sup>2</sup><br />
| [[12636/12635]]<br />
|-<br />
| S37/S39 = ([[1369/1368]])/([[1521/1520]])<br />
| ([[10/9]])/([[39/37]])<sup>2</sup><br />
| [[13690/13689]]<br />
|-<br />
| S41/S43 = ([[1681/1680]])/([[1849/1848]])<br />
| ([[11/10]])/([[43/41]])<sup>2</sup><br />
| [[18491/18490]]<br />
|-<br />
| S45/S47 = ([[2025/2024]])/([[2209/2208]])<br />
| ([[12/11]])/([[47/45]])<sup>2</sup><br />
| [[24300/24299]]<br />
|-<br />
| S46/S48 = ([[2116/2115]])/([[2304/2303]])<br />
| ([[49/45]])/([[24/23]])<sup>2</sup><br />
| [[25921/25920]]<br />
|-<br />
| S49/S51 = ([[2401/2400]])/([[2601/2600]])<br />
| ([[13/12]])/([[51/49]])<sup>2</sup><br />
| [[31213/31212]]<br />
|-<br />
| S52/S54 = ([[2704/2703]])/([[2916/2915]])<br />
| ([[55/51]])/([[27/26]])<sup>2</sup><br />
| [[37180/37179]]<br />
|-<br />
| S66/S68 = ([[4356/4355]])/([[4624/4623]])<br />
| ([[69/65]])/([[34/33]])<sup>2</sup><br />
| [[75141/75140]]<br />
|-<br />
| S78/S80 = ([[6084/6083]])/([[6400/6399]])<br />
| ([[81/77]])/([[40/39]])<sup>2</sup><br />
| [[123201/123200]]<br />
|}<br />
<br />
(Note that while a lot of these have pages, not all of them do, although that doesn't mean they shouldn't. A noticeable streak of commas currently without pages correspond to when dividing a superparticular interval implicates intervals from a higher [[prime limit]], as a surprising amount of 23-limit semiparticulars shown here already have pages.)<br />
<br />
== {{nowrap|S''k''² * S(''k'' + 1)}} and {{nowrap|S(''k'' − 1) * S''k''²}} (lopsided commas) ==<br />
=== Significance ===<br />
1. Tempering any two consecutive square-particulars, S''k'' and S(''k'' + 1), implies tempering the two associated lopsided commas as well as the associated [[triangle-particular]] and [[ultraparticular]], so the lopsided commas represent the general form of the highest-damage relations/consequences of doing so.<br />
<br />
2. If a comma (such as the diaschisma, [[2048/2025]]), admits an expression as a lopsided comma, it means that one is likely missing out on tempering opportunities by not also tempering the square-particulars composing it (such as [[256/255|S16]] and [[289/288|S17]] in the case of the diaschisma), often involving expanding the subgroup and adding a number of new equivalence relations (as previously explained) while simultaneously making the temperament more efficient and more precise.<br />
<br />
3. It is surprising that there are fairly simple general equivalence relations for these S-expressions, essentially being "free" to read off of an S-expression-based comma list, once you know the general form.<br />
<br />
=== Derivation of equivalence relation ===<br />
Using the clarity of [[S-expression/Advanced results#Using S-factorizations to understand the significance of S-expressions|S-factorizations]], we can show the interval relations implicated by these two new "lopsided" forms, which will make clear the reason for their name:<br />
<br />
S''k''<sup>2</sup> * S(''k''+1) = [''k''-1, ''k'', ''k''+1, ''k''+2]^(2[-1, 2, -1, 0] + [0, -1, 2, -1] = [-2, 4, -2, 0] + [0, -1, 2, -1] = [-2, 3, 0, -1]) implies:<br />
<br />
S''k''<sup>2</sup> * S(''k''+1) = (''k''/(''k''-1))<sup>2</sup> / ((''k''+2)/''k'') through [-2, 3, 0, -1] = [-2, 2, 0, 0] - [0, -1, 0, 1].<br />
<br />
S(''k''-1) * S''k''<sup>2</sup> = [''k''-2, ''k''-1, ''k'', ''k''+1]^([-1, 2, -1, 0] + 2[0, -1, 2, -1] = [-1, 2, -1, 0] + [0, -2, 4, -2] = [-1, 0, 3, -2]) implies:<br />
<br />
S(''k''-1) * S''k''<sup>2</sup> = (''k''/(''k''-2)) / ((''k''+1)/''k'')<sup>2</sup> through [-1, 0, 3, -2] = [-1, 0, 1, 0] - [0, 0, -2, 2].<br />
<br />
=== Tables ===<br />
Below are two tables of [[43-limit]] lopsided commas. First, the "top heavy" lopsided commas, where the squared interval is in the numerator, then the "bottom heavy" lopsided commas, where the squared interval is in the denominator. These tables are so big because these commas are quite large so the more interesting commas appear later. For this reason and for completeness, the tables show up to until a little past the largest known lopsided commas that have their own page: the [[olympia]] and the [[phaotic comma]].<br />
<br />
==== Top-heavy lopsided commas ====<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Square Relation<br />
! Ratio<br />
|-<br />
| S2<sup>2</sup>*S3 = [[3/2]] * [[4/3]]<br />
| ([[2/1]])<sup>2</sup> / ([[2/1]])<br />
| [[2/1]]<br />
|-<br />
| S3<sup>2</sup>*S4 = [[6/5]] * [[9/8]]<br />
| ([[3/2]])<sup>2</sup> / ([[5/3]])<br />
| [[27/20]]<br />
|-<br />
| S4<sup>2</sup>*S5 = [[10/9]] * [[16/15]]<br />
| ([[4/3]])<sup>2</sup> / ([[3/2]])<br />
| [[32/27]]<br />
|-<br />
| S5<sup>2</sup>*S6 = [[15/14]] * [[25/24]]<br />
| ([[5/4]])<sup>2</sup> / ([[7/5]])<br />
| [[125/112]]<br />
|-<br />
| S6<sup>2</sup>*S7 = [[21/20]] * [[36/35]]<br />
| ([[6/5]])<sup>2</sup> / ([[4/3]])<br />
| [[27/25]]<br />
|-<br />
| S7<sup>2</sup>*S8 = [[28/27]] * [[49/48]]<br />
| ([[7/6]])<sup>2</sup> / ([[9/7]])<br />
| [[343/324]]<br />
|-<br />
| S8<sup>2</sup>*S9 = [[36/35]] * [[64/63]]<br />
| ([[8/7]])<sup>2</sup> / ([[5/4]])<br />
| [[256/245]]<br />
|-<br />
| S9<sup>2</sup>*S10 = [[45/44]] * [[81/80]]<br />
| ([[9/8]])<sup>2</sup> / ([[11/9]])<br />
| [[729/704]]<br />
|-<br />
| S10<sup>2</sup>*S11 = [[55/54]] * [[100/99]]<br />
| ([[10/9]])<sup>2</sup> / ([[6/5]])<br />
| [[250/243]]<br />
|-<br />
| S11<sup>2</sup>*S12 = [[66/65]] * [[121/120]]<br />
| ([[11/10]])<sup>2</sup> / ([[13/11]])<br />
| [[1331/1300]]<br />
|-<br />
| S12<sup>2</sup>*S13 = [[78/77]] * [[144/143]]<br />
| ([[12/11]])<sup>2</sup> / ([[7/6]])<br />
| [[864/847]]<br />
|-<br />
| S13<sup>2</sup>*S14 = [[91/90]] * [[169/168]]<br />
| ([[13/12]])<sup>2</sup> / ([[15/13]])<br />
| [[2197/2160]]<br />
|-<br />
| S14<sup>2</sup>*S15 = [[105/104]] * [[196/195]]<br />
| ([[14/13]])<sup>2</sup> / ([[8/7]])<br />
| [[343/338]]<br />
|-<br />
| S15<sup>2</sup>*S16 = [[120/119]] * [[225/224]]<br />
| ([[15/14]])<sup>2</sup> / ([[17/15]])<br />
| [[3375/3332]]<br />
|-<br />
| S16<sup>2</sup>*S17 = [[136/135]] * [[256/255]]<br />
| ([[16/15]])<sup>2</sup> / ([[9/8]])<br />
| [[2048/2025]]<br />
|-<br />
| S17<sup>2</sup>*S18 = [[153/152]] * [[289/288]]<br />
| ([[17/16]])<sup>2</sup> / ([[19/17]])<br />
| [[4913/4864]]<br />
|-<br />
| S18<sup>2</sup>*S19 = [[171/170]] * [[324/323]]<br />
| ([[18/17]])<sup>2</sup> / ([[10/9]])<br />
| [[1458/1445]]<br />
|-<br />
| S19<sup>2</sup>*S20 = [[190/189]] * [[361/360]]<br />
| ([[19/18]])<sup>2</sup> / ([[21/19]])<br />
| [[6859/6804]]<br />
|-<br />
| S20<sup>2</sup>*S21 = [[210/209]] * [[400/399]]<br />
| ([[20/19]])<sup>2</sup> / ([[11/10]])<br />
| [[4000/3971]]<br />
|-<br />
| S21<sup>2</sup>*S22 = [[231/230]] * [[441/440]]<br />
| ([[21/20]])<sup>2</sup> / ([[23/21]])<br />
| [[9261/9200]]<br />
|-<br />
| S22<sup>2</sup>*S23 = [[253/252]] * [[484/483]]<br />
| ([[22/21]])<sup>2</sup> / ([[12/11]])<br />
| [[1331/1323]]<br />
|-<br />
| S23<sup>2</sup>*S24 = [[276/275]] * [[529/528]]<br />
| ([[23/22]])<sup>2</sup> / ([[25/23]])<br />
| [[12167/12100]]<br />
|-<br />
| S24<sup>2</sup>*S25 = [[300/299]] * [[576/575]]<br />
| ([[24/23]])<sup>2</sup> / ([[13/12]])<br />
| [[6912/6877]]<br />
|-<br />
| S25<sup>2</sup>*S26 = [[325/324]] * [[625/624]]<br />
| ([[25/24]])<sup>2</sup> / ([[27/25]])<br />
| [[15625/15552]]<br />
|-<br />
| S26<sup>2</sup>*S27 = [[351/350]] * [[676/675]]<br />
| ([[26/25]])<sup>2</sup> / ([[14/13]])<br />
| [[4394/4375]]<br />
|-<br />
| S27<sup>2</sup>*S28 = [[378/377]] * [[729/728]]<br />
| ([[27/26]])<sup>2</sup> / ([[29/27]])<br />
| [[19683/19604]]<br />
|-<br />
| S28<sup>2</sup>*S29 = [[406/405]] * [[784/783]]<br />
| ([[28/27]])<sup>2</sup> / ([[15/14]])<br />
| [[10976/10935]]<br />
|-<br />
| S29<sup>2</sup>*S30 = [[435/434]] * [[841/840]]<br />
| ([[29/28]])<sup>2</sup> / ([[31/29]])<br />
| [[24389/24304]]<br />
|-<br />
| S30<sup>2</sup>*S31 = [[465/464]] * [[900/899]]<br />
| ([[30/29]])<sup>2</sup> / ([[16/15]])<br />
| [[3375/3364]]<br />
|-<br />
| S31<sup>2</sup>*S32 = [[496/495]] * [[961/960]]<br />
| ([[31/30]])<sup>2</sup> / ([[33/31]])<br />
| [[29791/29700]]<br />
|-<br />
| S32<sup>2</sup>*S33 = [[528/527]] * [[1024/1023]]<br />
| ([[32/31]])<sup>2</sup> / ([[17/16]])<br />
| [[16384/16337]]<br />
|-<br />
| S33<sup>2</sup>*S34 = [[561/560]] * [[1089/1088]]<br />
| ([[33/32]])<sup>2</sup> / ([[35/33]])<br />
| [[35937/35840]]<br />
|-<br />
| S34<sup>2</sup>*S35 = [[595/594]] * [[1156/1155]]<br />
| ([[34/33]])<sup>2</sup> / ([[18/17]])<br />
| [[9826/9801]]<br />
|-<br />
| S35<sup>2</sup>*S36 = [[630/629]] * [[1225/1224]]<br />
| ([[35/34]])<sup>2</sup> / ([[37/35]])<br />
| [[42875/42772]]<br />
|-<br />
| S36<sup>2</sup>*S37 = [[666/665]] * [[1296/1295]]<br />
| ([[36/35]])<sup>2</sup> / ([[19/18]])<br />
| [[23328/23275]]<br />
|-<br />
| S37<sup>2</sup>*S38 = [[703/702]] * [[1369/1368]]<br />
| ([[37/36]])<sup>2</sup> / ([[39/37]])<br />
| [[50653/50544]]<br />
|-<br />
| S38<sup>2</sup>*S39 = [[741/740]] * [[1444/1443]]<br />
| ([[38/37]])<sup>2</sup> / ([[20/19]])<br />
| [[6859/6845]]<br />
|-<br />
| S39<sup>2</sup>*S40 = [[780/779]] * [[1521/1520]]<br />
| ([[39/38]])<sup>2</sup> / ([[41/39]])<br />
| [[59319/59204]]<br />
|-<br />
| S40<sup>2</sup>*S41 = [[820/819]] * [[1600/1599]]<br />
| ([[40/39]])<sup>2</sup> / ([[21/20]])<br />
| [[32000/31941]]<br />
|-<br />
| S41<sup>2</sup>*S42 = [[861/860]] * [[1681/1680]]<br />
| ([[41/40]])<sup>2</sup> / ([[43/41]])<br />
| [[68921/68800]]<br />
|-<br />
| S42<sup>2</sup>*S43 = [[903/902]] * [[1764/1763]]<br />
| ([[42/41]])<sup>2</sup> / ([[22/21]])<br />
| [[18522/18491]]<br />
|-<br />
| S43<sup>2</sup>*S44 = [[946/945]] * [[1849/1848]]<br />
| ([[43/42]])<sup>2</sup> / ([[45/43]])<br />
| [[79507/79380]]<br />
|-<br />
| S44<sup>2</sup>*S45 = [[990/989]] * [[1936/1935]]<br />
| ([[44/43]])<sup>2</sup> / ([[23/22]])<br />
| [[42592/42527]]<br />
|-<br />
| S46<sup>2</sup>*S47 = [[1081/1080]] * [[2116/2115]]<br />
| ([[46/45]])<sup>2</sup> / ([[24/23]])<br />
| [[12167/12150]]<br />
|-<br />
| S49<sup>2</sup>*S50 = [[1225/1224]] * [[2401/2400]]<br />
| ([[49/48]])<sup>2</sup> / ([[51/49]])<br />
| [[117649/117504]]<br />
|-<br />
| S50<sup>2</sup>*S51 = [[1275/1274]] * [[2500/2499]]<br />
| ([[50/49]])<sup>2</sup> / ([[26/25]])<br />
| [[31250/31213]]<br />
|-<br />
| S52<sup>2</sup>*S53 = [[1378/1377]] * [[2704/2703]]<br />
| ([[52/51]])<sup>2</sup> / ([[27/26]])<br />
| [[70304/70227]]<br />
|-<br />
| S55<sup>2</sup>*S56 = [[1540/1539]] * [[3025/3024]]<br />
| ([[55/54]])<sup>2</sup> / ([[57/55]])<br />
| [[166375/166212]]<br />
|-<br />
| S56<sup>2</sup>*S57 = [[1596/1595]] * [[3136/3135]]<br />
| ([[56/55]])<sup>2</sup> / ([[29/28]])<br />
| [[87808/87725]]<br />
|-<br />
| S58<sup>2</sup>*S59 = [[1711/1710]] * [[3364/3363]]<br />
| ([[58/57]])<sup>2</sup> / ([[30/29]])<br />
| [[48778/48735]]<br />
|-<br />
| S63<sup>2</sup>*S64 = [[2016/2015]] * [[3969/3968]]<br />
| ([[63/62]])<sup>2</sup> / ([[65/63]])<br />
| [[250047/249860]]<br />
|-<br />
| S64<sup>2</sup>*S65 = [[2080/2079]] * [[4096/4095]]<br />
| ([[64/63]])<sup>2</sup> / ([[33/32]])<br />
| [[131072/130977]]<br />
|-<br />
| S66<sup>2</sup>*S67 = [[2211/2210]] * [[4356/4355]]<br />
| ([[66/65]])<sup>2</sup> / ([[34/33]])<br />
| [[71874/71825]]<br />
|-<br />
| S70<sup>2</sup>*S71 = [[2485/2484]] * [[4900/4899]]<br />
| ([[70/69]])<sup>2</sup> / ([[36/35]])<br />
| [[42875/42849]]<br />
|-<br />
| S75<sup>2</sup>*S76 = [[2850/2849]] * [[5625/5624]]<br />
| ([[75/74]])<sup>2</sup> / ([[77/75]])<br />
| [[421875/421652]]<br />
|-<br />
| S76<sup>2</sup>*S77 = [[2926/2925]] * [[5776/5775]]<br />
| ([[76/75]])<sup>2</sup> / ([[39/38]])<br />
| [[219488/219375]]<br />
|-<br />
| S78<sup>2</sup>*S79 = [[3081/3080]] * [[6084/6083]]<br />
| ([[78/77]])<sup>2</sup> / ([[40/39]])<br />
| [[59319/59290]]<br />
|}<br />
<br />
==== Bottom-heavy lopsided commas ====<br />
{| class="wikitable center-all<br />
|-<br />
! S-expression<br />
! Square Relation<br />
! Ratio<br />
|-<br />
| S3<sup>2</sup>*S2 = [[3/2]] * [[9/8]]<br />
| ([[3/1]]) / ([[4/3]])<sup>2</sup><br />
| [[27/16]]<br />
|-<br />
| S4<sup>2</sup>*S3 = [[6/5]] * [[16/15]]<br />
| ([[2/1]]) / ([[5/4]])<sup>2</sup><br />
| [[32/25]]<br />
|-<br />
| S5<sup>2</sup>*S4 = [[10/9]] * [[25/24]]<br />
| ([[5/3]]) / ([[6/5]])<sup>2</sup><br />
| [[125/108]]<br />
|-<br />
| S6<sup>2</sup>*S5 = [[15/14]] * [[36/35]]<br />
| ([[3/2]]) / ([[7/6]])<sup>2</sup><br />
| [[54/49]]<br />
|-<br />
| S7<sup>2</sup>*S6 = [[21/20]] * [[49/48]]<br />
| ([[7/5]]) / ([[8/7]])<sup>2</sup><br />
| [[343/320]]<br />
|-<br />
| S8<sup>2</sup>*S7 = [[28/27]] * [[64/63]]<br />
| ([[4/3]]) / ([[9/8]])<sup>2</sup><br />
| [[256/243]]<br />
|-<br />
| S9<sup>2</sup>*S8 = [[36/35]] * [[81/80]]<br />
| ([[9/7]]) / ([[10/9]])<sup>2</sup><br />
| [[729/700]]<br />
|-<br />
| S10<sup>2</sup>*S9 = [[45/44]] * [[100/99]]<br />
| ([[5/4]]) / ([[11/10]])<sup>2</sup><br />
| [[125/121]]<br />
|-<br />
| S11<sup>2</sup>*S10 = [[55/54]] * [[121/120]]<br />
| ([[11/9]]) / ([[12/11]])<sup>2</sup><br />
| [[1331/1296]]<br />
|-<br />
| S12<sup>2</sup>*S11 = [[66/65]] * [[144/143]]<br />
| ([[6/5]]) / ([[13/12]])<sup>2</sup><br />
| [[864/845]]<br />
|-<br />
| S13<sup>2</sup>*S12 = [[78/77]] * [[169/168]]<br />
| ([[13/11]]) / ([[14/13]])<sup>2</sup><br />
| [[2197/2156]]<br />
|-<br />
| S14<sup>2</sup>*S13 = [[91/90]] * [[196/195]]<br />
| ([[7/6]]) / ([[15/14]])<sup>2</sup><br />
| [[686/675]]<br />
|-<br />
| S15<sup>2</sup>*S14 = [[105/104]] * [[225/224]]<br />
| ([[15/13]]) / ([[16/15]])<sup>2</sup><br />
| [[3375/3328]]<br />
|-<br />
| S16<sup>2</sup>*S15 = [[120/119]] * [[256/255]]<br />
| ([[8/7]]) / ([[17/16]])<sup>2</sup><br />
| [[2048/2023]]<br />
|-<br />
| S17<sup>2</sup>*S16 = [[136/135]] * [[289/288]]<br />
| ([[17/15]]) / ([[18/17]])<sup>2</sup><br />
| [[4913/4860]]<br />
|-<br />
| S18<sup>2</sup>*S17 = [[153/152]] * [[324/323]]<br />
| ([[9/8]]) / ([[19/18]])<sup>2</sup><br />
| [[729/722]]<br />
|-<br />
| S19<sup>2</sup>*S18 = [[171/170]] * [[361/360]]<br />
| ([[19/17]]) / ([[20/19]])<sup>2</sup><br />
| [[6859/6800]]<br />
|-<br />
| S20<sup>2</sup>*S19 = [[190/189]] * [[400/399]]<br />
| ([[10/9]]) / ([[21/20]])<sup>2</sup><br />
| [[4000/3969]]<br />
|-<br />
| S21<sup>2</sup>*S20 = [[210/209]] * [[441/440]]<br />
| ([[21/19]]) / ([[22/21]])<sup>2</sup><br />
| [[9261/9196]]<br />
|-<br />
| S22<sup>2</sup>*S21 = [[231/230]] * [[484/483]]<br />
| ([[11/10]]) / ([[23/22]])<sup>2</sup><br />
| [[2662/2645]]<br />
|-<br />
| S23<sup>2</sup>*S22 = [[253/252]] * [[529/528]]<br />
| ([[23/21]]) / ([[24/23]])<sup>2</sup><br />
| [[12167/12096]]<br />
|-<br />
| S24<sup>2</sup>*S23 = [[276/275]] * [[576/575]]<br />
| ([[12/11]]) / ([[25/24]])<sup>2</sup><br />
| [[6912/6875]]<br />
|-<br />
| S25<sup>2</sup>*S24 = [[300/299]] * [[625/624]]<br />
| ([[25/23]]) / ([[26/25]])<sup>2</sup><br />
| [[15625/15548]]<br />
|-<br />
| S26<sup>2</sup>*S25 = [[325/324]] * [[676/675]]<br />
| ([[13/12]]) / ([[27/26]])<sup>2</sup><br />
| [[2197/2187]]<br />
|-<br />
| S27<sup>2</sup>*S26 = [[351/350]] * [[729/728]]<br />
| ([[27/25]]) / ([[28/27]])<sup>2</sup><br />
| [[19683/19600]]<br />
|-<br />
| S28<sup>2</sup>*S27 = [[378/377]] * [[784/783]]<br />
| ([[14/13]]) / ([[29/28]])<sup>2</sup><br />
| [[10976/10933]]<br />
|-<br />
| S29<sup>2</sup>*S28 = [[406/405]] * [[841/840]]<br />
| ([[29/27]]) / ([[30/29]])<sup>2</sup><br />
| [[24389/24300]]<br />
|-<br />
| S30<sup>2</sup>*S29 = [[435/434]] * [[900/899]]<br />
| ([[15/14]]) / ([[31/30]])<sup>2</sup><br />
| [[6750/6727]]<br />
|-<br />
| S31<sup>2</sup>*S30 = [[465/464]] * [[961/960]]<br />
| ([[31/29]]) / ([[32/31]])<sup>2</sup><br />
| [[29791/29696]]<br />
|-<br />
| S32<sup>2</sup>*S31 = [[496/495]] * [[1024/1023]]<br />
| ([[16/15]]) / ([[33/32]])<sup>2</sup><br />
| [[16384/16335]]<br />
|-<br />
| S33<sup>2</sup>*S32 = [[528/527]] * [[1089/1088]]<br />
| ([[33/31]]) / ([[34/33]])<sup>2</sup><br />
| [[35937/35836]]<br />
|-<br />
| S34<sup>2</sup>*S33 = [[561/560]] * [[1156/1155]]<br />
| ([[17/16]]) / ([[35/34]])<sup>2</sup><br />
| [[4913/4900]]<br />
|-<br />
| S35<sup>2</sup>*S34 = [[595/594]] * [[1225/1224]]<br />
| ([[35/33]]) / ([[36/35]])<sup>2</sup><br />
| [[42875/42768]]<br />
|-<br />
| S36<sup>2</sup>*S35 = [[630/629]] * [[1296/1295]]<br />
| ([[18/17]]) / ([[37/36]])<sup>2</sup><br />
| [[23328/23273]]<br />
|-<br />
| S37<sup>2</sup>*S36 = [[666/665]] * [[1369/1368]]<br />
| ([[37/35]]) / ([[38/37]])<sup>2</sup><br />
| [[50653/50540]]<br />
|-<br />
| S38<sup>2</sup>*S37 = [[703/702]] * [[1444/1443]]<br />
| ([[19/18]]) / ([[39/38]])<sup>2</sup><br />
| [[13718/13689]]<br />
|-<br />
| S39<sup>2</sup>*S38 = [[741/740]] * [[1521/1520]]<br />
| ([[39/37]]) / ([[40/39]])<sup>2</sup><br />
| [[59319/59200]]<br />
|-<br />
| S40<sup>2</sup>*S39 = [[780/779]] * [[1600/1599]]<br />
| ([[20/19]]) / ([[41/40]])<sup>2</sup><br />
| [[32000/31939]]<br />
|-<br />
| S41<sup>2</sup>*S40 = [[820/819]] * [[1681/1680]]<br />
| ([[41/39]]) / ([[42/41]])<sup>2</sup><br />
| [[68921/68796]]<br />
|-<br />
| S42<sup>2</sup>*S41 = [[861/860]] * [[1764/1763]]<br />
| ([[21/20]]) / ([[43/42]])<sup>2</sup><br />
| [[9261/9245]]<br />
|-<br />
| S43<sup>2</sup>*S42 = [[903/902]] * [[1849/1848]]<br />
| ([[43/41]]) / ([[44/43]])<sup>2</sup><br />
| [[79507/79376]]<br />
|-<br />
| S44<sup>2</sup>*S43 = [[946/945]] * [[1936/1935]]<br />
| ([[22/21]]) / ([[45/44]])<sup>2</sup><br />
| [[42592/42525]]<br />
|-<br />
| S45<sup>2</sup>*S44 = [[990/989]] * [[2025/2024]]<br />
| ([[45/43]]) / ([[46/45]])<sup>2</sup><br />
| [[91125/90988]]<br />
|-<br />
| S48<sup>2</sup>*S47 = [[1128/1127]] * [[2304/2303]]<br />
| ([[24/23]]) / ([[49/48]])<sup>2</sup><br />
| [[55296/55223]]<br />
|-<br />
| S50<sup>2</sup>*S49 = [[1225/1224]] * [[2500/2499]]<br />
| ([[25/24]]) / ([[51/50]])<sup>2</sup><br />
| [[15625/15606]]<br />
|-<br />
| S51<sup>2</sup>*S50 = [[1275/1274]] * [[2601/2600]]<br />
| ([[51/49]]) / ([[52/51]])<sup>2</sup><br />
| [[132651/132496]]<br />
|-<br />
| S54<sup>2</sup>*S53 = [[1431/1430]] * [[2916/2915]]<br />
| ([[27/26]]) / ([[55/54]])<sup>2</sup><br />
| [[39366/39325]]<br />
|-<br />
| S56<sup>2</sup>*S55 = [[1540/1539]] * [[3136/3135]]<br />
| ([[28/27]]) / ([[57/56]])<sup>2</sup><br />
| [[87808/87723]]<br />
|-<br />
| S57<sup>2</sup>*S56 = [[1596/1595]] * [[3249/3248]]<br />
| ([[57/55]]) / ([[58/57]])<sup>2</sup><br />
| [[185193/185020]]<br />
|-<br />
| S62<sup>2</sup>*S61 = [[1891/1890]] * [[3844/3843]]<br />
| ([[31/30]]) / ([[63/62]])<sup>2</sup><br />
| [[59582/59535]]<br />
|-<br />
| S64<sup>2</sup>*S63 = [[2016/2015]] * [[4096/4095]]<br />
| ([[32/31]]) / ([[65/64]])<sup>2</sup><br />
| [[131072/130975]]<br />
|-<br />
| S65<sup>2</sup>*S64 = [[2080/2079]] * [[4225/4224]]<br />
| ([[65/63]]) / ([[66/65]])<sup>2</sup><br />
| [[274625/274428]]<br />
|-<br />
| S68<sup>2</sup>*S67 = [[2278/2277]] * [[4624/4623]]<br />
| ([[34/33]]) / ([[69/68]])<sup>2</sup><br />
| [[157216/157113]]<br />
|-<br />
| S74<sup>2</sup>*S73 = [[2701/2700]] * [[5476/5475]]<br />
| ([[37/36]]) / ([[75/74]])<sup>2</sup><br />
| [[50653/50625]]<br />
|-<br />
| S76<sup>2</sup>*S75 = [[2850/2849]] * [[5776/5775]]<br />
| ([[38/37]]) / ([[77/76]])<sup>2</sup><br />
| [[219488/219373]]<br />
|-<br />
| S77<sup>2</sup>*S76 = [[2926/2925]] * [[5929/5928]]<br />
| ([[77/75]]) / ([[78/77]])<sup>2</sup><br />
| [[456533/456300]]<br />
|-<br />
| S80<sup>2</sup>*S79 = [[3160/3159]] * [[6400/6399]]<br />
| ([[40/39]]) / ([[81/80]])<sup>2</sup><br />
| [[256000/255879]]<br />
|}<br />
<br />
== Equivalent S-expressions ==<br />
=== Significance and meaning ===<br />
All S-expressions have other equivalent S-expressions, however when the equivalence makes one comma a member of two of the infinite families discussed on this page, or otherwise makes it equal to a product or ratio between two such commas, this often means exceptional and nontrivial ("deep") tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending how you temper further. Generally we exclude 1/n-square-particulars, only noting up to 1/3-square-particulars, because equivalent 1/n-square-particular expressions become very common as you allow higher n, but are still quite rare for small n.<br />
<br />
=== A useful general rule ===<br />
While there are likely arbitrarily many ways of rewriting S-expressions due to the redundancy in representation, the following equivalence is, due to its simplicity and elegance, arguably most likely to be useful:<br />
<br />
<math><br />
\large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)<br />
</math><br />
<br />
This is important to note because using this simple rule we can derive an infinite amount of trivially and obviously equivalent S-expressions that should be discarded from [[#Examples]]. See [[S-expression/Advanced results]] for mathematical details.<br />
<br />
=== Examples ===<br />
Here is an incomplete list of examples (feel free to expand with any equivalences you find that you think are valuable).<br />
<br />
'''Importantly:''' examples that can ''easily'' (with one or two algebraic rewriting steps) be shown to result from the aforementioned [[#A useful general rule|useful general rule]] are considered invalid/trivial.<br />
<br />
{| class="wikitable center-all<br />
|-<br />
! Comma<br />
! S-expressions<br />
|-<br />
| [[64/63]]<br />
| (S4*S5*S6)/S3 = S4/(S6*S7) = S8<br />
|-<br />
| [[81/80]]<br />
| S6/S8 = S9<br />
|-<br />
| [[176/175]]<br />
| S8/S10 = S22*S23*S24<br />
|-<br />
| [[243/242]]<br />
| S9/S11 = S15/([[3025/3024|S22/S24 = S55]])<br />
|-<br />
| [[325/324]]<br />
| S10/S12 = S25*S26<br />
|-<br />
| [[540/539]]<br />
| S12/S14 = (S9*S10)/S7 = (S6/S7)/(S8/S10)<br />
|-<br />
| [[676/675]]<br />
| S13/S15 = S26<br />
|-<br />
| [[1225/1224]]<br />
| S35 = S49*S50<br />
|-<br />
| [[3025/3024]]<br />
| S22/S24 = S55 = S25/S27 * S99<br />
|-<br />
| [[2601/2600]]<br />
| S17/(S25*S26) = S51<br />
|-<br />
| [[9801/9800]]<br />
| S99 = S33/S35<br />
|-<br />
| [[25921/25920]]<br />
| S161 = S46/S48<br />
|-<br />
| [[123201/123200]]<br />
| S351 = S78/S80<br />
|}<br />
<br />
Note: Where a comma written in the form a/b is used in an S-expression, this means to replace that comma with any equivalent S-expression. This is done in the case of [[3025/3024]] as there are many S-expressions for it so restating them each time it appears seems inconvenient.<br />
<br />
A proof that every positive rational number (and thus every JI interval) can be written as an S-expression follows.<br />
<br />
It suffices to show every superparticular number including 2/1 has an expression using square-particulars:<br />
<br />
<math>\displaystyle<br />
\begin{align}<br />
& 2/1 = S_2 \cdot S_2 \cdot S_3\ ,\\<br />
& 3/2 = S_2 \cdot S_3\ ,\\<br />
& 4/3 = S_2\ ,\\<br />
& \frac{a/(a - 1)}{(b + 1)/b} = \prod_{k=a}^b \left( S_k = \frac{k/(k - 1)}{(k + 1)/k} \right) \\<br />
& \ \ \ = \frac{a/(a - 1)}{(a + 1)/a} \cdot \frac{(a + 1)/a}{(a + 2)/(a + 1)} \cdot \frac{(a + 2)/(a + 1)}{(a + 3)/(a + 2)} \cdot\ \ldots \cdot \frac{b/(b - 1)}{(b + 1)/b} = \frac{a/(a - 1)}{(b + 1)/b} \\<br />
& \implies \frac{a/(a - 1)}{(b + 1)/b} = S_a \cdot S_{a + 1} \cdot S_{a + 2} \cdot\ \ldots \cdot S_b \\<br />
& \implies \frac{S_2 \cdot S_2 \cdot S_3}{\prod_{a = 2}^k S_a} = 2 \cdot \left( \frac{2/(2 - 1)}{(k + 1)/k} \right)^{-1} = 2 \cdot \left( \frac{(k + 1)/k}{2} \right) = (k + 1)/k<br />
\end{align}<br />
</math><br />
<br />
From here it should not be hard to see how to make any positive rational number. For 11/6, for example, we can do (11/10)(10/9)(9/8)…(2/1) = 11 and then divide that by (6/5)(5/4)(4/3)(3/2)(2/1), meaning 11/6 = (11/10)(10/9)(9/8)(8/7)(7/6) because of the cancellations, then each of those superparticulars we replace with the corresponding S-expression to get the final S-expression. This final S-expression is likely to be far from the most efficient or interesting expression; the redundancy in S-expressions is a strength and feature, as it tells us that there are more than the trivial connections between commas and intervals and that S-expressions can be wielded as a mathematical tool/language to investigate and identify them.<br />
<br />
== Glossary ==<br />
<br />
; Superparticular<br />
: The interval/comma between two consecutive harmonics. See [[superparticular]].<br />
: These are of the form (''k'' + 1)/''k''.<br />
<br />
; Square-particular<br />
: A superparticular interval/comma whose numerator is a square number. A shorthand (nick)name for square superparticular.<br />
: These are of the form ''k''<sup>2</sup>/(''k''<sup>2</sup> - 1) = S''k''.<br />
<br />
; Triangle-particular<br />
: A superparticular interval/comma whose numerator is a [[triangular number]]. A shorthand (nick)name for triangular superparticular. An alternative name for 1/2-square-particular.<br />
: These are of the form (''k''<sup>2</sup> + k)/(''k''<sup>2</sup> + k - 2). (This always simplifies to a superparticular.)<br />
<br />
; 1/''n''-square-particular<br />
: A comma which is the product of ''n'' consecutive square-particulars and which can therefore be expressed as the ratio between two superparticulars.<br />
: These are of the form S''a''*S(''a''+1)*…*S''b'' = (''a''/(''a'' - 1))/((''b'' + 1)/''b'') = ''ab''/((''a'' - 1)(''b'' + 1)).<br />
: Replacing/substituting ''a'' with ''k'' and ''b'' with ''k'' + ''n'' - 1 gives us an equivalent expression that includes the number of square-particulars ''n'':<br />
: S''k''*S(''k''+1)*…*S(''k''+''n''-1) = (''k''/(''k'' - 1))/((''k'' + ''n'')/(''k'' + ''n'' - 1)) = ''k''(''k'' + ''n'' - 1)/((''k'' - 1)(''k'' + ''n''))<br />
: For ''b'' = ''a'' + 1 these can also be called triangle-particulars, in which case they are always superparticular.<br />
: These have implications for whether consistency in the (''n''+''k'')=(''b''+1)-[[odd-limit]] is ''potentially'' possible in a given temperament; see the [[#Sk*S(k + 1)*…*S(k + n - 1) (1/n-square-particulars)|section on 1/n-square-particulars]].<br />
<br />
; Odd-particular<br />
: An interval/comma between two consecutive odd harmonics. The odd analogue of superparticular.<br />
: These are of the form (2''k'' + 1)/(2''k'' - 1).<br />
<br />
; Throdd-particular<br />
: An interval/comma between two harmonics 3 apart which is not superparticular.<br />
: These are of the form (3''k'' + 1)/(3''k'' - 2) or (3''k'' + 2)/(3''k'' - 1).<br />
<br />
; Quodd-particular<br />
: An interval/comma between two harmonics 4 apart which is not superparticular or odd-particular.<br />
: These are of the form (4''k'' + 1)/(4''k'' - 3) or (4''k'' + 3)/(4''k'' - 1).<br />
<br />
; ''n''-odd-particular<br />
: An interval/comma between two coprime harmonics ''n'' apart (also called as [[Delta-N ratio|delta-''n'' ratio]]). It is the generalization of superparticular, odd-particular, throdd-particular, and quodd-particular.<br />
: If ''n'' is a prime, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is not superparticular. For example, 5-odd-particular intervals are of the form (5''k'' + 1)/(5''k'' - 4), (5''k'' + 2)/(5''k'' - 3), (5''k'' + 3)/(5''k'' - 2) or (5''k'' + 4)/(5''k'' - 1).<br />
: If ''n'' is a composite, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is neither superparticular nor of ''m''-odd-particular intervals where ''m'' is any other divisor of ''n''. For example, 6-odd-particular intervals are of the form (6''k'' + 1)/(6''k'' - 5) or (6''k'' + 5)/(6''k'' - 1).<br />
<br />
; Ultraparticular<br />
: An interval/comma which is the ratio of two consecutive square-particulars.<br />
: These are of the form S''k''/S(''k'' + 1).<br />
<br />
; Semiparticular<br />
: A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:<br />
: These are of the form S''k''/S(''k'' + 2).<br />
<br />
; S-expression<br />
: An expression using the S''k'' shorthand notation corresponding strictly to multiplying and dividing only (arbitrary) square-particulars. S-expressions include singular square superparticulars and expressions for other superparticulars in terms of square superparticulars.<br />
<br />
; S-factorization<br />
: An expression that takes a list of consecutive integer harmonics including the ''k''th harmonic and raises them to integer powers, similar to a [[smonzo]] but uniquely suited to analysing S-expressions.<br />
: For example: S''k'' = [''k''-1, ''k'', ''k''+1]^[-1, 2, -1] because S''k'' = (''k''-1)<sup>-1</sup>''k''<sup>2</sup>(''k''+1)<sup>-1</sup>.<br />
<br />
; S-comma<br />
: Any comma within one of the infinite families of commas discussed here, excluding 1/n-square-particulars for n>5 (so square-particulars, triangle-particulars, 1/3-square-particulars, 1/4-square-particulars and 1/5-square-particulars are included, but not anything beyond; this bound is used for exclusion (rather than n>3) to allow the utility of 1/5-square-particulars in avoiding twin primes by equating superparticular intervals on either side of the twin primes).<br />
<br />
; Indirect S-comma<br />
: Any comma that is the product or ratio of two S-commas. These appear frequently as S-expressions for commas that are more challenging/nontrivial to represent from the perspective of S-expressions, for example the [[schisma]] admits at least three such representations!<br />
<br />
== See further ==<br />
* [[S-expression/Advanced results|Advanced results]] – for the harder-to-reach algebra<br />
<br />
== Notes ==<br />
<references group="note" /><br />
<br />
[[Category:Elementary math]]<br />
[[Category:Pages with proofs]]<br />
[[Category:Ratio]]<br />
[[Category:Terms]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Template:Interval_table&diff=224387Template:Interval table2026-02-20T11:34:46Z<p>ArrowHead294: </p>
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<div>{{Texmap}}<br />
{{Beginner}}<br />
<br />
== Interval math ==<br />
Xen discussion uses two kinds of units:<br />
* ''Frequency ratios''<br />
** The [[frequency]] is the absolute pitch of any given tone, usually measured in [[hertz]] (Hz). The ratio between frequencies is just a number. Equal intervals have the same [[frequency ratio]].<br />
* ''Logarithmic units'' such as [[cents]] and [[edo]] steps that treat intervals we hear as equal as the same additive unit<br />
<br />
To stack two intervals, we use different types of operations for the two kinds of units. To stack two intervals written as ratios, we ''multiply'', whereas to stack two intervals written as cents or edo steps, we ''add'' the intuitive way. To "unstack" an interval from another interval, we ''divide'' the respective ratios and ''subtract'' logarithmic units. To convert between cents and ratios we use the following formulas:<br />
<br />
<math><br />
\begin{align}<br />
\text{cents} &= 1200 \cdot \log_{2} \left( \text{ratio} \right) \\<br />
\text{ratio} &= 2^{\left( \text{cents}/1200 \right)}<br />
\end{align}<br />
</math><br />
<br />
The unison has frequency ratio 1/1 and is 0 cents. The octave has frequency ratio 2/1 and is exactly 1200 cents. A standard semitone (in 12edo) has frequency ratio <math>\sqrt[12]{2}</math> and is exactly 100 cents (by definition).<br />
<br />
The notation ''m''\''n'' means ''m'' steps of ''n''-edo. For example, 12edo's perfect fifth can be denoted as 7\12, meaning "7 steps of 12-tone equal temperament".<br />
<br />
A common operation in xen math is the [[mediant]]. The mediant of two fractions, ''a''/''b'' and ''c''/''d'', is the "freshman sum" {{sfrac|''a'' + ''b''|''c'' + ''d''}}, which is always between ''a''/''b'' and ''c''/''d''. For example, the mediant of 4/3, the just perfect fourth, and 5/4, the just major third, is 9/7, the supermajor third. If two fractions are in lowest terms, their mediant is the simplest fraction that is strictly between both. The mediant is commonly used for both JI ratios and edo intervals.<br />
<br />
Another important operation is [[octave reduction|reduction]]. To reduce an interval a by an interval b means to stack or "unstack" ''b'' from ''a'' until ''a'' is at least the unison and less than ''b''. For example, 3/1 reduced by 2/1 is 3/2.<br />
<br />
== Basic JI ==<br />
[[Just intonation]] (JI) is the set of intervals that are tuned to rational frequency ratios, ones can be written as fractions of whole numbers.<br />
<br />
The easiest way to get concordance (smoothness, blending and buzzing) is to use low-numbered JI ratios in your interval or chord, for example the just perfect fifth [[3/2]], the just major third [[5/4]], and the lesser septimal tritone [[7/5]]. When pure JI ratios are used, a psychoacoustic effect called JI buzz occurs. When the overall chord is low number JI, such as 8:9:10:11:12:13:14, the result is very concordant.<br />
<br />
No edo interval except for the octave (2/1) and stacks of it is exact JI. A JI ratio might be far from a 12edo interval; for example 7/4 is 969 cents. This is another reason why JI is a common approach to xen.<br />
<br />
As stacking JI ratios involves multiplying, primes are important as the simplest building blocks of arbitrary JI ratios. So we can write every ratio as a vector called a ''monzo'', a list of powers for primes. We can visualize each ratio as living in some JI lattice (the set of all intervals built by stacking a finite set of basic intervals).<br />
<br />
There are many approaches to JI music: lattice-based JI, constant structure scales, free JI, primodality, tonality diamonds, combination product sets…<br />
<br />
JI is usually less mathy than RTT.<br />
<br />
The approach that RTT cares about the most is lattice-based JI. A JI lattice, or a subgroup, is built by stacking a finite set of JI intervals, usually primes such as 2, 3, 5, and 7.<br />
<br />
There are two ways the term ''[[limit]]'' is used.<br />
* The ''[[harmonic limit|p-prime-limit]]'' is the lattice built by multiplying the primes at most ''p'', possibly multiple times. We write a JI lattice by writing the basic intervals separated by periods. For example, {{nowrap|6/5 {{=}} 2 × 3 / 5}} is in the 5-limit, or the 2.3.5 subgroup, and so is {{nowrap|45/32 {{=}} (3 × 3 × 5) / (2 × 2 × 2 × 2 × 2)}}.<br />
* The ''[[odd limit|q-odd-limit]]'' is the set of all JI ratios where the larger of the numerator and denominator after removing factors of 2 from the JI ratios is at most the odd number ''q''. For example, 7/6, 13/5, 11/10, and 16/15 are all in the 15-odd-limit.<br />
<br />
== Basic RTT ==<br />
''Assuming several things from common 12edo practice'', JI has several disadvantages. To get infinite modulation and have exactly the same chords on every note, we need infinitely many notes unlike the finitely many notes of 12edo. JI with such infinite modulation and regularity also has small intervals that may be undesirable, called commas. This is the problem that [[regular temperament theory]] (RTT) exists to solve. Regular temperaments equate certain intervals by considering the difference between them as a comma and "[[tempering out]]" the difference. However, note that one need not treat JI like one would an edo, and that some regular temperament tunings are infinite and don't provide the advantages of finiteness.<br />
<br />
From the perspective of an edo user, another problem RTT solves is that there are very few small edos and they do not constitute that wide a palette. Especially in larger edos, RTT provides a way of not being overwhelmed with dozens of notes.<br />
<br />
RTT views edos as regular temperaments. Under this view, edos simplify the infinite JI space to a finite set, deforming the intervals so that certain chosen intervals vanish. We can also approach simplifying JI ratios from edos themselves, namely how edos approximate each prime. This is a vector called a [[val]]. Vals map primes to a set number of edo steps and thus tell us how many edo steps each interval in JI is mapped to. The usual 12edo val (called the 12edo [[patent val]]) in the 5-limit is {{val| 12 19 28 }}, as the 12edo intervals that are closest to 2/1, 3/1 and 5/1 are 12, 19 and 28 steps respectively.<br />
<br />
There are various temperaments in xen with varying levels of practicality. The most important one to know is probably [[meantone]] temperament, which equates four fifths ({{nowrap|(3/2)<sup>4</sup> {{=}} 81/16}}) with a major third plus two octaves ({{nowrap|(5/4) × 4 = 5 {{=}} 80/16}}), which is encoded by tempering out the syntonic comma [[81/80]] (monzo {{monzo| -4 4 -1 }}). <br />
<br />
A val tempers out a comma if, when you construct the comma from primes according to their tunings in the val, the result is 0 cents or the unison. For example, 12edo is a meantone edo because:<br />
* The patent val for 12edo in the 5-limit is {{val| 12 19 28 }}.<br />
* The comma 81/80 has monzo {{monzo| -4 4 -1 }}.<br />
* Constructing the tuning of a comma from mappings of primes involves multiplying each entry in the val to a corresponding entry in the comma's monzo, and then adding the resulting numbers together; this operation is called a "dot product".<br />
** {{nowrap|12 × (−4) {{=}} −48}}, corresponding to going down 4 octaves.<br />
** {{nowrap|19 × 4 {{=}} 76}}, corresponding to going up 4 perfect twelfths (or, to going up 4 octaves and 4 fifths).<br />
** {{nowrap|28 × (−1) = −28}}, corresponding to dividing by 5 (going down two octaves and a major third).<br />
** {{nowrap|(76 − 48) − 28 {{=}} 0}}<br />
* Since the result is 0, 12edo supports meantone. In RTT math, this can be written as:<br />
<br />
<math>\vmp{12 & 19 & 28}{-4 & 4 & -1} = 12 \times \left(-4\right) + 19 \times 4 + 28 \times \left(-1\right) = -48 + 76 - 28 = 0.</math><br />
<br />
== MOS scales ==<br />
[[Mos]] (moment of symmetry) scales are one way to generalize the diatonic scale; the diatonic scale is a mos scale. They are scales with two step sizes (large (L) and small (s)) with a uniquely elegant combination of properties:<br />
<br />
For every number of steps, the scale has at most two interval sizes with that number of steps. The scale can be made by stacking a certain fixed interval called the ''[[periods and generators|generator]]'' (and reducing by an interval called the ''[[periods and generators|period]]'', usually the octave or some equal division of it such as 1\2 or 1\3), over and over, stopping at some point where there are two step sizes distributed as evenly as possible.<br />
<br />
Every mos scale with ''m'' large steps and ''n'' small steps is a mode of some pattern. This is why you only need to write ''m''L&nbsp;''n''s for an octave-equivalent mos scale and specify the mode (using [[UDP]] for example). For example, every 5L&nbsp;3s mos scale is a mode of the pattern LLsLLsLs.<br />
<br />
An important way that mos scales vary is [[hardness]], defined as the size (in cents) of the L divided by the size (in cents) of the s step. Hardness can range from 1 to infinity. The larger the hardness, the harder the mos tuning; the smaller (closer to 1) the hardness, the softer the tuning. The two extremes are where the mos pattern no longer holds; 1 is where L and s steps are equal, and infinity is where s is so small that it disappears.<br />
<br />
Any given mos pattern is available in more than one edo, and the basic tuning of a mos pattern gives the smallest edo that provides that mos pattern. To adjust the hardness of a mos provided by an edo, we can add two edos, obtaining an edo where the hardness is the mediant of the two original edos'. For a diatonic example, 12edo has basic ({{nowrap|L/s {{=}} 2/1}}) diatonic, 17edo has hard ({{nowrap|L/s {{=}} 3/1}}) diatonic, and 19edo has soft ({{nowrap|L/s {{=}} 3/2}}) diatonic. {{nowrap|12 + 19 {{=}} 31}}, and 31edo diatonic has hardness {{nowrap|{{sfrac|2 + 3|1 + 2}} {{=}} 5/3}}.<br />
<br />
The generator size and the period thus determine the mos scales that can be obtained. Hardness varies with generator size within a mos's range.<br />
<br />
Every mos scale pattern has a generator range. Since the familiar diatonic scale is a mos 5L&nbsp;2s, here is an important fact to know: If the period is the octave and the generator is a fifth between 4\7 (686{{c}}) and 3\5 (720{{c}}), the resulting pattern is the diatonic mos.<br />
<br />
[[TAMNAMS]] is a common method for naming intervals of a mos scale.<br />
<br />
The table below shows the tuning spectrum for the diatonic scale and the temperaments each subset is associated with:<br />
<br />
{| class="wikitable" style="margin: auto auto auto auto;"<br />
|+ style="font-size: 105%;" | Tuning ranges of the diatonic mos<br />
|-<br />
! colspan="3" | Range !! rowspan="2" colspan="2" | Temperaments encompassed<br />
|-<br />
! Edo !! Cents !! Hardness<br />
|-<br />
| 7edo to 33edo || 685.714 to 690.909 || 1/1 to 5/4 || colspan="2" | [[Deeptone]]<br />
|-<br />
| 33edo to 19edo || 690.909 to 694.737 || 5/4 to 3/2 || rowspan="2" | Meantone || [[Flattertone]], [[flattone]]<br />
|-<br />
| 19edo to 12edo || 694.737 to 700.000 || 3/2 to 2/1 || [[Septimal meantone]]<br />
|-<br />
| 12edo to 29edo || 700.000 to 703.448 || 2/1 to 5/2 || colspan="2" | [[Schismic]]<br />
|-<br />
| 29edo to 17edo || 703.448 to 705.882 || 5/2 to 3/1 || colspan="2" | [[Pepperoni]], [[leapday]]<br />
|-<br />
| 17edo to 22edo || 705.882 to 709.091 || 3/1 to 4/1 || rowspan="3" | Superpyth || [[Quasisuper]]<br />
|-<br />
| 22edo to 27edo || 709.091 to 711.111 || 4/1 to 5/1 || [[Superpyth]]<br />
|-<br />
| 27edo to 5edo || 711.111 to 720.000 || 5/1 to &infin; || [[Ultrapyth]]<br />
|}<br />
<br />
== Edos ==<br />
* [[5edo]]: Equalized pentatonic ("equipentatonic").<br />
* [[7edo]]: Equalized diatonic ("equiheptatonic").<br />
* [[9edo]]: The simplest edo with a [[2L&nbsp;5s]] mos (sssLssL). This mos is of interest because it can be viewed as a tuning of the diatonic scale where whole steps are smaller than half steps (an "antidiatonic" scale). The corresponding temperament is [[mavila]], which is llike meantone except major and minor intervals are swapped. Some larger edos like 16edo and 23edo tune it better, though mavila has poor accuracy in general since the fifth is very flat.<br />
* [[11edo]]: Stretched 12edo, has [[4L&nbsp;3s]] mos (LLsLsLs) which is a stretched diatonic.<br />
* [[13edo]]: Compressed 12edo having the [[5L&nbsp;3s]] mos (LLsLLsLs) which is a compressed version of the diatonic scale.<br />
* [[15edo]]: The smallest edo with a [[5L&nbsp;5s]] mos (LsLsLsLsLs) commonly called the blackwood scale. Also the smallest with a [[7L&nbsp;1s]] mos (LLLLsLLL). Both scales are known for supporting relatively familiar major and minor chords with relatively unfamiliar melodic structures.<br />
* [[16edo]]: Has 2L&nbsp;5s (sssLssL) and [[7L&nbsp;2s]] (LLLsLLLLs), generated by the mavila temperament, for which it is a more accurate tuning than 9edo.<br />
* [[17edo]]: The smallest edo after 12edo with a diatonic scale, and the smallest after 12edo to provide perfect fifths which are consonant for most purposes. Its major intervals are sharper and its minor intervals flatter than in 12edo, so it is often said to have a dramatic sound. First neutral diatonic edo (providing neutral seconds, thirds, sixths, and sevenths).<br />
* [[18edo]]: Has two fifths, 733{{c}} and 667{{c}}, that are nearly equally off from [[3/2]].<br />
* [[19edo]]: The smallest edo after 12edo which supports [[meantone]]. Just major and minor thirds are better approximated than in 12edo, but perfect fifths are represented significantly worse. First [[interordinal]] diatonic edo (interordinals are semifourths, semisixths, semitenths, and semitwelfths).<br />
* [[22edo]]: Diatonic mos with a fifth significantly sharper than just, so that it has supermajor and subminor thirds (approximately [[9/7]] and [[7/6]]) for its major and minor thirds. Has a 5-limit major third (approximate [[5/4]]) which ''cannot'' be reached by stacking four fifths. Supports [[superpyth]] and 7L&nbsp;1s.<br />
* [[23edo]]: The largest edo without a diatonic, 5edo, or 7edo fifth. Supports mavila like 9edo and 16edo with the flat fifth.<br />
* [[24edo]]: Has both neutral thirds (and other neutral intervals) and semifourths (and other interordinals), each of these lending itself to different harmony. Has 12edo mos scales as well as new ones.<br />
* [[26edo]]: Even softer diatonic mos than 19edo, so much that the diatonic major third is nearly exactly [[26/21]] and the diatonic minor second is nearly exactly [[13/12]]. The [[7/4]] is also nearly exact, and the edo also has a good [[10/9]], [[14/11]], and [[11/8]].<br />
* [[27edo]]: Even harder diatonic mos than 22edo; the fifth is approximately about as sharp (by 9.2{{c}}) as 26edo's is flat (by 9.6{{c}}). It has 12edo's [[5/4]], a near-exact [[7/6]], and an approximate [[16/13]] neutral third.<br />
* [[29edo]]: First edo with a perfect fifth closer to just intonation than 12edo. The minor third is extremely close to just [[13/11]]. It offers a tuning of 7L&nbsp;1s with more consonant fifths than 15edo or 22edo before it. Its diatonic scale has similar melodic properties to 17edo, although subtler.<br />
* [[31edo]]: One of the most popular meantone edos. Close to historical [[quarter-comma meantone]]. Not only is its major third close to just [[5/4]], it also matches the harmonic seventh [[7/4]] well.<br />
* [[34edo]]: Good for the 5-limit (2.3.5), as it does not temper out 81/80 and has a good 5/4. Also contains all notes of 17edo.<br />
* [[36edo]]: Good for primes [[3/2|3]] and [[7/4|7]].<br />
* [[37edo]]: Good for primes [[5/4|5]], [[7/4|7]], [[11/8|11]] and [[13/8|13]], but renders 3/2 sharp, even more so than 27edo.<br />
* [[41edo]]: Often considered remarkably good for the primes up to 11. Good 3; flat 5 and 7; sharp 11 and 13. Known for the [[Kite guitar]].<br />
* [[46edo]]: Neogothic 3; sharp 5; flat 7, 11, and 13; good 17. Supports [[parapyth]]. Often compared to 41edo; some favor one, some the other.<br />
* [[53edo]]: Is a stack of near-just 3/2's which also approximates primes 5, 7, 13, and 19.<br />
* [[72edo]]: A notable subdivision of 12edo that is a very strong 11-limit (primes 2, 3, 5, 7, 11) temperament for its size.<br />
* [[87edo]]: Even better in 2.3.5.11.13 than 72edo is in the 11-limit, and a consistent and precise edo for approximating harmonics 8 to 16, but ratios with 7 suffer due to the 7 being flat and the 3 being sharp.<br />
* [[311edo]]: An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41. The final boss of RTT edos.<br />
<br />
[[Category:Overview]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Quasisuper&diff=224309Quasisuper2026-02-19T13:07:31Z<p>ArrowHead294: </p>
<hr />
<div>{{Infobox regtemp<br />
| Title = Quasisuper;&nbsp;quasisupra<br />
| Subgroups = 2.3.5.7, 2.3.5.7.11<br />
| Comma basis = [[64/63]], [[2430/2401]] (7-limit);<br>[[64/63]], [[99/98]], [[121/120]] (11-limit)<br />
| Edo join 1 = 17c | Edo join 2 = 22<br />
| Mapping = 1; 1 -13 -2 -6<br />
| Generators = 3/2<br />
| Generators tuning = 708.3<br />
| Optimization method = CWE<br />
| MOS scales = [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[5L&nbsp;12s]], [[17L&nbsp;5s]]<br />
| Pergen = (P8, P5)<br />
| Color name = Sasaguti<br />
| Odd limit 1 = 9 | Mistuning 1 = 13.7 | Complexity 1 = 17<br />
| Odd limit 2 = 11-limit 15 | Mistuning 2 = 14.9 | Complexity 2 = 17<br />
}}<br />
<br />
'''Quasisuper''' is an alternative [[extension]] of the [[archy]] [[chain of fifths]] to [[superpyth]]. Like superpyth, it is a [[regular temperament|temperament]] generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]][[~]][[9/8]], [[tempering out]] [[64/63]]. The difference is that this extension maps [[prime interval|prime]] [[5/1|5]] to −13 [[generator]]s, as a double-diminished fifth (C–G𝄫). This extension works in the range [[17edo|17c-edo]] to [[22edo|22-edo]]. In contrast, full 7-limit [[superpyth]] does not work in this range, as tunings with a flatter fifth than 22edo swap the sizes of [[7/5]] and [[10/7]]. This extension may be preferred over superpyth due to having a softer [[5L 2s|diatonic]] scale, with a small step of around 60 [[cent]]s compared to about 50 cents in regular 7-limit superpyth. <br />
<br />
The best extension to the [[11-limit]], '''quasisupra''', maps prime [[11/1|11]] to −6 generators as a diminished fifth (C–G♭), tempering out [[99/98]] as well as [[121/120]] and [[540/539]]. Removing prime 5 from quasisupra results in a 2.3.7.11-subgroup restriction, called '''supra''', which is notable for its simplicity. Finally, taking every other step of supra gives a 2.9.7.11-subgroup restriction, called [[machine]]. <br />
<br />
For technical data see [[Archytas clan #Quasisuper]] and [[Archytas clan #Supra|#Supra]].<br />
<br />
== Interval chain ==<br />
In the following tables, odd harmonics and subharmonics 1–11 are in '''bold'''.<br />
<br />
<div><div style="display: inline-grid; margin-right: 25px;"><br />
{| class="wikitable center-1 right-2 right-4"<br />
|+ style="font-size: 105%;" | Supra (2.3.7.11)<br />
|-<br />
! # !! Cents* !! Approximate ratios<br />
|-<br />
| 0 || 0.0 || '''1/1'''<br />
|-<br />
| 1 || 707.5 || '''3/2'''<br />
|-<br />
| 2 || 215.0 || '''8/7''', '''9/8'''<br />
|-<br />
| 3 || 922.5 || 12/7<br />
|-<br />
| 4 || 430.0 || 9/7, 14/11<br />
|-<br />
| 5 || 1137.5 || 21/11, 27/14, 64/33<br />
|-<br />
| 6 || 645.0 || '''16/11'''<br />
|-<br />
| 7 || 152.5 || 12/11<br />
|}<br />
<nowiki/>* In 2.3.7.11-subgroup [[CWE]] tuning, <br>octave reduced<br />
</div></div><br />
<div><div style="display: inline-grid;"><br />
{| class="wikitable center-1 right-2"<br />
|+ style="font-size: 105%;" | Quasisuper/quasisupra<br />
|-<br />
! #<br />
! Cents*<br />
! Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| 708.3<br />
| '''3/2'''<br />
|-<br />
| 2<br />
| 216.6<br />
| '''8/7''', '''9/8'''<br />
|-<br />
| 3<br />
| 925.0<br />
| 12/7<br />
|-<br />
| 4<br />
| 433.3<br />
| 9/7, 14/11<br />
|-<br />
| 5<br />
| 1141.6<br />
| 21/11, 27/14<br />
|-<br />
| 6<br />
| 649.9<br />
| '''16/11''', 22/15<br />
|-<br />
| 7<br />
| 158.2<br />
| 11/10, 12/11<br />
|-<br />
| 8<br />
| 866.6<br />
| 18/11<br />
|-<br />
| 9<br />
| 374.9<br />
| 27/22, 56/45<br />
|-<br />
| 10<br />
| 1083.2<br />
| 28/15<br />
|-<br />
| 11<br />
| 591.5<br />
| 7/5<br />
|-<br />
| 12<br />
| 99.8<br />
| 16/15<br />
|-<br />
| 13<br />
| 808.2<br />
| '''8/5'''<br />
|-<br />
| 14<br />
| 316.5<br />
| 6/5<br />
|-<br />
| 15<br />
| 1024.8<br />
| 9/5<br />
|-<br />
| 16<br />
| 533.1<br />
| 27/20<br />
|-<br />
| 17<br />
| 41.4<br />
| 81/80, 56/55<br />
|}<br />
<nowiki/>* in 11-limit CWE tuning, octave reduced<br />
</div></div><br />
<br />
== Scales ==<br />
=== Scala files ===<br />
* [[Supra7]] – in 56edo tuning<br />
* [[Supra12]] – in 56edo tuning<br />
* [[12-22a]] – in 22edo tuning<br />
<br />
== Tunings ==<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings<br />
|-<br />
! rowspan="2" | <br />
! colspan="3" | Euclidean<br />
|-<br />
! Constrained<br />
! Constrained & skewed<br />
! Destretched<br />
|-<br />
! Tenney<br />
| CTE: ~3/2 = 708.7690{{c}}<br />
| CWE: ~3/2 = 708.3716{{c}}<br />
| POTE: ~3/2 = 708.2385{{c}}<br />
|}<br />
<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings<br />
|-<br />
! rowspan="2" | <br />
! colspan="3" | Euclidean<br />
|-<br />
! Constrained<br />
! Constrained & skewed<br />
! Destretched<br />
|-<br />
! Tenney<br />
| CTE: ~3/2 = 708.7131{{c}}<br />
| CWE: ~3/2 = 708.3200{{c}}<br />
| POTE: ~3/2 = 708.2046{{c}}<br />
|}<br />
<br />
=== Tuning spectrum ===<br />
{| class="wikitable center-all left-4"<br />
|-<br />
! Edo<br>generator<br />
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*<br />
! Generator (¢)<br />
! Comments<br />
|-<br />
| [[12edo|7\12]]<br />
| <br />
| 700.000<br />
| 12cc val<br />
|-<br />
| <br />
| 3/2<br />
| 701.955<br />
| Pythagorean tuning<br />
|-<br />
| <br />
| 21/11<br />
| 703.893<br />
| <br />
|-<br />
| <br />
| 11/7<br />
| 704.377<br />
| <br />
|-<br />
| '''[[17edo|10\17]]'''<br />
| <br />
| '''705.882'''<br />
| 17c val, '''lower bound of 7-, 9-, and 11-odd-limit diamond monotone'''<br />
|-<br />
| <br />
| 11/9<br />
| 706.574<br />
| <br />
|-<br />
| <br />
| 21/20<br />
| 707.039<br />
| <br />
|-<br />
| [[56edo|33\56]]<br />
| <br />
| 707.143<br />
| 56cd val<br />
|-<br />
| <br />
| 11/6<br />
| 707.234<br />
| <br />
|-<br />
| <br />
| 7/5<br />
| 707.501<br />
| <br />
|-<br />
| [[39edo|23\39]]<br />
| <br />
| 707.692<br />
| 39d val<br />
|-<br />
| <br />
| 9/5<br />
| 707.840<br />
| <br />
|-<br />
| <br />
| 15/14<br />
| 708.056<br />
| <br />
|-<br />
| <br />
| 11/8<br />
| 708.114<br />
| <br />
|-<br />
| [[61edo|36\61]]<br />
| <br />
| 708.197<br />
| 61d val<br />
|-<br />
| <br />
| 5/3<br />
| 708.260<br />
| <br />
|-<br />
| <br />
| 5/4<br />
| 708.745<br />
| <br />
|-<br />
| <br />
| 9/7<br />
| 708.771<br />
| <br />
|-<br />
| '''[[22edo|13\22]]'''<br />
| <br />
| '''709.091'''<br />
| '''Upper bound of 7-, 9-, and 11-odd-limit diamond monotone'''<br />
|-<br />
| <br />
| 11/10<br />
| 709.286<br />
| <br />
|-<br />
| <br />
| 15/8<br />
| 709.311<br />
| <br />
|-<br />
| <br />
| 15/11<br />
| 710.508<br />
| <br />
|-<br />
| <br />
| 7/6<br />
| 711.043<br />
| <br />
|-<br />
| [[27edo|16\27]]<br />
| <br />
| 711.111<br />
| 27c val<br />
|-<br />
| <br />
| 7/4<br />
| 715.587<br />
| <br />
|-<br />
| [[5edo|3\5]]<br />
| <br />
| 720.000<br />
| 5c val<br />
|-<br />
| <br />
| 21/16<br />
| 729.219<br />
| <br />
|}<br />
<nowiki/>* Besides the octave<br />
<br />
[[Category:Quasisuper| ]] <!-- main article --><br />
[[Category:Rank-2 temperaments]]<br />
[[Category:Archytas clan]]<br />
[[Category:Nuwell temperaments]]<br />
[[Category:Hemimage temperaments]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Pajara&diff=224239Pajara2026-02-18T11:41:46Z<p>ArrowHead294: </p>
<hr />
<div>{{interwiki<br />
| en = Pajara<br />
| de = Pajara<br />
| es = <br />
| ja = <br />
}}<br />
{{Infobox regtemp<br />
| Title = Pajara<br />
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.17<br />
| Comma basis = [[50/49]], [[64/63]] (7-limit);<br>[[50/49]], [[64/63]], [[99/98]] (11-limit);<br>[[50/49]], [[64/63]], [[85/84]], [[99/98]]<br>(2.3.5.7.11.17)<br />
| Edo join 1 = 12 | Edo join 2 = 22<br />
| Mapping = 2; 1 -2 -2 -6 1<br />
| Generators = 3/2 | Generators tuning = 707.4 | Optimization method = CWE<br />
| MOS scales = [[2L&nbsp;8s]], [[10L&nbsp;2s]], [[12L&nbsp;10s]]<br />
| Pergen = (P8/2, P5)<br />
| Odd limit 1 = 9 | Mistuning 1 = 17.5 | Complexity 1 = 10<br />
| Odd limit 2 = 2.3.5.7.11.17 21 | Mistuning 2 = 22.4 | Complexity 2 = 22<br />
}}<br />
<br />
'''Pajara''' (pronounced /pəˈd͡ʒɑːrə/, with the J as in "jar") is a [[regular temperament|temperament]] with a half-octave [[period]] that represents both [[7/5]] and [[10/7]], so [[50/49]] is [[tempering out|tempered out]] and it is in the [[jubilismic clan]]. The [[generator]] is a [[3/2|perfect fifth]] in the neighborhood of 707–711&nbsp;[[cent]]s, or that minus a half-octave period, which is a semitone representing [[15/14]] and [[16/15]]. One period minus 2 such semitones is [[~]][[5/4]], which, if you work it out, implies that [[2048/2025]] is tempered out, so pajara is also in the [[diaschismic family]]. In fact, it shares the same structure as 5-limit [[diaschismic]]. Finally, two 4/3's (or an octave minus two semitones) represents 7/4 as well as 16/9, so [[64/63]] is tempered out and pajara is in the [[archytas clan]]. Tempering out any two of these commas (among others) produces the unique temperament pajara. <br />
<br />
Pajara has fairly low accuracy overall, due to the ~5/4 and ~7/4 necessarily being separated by 600 cents via vanishing of [[50/49]]. However, if one accepts the accuracy of [[12edo]] in the 5-limit, they would probably accept the accuracy of pajara as well. The vanishing of [[50/49]] means that [[49/48]] and [[25/24]] are tempered to the same interval, and allows for a simple alteration to produce the subharmonic sixth chord [[70:84:105:120|1/(12:10:8:7)]] with 6/5 and 12/7 by flattening the third and seventh the same amount from the harmonic seventh chord, [[4:5:6:7]]. <br />
<br />
Pajara has [[mos scale]]s of 10, 12, and 22 notes. The 10-note mos, Pajara[10], is notable for sharing a number of desirable properties with [[5L 2s|diatonic]], while having fundamentally different categories; for example, the ~7/4 is a now major 8-step, rather than a minor 6-step. This mos and the LsssLsssss [[modmos]] are called the ''symmetric'' and ''pentachordal'' decatonic scales and were independently invented/discovered by [[Paul Erlich]]<ref>Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]</ref> and [[Gene Ward Smith]]. They are often thought of as subsets of [[22edo]], without much loss of generality and accuracy.<br />
<br />
As does all diaschismic temperaments, pajara has a natural extension to prime [[17/1|17]], obtained by tempering out [[136/135]], [[256/255]], and [[289/288]]. This extension notably also tempers out [[120/119]], which equates the 1/(12:10:8:7) utonal tetrad with the otonal [[10:12:15:17]].<br />
<br />
See [[Diaschismic family #Pajara]] for technical data. See [[Pajara extensions]] for a discussion on the 11-limit extensions. <br />
<br />
== Interval chains ==<br />
There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but 22edo is the only [[11-odd-limit]] [[diamond monotone]] tuning, where primes [[3/1|3]] and [[5/1|5]] are less accurate than in optimal tunings of canonical 11-limit pajara.<br />
<br />
In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''. <br />
<br />
{| class="wikitable center-1 right-2 right-4"<br />
|+ style="font-size: 105%;" | Pajara (12&nbsp;&amp;&nbsp;22)<br />
|-<br />
! rowspan="2" | #<br />
! colspan="2" | Period 0<br />
! colspan="2" | Period 1<br />
|-<br />
! Cents*<br />
! Approximate ratios<br />
! Cents*<br />
! Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
| 600.0<br />
| 7/5, 10/7<br />
|-<br />
| 1<br />
| 707.2<br />
| '''3/2'''<br />
| 107.2<br />
| 15/14, 16/15, 21/20<br />
|-<br />
| 2<br />
| 214.4<br />
| '''8/7''', '''9/8'''<br />
| 814.4<br />
| '''8/5'''<br />
|-<br />
| 3<br />
| 921.5<br />
| 12/7<br />
| 321.5<br />
| 6/5<br />
|-<br />
| 4<br />
| 428.7<br />
| 9/7, 14/11<br />
| 1028.7<br />
| 9/5, 20/11<br />
|-<br />
| 5<br />
| 1135.9<br />
| 21/11, 27/14, 48/25, <br>64/33, 96/49<br />
| 535.9<br />
| 15/11, 27/20<br />
|-<br />
| 6<br />
| 643.1<br />
| '''16/11'''<br />
| 43.1<br />
| 45/44, 56/55, 81/80<br />
|}<br />
<br />
{| class="wikitable center-1 right-2 right-4"<br />
|+ style="font-size: 105%;" | Pajarous (10&nbsp;&amp;&nbsp;22)<br />
|-<br />
! rowspan="2" | #<br />
! colspan="2" | Period 0<br />
! colspan="2" | Period 1<br />
|-<br />
! Cents*<br />
! Approximate ratios<br />
! Cents*<br />
! Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
| 600.0<br />
| 7/5, 10/7<br />
|-<br />
| 1<br />
| 709.6<br />
| '''3/2'''<br />
| 109.6<br />
| 15/14, 16/15, 21/20<br />
|-<br />
| 2<br />
| 219.1<br />
| '''8/7''', '''9/8'''<br />
| 819.1<br />
| '''8/5'''<br />
|-<br />
| 3<br />
| 928.7<br />
| 12/7<br />
| 328.7<br />
| 6/5, 11/9<br />
|-<br />
| 4<br />
| 438.2<br />
| 9/7<br />
| 1038.2<br />
| 9/5, 11/6<br />
|-<br />
| 5<br />
| 1147.8<br />
| 27/14, 48/25, 55/28, <br>88/45, 96/49<br />
| 547.8<br />
| '''11/8''', 27/20<br />
|-<br />
| 6<br />
| 657.3<br />
| 22/15<br />
| 57.3<br />
| 22/21, 33/32, 81/80<br />
|}<br />
<nowiki/>* In 11-limit CWE tuning, octave-reduced<br />
<br />
== Scales ==<br />
=== 10-note (proper) ===<br />
{{Main| 2L&nbsp;8s }}<br />
<br />
The true mos is called the ''symmetric'' decatonic scale, because it repeats exactly at the half-octave, so the symmetric scale starting from {{nowrap|7/5~10/7}} is the same as the symmetric scale starting from 1/1. The near-mos, LsssLsssss, in which only the 5-step interval violates the rule of no more than 2 intervals per class, is called the ''pentachordal'' decatonic, because it consists of two identical [[pentachord]]s plus a split {{nowrap|9/8~8/7}} whole tone to complete the octave.<br />
<br />
=== 12-note (proper) ===<br />
{{Main| 10L&nbsp;2s }}<br />
<br />
=== Scala files ===<br />
* [[12-22h]]<br />
<br />
== Tunings ==<br />
As with [[archy]], there is a tradeoff in pajara between accuracy of 3 and accuracy of 7. Unlike tunings of archy which the fifth is around 710–712{{c}}, however, pajara is conventionally tuned flat of 22edo, since tunings sharp of about 710{{c}} lose a large degree of accuracy in 5/4 and especially 6/5. <br />
<br />
=== Norm-based tunings ===<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings<br />
|-<br />
! rowspan="2" | <br />
! colspan="3" | Euclidean<br />
|-<br />
! Constrained<br />
! Constrained & skewed<br />
! Destretched<br />
|-<br />
! Tenney<br />
| CTE: ~3/2 = 708.3557{{c}}<br />
| CWE: ~3/2 = 707.3438{{c}}<br />
| POTE: ~3/2 = 707.0477{{c}}<br />
|}<br />
<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings<br />
|-<br />
! rowspan="2" | <br />
! colspan="3" | Euclidean<br />
|-<br />
! Constrained<br />
! Constrained & skewed<br />
! Destretched<br />
|-<br />
! Tenney<br />
| CTE: ~3/2 = 708.1993{{c}}<br />
| CWE: ~3/2 = 707.1826{{c}}<br />
| POTE: ~3/2 = 706.8851{{c}}<br />
|}<br />
<br />
=== Tuning spectrum ===<br />
{| class="wikitable center-all left-4"<br />
|-<br />
! Edo<br>generator<br />
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval]])<br />
! Generator (¢)<br />
! Comments<br />
|-<br />
| 7\12<br />
| <br />
| 700.000<br />
| Lower bound of 9- and 11-odd-limit diamond monotone<br />
|-<br />
| <br />
| 3/2<br />
| 701.955<br />
| <br />
|-<br />
| 34\58<br />
| <br />
| 703.448<br />
| 58ddee val<br />
|-<br />
| 27\46<br />
| <br />
| 704.348<br />
| 46de val<br />
|-<br />
| <br />
| 11/7<br />
| 704.377<br />
| <br />
|-<br />
| <br />
| 9/5<br />
| 704.399<br />
| <br />
|-<br />
| 47\80<br />
| <br />
| 705.000<br />
| 80ddee val<br />
|-<br />
| <br />
| 5/3<br />
| 705.214<br />
| 5- and 15-odd-limit minimax<br />
|-<br />
| 20\34<br />
| <br />
| 705.882<br />
| 34d val<br />
|-<br />
| <br />
| 11/9<br />
| 706.574<br />
| <br />
|-<br />
| 53\90<br />
| <br />
| 706.667<br />
| 90dde val<br />
|-<br />
| <br />
| 5/4<br />
| 706.843<br />
| 7- and 11-limit POTT<br />
|-<br />
| 33\56<br />
| <br />
| 707.143<br />
| 56d val<br />
|-<br />
| <br />
| 11/6<br />
| 707.234<br />
| <br />
|-<br />
| <br />
| 15/11<br />
| 707.390<br />
| <br />
|-<br />
| 46\78<br />
| <br />
| 707.692<br />
| 78dd val<br />
|-<br />
| <br />
| 11/8<br />
| 708.114<br />
| <br />
|-<br />
| <br />
| 11/10<br />
| 708.749<br />
| 11-odd-limit minimax<br />
|-<br />
| <br />
| 9/7<br />
| 708.771<br />
| <br />
|-<br />
| 13\22<br />
| <br />
| 709.091<br />
| Upper bound of 11-odd-limit diamond monotone<br />
|-<br />
| <br />
| 7/6<br />
| 711.043<br />
| 7-odd-limit minimax<br />
|-<br />
| 32\54<br />
| <br />
| 711.111<br />
| 54e val<br />
|-<br />
| <br />
| 15/8<br />
| 711.731<br />
| <br />
|-<br />
| 19\32<br />
| <br />
| 712.500<br />
| 32e val<br />
|-<br />
| 25\42<br />
| <br />
| 714.286<br />
| 42cee val<br />
|-<br />
| <br />
| 7/4<br />
| 715.587<br />
| <br />
|-<br />
| 6\10<br />
| <br />
| 720.000<br />
| 10e val, upper bound of 9-odd-limit diamond monotone<br />
|}<br />
<br />
== Music ==<br />
; [[Jake Freivald]]<br />
* [https://soundcloud.com/jdfreivald/chord-sequence-in-paul-erlichs ''Chord Sequence in Paul Erlich's Decatonic Major''] (2014) – in Pajara[10], 22edo tuning<br />
<br />
; [[Joel Grant Taylor]]<br />
* [https://web.archive.org/web/20201127012345/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 ''Dirge''] – in the hexachordal dodecatonic modmos, [[12-22h]]<br />
* [https://web.archive.org/web/20201127012408/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 ''Sonatina''] – ditto<br />
<br />
; [[Chris Vaisvil]]<br />
* ''Smoke Filled Bar'' (2012) – [https://www.chrisvaisvil.com/smoke-filled-bar/ blog] | [https://web.archive.org/web/20230530093324/http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 play] – in 12-22h.<br />
<br />
== References ==<br />
<references/><br />
<br />
[[Category:Pajara| ]] <!-- main article --><br />
[[Category:Rank-2 temperaments]]<br />
[[Category:Archytas clan]]<br />
[[Category:Diaschismic family]]<br />
[[Category:Jubilismic clan]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Detempering&diff=224238Detempering2026-02-18T11:37:22Z<p>ArrowHead294: </p>
<hr />
<div>In [[regular temperament theory]], '''detempering''' is the process of taking a tempered [[tuning system]] and replacing each of its pitches with one or more pitches from its [[preimage]], that is, the just or tempered pitches that the pitch represents. It is the opposite of [[tempering out|tempering]]. Specifically, a '''detempered system''' (aka '''detemperament''' or '''detempering''') has each pitch of a tempered system (according to a fixed regular temperament) replaced with some set of interpretations of the pitch under the temperament mapping. If exactly one interpretation is used for each degree of a scale, then the detempered scale is called a '''one-to-one detempering'''. Ideally the resultant detempered scale will have a compact lattice. A higher rank temperament is also called a detempering of a lower-rank temperament if the lower-rank temperament results from tempering out one or more commas in the higher-rank temperament. For example, meantone is a detempering of 12edo.<br />
<br />
Detempering is one way among many to create a [[neji]], or a JI scale approximating a given scale.<br />
== One-to-one detemperings of equal temperaments ==<br />
The following are two equivalent definitions for one-to-one detemperings of an [[equal temperament]]:<br />
# A JI scale is a ''one-to-one detempering'' of an ET if each note of the equal temperament is matched to exactly one JI note which tempers to the note.<br />
# A JI scale ''S'' is ''epimorphic'' if on the [[JI subgroup]] <math>A \leq \mathbb{Q}_{>0}</math> generated by the intervals of ''S'', there exists a [[val]] {{nowrap|''v'': ''A'' → ℤ}} (which can be called an '''epimorphism''') such that {{nowrap|''v''(''S''[''i'']) {{=}} ''i''}} for all {{nowrap|''i'' ∈ ℤ}}.<br />
<br />
The two terms are equivalent because if a detempering of an ''n''-note equal temperament ''v'' is one-to-one, then second definition follows by the additivity of ''v'', and given the second definition, injectivity is immediate.<br />
<br />
The property is strictly stronger than [[constant structure]] (CS). When one assumes ''S'' is a CS but not that it is a one-to-one detempering, there is a unique set map <math>v : \{\text{intervals of $S$}\} \to \mathbb{Z}</math> that witnesses that ''S'' is a CS and satisfies {{nowrap|''v''(''S''[''i'']) {{=}} ''i''}} for all ''i''. Thus a CS scale ''S'' is a one-to-one detempering if and only if this mapping ''v'' extends to a linear map on the entirety of ''A''.<br />
<br />
The second definition extends naturally to asking whether a higher-dimensional mapping <math>S:\mathbb{Z}^n \to P</math> for an arbitrary codomain <math>P</math> of relative pitches is epimorphic, in the same sense of there existing an abelian group <math>A</math> and a linear map <math>v : A \to \mathbb{Z}^n</math> such that <math>v(S(x)) = x.</math> This can be of practical interest: one might ask whether an isomorphic keyboard mapping <math>S : \mathbb{Z}^2 \to P</math> is epimorphic.<br />
<br />
Temperaments [[support]]ed by vals for one-to-one detemperings have occasionally been considered. Some [[temperament]]s (including [[val]]s for small edos) can be viewed this way for small one-to-one detemperings despite their relatively low accuracy:<br />
<br />
* The 2.3.5 temperament [[dicot]] supports [[nicetone]] (3L&nbsp;2m&nbsp;2s), [[blackdye]] (5L&nbsp;2m&nbsp;3s) and superzarlino (a 17-note epimorphic scale) scale structures.<br />
* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L&nbsp;3m&nbsp;2s), [[diasem]] (5L&nbsp;2m&nbsp;2s), and other scales in the [[Generator sequence|Tas series]].<br />
<br />
=== Example ===<br />
Consider the Ptolemaic diatonic scale, {{nowrap|{9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1}<nowiki/>}}, which is nicetone with {{nowrap|L {{=}} 9/8|M {{=}} 10/9}}, and {{nowrap|s {{=}} 16/15}}. This scale is epimorphic because we can apply {{val| 7 11 16 }}, the [[7edo]] [[patent val]], to map the intervals into the number of scale steps:<br />
<br />
<math><br />
\left(\begin{array} {rrr}<br />
7 & 11 & 16<br />
\end{array} \right)<br />
\left(\begin{array}{rrrrrrr}<br />
-3 & -2 & 2 & -1 & 0 & -3 & 1 \\<br />
2 & 0 & -1 & 1 & -1 & 1 & 0 \\<br />
0 & 1 & 0 & 0 & 1 & 1 & 0<br />
\end{array}\right)<br />
=<br />
\left(\begin{array}{rrrrrrr}<br />
1 & 2 & 3 & 4 & 5 & 6 & 7<br />
\end{array}\right)<br />
</math><br />
<br />
where the columns of the 3×7 matrix are the scale intervals written in [[monzo]] form. Hence, 7edo (equipped with its patent val) is a val associated with the the Ptolemaic diatonic scale. Indeed, 7edo supports dicot temperament.<br />
<br />
=== Facts ===<br />
==== Definition: constant structure (CS) ====<br />
Given a [[periodic scale]] <math>S : \mathbb{Z} \to (0,\infty)</math> (with codomain written as ratios from {{nowrap|''S''(0) {{=}} 1}} in the linear frequency domain), let <math>C_k = \{ S[i+k]/S[i] : i \in \mathbb{Z}\}</math> be the [[interval class|set of ''k''-steps]] of ''S''. Then ''S'' ''is a [[constant structure]]'' (CS) if for any <math>i, j \in \mathbb{Z}, i \neq j,</math> we have <math>C_i \cap C_j = \varnothing.</math><br />
<br />
==== One-to-one detemperings of ETs are CSes ====<br />
{{proof|contents=<br />
Let {{nowrap|''v'': ''A'' → ℤ}} be the val associated with ''s''. Let <math>x \in C_j.</math> Then there exists <math>i > 0</math> such that <math>S[i+j]/S[i] = x.</math> Suppose by way of contradiction there exist <math>k \neq j</math> and <math>i > 0</math> such that <math>S[i+k]/S[i] = x.</math><br />
<br />
Then <math>v(x) = v(S[i+j]/S[i]) = v(S[i+j]) - v(S[i]) = i + j - i = j,</math> but also <math>v(x) = v(S[i^\prime+k]/S[i^\prime]) = v(S[i^\prime+k]) - v(S[i^\prime]) = k,</math> a contradiction.<br />
}}<br />
<br />
==== If the steps of a CS scale are linearly independent, then the scale is a one-to-one detempering of an ET ====<br />
Theorem: Suppose ''S'' is a 2/1-equivalent increasing constant structure JI scale of length ''n''. Let <math>C_1</math> be the set of 1-steps of ''S'', and suppose that <math>C_1</math> is a basis for the JI subgroup ''A'' generated by it. Then there exists an val <math> v: A \to \mathbb{Z}</math> which is a val of ''n''-edo (and a similar statement holds for other equaves).<br />
<br />
(The condition of <math>C_1</math> being a basis rather than merely a generating set cannot be omitted, since the scale {{nowrap|{5/4, 32/25, 2/1}<nowiki/>}} is a CS but not a one-to-one detempering. The converse of this conditional also fails, as {{nowrap|{9/8, 5/4, 3/2, 25/16, 2/1}<nowiki/>}} is epimorphic under [[5edo]]'s [[patent val]].)<br />
<br />
{{proof|contents=<br />
Define the linear map <math>v:A \to \mathbb{Z}</math> by defining <math>v(\mathbf{s}) = 1</math> for any step <math>\mathbf{s} \in C_1</math> and extending uniquely by linearity. Then for any <math>i \in \mathbb{Z}</math> we have <math>v(S[i]) = v(S[i]/S[i-1]\cdots S[1]) = v(S[i]/S[i-1]) + \cdots + v(S[1]) = i,</math> whence ''v'' is a one-to-one detempering. That <math>v(2) = n</math> is also automatic.<br />
}}<br />
<br />
=== Terminology ===<br />
As it is a common concept, one-to-one detemperings of ETs have also been called by a number of other names in xen theory, including ''transversal'', ''epimorphic scale'', and ''strong CS''.<br />
[[Category:Scale]]<br />
<br />
== Lifting ==<br />
The term ''lifting'' can be used as a [[JI-agnostic]] alternative to ''detempering''.<br />
<br />
In this sense, [[diasem]] (LMLSLMLSL) is a lifting of [[semiquartal]] (LSLSLSLSL) which "detempers" the S step of semiquartal into two steps sizes M and S.<br />
<br />
== Examples ==<br />
* [[Ringer scale]]s<br />
* [[Fantasy detempers]]<br />
* [[87edo/13-limit detempering]]<br />
* [[Diasem]]<br />
* [[Beautiful 27]]<br />
<br />
[[Category:Regular temperament theory]]<br />
[[Category:Terms]]<br />
[[Category:Method]]<br />
[[Category:Detempering| ]] <!-- main article --><br />
[[Category:Well temperament]]</div>ArrowHead294https://en.xen.wiki/index.php?title=Porcupine&diff=224236Porcupine2026-02-18T11:30:15Z<p>ArrowHead294: </p>
<hr />
<div>{{Interwiki<br />
| en = Porcupine<br />
| de = Porcupine<br />
| es = <br />
| ja = <br />
}}<br />
{{Infobox regtemp<br />
| Title = Porcupine<br />
| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.7.11<br />
| Comma basis = [[250/243]] (2.3.5);<br>[[55/54]], [[100/99]] (2.3.5.11);<br>[[55/54]], [[64/63]], [[100/99]] (2.3.5.7.11)<br />
| Mapping = 1; -3 -5 6 -4<br />
| Edo join 1 = 15 | Edo join 2 = 22<br />
| Generators = 11/10<br />
| Generators tuning = 163<br />
| Optimization method = CWE<br />
| MOS scales = [[1L&nbsp;6s]], [[7L&nbsp;1s]], [[7L&nbsp;8s]]<br />
| Pergen = (P8, P4/3)<br />
| Color name = Triyoti<br />
| Odd limit 1 = 5 | Mistuning 1 = 9.8 | Complexity 1 = 7<br />
| Odd limit 2 = 11-limit 15 | Mistuning 2 = 19.9 | Complexity 2 = 15<br />
}}<br />
[[File:porcupine.png|thumb|Porcupine equates three minor thirds (6/5, in red) with two perfect fourths (4/3, in green). To do so, it tempers out 250/243, which implies a generator of a flat 10/9.|600x600px]]<br />
[[File:porcupinesymmetricminor22edo.mp3|thumb|Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]]<br />
<br />
'''Porcupine''' is a [[regular temperament|temperament]] that is [[generator|generated]] by a minor whole tone which is tuned flat to around 160–165&nbsp;[[cent]]s, so that the porcupine [[comma]] ([[250/243]]) is [[tempering out|tempered out]]. Two generators (stacked) represent [[6/5]], and three represent [[4/3]]; from this, the generator itself represents a (very flat) [[10/9]]. This is in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] so that two of them reach a ''major'' third of [[5/4]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales. <br />
<br />
One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's is the Pythagorean minor seventh [[16/9]], which is flat of 9/5 by [[81/80]]. Thus, it can be determined that porcupine equates the syntonic comma 81/80 with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as [[meantone]] and [[schismic]]). <br />
<br />
Porcupine can be thought of as a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament (sometimes called ''porkypine'') without much additional damage compared to the 5-limit; the generator here represents not only 10/9, but also [[11/10]] and [[12/11]] (equivalently, [[55/54]], [[100/99]], and [[121/120]] are tempered out), with the consequence that the [[11/9]] interval, usually considered a neutral third, is in porcupine identical to the 6/5 minor third, due to the extreme flatness of 10/9. This also means that [[27/20]], the 5-limit "acute fourth", is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone), found at −4 generators (tuned to about 540–560 cents). This is because as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup at a certain standard of accuracy.<br />
<br />
It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960–990{{c}}), has already been flattened to merge it with (6/5)<sup>3</sup>, and therefore can be equated to [[7/4]]. This makes porcupine a weak extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to [[9/7]], and its fifth is tuned sharp, ranging from around 705–720{{c}}, with the best tunings around 711–712{{c}}, which roughly splits the damage on 7/4 and 9/7.<br />
<br />
See [[Porcupine family #Porcupine]] for technical data and alternative 7-limit extensions. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s.<br />
<br />
== Interval chain ==<br />
{{Main| Porcupine intervals }}<br />
<br />
In the following table, odd harmonics 1–11 are in '''bold'''. <br />
<br />
{| class="wikitable center-all right-2 left-3 right-7 left-8"<br />
|-<br />
! colspan="5" | Up from the tonic, and fourthward<br />
! colspan="5" | Down from the octave, and fifthward<br />
|-<br />
! #<br />
! Cents*<br />
! Ratios<br />
! Porcupine<br>notation<br />
! Ups and downs<br>notation<br />
! #<br />
! Cents*<br />
! Ratios<br />
! Porcupine<br>notation<br />
! Ups and downs<br>notation<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
| P1<br />
| P1<br />
| 0<br />
| 1200.0<br />
| '''2/1'''<br />
| P8<br />
| P8<br />
|-<br />
| 1<br />
| 162.8<br />
| 10/9, 11/10, 12/11<br />
| P2<br />
| vM2 = ^^m2<br />
| −1<br />
| 1037.2<br />
| 9/5, 11/6, 20/11<br />
| P7<br />
| ^m7 = vvM7<br />
|-<br />
| 2<br />
| 325.6<br />
| 6/5, 11/9<br />
| m3<br />
| ^m3 = vvM3<br />
| −2<br />
| 874.4<br />
| 5/3, 18/11<br />
| M6<br />
| vM6 = ^^m6<br />
|-<br />
| 3<br />
| 488.4<br />
| 4/3<br />
| m4<br />
| P4<br />
| −3<br />
| 711.6<br />
| '''3/2'''<br />
| M5<br />
| P5<br />
|-<br />
| 4<br />
| 651.3<br />
| 16/11, 22/15<br />
| m5<br />
| v5 = ^^d5<br />
| −4<br />
| 548.7<br />
| '''11/8''', 15/11<br />
| M4<br />
| ^4 = vvA4<br />
|-<br />
| 5<br />
| 814.1<br />
| 8/5<br />
| m6<br />
| ^m6 = vvM6<br />
| −5<br />
| 385.9<br />
| '''5/4'''<br />
| M3<br />
| vM3 = ^^m3<br />
|-<br />
| 6<br />
| 976.9<br />
| '''7/4''', 16/9<br />
| d7<br />
| m7<br />
| −6<br />
| 223.1<br />
| 8/7, '''9/8'''<br />
| A2<br />
| M2<br />
|-<br />
| 7<br />
| 1139.7<br />
| 35/18, 48/25, 64/33<br />
| d8<br />
| v8 = ^^d8<br />
| −7<br />
| 60.3<br />
| 25/24, 33/32, 36/35<br />
| A1<br />
| ^1 = vvA1<br />
|-<br />
| 8<br />
| 102.5<br />
| 16/15, 21/20<br />
| d2<br />
| ^m2 = vvM2<br />
| −8<br />
| 1097.5<br />
| 15/8, 40/21<br />
| A7<br />
| vM7 = ^^m7<br />
|-<br />
| 9<br />
| 265.3<br />
| 7/6<br />
| d3<br />
| m3<br />
| −9<br />
| 934.7<br />
| 12/7<br />
| A6<br />
| M6<br />
|-<br />
| 10<br />
| 428.2<br />
| 14/11<br />
| d4<br />
| v4 = ^^d4<br />
| −10<br />
| 771.8<br />
| 11/7<br />
| A5<br />
| ^5 = vvA5<br />
|-<br />
| 11<br />
| 591.0<br />
| 7/5<br />
| d5<br />
| ^d5 = vv5<br />
| −11<br />
| 609.0<br />
| 10/7<br />
| A4<br />
| vA4 = ^^4<br />
|-<br />
| 12<br />
| 753.8<br />
| 14/9<br />
| d6<br />
| m6<br />
| −12<br />
| 446.2<br />
| 9/7<br />
| A3<br />
| M3<br />
|}<br />
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced<br />
<br />
In the ups and downs notation, the [[enharmonic unison]] is the trudsharp, the triple-down augmented unison. The porcupine notation does not have an enharmonic unison.<br />
<br />
Besides the specific tuning shown here, there is a range of acceptable porcupine tunings that includes generators as small as 160{{c}} ([[15edo]]) and as large as 165.5{{c}} ([[29edo]]). However, the 29edo patent val does not support full 11-limit porcupine proper, since it does not temper out [[64/63]].<br />
<br />
== Chords and harmony ==<br />
{{Main| Chords of porcupine }}<br />
<br />
[[12/11]], [[11/10]], and [[10/9]] are all represented by the same interval, the generator. This makes chords such as 8:9:10:11:12 exceptionally common and easy to find.<br />
<br />
{| class="wikitable"<br />
|-<br />
| [[File:OtonalPentad_JI.mp3]]<br />
| [[File:OtonalPentad_22edo.mp3]]<br />
| [[File:OtonalPentad_29edo.mp3]]<br />
|-<br />
| 8:9:10:11:12 chord, in just intonation.<br>All intervals are slightly different.<br />
| Porcupine-tempered 8:9:10:11:12 chord, in [[22edo]].<br>Except the first, the intervals are the same.<br />
| Porcupine-tempered 8:9:10:11:12 chord, in [[29edo]].<br>Except the first, the intervals are the same.<br />
|}<br />
<br />
The interval representing both [[25/24]] and [[81/80]] can be found in this interval chain at −7 steps, and ranges from about 45 to 80{{c}} depending on the tuning. This can be considered the "chroma" of porcupine temperament.<br />
<br />
== Scales ==<br />
[[File:porcupine8.jpg|thumb|Porcupine[8]]] <br />
<br />
{{Main| Porcupine scales }}<br />
<br />
; Mos scales, tuning optimized on the 2.3.5.11 subgroup<br />
* [[Porkypine7]]<br />
* [[Porkypine8]]<br />
* [[Porkypine15]]<br />
<br />
; Mos scales, 8/5.12/7 [[Eigenmonzo|eigenmonzo (unchanged interval)]] tuning: <br />
* [[Porcupinewoo15]]<br />
* [[Porcupinewoo22]]<br />
<br />
== Tunings ==<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings<br />
|-<br />
! rowspan="2" | <br />
! colspan="3" | Euclidean<br />
|-<br />
! Constrained !! Constrained & skewed !! Destretched<br />
|-<br />
! Equilateral<br />
| CEE: ~10/9 = 163.6049{{c}}<br />
| CSEE: ~10/9 = 163.2835{{c}}<br />
| POEE: ~10/9 = 163.9280{{c}}<br />
|-<br />
! Tenney<br />
| CTE: ~10/9 = 164.1659{{c}}<br />
| CWE: ~10/9 = 164.0621{{c}}<br />
| POTE: ~10/9 = 163.9504{{c}}<br />
|-<br />
! Benedetti, <br>Wilson<br />
| CBE: ~10/9 = 164.3761{{c}}<br />
| CSBE: ~10/9 = 164.3761{{c}}<br />
| POBE: ~10/9 = 164.1610{{c}}<br />
|}<br />
<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.11-subgroup norm-based tunings<br />
|-<br />
! rowspan="2" | <br />
! colspan="3" | Euclidean<br />
|-<br />
! Constrained !! Constrained & skewed !! Destretched<br />
|-<br />
! Equilateral<br />
| CEE: ~11/10 = 163.1459{{c}}<br />
| CSEE: ~11/10 = 162.8445{{c}}<br />
| POEE: ~11/10 = 164.1867{{c}}<br />
|-<br />
! Tenney<br />
| CTE: ~11/10 = 163.8867{{c}}<br />
| CWE: ~11/10 = 163.9951{{c}}<br />
| POTE: ~11/10 = 164.0777{{c}}<br />
|-<br />
! Benedetti, <br>Wilson<br />
| CBE: ~11/10 = 164.2393{{c}}<br />
| CSBE: ~11/10 = 164.4623{{c}}<br />
| POBE: ~11/10 = 164.2221{{c}}<br />
|}<br />
<br />
{| class="wikitable mw-collapsible mw-collapsed"<br />
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings<br />
|-<br />
! rowspan="2" | <br />
! colspan="3" | Euclidean<br />
|-<br />
! Constrained !! Constrained & skewed !! Destretched<br />
|-<br />
! Equilateral<br />
| CEE: ~11/10 = 162.4448{{c}}<br />
| CSEE: ~11/10 = 162.2333{{c}}<br />
| POEE: ~11/10 = 162.2522{{c}}<br />
|-<br />
! Tenney<br />
| CTE: ~11/10 = 163.1055{{c}}<br />
| CWE: ~11/10 = 162.8156{{c}}<br />
| POTE: ~11/10 = 162.7474{{c}}<br />
|-<br />
! Benedetti, <br>Wilson<br />
| CBE: ~11/10 = 163.5299{{c}}<br />
| CSBE: ~11/10 = 163.2310{{c}}<br />
| POBE: ~11/10 = 163.0304{{c}}<br />
|}<br />
<br />
=== Tuning spectrum ===<br />
{| class="wikitable center-all left-4"<br />
|-<br />
! EDO<br>generator<br />
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*<br />
! Generator (¢)<br />
! Comments<br />
|-<br />
| '''[[8edo|1\8]]'''<br />
| <br />
| '''150.000'''<br />
| '''Lower bound of 5-odd-limit diamond monotone'''<br />
|-<br />
| <br />
| [[12/11]]<br />
| 150.637<br />
| Lower bound of 11- and 15-odd-limit diamond tradeoff<br />
|-<br />
| <br />
| [[6/5]]<br />
| 157.821<br />
| 1/2-comma; lower bound of 5-, 7-, and 9-odd-limit diamond tradeoff<br />
|-<br />
| '''[[15edo|2\15]]'''<br />
| <br />
| '''160.000'''<br />
| '''Lower bound of 7- to (11-limit) 15-odd-limit diamond monotone'''<br />
|-<br />
| <br />
| [[7/4]]<br />
| 161.471<br />
| <br />
|-<br />
| [[52edo|7\52]]<br />
| <br />
| 161.538<br />
| 52b val<br />
|-<br />
| <br />
| [[14/11]]<br />
| 161.751<br />
| <br />
|-<br />
| <br />
| [[7/5]]<br />
| 162.047<br />
| <br />
|-<br />
| [[37edo|5\37]]<br />
| <br />
| 162.162<br />
| <br />
|-<br />
| <br />
| [[16/11]]<br />
| 162.171<br />
| <br />
|-<br />
| [[96edo|13\96]]<br />
| <br />
| 162.500<br />
| 96b val<br />
|-<br />
| [[59edo|8\59]]<br />
| <br />
| 162.712<br />
| <br />
|-<br />
| <br />
| [[8/5]]<br />
| 162.737<br />
| 2/5-comma, 5-odd and 7-odd minimax<br />
|-<br />
| <br />
| [[28/15]]<br />
| 162.897<br />
| <br />
|-<br />
| <br />
| [[7/6]]<br />
| 162.986<br />
| <br />
|-<br />
| '''[[22edo|3\22]]'''<br />
| <br />
| '''163.636'''<br />
| '''Upper bound of 7- to (11-limit) 15-odd-limit diamond monotone'''<br />
|-<br />
| <br />
| [[14/9]]<br />
| 163.743<br />
| 9-, 11-, and (11-limit) 15-odd-limit minimax<br />
|-<br />
| <br />
| [[16/15]]<br />
| 163.966<br />
| 3/8-comma<br />
|-<br />
| [[51edo|7\51]]<br />
| <br />
| 164.706<br />
| 51d val<br />
|-<br />
| <br />
| [[11/10]]<br />
| 165.004<br />
| <br />
|-<br />
| [[29edo|4\29]]<br />
| <br />
| 165.517<br />
| 29d val<br />
|-<br />
| <br />
| [[22/15]]<br />
| 165.762<br />
| <br />
|-<br />
| <br />
| [[4/3]]<br />
| 166.015<br />
| 1/3-comma; upper bound of 5- and 7-odd-limit diamond tradeoff<br />
|-<br />
| [[36edo|5\36]]<br />
| <br />
| 166.667<br />
| 36cde val<br />
|-<br />
| '''[[7edo|1\7]]'''<br />
| <br />
| '''171.429'''<br />
| '''Upper bound of 5-odd-limit diamond monotone'''<br />
|-<br />
| <br />
| [[11/9]]<br />
| 173.704<br />
| <br />
|-<br />
| <br />
| [[10/9]]<br />
| 182.404<br />
| Untempered generator; upper bound of 9- to 15-odd-limit diamond tradeoff<br />
|}<br />
<nowiki/>* Besides the octave<br />
<br />
== History ==<br />
Porcupine temperament/scales were discovered by [[Dave Keenan]], but did not have a name until [[Herman Miller]] mentioned that his ''Mizarian Porcupine Overture'' in 15et had a section that pumps the 250/243 comma. Although this music did not use a porcupine mos or [[modmos]] (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of [[MOS]] scales was fully evident. It was clear that even though Herman's piece was in 15edo, 22edo was a porcupine tuning par excellence, and that was an interesting development in itself.<br />
<br />
== See also ==<br />
* [[Porcupine notation]]<br />
* [[Porcupine modes]]<br />
* [[Porcupine temperament modal harmony]]<br />
* [[Porcupine Album Project]]<br />
<br />
== Music ==<br />
=== 20th century ===<br />
; [[Herman Miller]]<br />
* [https://sites.google.com/site/teamouse/home#TOC-Mizarian-music ''Mizarian Porcupine Overture''] (1999) – [https://web.archive.org/web/20201127014859/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/MizarianPorcupineOverture.mp3 play] – in [[15edo]] tuning, namesake of the temperament<br />
<br />
=== 21st century ===<br />
; [[Flora Canou]]<br />
* [https://soundcloud.com/floracanou/april-porkfest?in=floracanou/sets/totmc-suite "April Porkfest"] from [https://soundcloud.com/floracanou/sets/totmc-suite ''TOTMC Suite''] (2023–2025) – in 11-limit CTE tuning<br />
<br />
; [[User:CellularAutomaton|CellularAutomaton]]<br />
* [https://cellularautomaton.bandcamp.com/track/minnow ''Minnow''] (2024) – in [[29edo]] tuning<br />
<br />
; [[Paul Erlich]]<br />
* [https://web.archive.org/web/20070928093239/http://66.98.148.43/~xenharmo/mp3/erlich/glassic.mp3 ''Glassic''] – in [[22edo]] tuning (at least the beginning part is in porcupine.)<br />
<br />
; [[Jake Freivald]]<br />
* [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/porcupine-comma-pump.mp3 ''Porcupine Comma Pump'']{{dead link}}<br />
<br />
; [[Cody Hallenbeck]]<br />
* ''Porcupine Walk'' (2019)<br />
** [https://soundcloud.com/cody-hallenbeck/porcupine-walk 15edo version] · [https://soundcloud.com/cody-hallenbeck/porcupine-walk-22edo 22edo version]<br />
<br />
; [[Lillian Hearne]]<br />
* [https://soundcloud.com/lillianhearne/mass-in-22edo-sanctus ''Sanctus''] (2015)<br />
<br />
; [[Andrew Heathwaite]]<br />
* [https://soundclick.com/share.cfm?id=8839060 ''being a''] (2010) – in Porcupine[8], mode 1|6, 22edo tuning<br />
<br />
; [[Jollybard]]<br />
* [https://soundcloud.com/jollybard/porcupeen ''Porcupeen''] (2017)<br />
* [https://jollybard.bandcamp.com/track/porcupine "Porcupine"], from ''pato, with friends'' (2019)<br />
<br />
; [[Igliashon Jones]]<br />
* [https://cityoftheasleep.bandcamp.com/track/second-breakfast-15edo ''Second Breakfast (15edo)''] (2018){{dead link}}<br />
<br />
; [[Löis Lancaster]]<br />
* [https://soundcloud.com/lois-lancaster/porcupine-experience ''Porcupine Experience''] (2012) – in 22edo tuning<br />
<br />
; [[John Moriarty]]<br />
* [https://www.youtube.com/watch?v=se79rdp705Y ''Flying Straight Down''] (2020) – in 22edo tuning<br />
<br />
; [[Omega9]]<br />
* [https://www.youtube.com/watch?v=DSao0Yg3Tck ''Life on Mars''] (2014)<br />
<br />
; [[Petr Pařízek]]<br />
* [[:File:AmongOtherThings2.mp3|''Among Other Things 2'']]<br />
<br />
; [[Ray Perlner]]<br />
* [https://www.youtube.com/watch?v=8reCr2nDGbw ''Porcupine Lullaby''] (2020) – in 37edo tuning<br />
* [https://www.youtube.com/playlist?list=PLkW9S8bpltfw464vJg3CAJJbV4IR6ggPd ''Porcupine{{lbrack}}7{{rbrack}} Modal Fugues''] – 7-piece playlist<br />
<br />
; [[Gene Ward Smith]] and {{w|Modest Mussorgsky}}<br />
* [https://www.archive.org/download/NightOnPorcupineMountain/Genewardsmithmussorgsky-NightOnPorcupineMountain.mp3 ''Night on Porcupine Mountain''] (archived 2010) – in 22edo tuning<br />
<br />
; [[Chris Vaisvil]]<br />
* ''Gently Playing With Miller's Porcupine'' (2011) – [https://www.chrisvaisvil.com/four-pieces-in-porcupine-temperament/ blog] | [https://web.archive.org/web/20231228102528/http://micro.soonlabel.com/15-ET/daily20110619_millers_porcupine_7a.mp3 play] – in Porcupine[7], mode 3|3, 15edo tuning<br />
* [https://web.archive.org/web/20231121064756/http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-indian.mp3 ''15 Porcupines in India''] – sarangi, tambura and sitar improvisation<br />
* [https://web.archive.org/web/20240118050711/http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-piano.mp3 ''15 Quills''] – piano solo<br />
* [https://web.archive.org/web/20231121043724/http://micro.soonlabel.com/15-ET/daily20111231-porcupine15-prickly-side-of-love.mp3 ''Prickly Side of Love''] – rock band with vocals<br />
* [https://web.archive.org/web/20221221154102/http://micro.soonlabel.com/15-ET/daily20120102-porcupine-organ.mp3 ''Porcupine Organ Composition'']<br />
<br />
; [[Nick Vuci]]<br />
* [https://en.xen.wiki/images/0/0b/NickVuci-20230426-22edo-PorcupinePrelude1.mp3 ''Porcupine Prelude 1''] – in 22edo tuning<br />
* [https://en.xen.wiki/images/3/39/NickVuci-20230518-22edo-PorcupinePrelude2.mp3 ''Porcupine Prelude 2''] – in 22edo tuning<br />
* [https://en.xen.wiki/images/b/bd/NickVuci-20230521-22edo-PorcupinePrelude3.mp3 ''Porcupine Prelude 3''] – in 22edo tuning<br />
* [https://en.xen.wiki/images/0/0b/NickVuci-20230523-22edo-Praeambulum.mp3 ''Porcupine Praeambulum''] – in 22edo tuning<br />
* [https://en.xen.wiki/images/2/26/NickVuci-20230531-22edo-PorcupineChoraleWithPrelude.mp3 ''Porcupine Chorale with Prelude "Nature's Lament"''] – in 22edo tuning<br />
<br />
; [[Well-Tempered Fox]]<br />
* [https://www.youtube.com/watch?v=INM6J9pS_xE ''Porcupine Major Overture''] (2015) – in 22edo tuning<br />
* [https://soundcloud.com/pianodog/waltzing-in-candyland-15-edo ''Waltzing in Candyland''] (2015) – in Porcupine[8], 15edo tuning<br />
<br />
; [[Juhani Nuorvala]]<br />
* [https://www.youtube.com/watch?v=aAHkjOvplVg ''Kellot (Bells)''] (2025) – in 96edo tuning<br />
<br />
[[Category:Porcupine| ]] <!-- Main article --><br />
[[Category:Rank-2 temperaments]]<br />
[[Category:Porcupine family]]<br />
[[Category:Archytas clan]]<br />
[[Category:Keemic temperaments]]<br />
[[Category:Listen]]</div>ArrowHead294