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| Within every [[EDO|edo]] one can specify a '''maximally even''' (ME) scale for every smaller edo. The maximally even scale is the closest the parent edo can get to representing the smaller edo. Mathematically, ME scales of n notes in m edo are any [[mode|mode]] of the sequence ME(n, m) = [floor(i*m/n) | i=1..n], where the [https://en.wikipedia.org/wiki/Floor_and_ceiling_functions "floor"] function rounds down to the nearest integer.
| | {{Distinguish|Distributional evenness}} |
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| | {{Wikipedia}} |
| | A [[periodic scale|periodic]] [[binary scale]] is '''maximally even''' ('''ME''') with respect to an [[equal-step tuning]] if it is the result of rounding a smaller equal tuning to the nearest notes of the parent equal tuning with the same equave. Equivalently, a scale is maximally even if its two [[step]] sizes are [[Distributional evenness|evenly distributed]] within its [[step pattern]] and differ by exactly one step of the parent tuning. In other words, such a scale satisfies the property of '''maximal evenness'''. The first condition implies that ME scales are [[MOS scale]]s, and the second condition implies that the scale's [[step ratio]] is [[superparticular]]. |
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| The maximally even scale will be one:
| | In particular, within every [[edo]], one can specify such a scale for every smaller number of notes. An ''m''-note maximally even scale in ''n''-edo is the closest ''n''-edo can get to representing ''m''-edo. |
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| a. which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent edo). | | == Mathematics == |
| | === Definition === |
| | Mathematically, if {{nowrap|0 < ''n'' < ''m''}}, a ''maximally even (sub)set of size n'' in '''Z'''/''m'''''Z''' is any translate of the set |
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| b. whose steps are distributed as evenly as possible.
| | <math>\operatorname{ME}(n, m) = \left\{ m\mathbb{Z} + \ceil{\frac{im}{n}} : i \in \{0, ..., n-1\} \right\} \subseteq \mathbb{Z}/m\mathbb{Z},</math> |
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| (a) and (b) above imply that the ME scale will be a [[MOSScales|moment of symmetry scale]].
| | where the {{w|ceiling function}} fixes integers and rounds up non-integers to the next higher integer. It can be proven that when ''n'' does not divide ''m'', ME(''n'', ''m'') is a [[MOS scale|MOS subset]] of '''Z'''/''m'''''Z''' where the two step sizes differ by exactly 1, and that the set of degrees where each step size occurs is itself maximally even in '''Z'''/''n'''''Z''', satisfying the informal definition above. ME(''n'', ''m'') is the lexicographically first mode among its rotations, and combined with the fact that it is a MOS, this implies that ME(''n'', ''m'') is the brightest mode in the MOS sense. |
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| The probably most popular heptatonic ME scale is the major scale of [[12edo|12edo]]: <span style="font-family: monospace; ">2 2 1 2 2 2 1</span>, but also every [http://en.wikipedia.org/wiki/Diatonic_scale diatonic scale] of 12edo is maximally even. Some more detailed examples follow.
| | It is easy to show that replacing ceil() with round() (rounding half-integers up) gives an equivalent definition; floor() does too, since ME(''n'', ''m'') is a MOS and thus [[chirality|achiral]]. |
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| A variant of the maximal evenness scale is the Irvian mode, which originates from leap year arrangements of Sym454 calendar (see below).
| | === Complement of a maximally even subset is maximally even === |
| | Proof sketch: We may assume that {{nowrap|gcd(''n'', ''m'') {{=}} 1}}; there are two cases. |
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| == Irvian mode and a relation to a proposed calendar reform ==
| | Case 1: ME(''n'', ''m'') where {{nowrap|''n'' < {{frac|''m''|2}}}}. This is a maximally even subset of Z/mZ with step sizes {{nowrap|L > s > 1}}, which determines the locations of step sizes of 2 in the complement. The rest of the complement's step sizes are 1. The sizes of the chunks of 1 are {{nowrap|L − 2}} and {{nowrap|s − 2}} (0 is a valid chunk size), and the sizes form a maximally even MOS. |
| In 2004, Dr. Irvin Bromberg of University of Toronto developed a calendar called Sym454, and a leap year pattern for the calendar that is symmetrical and as smoothly spread as possible. The calendar is proposed as a variant to replace Gregorian calendar's uneven distribution of dates. The goal of the initial pattern was to minimize divergence of calendar days from cardinal dates such as equinoxes, solstices, and "new year moments", however the pattern also has an interpretation in terms of MOS scale making and keyboard mapping.
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| Such a pattern produces a specific maximum evenness mode, which is named '''Irvian mode'''. A stand-alone leap week at the end of year in Sym454 lore is called Irvember, and therefore the constructed name of the mode would be Irvian.
| | Case 2: ME(''n'', ''m'') where {{nowrap|''n'' > {{frac|''m''|2}}}}. This has step sizes 1 and 2. The chunks of 1 (of nonzero size since {{nowrap|''n'' > {{frac|''m''|2}}}}) occupy a maximally even subset of the slots of ME(''n'', ''m'') (*). Now replace each 1 with "|" and each 2 with "$|". |
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| The pattern is defined by the following:<blockquote>'''Year is leap if the remainder of (L x Y + K)/ C is less than L.'''
| | (e.g.) {{nowrap|2112111 → <nowiki>$|||$||||</nowiki>}} |
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| L = number of leap years per cycle,
| | Consider the resulting binary word of "|" and "$". The "|"s form chunks of sizes that differ by 1 and are distributed in a MOS way by (*). The desired complement, occupied by the "$"'s, thus forms a maximally even subset. |
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| Y = number of the year
| | == Sound perception == |
| | The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31st of an octave instead of one 13th). |
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| C = number of years per cycle
| | The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's {{nowrap|3 3 3 4}} will sound more like 4edo than its {{nowrap|1 1 1 1 1 1 1 1 1 1 1 2}} will sound like 12edo. |
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| K = (C-1)/2 if odd, can choose between (C-1)/2 and C/2 if even</blockquote>The current, "canonical" usage of the cycle is that of 52 leap week years in 293 years - year is leap if (52 x Year + 146)/293 is less than 52. Musically, this would correspond to a [[33L 19s|33L 19]]<nowiki/>s MOS scale. In addition, if the remainder of the leap year is less than the count of long intervals in the MOS, the next year will be in a long interval, otherwise in a short interval. For example here, this means if remainder is less than 33, next leap year (or key) will be 6 years later (6 steps above), otherwise 5 years later.
| | Maximally even sets tend to be familiar and musically relevant scale collections. Examples: |
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| Even-length symmetrical cycles with an irreducible (that is odd) number of years per cycle have a feature where they aren't 100% symmetrical - two middle years follow a pattern of non-leap - leap. If the K is chosen as (C-1)/2 instead of C/2, the sequence will be leap, nonleap.
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| Every Irvian mode is a maximal evenness scale, but not every maximal evenness scale is Irvian.
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| === Example on a standard 12edo piano ===
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| The 12edo piano key layout, which is predominantly use in the world today, is an example of an Irvian mode that is subject to even-length leap rule modification.<blockquote>'''Year is leap if the remainder of (7 x Year + 6) / 12 is less than 7.'''</blockquote>Such a pattern generates keys number '''1-3-5-6-8-10-12-1''' to be the keys on the scale, which is a '''5L 2s''' scale in a pattern of '''LLsLLLs'''. White keys are leap years, and black keys are common years.
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| Years 1,3,6,8,10, that is notes C, D, F, G, A have a long interval - a tone - after them, while E and B, with remainder of 6, have a semitone. When started on C turns out to be plain C major. In this case, the accumulator K is taken to be C/2 instead of (C-1)/2 as with odd cycles, therefore middle of the cycle is nonleap-leap, that is F and F#. Choosing 5 instead of 6 for the K would produce a Lydian scale on C, or a F major scale - patterns of keys are reversed.
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| === 17edo ===
| | * The maximally even heptatonic set of [[19edo]] is, like the one in [[12edo]], a [[5L 2s|diatonic scale]]. |
| [[3L 4s]]:<blockquote>'''Year is leap if the remainder of (7 x Year + 8) / 17 is less than 7'''</blockquote>1-3-6-8-10-13-15 | | * The maximally even heptatonic sets of [[17edo]] and [[24edo]], in contrary, are [[4L 3s|mosh scales]] (Neutrominant[7]). |
| | * The maximally even heptatonic set of [[22edo]] is Porcupine[7] (the superpythagorean diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is the octatonic scale of Hedgehog, the maximally even nonatonic set of 22edo is Orwell[9], (as well as 13-tonic being an Orwell[13]), while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara. |
| | * The maximally even 13-element set in 24edo is Ivan Wyschnegradsky's diatonicized chromatic scale. |
| | * The maximally even sets in edos 40 and higher have step sizes so close together that they can sound like [[circulating temperament]]s with the right timbre. |
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| s L s s L s L.
| | [[Irvian mode]] is a specific mode of the scale, where the notes are also symmetrically arranged. For example, the major mode of the basic diatonic scale from 12edo, <code>2 2 1 2 2 2 1</code>, is not only a maximally even scale, but also the Irvian mode of such scale. Such a mode is best shown in odd EDOs, which truly have a "middle" note owing to being odd, and therefore allowing for true symmetric arrangements of notes. |
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| Starting from the other key, it's bayati 3232322. 17edo is the only temperament where bayati is parallel to the Irvian mode.
| | === Quasi-equal scales === |
| | Note that "maximally even" is equivalent to "quasi-equal-interval-symmetrical" in [[Joel Mandelbaum]]'s 1961 thesis [http://www.anaphoria.com/mandelbaum.html Multiple Divisions of the Octave and the Tonal Resources of 19-Tone Temperament]. Previous versions of this article have conflated "quasi-equal" with "quasi-equal-interval symmetrical". In fact, "quasi-equal" scales, according to Mandelbaum, meet the first criterion listed above, but not necessarily the second. |
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| [[7L 3s]]:<blockquote>'''Year is leap if the remainder of (10 x Year + 8) / 17 is less than 10.'''</blockquote>0-2-4-5-7-9-11-12-14-16-17
| | Examples of quasi-equal scales include [[equipentatonic]] and [[equiheptatonic]] scales among others. |
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| L L s L L L s L L s
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| [[Maqam|Maqamic]] alternative as listed on the 17edo page:
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| 0-2-4-6-7-9-11-12-14-16-17
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| L L L s L L s L L s
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| Such a scale ends up skipping the perfect fifth. Starting on a different note, the scale can be made to have a perfect fifth, for example:
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| 0-1-3-5-7-8-10-12-13-15-17
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| s L L L s L L s L L
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| However, such note arrangements are not Irvian, although they are maximal evenness.
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| === 22edo ===
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| <blockquote>'''Year is leap if the remainder of (13 x Year + 11) / 22 is less than 13.'''</blockquote>Orwell[13]:
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| 0-2-4-5-7-9-10-12-14-16-17-19-21-0, proper Irvian mapping as directly taken from the formula.
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| Following mappings are ME but not Irvian:
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| 0-2-3-5-7-8-10-12-14-15-17-19-20-22, as mentioned on the [[22edo]] page.
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| Alternatives that do not skip the perfect fifth:
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| 0-2-3-5-7-8-10-12-13-15-17-19-20-22
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| 0-1-3-5-6-8-10-12-13-15-17-18-20-22
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| As it is tenuous to write out all the notes, this is a table of a few possible Irvian modes of 22edo:
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| {| class="wikitable"
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| !Name
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| !Formula core
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| |-
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| |Porcupine[15]
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| |(15 x Year + 11) / 22
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| |-
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| |Superpyth[5]
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| |(5 x Year + 11) / 22
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| |-
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| |Porcupine[7]
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| |(7 x Year + 11) / 22
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| |}
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| === 31edo ===
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| 31 edo contains the following Irvian modes, derived from ME [[31edo MOS scales]]:
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| {| class="wikitable"
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| |+31edo Irvian modes
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| !Name
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| !Formula core
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| !Key layout
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| |Würschmidt[3]
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| |(3 x Year + 15) / 31
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| |6-16-26
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| |-
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| |Myna[4]
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| |(4 x Year + 15) / 31
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| |4-12-20-28
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| |-
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| |Mothra[5]
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| |(5 x Year + 15) / 31
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| |4-10-16-22-28
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| |-
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| |Hemithirds[6]
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| |(6 x Year + 15) / 31
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| |3-8-13-19-24-29
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| |-
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| |Mohajira[7]
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| |(7 x Year + 15) / 31
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| |3-7-12-16-20-25-29
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| |-
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| |Nusecond[8]
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| |(8 x Year + 15) / 31
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| |2-6-10-14-18-22-26-30
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| |-
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| |Orwell[9]
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| |(9 x Year + 15) / 31
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| |2-6-9-13-16-19-23-26-30
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| |-
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| |Miracle[10]
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| |(10 x Year + 15) / 31
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| |2-5-8-11-14-18-21-24-27-30
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| |}
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| and so on.
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| == Sound perception ==
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| The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31st of an octave instead of one 13th).
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| The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo.
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| Maximally even sets tend to be familiar and musically relevant scale collections. Examples:
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| <ul><li>The maximally even heptatonic set of [[19edo|19edo]] is, like the one in 12edo, a diatonic scale.</li><li>The maximally even heptatonic sets of [[17edo|17edo]] and [[24edo|24edo]], in contrary, are Maqamic[7].</li><li>The maximally even heptatonic set of [[22edo|22edo]] is Porcupine[7] (the superpythagorean diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is the octatonic scale of Hedgehog, the maximally even nonatonic set of 22edo is Orwell[9], while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara.</li><li>The maximally even 13-element set in 24edo is Ivan Wyschnegradsky's diatonicized chromatic scale.</li><li>The maximally even sets in edos 40 and higher have step sizes so close together that they can sound like [[Circulating temperament|circulating temperaments]] with the right timbre.
| | == Discovery of temperaments with a given generator == |
| </li></ul>
| | Maximum evenness scales' generator and amount of notes follow the formula {{nowrap|''LU'' (mod ''N'') {{=}} 1}}, where ''L'' is the note amount per period, ''U'' is the generator, and ''N'' is the EDO's cardinality. Note that ''L'' and ''U'' have to be coprime for the period to be 1 octave. |
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| Note that "maximally even" is equivalent to "quasi-equal-interval-symmetrical" in [[Joel_Mandelbaum|Joel Mandelbaum]]'s 1961 thesis [http://www.anaphoria.com/mandelbaum.html Multiple Divisions of the Octave and the Tonal Resources of 19-Tone Temperament]. Previous versions of this article have conflated "quasi-equal" with "quasi-equal-interval symmetrical". In fact, "quasi-equal" scales, according to Mandelbaum, meet the first criterion listed above, but not necessarily the second.
| | As such, it's possible to discover a temperament with a given generator in a given EDO simply by [[temperament merging]] the amount of notes with the EDO's cardinality. |
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| == Trivia == | | === Example 1: 12edo's diatonic === |
| * Maximally even heptatonic scale of [[19edo]] is the leap year arrangement of the [[Wikipedia:Hebrew calendar|Hebrew calendar]].
| | Generator of 12edo's diatonic is 7\12, as is the amount of notes. As such, we simply carry out {{nowrap|7 & 12}} to find the desired temperament. In 5-limit, that's meantone, tempering out 81/80, and consistent with world musical practices today. |
| * Maximally even octatonic scale of [[33edo]] is a leap year arrangement of the Dee calendar and the tabular, evened version of the [[Wikipedia:Iranian calendars|Persian calendar]].
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| == External links ==
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| * [https://individual.utoronto.ca/kalendis/leap/52-293-sym454-leap-years.htm 52/293 Symmetry454 Leap Years]
| | === Example 2: 37edo's 11/8 === |
| | Let's say we want to see what would repeatedly stacking 11th harmonic do well in all of 11-limit, in an EDO that presents it well. |
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| * [http://individual.utoronto.ca/kalendis/leap/index.htm#slc Solar Calendar Leap Rules - subsection Symmetrical Leap Cycles] | | 11/8 amounts to 17 steps of 37edo, and the solution to the problem {{nowrap|17*''x'' (mod 1) {{=}} 37}} is 24, meaning if the generator is 11/8, we are dealing with a 24 tone maximally even scale. As such, the temperament we are looking for is {{nowrap|24 & 37}}, which can be interpreted as [[freivald]] or [[emka]]. |
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| | === Example 3: On-request maximum evenness scales === |
| | Let's say we want to see what rank two temperament does Sym454 leap rule represent, 62\293 generator with 52/293 note count. |
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| [[Category:Scale theory]]
| | We simply merge {{nowrap|52 & 293}} in a selected limit to get our answer. Let's say 17 limit, we get a {{nowrap|52 & 243c}} temperament with a comma list 225/224, 715/714, 2880/2873, 22750/22627 and 60112/60025. |
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| | [[Category:Scale]] |
| [[Category:Todo:cleanup]] | | [[Category:Todo:cleanup]] |