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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | | de = Reguläre_Temperatur |
| : This revision was by author [[User:clumma|clumma]] and made on <tt>2014-12-14 22:20:08 UTC</tt>.<br>
| | | en = Regular temperament |
| : The original revision id was <tt>535151684</tt>.<br>
| | | es = |
| : The revision comment was: <tt>GTFO</tt><br>
| | | ja = レギュラーテンペラメント |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
| | {{Beginner|Mathematical theory of regular temperaments}} |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"><span style="display: block; text-align: right;">Other languages: [[xenharmonie/Verallgemeinerte reguläre Temperatur|Deutsch]]
| | {{Wikipedia}} |
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| | A '''regular temperament''' ('''RT''') is an abstract [[tuning system]] that looks the same no matter which pitch you start from (or consider the [[tonic]]). In other words, unlimited free modulation is possible: any [[interval]] can be stacked as many times as you like. A regular temperament is [[generate]]d by a set of generating intervals, usually one of which is considered the [[period]], and any note which is part of the regular temperament can be reached by stacking whole numbers of these generating intervals above a defined root note. For example, [[meantone]] temperament is generated by the [[2/1|octave]] and a tempered (detuned) version of the [[3/2|perfect fifth]], with the octave usually being considered the period, and every interval in meantone can be expressed as an integer number of octaves plus an integer number of fifths. In meantone, a {{W|major second}} is equal to two perfect fifths minus an octave, and a {{W|major third}} is four perfect fifths minus two octaves. Regular temperaments theoretically have an infinite number of notes, and besides [[equal temperament]]s, regular temperaments usually<ref group="note">This is true if there exist two generators such that size in [[cent]]s of one generator divided by that of the other is an {{W|irrational number}}. This is not true for tunings where every generator is a whole number of steps of some [[edo]] or other [[equal-step tuning]].</ref> have an infinite number of notes in between ''any two other notes''. |
| =Characterizing an abstract regular temperament=
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| An //abstract regular temperament// is a [[regular temperament]] considered apart from any consideration of tuning, which classifies all of its tunings as tunings of the single abstract temperament. Various methods have been proposed for uniquely characterizing an abstract regular temperament. Among these are
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| * **The [[Wedgies and Multivals|wedgie]]**
| | In addition to unlimited modulation, regular temperaments are by definition thought of as being approximations of some system of pure or target intervals, very often a [[just intonation]] (JI) [[subgroup]]. Each abstract interval is interpreted as a tempered, or detuned, version of the target interval (more accurately, a set of target intervals). For example, the octave in meantone represents the just ratio [[2/1]], the perfect fifth [[3/2]], and the major third [[5/4]]. Certain intervals are tempered to the [[1/1|unison]], or [[tempering out|tempered out]]; in a regular temperament, these intervals are known as [[comma]]s. In meantone, since stacking up four perfect fifths, down two octaves, and down a major third reaches the unison, we get that {{nowrap|(3/2)<sup>4</sup> / (2/1)<sup>2</sup> / (5/4) {{=}} [[81/80]]}} is tempered out, and thus 81/80 is a comma of meantone. Any two just intervals separated by a comma of a temperament, for example [[9/8]] and [[10/9]] in meantone, are mapped to the same tempered interval in the temperament, in this case a major second. A temperament only qualifies as a regular temperament if this interpretation works in a perfectly consistent way: The product of two tempered intervals must always be the tempered version of the product of the JI intervals; for example, if the ratios 3/2 and 5/4 are in the target interval set, then ~3/2 × ~5/4 = ~[[15/8]] must always be true. ("~" denotes tempered.) In any temperament, each target interval is mapped to a unique tempered interval, though a tempered interval can represent multiple target intervals. |
| This uses [[http://en.wikipedia.org/wiki/Exterior_algebra|multilinear algebra]] to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the [[Interior product|interior product]] of a [[Wedgies and Multivals|wedgie]] for a p-limit temperament with the p-limit monzos.
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| For example, using "∨" to represent the interior product, we have mir = <<6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir ∨ |1 0 0 0> is <0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1> which is <1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0> which is also <1 1 3 3|; <1 1 3 3| tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces an additional comma into a multival, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.
| | One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single step size. JI itself can be considered a [[trivial temperament]] where no tempering is happening: No commas are tempered out, and all of them are preserved as small pitch differences. Another example of a trivial temperament is [[single-pitch tuning]], where there are ''no'' generating intervals, and only a single pitch is available. In between JI and equal temperaments lies the cornucopia of temperaments discussed in [[Paul Erlich]]'s seminal work, [[:File:MiddlePath2015.pdf|''A Middle Path Between Just Intonation and the Equal Temperaments'']]. |
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| As explained on the [[Interior product#Applications|interior product]] page, if W is the r-wedgie defining the rank r temperament, then the tuning of map for the temperament can be defined via an (r-1)-multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q) • V.
| | == History == |
| | The roots of '''regular temperament theory''' ('''RTT''') can be traced back for centuries. The practice far predates the theory, and in particular [[meantone]] temperament has been known since the 15th century. Many early pioneers set the stage for the general theory to come: |
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| * **[[Normal lists|Normal val lists]]** | | * {{W|Nicola Vicentino}} (1511–1576): [[adaptive JI]], [[31edo|31et]] |
| Given a list of vals, we may [[Saturation|saturate]] it and reduce it using the [[Normal lists|Hermite normal form]] to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [<1 1 3 3|, <0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].
| | * {{W|Leonhard Euler}} (1707–1783): [[5-limit]] tonespace |
| | * {{W|Hermann von Helmholtz}} (1821–1894): psychoacoustics |
| | * {{W|R. H. M. Bosanquet}} (1841–1913): regular mapping, generalized keyboard |
| | * {{W|Shohe Tanaka}} (1862–1945): 5-limit tonespace (triangular projection) |
| | * [[Adriaan Fokker]] (1887–1972): [[Fokker block|periodicity blocks]] |
| | * [[Harry Partch]] (1901–1974): [[JI|extended JI]] |
| | * [[Erv Wilson]] (1928–2016): extended tonespace (and projections), [[mos]], scale tree |
| | * [[Easley Blackwood]] (1933–2023): Blackwood[10], syntonic comma vanishing relation as equation |
| | * [[George Secor]] (1943–2020): miracle temperament |
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| * **The [[Tenney-Euclidean Tuning|Frobenius projection map]]**
| | A significant amount of this theory's early development occurred online via the {{w|Yahoo! Groups}} service. The groundwork was laid by [[Paul Erlich]], [[Graham Breed]], [[Dave Keenan]], [[Herman Miller]], and [[Paul Hahn]] in the late 1990's. |
| Given any list of monzos, or any list of vals, we may compute the associated Frobenius projection map. This corresponds uniquely with an abstract regular temperament. The intervals of the abstract temperament may be defined via multiplication by the projection map, leading to [[fractional monzos]] which are actually the tunings of these intervals in [[Fractional monzos|Frobenius tuning]]. However, using the Frobenius projection map to define the abstract temperament by no means commits us to Frobenius tuning.
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| * **[[Just intonation subgroups]] and [[Transversal|transversals]]**
| | In 2001 [[Gene Ward Smith]] joined Yahoo! Groups and immediately began making major contributions to the conversation, introducing new terminology and higher-level math. He and his closer collaborators such as [[Mike Battaglia]] also did much of the work to document RTT on this wiki. |
| A relatively concrete approach, but one which is not canonically defined, is to define a [[transversal]] for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament.
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| For example, for [[Gamelismic clan|miracle temperament]] [2, 15/14] defines a rank two 7-limit subgroup whose [[Normal lists|normal interval list]] is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.
| | In 2009 [[Kite Giedraitis]] began developing his own approach to RTT, including some noteworthy innovations. |
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| * **[[Normal lists|Normal comma lists]]**
| | == FAQ == |
| The normal comma list uniquely defines the abstract temperament, and has the advantage of showing family relationships even more clearly than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list.
| | === Why would I want to use a regular temperament? === |
| | Regular temperaments are of most use to musicians who want their music to sound as much as possible like stacking-based [[just intonation]], but without the difficulties normally associated with it, such as [[wolf interval]]s, [[comma]]s, and [[comma pump]]s. Specifically, if your chord progression [[pump]]s a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral thirds, without caring much what ratio they are tuned to. Thus one might use rastmic even though no commas are pumped. |
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| * **[[http://en.wikipedia.org/wiki/Row_echelon_form|Reduced row echelon form]]**
| | === How does regular temperament theory help me compose music? === |
| If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.
| | The skill of music composition is acquired by studying the disciplines such as {{w|harmony}}, {{w|musical form|form}}, {{w|orchestration}}, in addition to extensive listening. One common misconception is that learning regular temperament theory can be a substitute for any of those. Regular temperament theory does indeed present you with numerous tuning systems, and provide the tools to help you compare and choose between them based on some common goals. It also tells you how harmonic resources are available in each tuning system, though the question of putting them together to a piece of work is really up to you to experiment with. In other words, one may think of the relationship between regular temperament theory and composition as this: regular temperament theory tells you how to ''choose'' a tuning, while composition regards how to ''use'' a chosen tuning. |
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| For example, if we feed [<22 35 51 62|, <31 49 72 87|, <84 133 195 236|] into a reduced row echelon form routine, we obtain [<1 0 3 1|, <0 1 -3/7 8/7|, <0 0 0 0|]. Stripping off the zero val in the final row, we get E = [<1 0 3 1|, <0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1>, and |-1 -1 0 1>E* = [0 1/7]. Multiply by |1 0 0 0>, the val for 2, and the result is |1 0 0 0>E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.
| | === What do I need to know to understand all the numbers on the pages for individual regular temperaments? === |
| | Although the concept of regular temperament is centuries old and predates much of modern mathematics, members of the Yahoo! Alternative Tuning List have developed a particular form of numerical shorthand for describing the properties of temperaments. The most important of these are [[val]]s ([[mapping]]s), [[monzo]]s and [[tempering out|tempering out comma]]s, which any student of the modern regular temperament paradigm should become familiar with. These concepts are rather straightforward and require little math to understand. |
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| =Translation between methods of specifying temperaments=
| | The [[rank]] of a temperament is its dimension. It equals the number of generators in the [[Just intonation subgroup|subgroup]] being used minus the number of independent commas that are tempered out. |
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| The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while it in some ways is not a very good system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in its full generality.
| | Another recent contribution to the field of temperament is the concept of [[optimization]], which can take many forms. The point of optimization is to minimize the difference between a temperament and JI by finding an optimal tuning for the generator. The most frequently used forms of optimization are [[POTE tuning|POTE]] ("Pure-Octave Tenney–Euclidean"), [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please") and more recently [[CWE]] ("Constained Weil–Euclidean"), which has become the new standard instead of POTE since POTE is meant to be an approximation. Optimization is rather intensive mathematically, but it is seldom left as an exercise to the reader; most temperaments are presented here in their optimal forms in terms of POTE and CTE generators. In addition, for each temperament there is a [[optimal ET sequence|sequence of equal temperaments]] showing possible [[equal-step tuning]]s in the order of better absolute accuracy to JI. |
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| ==Wedgies==
| | The most common browser tools used for finding optimal tunings (useful for investigating new temperaments) are [[Graham Breed]]'s [http://x31eq.com/temper/ Temperament Finder] and [[Sintel]]'s [https://sintel.pythonanywhere.com/ Temperament Calculator]; the former gives temperament names (usually consistent with the wiki) and implements a wide variety of features like finding related temperaments while the latter implements CWE and more complex types of subgroups (like allowing ratios as generators) and supports an alternative notation to [[warts]] that is more convenient for arbitrary subgroups. |
| To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (ie, the wedge product of r-1 elements representing primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain n choose r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arithmetic, not floating point numbers.
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| ==Frobenius projection maps==
| | Usually, temperaments have names coming from a wide array of [[temperament names|sources]], but they can also have systematic and rigorous names, from which the comma(s) can be deduced. The most common systematic temperament naming system on the wiki is [[Kite's color notation]]: {{nowrap|wa {{=}} 3-limit|yo {{=}} 5-over|gu {{=}} 5-under|zo {{=}} 7-over|and ru {{=}} 7-under}} (see also [[Kite's color notation/Temperament names]]). |
| To translate from the Frobenius matrix to the RREF, simply reduce the matrix to RREF form in the usual way. To translate from RREF to the Frobenius matrix, if E is the RREF form then the matrix is E`E. Here the definition for the pseudoinverse E` using only matrix inverse and transpose can be used.
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| ==The normal val list==
| | Yet another recent development is the concept of a [[pergen]], appearing in our [[Tour of regular temperaments]] as (P8, P5/2) or somesuch, which classifies temperaments by their period and the generator(s), giving ideas of how to notate these temperaments. For rank-2 temperaments we have developed a similar classification system called [[ploidacot]]. |
| To translate from the normal val list to the RREF, simply reduce the normal val list. To obtain the normal val list from the RREF, clear denominators from the rows of the RREF, saturate the result, reduce that to Hermite normal form and make the adjustment to normal val form.
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| ==The normal comma list== | | == Further reading == |
| To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection map by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one.
| | === Introductory materials === |
| | * ''[[A Middle Path]]'': this is [[Paul Erlich]]'s guide to RTT (regular temperament theory) |
| | * [[Dave Keenan & Douglas Blumeyer's guide to RTT]] |
| | * [[Keenan Pepper's explanation of vals]] |
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| Maple code for the parts of this which do not call the Maple functions for Hermite normal form, Smith normal form or the pseudoinverse can be found in the article [[Basic abstract temperament translation code]].
| | === Key regular temperament concepts === |
| | These topics are covered in the introductory materials above, but you can read about them here in more depth: |
| | * [[Monzo]] |
| | * [[Val]] |
| | * [[Mapping]] |
| | * [[Comma basis]] |
| | * [[Patent val]] |
| | * [[Tempering out]] |
| | * [[Rank and codimension]] |
| | * [[Tuning map]] |
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| =The Geometry of Regular Temperaments= | | === Lists of temperaments === |
| | Temperaments that approximate important harmonies relatively well with a small number of notes: |
| | * [[Low harmonic entropy linear temperaments]] |
| | * [[Middle Path table of 5-limit rank-2 temperaments]] |
| | * [[Middle Path table of 7-limit rank-2 temperaments]] |
| | * [[Middle Path table of 11-limit rank-2 temperaments]] |
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| Abstract regular temperaments can be identified with [[http://en.wikipedia.org/wiki/Rational_point|rational points]] on an [[http://en.wikipedia.org/wiki/Algebraic_variety|algebraic variety]] known as a [[http://en.wikipedia.org/wiki/Grassmannian|Grassmannian]]. In particular, if the number of primes in the p-limit is n, and the rank of the temperament is r, then the real Grassmannian **Gr**(r, n) has points identified with the r-dimensional subspaces of the n-dimensional real vector space **R**^n. This has an embedding into a real vector space known as the [[http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding|Plücker embedding]], which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank r in the p-limit may be defined as rational points on **Gr**(r, n), though we should note that most of these do not correspond to anything worth much as a temperament. In matrix terms, the real Grassmannian **Gr**(r, n) can be identified with real symmetric projection matrices with trace r. The rational symmetric projection matrices with trace r are precisely the Frobenius projections, so under this identification it is clear they represent rational points on **Gr**(r, n). A rational projection matrix of trace r which is not symmetric is still a tuning map; minimax and least squares tunings provide examples of this.
| | More comprehensive lists: |
| | * [[Bird's eye view of temperaments by accuracy]] (article): temperaments the Xen Wiki contributors find most useful for approximating JI - with edo tunings and note counts for the harmonies they target, and explanations of their structure |
| | * [[Survey of efficient temperaments by subgroup]] (table): good general-purpose temperaments, sorted by size (notes per equave) and by JI subgroup |
| | * [[Map of rank-2 temperaments]] (table): temperaments (some general, some niche) sorted by the size of their period and generator |
| | * [[Temperaments for MOS shapes]] (table): temperaments (some general, some niche) sorted by the scale shape they generate |
| | * [[Tour of regular temperaments]] (article): huge gallery of the dozens of families of temperaments that have been described; ''very technical - not for the faint of heart'' |
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| Grassmannians have the structure of a smooth, homogenous [[http://en.wikipedia.org/wiki/Metric_space|metric space]], and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian **Gr**(2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below.
| | === Other writings on temperaments === |
| | * [[Mike's lectures on regular temperament theory|Mike Battaglia's lectures on RTT]] |
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| [[image:dualzoom.gif]]</pre></div>
| | == Notes == |
| <h4>Original HTML content:</h4>
| | <references group="note"/> |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Regular temperament</title></head><body><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/Verallgemeinerte%20regul%C3%A4re%20Temperatur">Deutsch</a><br />
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| | == External links == |
| <!-- ws:start:WikiTextTocRule:14:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><a href="#Characterizing an abstract regular temperament">Characterizing an abstract regular temperament</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#Translation between methods of specifying temperaments">Translation between methods of specifying temperaments</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#The Geometry of Regular Temperaments">The Geometry of Regular Temperaments</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: -->
| | * [http://x31eq.com/paradigm.html Graham Breed's "The Regular Mapping Paradigm"] |
| <!-- ws:end:WikiTextTocRule:22 --><br />
| | * [https://youtu.be/ZoAuVgndmbU John Moriarty – Tuning Theory 2: Temperament ("Microtonal" Theory)], a video lecture |
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Characterizing an abstract regular temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->Characterizing an abstract regular temperament</h1>
| | [[Category:Regular temperament theory| ]] <!-- Main article --> |
| An <em>abstract regular temperament</em> is a <a class="wiki_link" href="/regular%20temperament">regular temperament</a> considered apart from any consideration of tuning, which classifies all of its tunings as tunings of the single abstract temperament. Various methods have been proposed for uniquely characterizing an abstract regular temperament. Among these are<br />
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| <ul><li><strong>The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a></strong></li></ul>This uses <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">multilinear algebra</a> to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the <a class="wiki_link" href="/Interior%20product">interior product</a> of a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for a p-limit temperament with the p-limit monzos.<br />
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| For example, using &quot;∨&quot; to represent the interior product, we have mir = &lt;&lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir ∨ |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1&gt; which is &lt;1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0&gt; which is also &lt;1 1 3 3|; &lt;1 1 3 3| tempers out the commas of miracle as well as 15/14 (or equivalently 16/15), sending them to the unison. The interior product forces an additional comma into a multival, lowering the rank by one. When we do this to a wedgie, we obtain a multival of rank one less, which has all the commas of the wedgie plus an additional comma, the interval we want to represent.<br />
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| As explained on the <a class="wiki_link" href="/Interior%20product#Applications">interior product</a> page, if W is the r-wedgie defining the rank r temperament, then the tuning of map for the temperament can be defined via an (r-1)-multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q) • V.<br />
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| <ul><li><strong><a class="wiki_link" href="/Normal%20lists">Normal val lists</a></strong></li></ul>Given a list of vals, we may <a class="wiki_link" href="/Saturation">saturate</a> it and reduce it using the <a class="wiki_link" href="/Normal%20lists">Hermite normal form</a> to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [&lt;1 1 3 3|, &lt;0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].<br />
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| <ul><li><strong>The <a class="wiki_link" href="/Tenney-Euclidean%20Tuning">Frobenius projection map</a></strong></li></ul>Given any list of monzos, or any list of vals, we may compute the associated Frobenius projection map. This corresponds uniquely with an abstract regular temperament. The intervals of the abstract temperament may be defined via multiplication by the projection map, leading to <a class="wiki_link" href="/fractional%20monzos">fractional monzos</a> which are actually the tunings of these intervals in <a class="wiki_link" href="/Fractional%20monzos">Frobenius tuning</a>. However, using the Frobenius projection map to define the abstract temperament by no means commits us to Frobenius tuning.<br />
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| <ul><li><strong><a class="wiki_link" href="/Just%20intonation%20subgroups">Just intonation subgroups</a> and <a class="wiki_link" href="/Transversal">transversals</a></strong></li></ul>A relatively concrete approach, but one which is not canonically defined, is to define a <a class="wiki_link" href="/transversal">transversal</a> for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament.<br />
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| For example, for <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a> [2, 15/14] defines a rank two 7-limit subgroup whose <a class="wiki_link" href="/Normal%20lists">normal interval list</a> is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.<br />
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| <ul><li><strong><a class="wiki_link" href="/Normal%20lists">Normal comma lists</a></strong></li></ul>The normal comma list uniquely defines the abstract temperament, and has the advantage of showing family relationships even more clearly than the normal val list. Intervals of the temperament may be defined after computing another means of representing the temperament such as the normal val list.<br />
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| <ul><li><strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Row_echelon_form" rel="nofollow">Reduced row echelon form</a></strong></li></ul>If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.<br />
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| For example, if we feed [&lt;22 35 51 62|, &lt;31 49 72 87|, &lt;84 133 195 236|] into a reduced row echelon form routine, we obtain [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|, &lt;0 0 0 0|]. Stripping off the zero val in the final row, we get E = [&lt;1 0 3 1|, &lt;0 1 -3/7 8/7|]. The monzo for 7/6 is |-1 -1 0 1&gt;, and |-1 -1 0 1&gt;E* = [0 1/7]. Multiply by |1 0 0 0&gt;, the val for 2, and the result is |1 0 0 0&gt;E*, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.<br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Translation between methods of specifying temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Translation between methods of specifying temperaments</h1>
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| The various methods for specifying an abstract regular temperament can be translated from one to another. Below we explain how to translate to and from reduced row echelon form (RREF.) The point of using RREF as the transportation hub is that while it in some ways is not a very good system for musical purposes, it is quick and easy to compute, with no requirement to use Smith or Hermite normal forms or to make use of the pseudoinverse in its full generality.<br />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Translation between methods of specifying temperaments-Wedgies"></a><!-- ws:end:WikiTextHeadingRule:4 -->Wedgies</h2>
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| To translate to wedgies from RREF simply take the wedge product of the rows of the RREF and then reduce the resulting multivector to a wedgie. To translate from wedgies to RREF, for a wedgie of rank r in n dimensions (where n = pi(p) is the number of primes in the p-limit) take a wedge product of basis vectors involving r-1 basis elements (ie, the wedge product of r-1 elements representing primes) and wedge these with the basis element for each prime, obtaining either 0 or an r-fold wedge product with sign +-1. Take the corresponding element of the wedgie times the +-1 sign (which is computed from the parity of the permutation of the r elements.) This gives a val; do this for every combination of r-1 basis elements to obtain n choose r-1 vals, and reduce the result to an RREF by the usual Gaussian reduction. If possible, this should be done using rational arithmetic, not floating point numbers.<br />
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| <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Translation between methods of specifying temperaments-Frobenius projection maps"></a><!-- ws:end:WikiTextHeadingRule:6 -->Frobenius projection maps</h2>
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| To translate from the Frobenius matrix to the RREF, simply reduce the matrix to RREF form in the usual way. To translate from RREF to the Frobenius matrix, if E is the RREF form then the matrix is E`E. Here the definition for the pseudoinverse E` using only matrix inverse and transpose can be used.<br />
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| <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Translation between methods of specifying temperaments-The normal val list"></a><!-- ws:end:WikiTextHeadingRule:8 -->The normal val list</h2>
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| To translate from the normal val list to the RREF, simply reduce the normal val list. To obtain the normal val list from the RREF, clear denominators from the rows of the RREF, saturate the result, reduce that to Hermite normal form and make the adjustment to normal val form.<br />
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| <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Translation between methods of specifying temperaments-The normal comma list"></a><!-- ws:end:WikiTextHeadingRule:10 -->The normal comma list</h2>
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| To translate from the normal comma list to the RREF, find the null space of the matrix of monzos of the normal comma list in form of a matrix, take the transpose of that matrix, and reduce to the RREF. If E is the RREF, to find the normal comma list first find the Frobenius projection map by computing I - M`M, where I is the identity matrix. Clear denominators from this, and saturate. Then reverse rows, reduce to Hermite normal form, reverse rows again, and adjust the result so that the monzos represent commas greater than one.<br />
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| Maple code for the parts of this which do not call the Maple functions for Hermite normal form, Smith normal form or the pseudoinverse can be found in the article <a class="wiki_link" href="/Basic%20abstract%20temperament%20translation%20code">Basic abstract temperament translation code</a>.<br />
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| <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="The Geometry of Regular Temperaments"></a><!-- ws:end:WikiTextHeadingRule:12 -->The Geometry of Regular Temperaments</h1>
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| Abstract regular temperaments can be identified with <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_point" rel="nofollow">rational points</a> on an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Algebraic_variety" rel="nofollow">algebraic variety</a> known as a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Grassmannian" rel="nofollow">Grassmannian</a>. In particular, if the number of primes in the p-limit is n, and the rank of the temperament is r, then the real Grassmannian <strong>Gr</strong>(r, n) has points identified with the r-dimensional subspaces of the n-dimensional real vector space <strong>R</strong>^n. This has an embedding into a real vector space known as the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding" rel="nofollow">Plücker embedding</a>, which if the point in question corresponds to a temperament is the wedgie of the temperament. Regular temperaments of rank r in the p-limit may be defined as rational points on <strong>Gr</strong>(r, n), though we should note that most of these do not correspond to anything worth much as a temperament. In matrix terms, the real Grassmannian <strong>Gr</strong>(r, n) can be identified with real symmetric projection matrices with trace r. The rational symmetric projection matrices with trace r are precisely the Frobenius projections, so under this identification it is clear they represent rational points on <strong>Gr</strong>(r, n). A rational projection matrix of trace r which is not symmetric is still a tuning map; minimax and least squares tunings provide examples of this.<br />
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| Grassmannians have the structure of a smooth, homogenous <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Metric_space" rel="nofollow">metric space</a>, and hence represent a distinctly geometric mathematical object. In the 5-limit, the Grassmannian <strong>Gr</strong>(2, 3), consisting of the planes through the origin in three dimensional space, may be identified with the projective plane, and hence 5-limit rank two temperaments may be pictured as points in a projective plane, as below.<br />
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A regular temperament (RT) is an abstract tuning system that looks the same no matter which pitch you start from (or consider the tonic). In other words, unlimited free modulation is possible: any interval can be stacked as many times as you like. A regular temperament is generated by a set of generating intervals, usually one of which is considered the period, and any note which is part of the regular temperament can be reached by stacking whole numbers of these generating intervals above a defined root note. For example, meantone temperament is generated by the octave and a tempered (detuned) version of the perfect fifth, with the octave usually being considered the period, and every interval in meantone can be expressed as an integer number of octaves plus an integer number of fifths. In meantone, a major second is equal to two perfect fifths minus an octave, and a major third is four perfect fifths minus two octaves. Regular temperaments theoretically have an infinite number of notes, and besides equal temperaments, regular temperaments usually[note 1] have an infinite number of notes in between any two other notes.
In addition to unlimited modulation, regular temperaments are by definition thought of as being approximations of some system of pure or target intervals, very often a just intonation (JI) subgroup. Each abstract interval is interpreted as a tempered, or detuned, version of the target interval (more accurately, a set of target intervals). For example, the octave in meantone represents the just ratio 2/1, the perfect fifth 3/2, and the major third 5/4. Certain intervals are tempered to the unison, or tempered out; in a regular temperament, these intervals are known as commas. In meantone, since stacking up four perfect fifths, down two octaves, and down a major third reaches the unison, we get that (3/2)4 / (2/1)2 / (5/4) = 81/80 is tempered out, and thus 81/80 is a comma of meantone. Any two just intervals separated by a comma of a temperament, for example 9/8 and 10/9 in meantone, are mapped to the same tempered interval in the temperament, in this case a major second. A temperament only qualifies as a regular temperament if this interpretation works in a perfectly consistent way: The product of two tempered intervals must always be the tempered version of the product of the JI intervals; for example, if the ratios 3/2 and 5/4 are in the target interval set, then ~3/2 × ~5/4 = ~15/8 must always be true. ("~" denotes tempered.) In any temperament, each target interval is mapped to a unique tempered interval, though a tempered interval can represent multiple target intervals.
One particularly simple kind of regular temperaments is equal temperaments, which represent all intervals by multiples of a single step size. JI itself can be considered a trivial temperament where no tempering is happening: No commas are tempered out, and all of them are preserved as small pitch differences. Another example of a trivial temperament is single-pitch tuning, where there are no generating intervals, and only a single pitch is available. In between JI and equal temperaments lies the cornucopia of temperaments discussed in Paul Erlich's seminal work, A Middle Path Between Just Intonation and the Equal Temperaments.
History
The roots of regular temperament theory (RTT) can be traced back for centuries. The practice far predates the theory, and in particular meantone temperament has been known since the 15th century. Many early pioneers set the stage for the general theory to come:
A significant amount of this theory's early development occurred online via the Yahoo! Groups service. The groundwork was laid by Paul Erlich, Graham Breed, Dave Keenan, Herman Miller, and Paul Hahn in the late 1990's.
In 2001 Gene Ward Smith joined Yahoo! Groups and immediately began making major contributions to the conversation, introducing new terminology and higher-level math. He and his closer collaborators such as Mike Battaglia also did much of the work to document RTT on this wiki.
In 2009 Kite Giedraitis began developing his own approach to RTT, including some noteworthy innovations.
FAQ
Why would I want to use a regular temperament?
Regular temperaments are of most use to musicians who want their music to sound as much as possible like stacking-based just intonation, but without the difficulties normally associated with it, such as wolf intervals, commas, and comma pumps. Specifically, if your chord progression pumps a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral thirds, without caring much what ratio they are tuned to. Thus one might use rastmic even though no commas are pumped.
How does regular temperament theory help me compose music?
The skill of music composition is acquired by studying the disciplines such as harmony, form, orchestration, in addition to extensive listening. One common misconception is that learning regular temperament theory can be a substitute for any of those. Regular temperament theory does indeed present you with numerous tuning systems, and provide the tools to help you compare and choose between them based on some common goals. It also tells you how harmonic resources are available in each tuning system, though the question of putting them together to a piece of work is really up to you to experiment with. In other words, one may think of the relationship between regular temperament theory and composition as this: regular temperament theory tells you how to choose a tuning, while composition regards how to use a chosen tuning.
What do I need to know to understand all the numbers on the pages for individual regular temperaments?
Although the concept of regular temperament is centuries old and predates much of modern mathematics, members of the Yahoo! Alternative Tuning List have developed a particular form of numerical shorthand for describing the properties of temperaments. The most important of these are vals (mappings), monzos and tempering out commas, which any student of the modern regular temperament paradigm should become familiar with. These concepts are rather straightforward and require little math to understand.
The rank of a temperament is its dimension. It equals the number of generators in the subgroup being used minus the number of independent commas that are tempered out.
Another recent contribution to the field of temperament is the concept of optimization, which can take many forms. The point of optimization is to minimize the difference between a temperament and JI by finding an optimal tuning for the generator. The most frequently used forms of optimization are POTE ("Pure-Octave Tenney–Euclidean"), TOP ("Tenney OPtimal", or "Tempered Octaves, Please") and more recently CWE ("Constained Weil–Euclidean"), which has become the new standard instead of POTE since POTE is meant to be an approximation. Optimization is rather intensive mathematically, but it is seldom left as an exercise to the reader; most temperaments are presented here in their optimal forms in terms of POTE and CTE generators. In addition, for each temperament there is a sequence of equal temperaments showing possible equal-step tunings in the order of better absolute accuracy to JI.
The most common browser tools used for finding optimal tunings (useful for investigating new temperaments) are Graham Breed's Temperament Finder and Sintel's Temperament Calculator; the former gives temperament names (usually consistent with the wiki) and implements a wide variety of features like finding related temperaments while the latter implements CWE and more complex types of subgroups (like allowing ratios as generators) and supports an alternative notation to warts that is more convenient for arbitrary subgroups.
Usually, temperaments have names coming from a wide array of sources, but they can also have systematic and rigorous names, from which the comma(s) can be deduced. The most common systematic temperament naming system on the wiki is Kite's color notation: wa = 3-limit, yo = 5-over, gu = 5-under, zo = 7-over, and ru = 7-under (see also Kite's color notation/Temperament names).
Yet another recent development is the concept of a pergen, appearing in our Tour of regular temperaments as (P8, P5/2) or somesuch, which classifies temperaments by their period and the generator(s), giving ideas of how to notate these temperaments. For rank-2 temperaments we have developed a similar classification system called ploidacot.
Further reading
Introductory materials
Key regular temperament concepts
These topics are covered in the introductory materials above, but you can read about them here in more depth:
Lists of temperaments
Temperaments that approximate important harmonies relatively well with a small number of notes:
More comprehensive lists:
Other writings on temperaments
Notes
- ↑ This is true if there exist two generators such that size in cents of one generator divided by that of the other is an irrational number. This is not true for tunings where every generator is a whole number of steps of some edo or other equal-step tuning.
External links
