Tour of regular temperaments: Difference between revisions

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Regular temperaments are non-Just tunings wherein the infinite number of intervals in p-limit Just intonation (or any subgroup thereof) are mapped to a smaller (though possibly still infinite) set of tempered intervals, by "tempering" (deliberately mistuning) some of the ratios such that a comma (or set of commas) "vanishes" by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas a tempered out. Temperaments effectively reduce the "dimensionality" of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals would be located by a four-dimensional set of coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which commas (and how many) are tempered out, and intervals can be located with a set of one-, two-, or three-dimensional coordinates (depending on the number of commas that have been tempered out, or in other words the "rank" of the temperament).

A rank r [[http://en.wikipedia.org/wiki/Regular_temperament|regular temperament]] in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r vals. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of [[comma|commas]] tempered out by the temperament, or a set of r independent [[Vals and Tuning Space|vals]] defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[comma pump examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.

==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 Just intonation, but without the difficulties normally associated with JI (such as wolf intervals, commas, and comma pumps). They 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, which 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.

==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 "short-hand" for describing the properties of temperaments. The most important of these are vals and commas, which any student of the modern regular temperament paradigm should become familiar with as a first order of business. These concepts are rather straight-forward and require very little math to understand.

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 two most frequently used forms of optimization are POTE ("Pure-Octave Tenney-Euclidean") and [[Top tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please"). Optimization is rather mathematically-intensive, but it is seldom (if ever) left as an exercise to the reader; most temperaments are presented here in their optimal forms.

=[[edo|Equal temperaments]]=

[[Equal Temperaments|Equal temperaments]] (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a "Rank 1") temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve "fun" results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.

=Rank 2 (including "linear") temperaments[[#lineartemperaments]]=

Regular temperaments of ranks two and three are cataloged [[Optimal patent val|here]]. Other pages listing them are [[Paul Erlich]]'s [[Catalog of five-limit rank two temperaments|catalog of 5-limit rank two temperaments]], and a [[Proposed names for rank 2 temperaments|page]] listing higher limit rank two temperaments.

P-limit Rank 2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators (though they may have any number of step-sizes). This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. Rank-2 temperaments can be reduced to a related rank-1 temperaments by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the Syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-TET by tempering out the Pythagorean comma.

As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the [[Normal lists|normal comma list]] of the various temperaments, where a **comma** is a small interval, not a square or cube or other power, which is tempered out by the temperament.

Meantone is a familiar historical temperament based on a chain of fifths (or fourths), but it is only one of many possibilities for temperaments based on a chain of generating intervals. These are referred to as "rank 2" temperaments, since they are based on a set of two linearly independent intervals. One of these intervals (typically an octave or fraction of an octave) can be selected as the "period", and another interval, usually chosen to be smaller than the period, is referred to as the "generator".

===[[Meantone family]]===

The meantone family tempers out 81/80, also called the syntonic comma. 81/80 manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as some sort of average or "mean" of the two tones. It has a flattened fifth (or sharpened fourth) as generator. Some meantone tunings are [[12edo]], [[19edo]], [[31edo]], [[43edo]], [[50edo]], [[55edo]] and [[81edo]]. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of syntonic comma (the 81/80 interval.)

===[[Schismatic family]]===

The schismatic family tempers out the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the syntonic comma (81/80). The 5-limit version of the temperament is a [[Microtempering|microtemperament]] which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity - whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4.

===[[Kleismic family]]===

The kleismic family of temperaments tempers out the kleisma of 15625/15552, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. The kleismic family includes [[15edo]], [[19edo]], [[34edo]], [[49edo]], [[53edo]], [[72edo]], [[87edo]] and [[140edo]] among its possible tunings.

===[[Magic family]]===

The magic family tempers out 3125/3072, known as the magic comma or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/2. The magic generator is itself an approximate 5/4.

The magic family includes [[16edo]], [[19edo]], [[22edo]], [[25edo]], and [[41edo]] among its possible tunings, with the latter being near-optimal.

===[[Diaschismic family]]===

The diaschismic family tempers out 2048/2025, the [[diaschisma]], which tempers things such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. A noted 7-limit extension to diaschismic is[[pajara| pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out, and of which [[22edo]] is an excellent tuning.

===[[Pelogic family]]===

This tempers out the pelogic comma, 135/128, also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, the accumulated flatness leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates 2L5s "anti-diatonic" scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo]], [[16edo]], [[23edo]], and [[25edo]].

===[[Porcupine family]]===

The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include [[15edo]], [[22edo]], [[37edo]], and [[59edo]].

===[[Wuerschmidt family]]===

The wuerschmidt family tempers out Wuerschmidt's comma, 393216/390625 = |17 1 -8&gt;. Wuerschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a perfect 5th two octaves up); that is, (5/4)^8 * (393216/390625) = 6. It tends to generate the same MOS's as [[magic family|magic temperament]], but is tuned slightly more accurately. Both [[31edo]] and [[34edo]] can be used as wuerschmit tunings, as can [[65edo]], which is quite accurate.

===[[Augmented family]]===

The augmented family tempers out the diesis of 128/125, the difference between three 5/4 major thirds and a 2/1 octave, and so identifies the major third with the third-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]], which is an excellent tuning for augmented. It is the temperament that results in what is commonly called the "augmented scale" (3L3s) in common 12-based music theory, as well as what is commonly called "Tcherepnin's scale" (3L6s).

===[[Dicot family]]===

The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24 (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into two neutral-sized intervals that are taken to approximate both. [[7edo]] makes for a "good" dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the "neutral" dicot 3rds span a 3/2. Tunings include [[7edo]], [[10edo]], and [[17edo]].

===[[Tetracot family]]===

The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by 20000/19683, the minimal diesis or tetracot comma. 7edo can also be considered a tetracot tuning, as can 20edo, 27edo, 34edo, and 41edo.

===[[Sensipent family]]===

This tempers out the sensipent comma, 78732/78125, also known as the medium semicomma.

===[[Semicomma family|Orwell and the semicomma family]]===

The semicomma (also known as **Fokker's comma)** 2109375/2097152 = |-21 3 7&gt; is tempered out by the members of the semicomma family. It doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as 7/6, leading to orwell temperament.

===[[Pythagorean family]]===

The Pythagorean family tempers out the Pythagorean comma, |-19 12 0&gt;. Since this is a 3-limit comma, it is also a 5-limit comma and can stand as parent to a 7-limit or higher family, in this case containing compton temperament and catler temperament. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate chains of 12-equal, offset from one another justly tuned 5/4.

===[[Archytas clan]]===

This clan tempers out the Archytas comma, 64/63, which is a triprime comma with factors of 2, 3 and 7. The clan consists of rank two temperaments, and should not be confused with the [[Archytas family]] of rank three temperaments.

===[[Apotome family]]===

This family tempers out the apotome, 2187/2048, which is a 3-limit comma.

===[[Gammic family]]===

The gammic family tempers out the gammic comma, |-29 -11 20&gt;. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is Neptune temperament.

===[[Minortonic family]]===

This tempers out the minortone comma, |-16 35 -17&gt;. The head of the family is minortonic temperament, with a generator of a minor tone (~10/9).

===[[Bug family]]===

This tempers out 27/25, the large limma or bug comma.

===[[Trienstonic clan]]===

This clan tempers out the septimal third-tone, 28/27, a triprime comma with factors of 2, 3 and 7.

===[[Sycamore family]]===

The sycamore family tempers out the sycamore comma, |-16 -6 11&gt; = 48828125/47775744, which is the amount by which five stacked chromatic semitones, 25/24, exceed 6/5, and hence also the amount six exceeds 5/4.

===[[Mutt family]]===

This tempers out the mutt comma, |-44 -3 21&gt;, leading to some strange properties.

===[[Escapade family]]===

This tempers out the escapade comma, |32 -7 -9&gt;, which is the difference between nine just major thirds and seven just fourths.

===[[Vulture family]]===

This tempers out the vulture comma, |24 -21 4&gt;.

===[[Vishnuzmic family]]===

This tempers out the vishnuzma, |23 6 -14&gt;, or the amount by which seven chromatic semitones (25/24) fall short of a perfect fourth (4/), or (4/3)/(25/24)^7.

===[[Luna Family]]===

This tempers out the Luna comma; |38 -2 -15&gt; (274877906944/274658203125)

===[[Gamelismic clan]]===

If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another [[Just intonation subgroups|subgroup]] of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. We can modify the definition of [[Normal lists|normal comma list]] for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.

Particularly noteworthy as member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds.

Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called "blackjack" and a 31-note scale called "canasta" have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.

===[[Sensamagic clan]]===

This clan tempers out 245/243, the sensamagic comma.

===[[Jubilismic clan]]===

This tempers out the jubilisma, 50/49, E.A. the difference between 10/7 and 7/5.

===[[Hemimean clan]]===

This tempers out the hemimean comma, 3136/3125, a no-threes comma.

===[[Mirkwai clan]]===

This tempers out the mirkwai comma, |0 3 4 -5&gt; = 16875/16807, a no-twos comma (ratio of odd numbers.)

===[[Quartonic temperaments]]===

These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35.

===[[Avicennmic temperaments]]===

These temper out the avicennma, |-9 1 2 1&gt; = 525/512, also known as Avicenna's enharmonic diesis.

===[[Starling temperaments]]===

Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.

===[[Marvel temperaments]]===

These temper out |-5 2 2 -1&gt; = 225/224, the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.

===[[Hemifamity temperaments]]===

The hemifamity temperaments temper out the hemifamity comma, |10 -6 1 -1&gt; = 5120/5103.

===[[Porwell temperaments]]===

The porwell temperaments temper out the porwell comma, |11 1 -3 -2&gt; = 6144/6125.

===[[Breedsmic temperaments]]===

A breedsmic temperament is one which tempers out the breedsma, 2401/2400. Some which do not get discussed elsewhere are collected on a page [[Breedsmic temperaments|here]].

===[[Ragismic microtemperaments]]===

A ragismic temperament is one which tempers out 4375/4374. These are not by any means all microtemperaments, but those which are not highly accurate are probably best discussed under another heading. Accurate ones include ennealimmal, supermajor, enneadecal, amity, mitonic, parakleismic, gamera and vishnu. Pontiac belongs on the list but falls under the schismatic family rubric.

===[[Turkish maqam music temperaments]]===

Vsrious theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish [[Arabic, Turkish, Persian|maqam music]] in a systematic way. This includes, in effect, certain linear temperaments.

=Rank 3 temperaments=

Even less familiar than rank 2 temperaments are the [[Planar Temperament|rank 3 temperaments]], based on a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank 3 temperaments by the commas they temper out.

===[[Marvel family]]===

The head of the marvel family is marvel, which tempers out 225/224. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.

===[[Starling family]]===

Starling tempers out 126/125, and like marvel it has the same generators as the 5-limit. An excellent tuning for starling is [[77edo]], but 31, 46 or 58 also work nicely.

===[[Gamelismic family]]===

Not to be confused with the gamelismic clan of rank two temperaments, the gamelismic family are those rank three temperaments which temper out the gamelisma, 1029/1024.

===[[Breed family]]===

Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749et will certainly do the trick. Breed has generators of 2, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7.

===[[Ragisma family]]===

The 7-limit rank three microtemperament which tempers out the ragisma, 4375/4374, extends to various higher limit rank three temperaments such as thor.

===[[Hemifamity family]]===

The hemifamity family of rank three temperaments tempers out the hemifamity comma, 5120/5103.

The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125.

===[[Horwell family]]===

The horwell family of rank three temperaments tempers out the horwell comma, 65625/65536.

===[[Hemimage family]]===

The hemimage family of rank three temperaments tempers out the hemimage comma, 10976/10935.

===[[Sensamagic family]]===

These temper out 245/243.

===[[Keemic family]]===

These temper out 875/864.

These temper out the senga, 686/675.

===[[Orwellismic family]]===

These temper out 1728/1715.

===[[Nuwell family]]===  

These temper out the nuwell comma, 2430/2401.

===[[Octagar family]]===

The octagar family of rank three temperaments tempers out the octagar comma, 4000/3969.

===[[Mirkwai family]]===

The mirkwai family of rank three temperaments tempers out the mirkwai comma, 16875/16807.

===[[Hemimean family]]===

The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125.

===[[Kleismic rank three family]]===

These are the rank three temperaments tempering out the kleisma, 15625/15552. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

These are the rank three temperaments tempering out the dischisma, 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

These are the rank three temperaments tempering out the didymus comma, 81/80. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

===[[Porcupine rank three family]]===

These are the rank three temperaments tempering out the porcupine comma or aximal diesis, 250/243.If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

===[[Archytas family]]===

Archytas temperament tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord.

===[[Jubilismic family]]===

Jubilismic temperament tempers out 50/49 and thereby identifies the two septimal tritones, 7/5 and 10/7.

===[[Semaphore family]]===  

Semaphore temperament tempers out 49/48 and thereby identifies the septimal minor third, 7/6 and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like "semi-fourth".

The quartonic temperament tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7.

===[[Werckismic temperaments]]===

These temper out the werckisma, 441/440.

===[[Swetismic temperaments]]===

These temper out the swetisma, 540/539.

===[[Lehmerismic temperaments]]===

These temper out the lehmerisma, 3025/3024.

===[[Kalismic temperaments]]===

These temper out the kalisma, 9801/9800.

=[[Rank four temperaments|Rank 4 temperaments]]=

Even less explored than rank three temperaments are rank four temperaments. In fact, unless one counts 7-limit JI they don't seem to have been explored at all. However, they could be used; for example [[Hobbits|hobbit scales]] can be constructed for them.

=[[Subgroup temperaments]]=  

=Commatic realms=  

By a //commatic realm// is meant the whole collection of regular temperaments of various ranks and for both full groups and [[Just intonation subgroups|subgroups]] tempering out a given comma. For some commas, looking at the full commatic realm seems the best approach to discussing associated temperaments.

==[[Orgonia]]==

By //Orgonia// is meant the commatic realm of the [[11-limit]] comma 65536/65219 = |16 0 0 -2 -3&gt;, the orgonisma.

==[[The Biosphere]]==

The Biosphere is the name given to the commatic realm of the 13-limit comma 91/90.

==[[The Archipelago]]==

The Archipelago is a name which has been given to the commatic realm of the [[13-limit]] comma 676/675.

* [[http://en.wikipedia.org/wiki/Regular_temperament|Regular temperaments - Wikipedia]]</pre></div>

<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tour of Regular Temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:164:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:164 --&gt;&lt;!-- ws:start:WikiTextTocRule:165: --&gt;&lt;a href="#Regular temperaments"&gt;Regular temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:165 --&gt;&lt;!-- ws:start:WikiTextTocRule:166: --&gt;&lt;!-- ws:end:WikiTextTocRule:166 --&gt;&lt;!-- ws:start:WikiTextTocRule:167: --&gt;&lt;!-- ws:end:WikiTextTocRule:167 --&gt;&lt;!-- ws:start:WikiTextTocRule:168: --&gt; | &lt;a href="#Equal temperaments"&gt;Equal temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:168 --&gt;&lt;!-- ws:start:WikiTextTocRule:169: --&gt; | &lt;a href="#Rank 2 (including &amp;quot;linear&amp;quot;) temperaments"&gt;Rank 2 (including &amp;quot;linear&amp;quot;) temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:169 --&gt;&lt;!-- ws:start:WikiTextTocRule:170: --&gt;&lt;!-- ws:end:WikiTextTocRule:170 --&gt;&lt;!-- ws:start:WikiTextTocRule:171: --&gt;&lt;!-- ws:end:WikiTextTocRule:171 --&gt;&lt;!-- ws:start:WikiTextTocRule:172: --&gt;&lt;!-- ws:end:WikiTextTocRule:172 --&gt;&lt;!-- ws:start:WikiTextTocRule:173: --&gt;&lt;!-- ws:end:WikiTextTocRule:173 --&gt;&lt;!-- ws:start:WikiTextTocRule:174: --&gt;&lt;!-- ws:end:WikiTextTocRule:174 --&gt;&lt;!-- ws:start:WikiTextTocRule:175: --&gt;&lt;!-- ws:end:WikiTextTocRule:175 --&gt;&lt;!-- ws:start:WikiTextTocRule:176: --&gt;&lt;!-- ws:end:WikiTextTocRule:176 --&gt;&lt;!-- ws:start:WikiTextTocRule:177: --&gt;&lt;!-- ws:end:WikiTextTocRule:177 --&gt;&lt;!-- ws:start:WikiTextTocRule:178: --&gt;&lt;!-- ws:end:WikiTextTocRule:178 --&gt;&lt;!-- ws:start:WikiTextTocRule:179: --&gt;&lt;!-- ws:end:WikiTextTocRule:179 --&gt;&lt;!-- ws:start:WikiTextTocRule:180: --&gt;&lt;!-- ws:end:WikiTextTocRule:180 --&gt;&lt;!-- ws:start:WikiTextTocRule:181: --&gt;&lt;!-- ws:end:WikiTextTocRule:181 --&gt;&lt;!-- ws:start:WikiTextTocRule:182: --&gt;&lt;!-- ws:end:WikiTextTocRule:182 --&gt;&lt;!-- ws:start:WikiTextTocRule:183: --&gt;&lt;!-- ws:end:WikiTextTocRule:183 --&gt;&lt;!-- ws:start:WikiTextTocRule:184: --&gt;&lt;!-- ws:end:WikiTextTocRule:184 --&gt;&lt;!-- ws:start:WikiTextTocRule:185: --&gt;&lt;!-- ws:end:WikiTextTocRule:185 --&gt;&lt;!-- ws:start:WikiTextTocRule:186: --&gt;&lt;!-- ws:end:WikiTextTocRule:186 --&gt;&lt;!-- ws:start:WikiTextTocRule:187: --&gt;&lt;!-- ws:end:WikiTextTocRule:187 --&gt;&lt;!-- ws:start:WikiTextTocRule:188: --&gt;&lt;!-- ws:end:WikiTextTocRule:188 --&gt;&lt;!-- ws:start:WikiTextTocRule:189: --&gt;&lt;!-- ws:end:WikiTextTocRule:189 --&gt;&lt;!-- ws:start:WikiTextTocRule:190: --&gt;&lt;!-- ws:end:WikiTextTocRule:190 --&gt;&lt;!-- ws:start:WikiTextTocRule:191: --&gt;&lt;!-- ws:end:WikiTextTocRule:191 --&gt;&lt;!-- ws:start:WikiTextTocRule:192: --&gt;&lt;!-- ws:end:WikiTextTocRule:192 --&gt;&lt;!-- ws:start:WikiTextTocRule:193: --&gt;&lt;!-- ws:end:WikiTextTocRule:193 --&gt;&lt;!-- ws:start:WikiTextTocRule:194: --&gt;&lt;!-- ws:end:WikiTextTocRule:194 --&gt;&lt;!-- ws:start:WikiTextTocRule:195: --&gt;&lt;!-- ws:end:WikiTextTocRule:195 --&gt;&lt;!-- ws:start:WikiTextTocRule:196: --&gt;&lt;!-- ws:end:WikiTextTocRule:196 --&gt;&lt;!-- ws:start:WikiTextTocRule:197: --&gt;&lt;!-- ws:end:WikiTextTocRule:197 --&gt;&lt;!-- ws:start:WikiTextTocRule:198: --&gt;&lt;!-- ws:end:WikiTextTocRule:198 --&gt;&lt;!-- ws:start:WikiTextTocRule:199: --&gt;&lt;!-- ws:end:WikiTextTocRule:199 --&gt;&lt;!-- ws:start:WikiTextTocRule:200: --&gt;&lt;!-- ws:end:WikiTextTocRule:200 --&gt;&lt;!-- ws:start:WikiTextTocRule:201: --&gt;&lt;!-- ws:end:WikiTextTocRule:201 --&gt;&lt;!-- ws:start:WikiTextTocRule:202: --&gt;&lt;!-- ws:end:WikiTextTocRule:202 --&gt;&lt;!-- ws:start:WikiTextTocRule:203: --&gt;&lt;!-- ws:end:WikiTextTocRule:203 --&gt;&lt;!-- ws:start:WikiTextTocRule:204: --&gt;&lt;!-- ws:end:WikiTextTocRule:204 --&gt;&lt;!-- ws:start:WikiTextTocRule:205: --&gt;&lt;!-- ws:end:WikiTextTocRule:205 --&gt;&lt;!-- ws:start:WikiTextTocRule:206: --&gt;&lt;!-- ws:end:WikiTextTocRule:206 --&gt;&lt;!-- ws:start:WikiTextTocRule:207: --&gt;&lt;!-- ws:end:WikiTextTocRule:207 --&gt;&lt;!-- ws:start:WikiTextTocRule:208: --&gt;&lt;!-- ws:end:WikiTextTocRule:208 --&gt;&lt;!-- ws:start:WikiTextTocRule:209: --&gt;&lt;!-- ws:end:WikiTextTocRule:209 --&gt;&lt;!-- ws:start:WikiTextTocRule:210: --&gt; | &lt;a href="#Rank 3 temperaments"&gt;Rank 3 temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:210 --&gt;&lt;!-- ws:start:WikiTextTocRule:211: --&gt;&lt;!-- ws:end:WikiTextTocRule:211 --&gt;&lt;!-- ws:start:WikiTextTocRule:212: --&gt;&lt;!-- ws:end:WikiTextTocRule:212 --&gt;&lt;!-- ws:start:WikiTextTocRule:213: --&gt;&lt;!-- ws:end:WikiTextTocRule:213 --&gt;&lt;!-- ws:start:WikiTextTocRule:214: --&gt;&lt;!-- ws:end:WikiTextTocRule:214 --&gt;&lt;!-- ws:start:WikiTextTocRule:215: --&gt;&lt;!-- ws:end:WikiTextTocRule:215 --&gt;&lt;!-- ws:start:WikiTextTocRule:216: --&gt;&lt;!-- ws:end:WikiTextTocRule:216 --&gt;&lt;!-- ws:start:WikiTextTocRule:217: --&gt;&lt;!-- ws:end:WikiTextTocRule:217 --&gt;&lt;!-- ws:start:WikiTextTocRule:218: --&gt;&lt;!-- ws:end:WikiTextTocRule:218 --&gt;&lt;!-- ws:start:WikiTextTocRule:219: --&gt;&lt;!-- ws:end:WikiTextTocRule:219 --&gt;&lt;!-- ws:start:WikiTextTocRule:220: --&gt;&lt;!-- ws:end:WikiTextTocRule:220 --&gt;&lt;!-- ws:start:WikiTextTocRule:221: --&gt;&lt;!-- ws:end:WikiTextTocRule:221 --&gt;&lt;!-- ws:start:WikiTextTocRule:222: --&gt;&lt;!-- ws:end:WikiTextTocRule:222 --&gt;&lt;!-- ws:start:WikiTextTocRule:223: --&gt;&lt;!-- ws:end:WikiTextTocRule:223 --&gt;&lt;!-- ws:start:WikiTextTocRule:224: --&gt;&lt;!-- ws:end:WikiTextTocRule:224 --&gt;&lt;!-- ws:start:WikiTextTocRule:225: --&gt;&lt;!-- ws:end:WikiTextTocRule:225 --&gt;&lt;!-- ws:start:WikiTextTocRule:226: --&gt;&lt;!-- ws:end:WikiTextTocRule:226 --&gt;&lt;!-- ws:start:WikiTextTocRule:227: --&gt;&lt;!-- ws:end:WikiTextTocRule:227 --&gt;&lt;!-- ws:start:WikiTextTocRule:228: --&gt;&lt;!-- ws:end:WikiTextTocRule:228 --&gt;&lt;!-- ws:start:WikiTextTocRule:229: --&gt;&lt;!-- ws:end:WikiTextTocRule:229 --&gt;&lt;!-- ws:start:WikiTextTocRule:230: --&gt;&lt;!-- ws:end:WikiTextTocRule:230 --&gt;&lt;!-- ws:start:WikiTextTocRule:231: --&gt;&lt;!-- ws:end:WikiTextTocRule:231 --&gt;&lt;!-- ws:start:WikiTextTocRule:232: --&gt;&lt;!-- ws:end:WikiTextTocRule:232 --&gt;&lt;!-- ws:start:WikiTextTocRule:233: --&gt;&lt;!-- ws:end:WikiTextTocRule:233 --&gt;&lt;!-- ws:start:WikiTextTocRule:234: --&gt;&lt;!-- ws:end:WikiTextTocRule:234 --&gt;&lt;!-- ws:start:WikiTextTocRule:235: --&gt;&lt;!-- ws:end:WikiTextTocRule:235 --&gt;&lt;!-- ws:start:WikiTextTocRule:236: --&gt;&lt;!-- ws:end:WikiTextTocRule:236 --&gt;&lt;!-- ws:start:WikiTextTocRule:237: --&gt;&lt;!-- ws:end:WikiTextTocRule:237 --&gt;&lt;!-- ws:start:WikiTextTocRule:238: --&gt;&lt;!-- ws:end:WikiTextTocRule:238 --&gt;&lt;!-- ws:start:WikiTextTocRule:239: --&gt;&lt;!-- ws:end:WikiTextTocRule:239 --&gt;&lt;!-- ws:start:WikiTextTocRule:240: --&gt; | &lt;a href="#Rank 4 temperaments"&gt;Rank 4 temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:240 --&gt;&lt;!-- ws:start:WikiTextTocRule:241: --&gt; | &lt;a href="#Subgroup temperaments"&gt;Subgroup temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:241 --&gt;&lt;!-- ws:start:WikiTextTocRule:242: --&gt; | &lt;a href="#Commatic realms"&gt;Commatic realms&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:242 --&gt;&lt;!-- ws:start:WikiTextTocRule:243: --&gt;&lt;!-- ws:end:WikiTextTocRule:243 --&gt;&lt;!-- ws:start:WikiTextTocRule:244: --&gt;&lt;!-- ws:end:WikiTextTocRule:244 --&gt;&lt;!-- ws:start:WikiTextTocRule:245: --&gt;&lt;!-- ws:end:WikiTextTocRule:245 --&gt;&lt;!-- ws:start:WikiTextTocRule:246: --&gt;&lt;!-- ws:end:WikiTextTocRule:246 --&gt;&lt;!-- ws:start:WikiTextTocRule:247: --&gt;