Tour of regular temperaments: Difference between revisions

<span style="display: block; text-align: right;">[[de:Reguläre_Temperaturen]]  

Regular temperaments are non-Just tunings in which the infinite number of intervals in [[Harmonic_Limit|p-limit]] [[Just_intonation|Just intonation]], or any [[Just_intonation_subgroups|subgroup]] thereof, are mapped to a smaller, though still infinite, set of [[tempering_out|tempered]] intervals. This is done by 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 are 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 located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, 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. The dimensionality is 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 commas tempered out by the temperament, or a set of r independent 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 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 shorthand for describing the properties of temperaments. The most important of these are vals (aka mappings) and 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 equals the number of primes in the subgroup minus the number of linearly independent (i.e. non-redundant) 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 two most frequently used forms of optimization are [[POTE_tuning|POTE]] ("Pure-Octave Tenney-Euclidean") and [[TOP_tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please"). 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.

Yet another recent development is the concept of a [[pergen]], appearing here as (P8, P5/2) or somesuch. Every rank-2, rank-3, rank-4, etc. temperament has a pergen, which specifies the period and the generator. Assuming the prime subgroup includes both 2 and 3, the period is either an octave or some fraction of it, and the generator is either some 3-limit interval or some fraction of one. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every strong extension of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but don't uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.

Each temperament has two names: a traditional name and a [[Color notation|color name]]. The traditional names are [[Temperament Names|arbitrary]], but the color names are systematic and rigorous, and the comma can be deduced from the color name. Wa = 3-limit, yo = 5-over, gu = 5-under, zo = 7-over, and ru = 7-under. See also [[Color notation/Temperament Names|Color Notation/Temperament Names]].

=Equal temperaments (Rank-1 temperaments)=

[[Equal-step tuning|Equal temperaments]] (abbreviated ET or tET) and [[EDO|equal divisions of the octave]] (abbreviated EDO or ED2) 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, or 12edo, 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-ET.

=Rank-2 (including linear) temperaments=

A p-limit rank-2 temperament maps all intervals of p-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a linear temperament. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma. 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-ET by tempering out the Pythagorean comma.

Regular temperaments of ranks two and three are cataloged on the [[Optimal patent val]] page. Rank-2 temperaments are also listed at [[Proposed names for rank 2 temperaments]] by their generator mappings, and at [[Map of rank-2 temperaments]] by their generator size. See also the [[pergen]]s page. There is also [[Graham Breed]]'s [http://x31eq.com/catalog2.html giant list of regular temperaments].

== 2.3 Families ==

These are families defined by a comma that uses only primes 2 and 3, i.e. a 3-limit comma. Every 3-limit comma is also a 5-limit comma. If prime 5 is assumed to be part of the subgroup, and no other comma is tempered out, the comma creates a rank-2 temperament. The comma can also stand as parent to a 7-limit or higher family when other commas are tempered out as well. Any temperament in these families can be thought of as consisting of mutiple "copies" of an EDO separated by a small comma. This small comma is represented in the pergen by ^1.

===[[Blackwood|Blackwood or Sawa family]] (P8/5, ^1)===

This family tempers out the limma, [8 -5 0> = 256/243, which implies 5-edo.

===[[Apotome family|Apotome or Lawa family]] (P8/7, ^1)===

This family tempers out the apotome, [-11 7 0> = 2187/2048, which implies 7-edo.

===[[Pythagorean family|Pythagorean or Lalawa family]] (P8/12, ^1)===

The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = [-19 12 0&gt;. This family includes the compton and catler temperaments. 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-edo, offset from one another justly tuned 5/4. Several 12-edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.

== 2.3.5 Families ==

These are families defined by a comma that uses only primes 2, 3 and 5. As we go up in 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.) The same comment applies to 7-limit temperaments and rank three, etc. Members of families and their relationships can be classified by the [[Normal_lists|normal comma list]] of the various temperaments. Families include weak extensions as well as strong, in other words, the pergen shown here may change.

===[[Meantone family|Meantone or Gu family]] (P8, P5) ===

The meantone family tempers out [[81/80]], also called the syntonic comma. This comma 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 a 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 the syntonic comma.

===[[Schismatic family|Schismatic or Layo family]] (P8, P5)===

The schismatic family tempers out the schisma of [-15 8 1> = [[32805/32768]], which is the amount by which the Pythagorean comma exceeds the syntonic comma. 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. Tunings include [[12edo]], [[29edo]], [[41edo]], [[53edo]], and [[118edo]].

===[[Pelogic family|Pelogic or Layobi family]] (P8, P5)===

This tempers out the pelogic comma, [-7 3 1> = [[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, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates [[2L 5s]] "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]].

===[[Father family|Father or Gubi family]] (P8, P5)===

This tempers out [[16/15]], the just diatonic semitone, and equates 5/4 with 4/3.

===[[Diaschismic family|Diaschismic or Sagugu family]] (P8/2, P5)===

The diaschismic family tempers out the [[diaschisma]], [11 -4 -2> or 2048/2025, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period of an approximate 45/32 or 64/45, 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]] temperament, where the intervals 50/49 and 64/63 are tempered out, of which [[22edo]] is an excellent tuning.

===[[Immunity family|Immunity or Sasa-yoyo family]] (P8, P4/2)===

This tempers out the immunity comma, [16 -13 2> (1638400/1594323).

===[[Bug family|Bug or Gugu family]] (P8, P4/2)===

This tempers out [[27/25]], the large limma or bug comma. The generator is an approximate 6/5 or 10/9.

===[[Dicot family|Dicot or Yoyo family]] (P8, P5/2)===

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]].

===[[Augmented_family|Augmented or Trigu  family]] (P8/3, P5)===

The augmented family tempers out the diesis of [7 0 -3> = [[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" ([[3L 3s]]) in common 12-based music theory, as well as what is commonly called "[http://www.tcherepnin.com/alex/basic_elem1.htm#9step Tcherepnin's scale]" ([[3L 6s]]).

===[[Porcupine family|Porcupine or Triyo family]] (P8, P4/3)===

The porcupine family tempers out [1 -5 3> = [[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]].

===[[Laconic family|Laconic or Latrigubi family]] (P8, P5/3)===

This tempers out the laconic comma, [-4 7 -3> (2187/2000), which is the difference between three 10/9's and one 3/2. Laconic is supported by [[16edo]], [[21edo]], and [[37edo]] (using the 37b mapping), among others.

===[[Dimipent family|Dimipent or Quadgu family]] (P8/4, P5)===

The dimipent (or diminished) family tempers out the major diesis or diminished comma, [3 4 -4> or [[648/625]], the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as [[12edo]].

===[[Tetracot family|Tetracot or Saquadyo family]] (P8, P5/4)===

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 [5 -9 4> (20000/19683), the minimal diesis or [[tetracot comma]]. [[7edo]] can also be considered a tetracot tuning, as can [[20edo]], [[27edo]], [[34edo]], and [[41edo]].

===[[Vulture family|Vulture or Sasa-quadyo family]] (P8, P12/4)===

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

===[[Comic family|Comic or Saquadyobi family]] (P8/2, M2/4)===

This tempers out the comic comma, [13 -14 4> = 5120000/4782969

===[[Pental family|Pental or Trila-quingu family]] (P8/5, P5)===

This tempers out the pental comma, 847288609443/838860800000 = [-28 25 -5&gt;.

===[[Amity family|Amity or Saquinyo family]] (P8, P11/5)===

This tempers out the [[amity comma]], 1600000/1594323 = [9 -13 5&gt;.

===[[Magic family|Magic or Laquinyo family]] (P8, P12/5)===

The magic family tempers out [-10 -1 5> (3125/3072), known as the [[magic comma]] or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/1. The 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.

===[[Fifive family|Fifive or Saquinbiyo family]] (P8/2, P5/5)===

This tempers out the fifive comma, [-1 -14 10> = 9765625/9565938.

===[[Qintosec family|Qintosec or Quadsa-quinbigu family]] (P8/5, P5/2)===

This tempers out the qintosec comma, 140737488355328/140126044921875 = [47 -15 -10&gt;.

===[[Trisedodge family|Trisedodge or Saquintrigu family]] (P8/5, P4/3)===

This tempers out the trisedodge comma, 30958682112/30517578125 = [19 10 -15&gt;.

===[[Kleismic family|Kleismic or Tribiyo family]] (P8, P12/6)===

The kleismic family of temperaments tempers out the [[kleisma]] [-6 -5 6> = 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.

===[[Semicomma_family|Orwell or Sepru, and the semicomma or Lasepyo family]] (P8, P12/7)===

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.

===[[Wesley family|Wesley or Lasepyobi family]] (P8, ccP4/7)===

This tempers out the wesley comma, [-13 -2 7> = 78125/73728. Seven generators equals a double-compound 4th of ~16/3.

===[[Sensipent family|Sensipent or Sepgu family]] (P8, ccP5/7)===

The sensipent (sensi) family tempers out the [[sensipent comma]], [2 9 -7> (78732/78125), also known as the medium semicomma. Seven generators equals a double-compound 5th of ~6/1.Tunings include [[8edo]], [[19edo]], [[46edo]], and [[65edo]].

===[[Vishnuzmic family|Vishnuzmic or Sasepbigu family]] (P8/2, P4/7)===

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 (4/3), or (4/3)/(25/24)^7.

===[[Mutt family|Mutt or Trila-septriyo family]] (P8/3, ccP4/7)===

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

===[[Würschmidt family|Würschmidt or Saquadbigu family]] (P8, ccP5/8)===

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

===[[Escapade family|Escapade or Sasa-tritrigu family]] (P8, P4/9)===

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

===[[Shibboleth family|Shibboleth or Tritriyo family]] (P8, ccP4/9)===

This tempers out the shibboleth comma, [-5 -10 9> = 1953125/1889568. Nine generators of ~6/5 equal a double compound 4th of ~16/3.

===[[Sycamore family|Sycamore or Laleyo family]] (P8, P5/11)===

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. Eleven of these generators equals ~3/2.

===[[Ditonmic family|Ditonmic or Lala-theyo family]] (P8, c⁴P4/13)===

This tempers out the ditonma, [-27 -2 13> = 1220703125/1207959552. Thirteen generators equals a quadruple-compound 4th.

===[[Luna family|Luna or Sasa-quintrigu family]] (P8, ccP4/15)===

This tempers out the luna comma, [38 -2 -15&gt; (274877906944/274658203125). Fifteen generators equals a double-compound 4th of ~16/3.

===[[Minortonic family|Minortonic or Trila-segu family]] (P8, ccP5/17)===

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). Seventeen generators equals a double-compound 5th 6/1.

===[[Maja family|Maja or Saseyo family]] (P8, c⁶P4/17)===

This tempers out the maja comma, [-3 -23 17> = 762939453125/753145430616. Seventeen generators equals a sextuple-compound 4th.

===[[Maquila family|Maquila or Trisa-segu family]] (P8, c⁷P5/17)===

This tempers out the maquila comma, 562949953421312/556182861328125 = [49 -6 -17&gt;. Seventeen generators equals a septuple-compound 5th.

===[[Gammic family|Gammic or Laquinquadyo family]] (P8, P5/20)===

The gammic family tempers out the gammic comma, [-29 -11 20&gt;. Nine generators of about 35¢ equals ~6/5, eleven equal ~5/4 and twenty equal ~3/2. 34-edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is Neptune temperament.

==2.3.7 Clans==

These are defined by a comma that uses only primes 2, 3 and 7, i.e. a no-fives comma. See also [[subgroup temperaments]].  

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. 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.

===[[Archytas clan|Archytas or Ru clan]] (P8, P5)===

This clan tempers out the Archytas comma, [[64/63]]. It equates 7/4 with 16/9. The clan consists of rank two temperaments, and should not be confused with the [[Archytas family]] of rank three temperaments. Its best downward extension is [[Superpyth]].

=== Laru clan (P8, P5) ===

This clan tempers out [-13 10 0 -1> =  50.7¢. It equates 7/4 to an augmented 6th. Its best downward extension is [[Meantone family|Septimal Meantone]].

===[[Garischismic temperaments|Garischismic or Sasaru clan]] (P8, P5)===

This clan tempers out the garischisma, [25 -14 0 -1&gt; = 33554432/33480783. It equates 8/7 to two apotomes ([-11 7> = 2187/2048). This clan includes [[Vulture family|vulture]], [[Breedsmic temperaments|newt]], [[Schismatic family|garibaldi]], [[Landscape microtemperaments|sextile]], and satin.

===[[Trienstonic clan|Trienstonic or Zo clan]] (P8, P5) ===

This clan tempers out the septimal third-tone [[28/27]], a low-accuracy temperament that equates 7/6 with 9/8.

===[[Slendro clan|Slendro (Semaphore) or Zozo clan]] (P8, P4/2)===

This clan tempers out the slendro diesis, [[49/48]]. Generator = 8/7 or 7/6. Its best downward extension is [[Godzilla]]. See also [[Semaphore]].

=== Sasa-zozo clan (P8, P5/2) ===

This clan tempers out [15 -13 0 2> = 12.2¢, and includes as a strong extension the [[Hemififths]] temperament.

===[[Gamelismic clan|Gamelismic or Latrizo clan]] (P8, P5/3)===

This clan tempers out the gamelisma, [-10 1 0 3> = 1029/1024, a no-fives comma. Three 8/7 generators equals a 5th. A particularly noteworthy 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, thus it's a weak extension. Its 21-note scale called "blackjack" and 31-note scale called "canasta" have some useful properties. It is the most efficient 11-limit temperament for many purposes, with a tuning close to 72-EDO.

=== Latriru clan (P8, P11/3) ===

This clan tempers out [-9 11 0 -3> = 15.0¢. Generator = 81/56. It includes as a strong extension the [[Liese]] temperament, which is in the Meantone family.

===[[Stearnsmic temperaments|Stearnsmic or Latribiru clan]] (P8/2, P4/3)===

Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6&gt; = 118098/117649.

=== Laquadru clan (P8, P11/4) ===

This clan tempers out [-3 9 0 -4> = 42.3¢. Generator = 9/7. It includes as a strong extension the [[Squares]] temperament, which is in the Meantone family.

=== Saquadru clan (P8, P12/4) ===

This clan tempers out [16 -3 0 -4> = 18.8¢. Generator = 21/16. It includes as a strong extension the [[Vulture family|Vulture]] temperament, which is in the Vulture family.

=== Saquinzo clan (P8, P12/5) ===

This clan tempers out [5 -12 0 5> = 20.7¢. It includes as a strong extension the [[Magic]] temperament, which is in the Magic family.

=== Sepru clan (P8, P12/7) ===

This clan tempers out [7 8 0 -7> = 33.8¢. Generator = 7/6. It includes as a strong extension the [[Orwell]] temperament, which is in the Semicomma family.

== 2.3.11 and 2.3.13 Clans ==

These are defined by a comma that uses only primes 2, 3 and either 11 or 13. See also [[subgroup temperaments]].

=== [[Rastmic temperaments|Rastmic or Neutral or Lulu clan]] (P8, P5/2) ===

This 2.3.11 clan tempers out 243/242 = [-1 5 0 0 -2>. Generator = 11/9. It includes as a strong extension the [[Dicot]] temperament, which is in the Dicot family.

=== [[Hemif|Hemif or Thuthu clan]] (P8, P5/2) ===

This 2.3.13 clan tempers out 512/507 = [9 -1 0 0 0 -2>. Generator = 16/13. It includes as a strong extension the [[Dicot]] temperament, which is in the Dicot family.

== 2.5.7 Clans ==

These are defined by a comma that uses only primes 2, 5 and 7, i.e. a  no-threes comma. See also [[subgroup temperaments]].

=== [[Jubilismic clan|Jubilismic or Biruyo Nowa clan]] (P8/2, M3) ===

This clan tempers out the jubilisma, [[50/49]], which is the difference between 10/7 and 7/5. The pergen's M3 generator equals 5/4. The half-octave period equals 7/5.

===[[Hemimean clan|Hemimean or Zozoquingu Nowa clan]] (P8, M2)===

This clan tempers out the hemimean comma, [6 0 -5 2> = 3136/3125. Two M2 generators equals 5/4, and five of them equals 7/4.

===[[Quince clan|Quince or Lasepzo-agugu Nowa clan]] (P8, M2/2)===

This clan tempers out the quince, [-15 0 -2 7&gt; = 823543/819200. Two generators equals 8/7 (a M2), and seven generators equals 8/5.

== 3.5.7 Clans ==

These are defined by a comma that uses only primes 3, 5 and 7, i.e. a no-twos comma (ratio of odd numbers). If no other commas are tempered out, It is a non-octave tuning, with a period of a perfect 12th or tritave, 3/1. See also [[subgroup temperaments]].

===[[Sensamagic clan|Sensamagic or Zozoyo Noca clan]] (P12, M3)===

This 3.5.7 clan tempers out the sensamagic comma [0 -5 1 2> = 245/243. The M3 generator = 9/7, and two generators equals 5/3.

===[[Mirkwai clan|Mirkwai or Quinru-aquadyo Noca clan]] (P12, cM7/4)===

This 3.5.7 clan tempers out the mirkwai comma, [0 3 4 -5&gt; = 16875/16807. Four generators = a compound major 7th = 27/7.

=Rank-3 temperaments=

Even less familiar than rank-2 temperaments are the [[Planar_Temperament|rank-3 temperaments]], generated by 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. They can also be identified by pergen. The 3rd element of a rank-3 pergen is usually a comma, but sometimes it's some fraction of a 5-limit or 7-limit interval.

== 2.3.5 Families ==

These are families defined by a comma that uses only primes 2, 3 and 5. Every suchcomma defines a rank-3 family, thus every comma in the list of rank-2 2.3.5 families could be included here. If nothing else is tempered out we have a 7-limit rank-3 temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

===[[Didymus rank three family|Didymus or Gu rank three family]] (P8, P5, ^1)===

These are the rank three temperaments tempering out the didymus or meantone comma, 81/80. The meantone comma equates every 5-limit interval to some 3-limit interval, therefore the generators are the same as for 2.3.7 JI: 2/1, 3/1 and 7/1. These may be reduced to 2/1, 3/2 and 7/4, and 7/4 may be reduced further to 64/63. Hence in the pergen,^1 = 64/63.

===[[Diaschismic rank three family|Diaschismic or Sagugu rank three family]] (P8/2, P5, /1)===

These are the rank three temperaments tempering out the dischisma, [11 -4 -2> = 2048/2025. /1 = 64/63.

===[[Porcupine rank three family|Porcupine or Triyo rank three family]] (P8, P4/3, /1)===

These are the rank three temperaments tempering out the porcupine comma or maximal diesis, [1 -5 3> = 250/243. /1 = 64/63.

===[[Kleismic rank three family|Kleismic or Tribiyo rank three family]] (P8, P12/6, /1)===

These are the rank three temperaments tempering out the kleisma, [-6 -5 6> = 15625/15552. /1 = 64/63.

== 2.3.7 Families ==

These are families defined by a comma that uses only primes 2, 3 and 7 (no-fives comma). If nothing else is tempered out, we have a rank-3 temperament of  2.3.5.7 in which one of the generators is ~5/1. This generator can be reduced to ~5/4, which may be reduced further to ~81/80. Hence in all the pergens below, ^1 = ~81/80.

===[[Archytas family|Archytas or Ru family]] (P8, P5, ^1)===

Archytas temperament tempers out 64/63. This comma equates every 7-limit interval to some 3-limit interval. If 81/80 were tempered out too, the otonal tetrad 4:5:6:7 would be identified with the dominant seventh chord.

===[[Garischismic temperaments|Garischismic or Sasaru family]] (P8, P5, ^1)===

A garischismic temperament is one which tempers out the garischisma, [25 -14 0 -1&gt; = 33554432/33480783.

===[[Semiphore family|Semiphore or Zozo family]] (P8, P4/2, ^1)===

Semiphore 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". See also [[Semaphore|Sem'''<u>a</u>'''phore]] and [[Slendro clan|Slendro]].

===[[Gamelismic family|Gamelismic or Latrizo family]] (P8, P5/3, ^1)===

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, [-10 1 0 3> = 1029/1024.  

===[[Stearnsmic temperaments|Stearnsmic or Latribiru family]] (P8/2, P4/3, ^1)===

Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6&gt; = 118098/117649.

== 2.3.5.7 Families ==

These are families defined by a comma that uses primes 2, 3, 5 and 7.

===[[Marvel family|Marvel or Ruyoyo family]] (P8, P5, ^1)===

The head of the marvel family is marvel, which tempers out [-5 2 2 -1&gt; = [[225/224]]. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle. Other family members 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.

The marvel comma equates every 7-limit interval to some 5-limit interval, therefore the generators are the same as for 5-limit JI: 2/1, 3/1 and 5/1. These may be reduced to 2/1, 3/2 and 5/4, and 5/4 may be reduced further to 81/80. Hence in the pergen, ^1 = 81/80.  

===[[Starling family|Starling or Zotrigu family]] (P8, P5, ^1)===

Starling tempers out the septimal semicomma or starling comma [1 2 -3 1> = [[126/125]], the difference between three 6/5s plus one 7/6, and an octave. Like marvel, it has the same generators as 5-limit JI. An excellent tuning for starling is [[77edo]], but 31, 46 or 58 also work nicely. Its family includes myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum. ^1 = 81/80.

===[[Sensamagic family|Sensamagic or Zozoyo family]] (P8, P5, ^1)===

These temper out [0 -5 1 2> = 245/243. ^1 = 64/63.

===[[Greenwoodmic temperaments|Greenwoodmic or Ruruyo family]] (P8, P5, ^1)===

These temper out the greenwoodma, [-3 4 1 -2&gt; = 405/392. ^1 = 64/63.

===[[Avicennmic temperaments|Avicennmic or Zoyoyo family]] (P8, P5, ^1)===

These temper out the avicennma, [-9 1 2 1&gt; = 525/512, also known as Avicenna's enharmonic diesis. ^1 = 81/80.

===[[Keemic family|Keemic or Zotriyo family]] (P8, P5, ^1)===

These temper out the keema [-5 -3 3 1> = 875/864. ^1 = 81/80.

===[[Orwellismic family|Orwellismic or Triru-agu family]] (P8, P5, ^1)===

These temper out [6, 3, -1, -3> = 1728/1715. ^1 = 64/63.

===[[Nuwell family|Nuwell or Quadru-ayo family]] (P8, P5, ^1)===

These temper out the nuwell comma, [1, 5, 1, -4> = 2430/2401. ^1 = 64/63.

===[[Ragisma family|Ragisma or Zoquadyo family]] (P8, P5, ^1)===

The 7-limit rank three microtemperament which tempers out the ragisma, [-1 -7 4 1&gt; = 4375/4374, extends to various higher limit rank three temperaments such as thor. 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. ^1 = 81/80.

===[[Hemifamity family|Hemifamity or Saruyo family]] (P8, P5, ^1)===

The hemifamity family of rank three temperaments tempers out the hemifamity comma, [10 -6 1 -1> = 5120/5103. ^1 = 81/80.

===[[Horwell family|Horwell or Lazoquinyo family]] (P8, P5, ^1)===

The horwell family of rank three temperaments tempers out the horwell comma, [-16 1 5 1> = 65625/65536. ^1 = 81/80.

===[[Hemimage family|Hemimage or Satrizo-agu family]] (P8, P5, ^1)===

The hemimage family of rank three temperaments tempers out the hemimage comma, [5 -7 -1 3> = 10976/10935. ^1 = 64/63.

===[[Tolermic family|Tolermic or Sazoyoyo family]] (P8, P5, ^1)===

These temper out the tolerma, [10 -11 2 1> = 179200/177147. ^1 = ~81/80.

===[[Mint family|Mint or Rugu family]] (P8, P5, ^1)===

The mint temperament is a low complexity, high error temperament, tempering out the septimal quarter-tone 36/35, equating 7/6 with 6/5, and 5/4 with 9/7. ^1 = 81/80 or 64/63.

===[[Septisemi temperaments|Septisemi or Zogu family]] (P8, P5, ^1)===

These are very low complexity temperaments tempering out the minor septimal semitone, [[21/20]] and hence equating 5/3 with 7/4. ^1 = 81/80.

===[[Jubilismic family|Jubilismic or Biruyo family]] (P8/2, P5, ^1)===

Jubilismic temperament tempers out 50/49 and thereby equates the two septimal tritones, 7/5 and 10/7. This is the period, and ^1 = ~81/80.

===[[Cataharry temperaments|Cataharry or Labirugu family]] (P8, P4/2, ^1)===

Cataharry temperaments temper out the cataharry comma, [-4 9 -2 -2&gt; = 19683/19600. Half of a 4th is ~81/70. ^1 = 81/80.

===[[Breed family|Breed or Bizozogu family]] (P8, P5/2, /1)===

Breed is a 7-limit microtemperament which tempers out [-5 -1 -2 4&gt; = 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749edo will certainly do the trick. Breed has generators of 2/1, a 49/40-cum-60/49 neutral third, and 64/63.

The mirwomo family of rank three temperaments tempers out the mirwomo comma, [-15 3 2 2> = 33075/32768. Half a fith is ~128/105 and ^1 = ~81/80.

===[[Landscape family|Landscape or Trizogugu family]] (P8/3, P5, ^1)===

The 7-limit rank three microtemperament which tempers out the lanscape comma, [-4 6 -6 3&gt; = 250047/250000, extends to various higher limit rank three temperaments such as tyr and odin. The period is ~63/50 and ^1 = 81/80.

The dimcomp family of rank three temperaments tempers out the dimcomp comma, [-1 -4 8 -4> = 390625/388962. The period = ~25/21.

===[[Sengic family|Sengic or Trizo-agugu family]] (P8, P5, vm3/2)===

These temper out the senga, [1 -3 -2 3> = 686/675. One generator = ~15/14, two = ~7/6 (a downminor 3rd), and three = ~6/5.

===[[Porwell family|Porwell or Sarurutrigu family]] (P8, P5, ^m3/2)===

The porwell family of rank three temperaments tempers out the porwell comma, [11 1 -3 -2> = 6144/6125. Two ~35/32 generators equal an upminor 3rd of ~6/5.

The octagar family of rank three temperaments tempers out the octagar comma, [5 -4 3 -2> = 4000/3969. Two ~80/63 generators equal an upminor 6th of ~8/5.

===[[Hemimean family|Hemimean or Zozoquingu family]] (P8, P5, vM3/2)===

The hemimean family of rank three temperaments tempers out the hemimean comma, [6 0 -5 2> = 3136/3125. Two ~28/25 generators equal a downmajor 3rd of ~5/4.

===[[Wizmic microtemperaments|Wizmic or Quinzo-ayoyo family]] (P8, P5, vm7/2)===

A wizmic temperament is one which tempers out the wizma, [ -6 -8 2 5 &gt; = 420175/419904. vm7 = 7/4.

The mirkwai family of rank three temperaments tempers out the mirkwai comma, [0 3 4 -5> = 16875/16807. Four ~7/5 generators equal a compound upmajor 7th =  ~27/7.

=[[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.

===[[Valinorismic temperaments|Valinorismic or Lorugugu temperaments]] ===

These temper out the valinorsma, [4 0 -2 -1 1> = 176/175.

===[[Rastmic temperaments|Rastmic or Lulu temperaments]]===

These temper out the rastma, [-1 5 0 0 -2> = 243/242. As a no-fives no-sevens rank-2 temperament, it's (P8, P5/2).

===[[Werckismic temperaments|Werckismic or Luzozogu temperaments]]===

These temper out the werckisma, [-3 2 -1 2 -1> = 441/440.

===[[Swetismic temperaments|Swetismic or Lururuyo temperaments]]===

These temper out the swetisma, [2 3 1 -2 -1> = 540/539.

===[[Lehmerismic temperaments|Lehmerismic or Loloruyoyo temperaments]]===

These temper out the lehmerisma, [-4 -3 2 -1 2> = 3025/3024.

===[[Kalismic temperaments|Kalismic or Bilorugu temperaments]]===

These temper out the kalisma, [-3 4 -2 -2 2> = 9801/9800.

=[[Subgroup temperaments]]=

=Commatic realms of 11-limit and 13-limit commas=

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|Orgonia or Satrilu-aruru]]==

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

==[[The Biosphere|The Biosphere or Thozogu]] ==

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

==[[The Archipelago|The Archipelago or Bithogu]]==

The Archipelago is a name which has been given to the commatic realm of the [[13-limit]] comma [2 -3 -2 0 0 2> = 676/675.

= Miscellaneous other temperaments =

===[[31 comma temperaments]]===

These all have period 1/31 of an octave.

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

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

===[[Very low accuracy temperaments]]===

All hope abandon ye who enter here.

===[[Very high accuracy temperaments]]===

===[[High badness temperaments]]===