Tour of Regular Temperaments

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Regular temperaments

Regular temperaments are non-Just tunings in which the infinite number of intervals in p-limit Just intonation, or any subgroup thereof, are mapped to a smaller, though still infinite, set of 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 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 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. 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 3rds, without caring much what ratio they are tuned to. Thus one might use Rastmic even though no commas are pumped.

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 ("Pure-Octave Tenney-Euclidean") and 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(s). Assuming the prime subgroup includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a 5th or some fraction of some 3-limit interval. 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 name. The traditional names are 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.

Equal temperaments (Rank-1 temperaments)

Equal temperaments (abbreviated ET or tET) and 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 pergens page. There is also Graham Breed's giant list of regular temperaments.

Families defined by a 2.3 (wa) comma

These are families defined by a comma that uses a wa or 3-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, or when prime 7 is assumed to be part of the subgroup. 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 or Sawa family (P8/5, ^1)

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

Apotome or Lawa family (P8/7, ^1)

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

Pythagorean or Lalawa family (P8/12, ^1)

The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = [-19 12 0>, which implies 12-edo. This family includes the compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave The 5-limit compton temperament can be thought of as multiple rings of 12-edo, offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12-edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.

Families defined by a 2.3.5 (ya) comma

These are families defined by a ya or 5-limit comma. 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 comma list of the various temperaments. Families include weak extensions as well as strong, in other words, the pergen shown here may change.

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 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 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.

Suprapyth or Sayo family (P8, P5)

The Suprapyth or Sayo family tempers out [12 -9 1> = 20480/19683, which equates 5/4 to a Pythagorean augmented 2nd. Being a fourthward comma, it tends to sharpen the 5th, hence it's "super-pythagorean". The best 7-limit extension adds the Archy or Ru comma to make the Superpyth temperament.

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. 5/4 is equated to 3 fourths minus 1 octave. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include 9edo, 16edo, 23edo, and 25edo.

Father or Gubi family (P8, P5)

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

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. 5/4 is equated to 3 periods minus 2 fifths. The major 2nd ~9/8 is divided in half, with each half equated to ~16/15. Diaschismic tunings include 12edo, 22edo, 34edo, 46edo, 56edo, 58edo and 80edo. An obvious 7-limit interpretation of the period is 7/5, which makes pajara temperament, where the intervals 50/49 and 64/63 are tempered out. 22edo is an excellent pajara tuning.

Bug or Gugu family (P8, P4/2)

This low-accuracy family of temperaments tempers out 27/25, the large limma or bug comma. The generator is an approximate 6/5 or 10/9 = ~250¢, two of which make ~4/3. 5/4 is equated to 1 octave minus 3 generators. An obvious 7-limit interpretation of the generator is 7/6, which makes Slendro aka Semaphore aka Zozo.

Immunity or Sasa-yoyo family (P8, P4/2)

This tempers out the immunity comma, [16 -13 2> (1638400/1594323). Its generator is ~729/640 = ~247¢, two of which make ~4/3. 5/4 is equated to 3 octaves minus 13 generators. An obvious 7-limit interpretation of the generator is 7/6, which leads to Slendro aka Semaphore aka Zozo.

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 a neutral-sized 3rd of ~350¢ that is 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. An obvious 2.3.11 nterpretation of the generator is ~11/9, which leads to Rastmic aka Neutral aka Lulu.

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 "Tcherepnin's scale" (3L 6s).

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. 5/4 is equated to 1 octave minus 5 generators. Some porcupine temperaments include 15edo, 22edo, 37edo, and 59edo. An important 7-limit extension also tempers out 64/63.

Laconic or Latrigubi family (P8, P5/3)

This low-accuracy family of temperaments tempers out the laconic comma, [-4 7 -3> (2187/2000), which is the difference between three 10/9's and one 3/2. The generator is ~10/9 = ~230¢. 5/4 is equated to 7 generators minus 1 octave. Laconic is supported by 16edo, 21edo, and 37edo (using the 37b mapping), among others. An obvious 7-limit interpretation of the generator is ~8/7, which leads to Gamelismic aka Latrizo.

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. 5/4 is equated to 1 fifth minus 1 period.

Negri or Laquadyo family (P8, P4/4)

This tempers out the negri comma, [-14 3 4>. Its generator is ~16/15, four of which make ~4/3. 5/4 is equated to 3 generators.

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. 5/4 is equated to 9 generators minus an octave. 7edo can also be considered a tetracot tuning, as can 20edo, 27edo, 34edo, and 41edo.

Vulture or Sasa-quadyo family (P8, P12/4)

This tempers out the vulture comma, [24 -21 4>. Its generator is ~320/243 = ~475¢, four of which make ~3/1. 5/4 is equated to 21 generators minus 8 octaves. An obvious 7-limit interpretation of the generator is 21/16, which makes Saquadru.

Comic or Saquadyobi family (P8/2, M2/4)

This tempers out the comic comma, [13 -14 4> = 5120000/4782969. Its generator is ~81/80 = 55¢. 5/4 is equated to 7 generators. An obvious 11-limit interpretation of the generator is 33/32, which makes Laquadlo.

Pental or Trila-quingu family (P8/5, P5)

This tempers out the pental comma, 847288609443/838860800000 = [-28 25 -5>. The period is 59049/51200, and 5 periods make an octave. The generator is a 5th, or equivalently, 3/5 of an 8ve minus a 5th. This alternate generator is only about 18¢, thus the scales have a very lopsided L/s ratio. 5/4 is equated to 2/5 of an octave minus 5 of these 18¢ generators. An obvious 7-limit interpretation of the generator is 8/7, which leads to Laquinzo.

Amity or Saquinyo family (P8, P11/5)

This tempers out the amity comma, 1600000/1594323 = [9 -13 5>. The generator is 243/200 = ~339.5¢, five of which make ~8/3. 5/4 is equated to 4 octaves minus 13 generators, or fifths minus 3 generators. An obvious 11-limit interpretation of the generator is 11/9, which makes Saquinlo. An obvious 13-limit interpretation of the generator is 39/32, which makes Lala-quintho.

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 or Saquinbiyo family (P8/2, P5/5)

This tempers out the fifive comma, [-1 -14 10> = 9765625/9565938. The period is ~4374/3125 = [1 7 -5>, two of which make an octave. The generator is ~27/25, five of which make ~3/2. 5/4 is equated to 7 generators minus 1 period.

Qintosec or Quadsa-quinbigu family (P8/5, P5/2)

This tempers out the qintosec comma, 140737488355328/140126044921875 = [47 -15 -10>. The period is ~524288/455625 = [19 -6 -4>, five of which equals an octave. The generator is ~16/15. A period plus a generator makes half a fifth. 5/4 is equated to 3 periods minus 3 generators. An obvious 7-limit interpretation of the period is 8/7.

Trisedodge or Saquintrigu family (P8/5, P4/3)

This tempers out the trisedodge comma, 30958682112/30517578125 = [19 10 -15>. The period is ~144/125 = 240¢. The generator is ~6/5. Six periods minus three generators equals ~4/3. 5/4 is equated to 2 generators minus 1 period. An obvious 7-limit interpretation of the period is 8/7.

Ampersand or Lala-tribiyo family (P8, P5/6)

This tempers out Ampersand's comma = 34171875/33554432 = [-25 7 6>. The generator is ~16/15, of which six make ~3/2. 5/4 is equated to 1 octave minus 7 generators. If the generator is also equated to ~15/14, and three generators to ~11/9, one gets the Miracle temperament.

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. 5/4 is equated to 5 generators minus 1 octave. The kleismic family includes 15edo, 19edo, 34edo, 49edo, 53edo, 72edo, 87edo and 140edo among its possible tunings.

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>, is tempered out by the members of the semicomma family. Its generator is ~75/64, seven of which equals ~3/1. 5/4 is equated to 1 octave minus 3 generators. This temperament doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as ~7/6, leading to Orwell or Sepru temperament.

Wesley or Lasepyobi family (P8, ccP4/7)

This tempers out the wesley comma, [-13 -2 7> = 78125/73728. The generator is ~125/96 = ~412¢. Seven generators equals a double-compound 4th of ~16/3. 5/4 is equated to 1 octave minus 2 generators. An obvious 7-limit interpretation of the generator is 9/7, leading to the Lasepru temperament. An obvious 3-limit interpretation of the generator is 81/64, implying 29-edo.

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. Its generator is ~162/125 = ~443¢. Seven generators equals a double-compound 5th of ~6/1. 5/4 is equated to 9 generators minus 3 octaves.Tunings include 8edo, 19edo, 46edo, and 65edo. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzo temperament.

Vishnuzmic or Sasepbigu family (P8/2, P4/7)

This tempers out the vishnuzma, [23 6 -14>, or the amount by which seven chromatic semitones (25/24) fall short of a perfect fourth (4/3), or (4/3)/(25/24)^7. The period is ~[-11 -3 7> and the generator is ~25/24. 5/4 is equated to 1 period minus 3 generators.

Mutt or Trila-septriyo family (P8/3, ccP4/7)

This tempers out the mutt comma, [-44 -3 21>, leading to some strange properties. Seven ~5/4 generators equals a double-compound 4th = ~16/3. The third-octave period is not 5/4, thus the generator is equivalently a period minus ~5/4, only about 14¢. The L/s ratio is very lopsided, and scales resemble a "fuzzy" augmented chord.

Würschmidt or Saquadbigu family (P8, ccP5/8)

The würschmidt (or wuerschmidt) family tempers out the Würschmidt comma, 393216/390625 = [17 1 -8>. 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 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 or Sasa-tritrigu family (P8, P4/9)

This tempers out the escapade comma, [32 -7 -9>, which is the difference between nine just major thirds and seven just fourths. The generator is ~[-14 3 4> = ~55¢, and nine of them equal ~4/3. Seven of them equal ~5/4, thus two of them equal ~16/15. An obvious 11-limit interpretation of the generator is 33/32, leading to the Trisa-tritrilu temperament.

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. 5/4 is equated to 3 octaves minus 10 generators.

Sycamore or Laleyo family (P8, P5/11)

The sycamore family tempers out the sycamore comma, [-16 -6 11> = 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 or Lala-theyo family (P8, c4P4/13)

This tempers out the ditonma, [-27 -2 13> = 1220703125/1207959552. Thirteen ~[-12 -1 6> generators of about 407¢ equals a quadruple-compound 4th. 5/4 is equated to 1 octave minus 2 generators. An obvious 3-limit interpretation of the generator is 81/64, which implies 53-edo, which is a good tuning for this high-accuracy family of temperaments.

Luna or Sasa-quintrigu family (P8, ccP4/15)

This tempers out the luna comma, [38 -2 -15> (274877906944/274658203125). The generator is ~{18 -1 -7> = ~193¢. Two generators equals ~5/4, and fifteen generators equals a double-compound 4th of ~16/3.

Minortonic or Trila-segu family (P8, ccP5/17)

This tempers out the minortone comma, [-16 35 -17>. 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. 5/4 is equated to 35 generators minus 5 octaves.

Maja or Saseyo family (P8, c6P4/17)

This tempers out the maja comma, [-3 -23 17> = 762939453125/753145430616. The generator is ~162/125 = ~453¢. Seventeen generators equals a sextuple-compound 4th. 5/4 is equated to 9 octaves minus 23 generators.

Maquila or Trisa-segu family (P8, c7P5/17)

This tempers out the maquila comma, 562949953421312/556182861328125 = [49 -6 -17>. The generator is ~512/375 = ~535¢. Seventeen generators equals a septuple-compound 5th. 5/4 is equated to 3 octaves minus 6 generators. An obvious 11-limit interpretation of the generator is 11/8, leading to the Lala-selo temperament. However, Lala-selo isn't nearly as accurate as Trisa-segu.

Gammic or Laquinquadyo family (P8, P5/20)

The gammic family tempers out the gammic comma, [-29 -11 20>. 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.

Clans defined by a 2.3.7 (za) comma

These are defined by a za or 7-limit-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 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 comma list for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.

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 the Laru comma [-13 10 0 -1> = 50.7¢. It equates 7/4 to an augmented 6th. Its best downward extension is Septimal Meantone.

Garischismic or Sasaru clan (P8, P5)

This clan tempers out the garischisma, [25 -14 0 -1> = 33554432/33480783. It equates 8/7 to two apotomes ([-11 7> = 2187/2048), and 7/4 to a double-diminished 8ve [23 -14>. This clan includes vulture, newt, garibaldi, sextile, and satin.

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, and 7/4 with 27/16.

Laruru clan (P8/2, P5)

This clan tempers out the Laruru comma [-7 8 0 -2> = 78¢. Two ~81/56 periods equal an 8ve. The generator is ~3/2, and four generators minus three periods equals ~7/4. The major 2nd ~9/8 is divided in half, with each half equated to ~28/27. See also the Diaschismatic or Sagugu temperament and the Jubalismic or Biruyo temperament.

Slendro (Semaphore) or Zozo clan (P8, P4/2)

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

Sasa-zozo clan (P8, P5/2)

This clan tempers out the Sasa-zozo comma [15 -13 0 2> = 12.2¢, and includes as a strong extension the Hemififths temperament. 7/4 is equated to 13 generators minus 3 octaves. An obvious 11-limit interpretation of the ~351¢ generator is 11/9, leading to the Lulu temperament.

Triru clan (P8/3, P5)

This clan tempers out the Triru comma, [-1 6 0 -3> = 105¢, a low-accuracy temperament. Three ~9/7 periods equals an 8ve. The generator is ~3/2, and two generators minus a period equals ~7/4. An obvious 5-limit interpretation of the ~400¢ period is 5/4, leading to the Augmented or Trigu temperament.

Trizo clan (P8, P5/3)

This clan tempers out the Trizo comma, [-2 -4 0 3> = 99¢, a low-accuracy temperament. Three ~7/6 generators equals a 5th, and four equal ~7/4. An obvious interpretation of the ~234¢ generator is 8/7, leading to the much more accurate Gamelismic or Latrizo temperament.

Gamelismic or Latrizo clan (P8, P5/3)

This clan tempers out the gamelisma, [-10 1 0 3> = 1029/1024. Three ~8/7 generators equals a 5th. 7/4 is equated to an 8ve minus a generator. 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 the Latriru comma [-9 11 0 -3> = 15.0¢. Generator = ~112/81 = ~566¢. Three generators equals ~8/3. 7/4 is equated to 11 generators minus 5 octaves. An obvious 2.3.5.7 interpretation of the generator is 7/5, leading to the Liese temperament, which is a weak extension of Meantone.

Stearnsmic or Latribiru clan (P8/2, P4/3)

Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6> = 118098/117649. The period is ~486/343 = ~600¢. The generator is ~9/7 = ~434¢, or alternatively one period minus ~9/7, which equals ~54/49 = ~166¢. Three of these alternate generators equals ~4/3. 7/4 is equated to 5 ~9/7 generators minus an octave. Equating the ~54/49 generator to ~10/9 creates a weak extension of the Porcupine or Triyo temperament, as does equating the period to ~7/5.

This clan tempers out the Laquadru comma [-3 9 0 -4> = 42.3¢. its generator is ~9/7. Four generators equals ~8/3. 7/4 is equated to 4 octaves minus 9 generators. This clan includes as a strong extension the Squares temperament, which is a weak extension of Meantone.

This clan tempers out the Saquadru comma [16 -3 0 -4> = 18.8¢. Its generator is ~21/16. Four generators makes ~3/1. 7/4 is equated to 2 octaves minus 3 generators. This clan includes as a strong extension the Vulture temperament, which is in the Vulture family.

Laquinzo clan (P8/5, P5)

This clan tempers out the Laquinzo comma [-14 0 0 5> = 44¢. Five ~8/7 periods equals an 8ve, and four periods equals ~7/4. The generator is ~3/2. Unlike the Blackwood or Sawa family, ~3/2 is not equated with three-fifths of an octave, resulting in very small intervals.

Quinru clan (P8, P5/5)

This clan tempers out the Quinru comma [3 7 0 -5> = 70¢. The ~54/49 generator is about 139¢. Two of them equal ~7/6, three equal ~9/7, five equal ~3/2, and seven equal ~7/4.

Saquinzo clan (P8, P12/5)

This clan tempers out the Saquinzo comma [5 -12 0 5> = 20.7¢. Its generator is ~243/196 = ~380¢. Five generators makes ~3/1. 7/4 is equated to 12 generators minus 3 octaves. An obvious 5-limit interpretation of the generator is 5/4, leading to the Magic temperament, which is in the Magic family.

Sepru clan (P8, P12/7)

This clan tempers out the Sepru comma [7 8 0 -7> = 33.8¢. Its generator is ~7/6. Seven generators equals ~3/1. 7/4 is equated to 8 generators minus 1 octave. This clan includes as a strong extension the Orwell temperament, which is in the Semicomma family.

Clans defined by a 2.3.11 (ila) comma

Rastmic or Neutral or Lulu clan (P8, P5/2)

This 2.3.11 clan tempers out 243/242 = [-1 5 0 0 -2>. Its generator is ~11/9. Two generators equals ~3/2. 11/8 is equated to 5 generators minus an octave. This clan includes as a strong extension the Dicot temperament, which is in the Dicot family.

This 2.3.11 clan tempers out the Laquadlo comma [-17 2 0 0 4>. Its half-ocave period is ~363/256, and its generator is ~33/32. Four generators equals ~9/8. 3/2 is equated to a period plus 2 generators, and 11/8 is equated to a period minus a generator. This clan includes as a strong extension the Comic aka Saquadyobi temperament, which is in the Comic family.

Clans defined by a 2.3.13 (tha) comma

Hemif or Thuthu clan (P8, P5/2)

This 2.3.13 clan tempers out 512/507 = [9 -1 0 0 0 -2>. Its generator is ~16/13. Two generators equals ~3/2. 13/8 is equated to 1 octave minus 1 generator. This clan includes as a strong extension the Dicot temperament, which is in the Dicot family.

Satritho clan (P8, P11/3)

This 2.3.13 clan tempers out the Satritho comma 512/507 = [0 -7 0 0 0 3>. Its generator is ~18/13. Three generators equals ~8/3. 13/8 is equated to 7 generators minus three octaves. This clan is related to the Latriru clan.

Clans defined by a 2.5.7 (yaza nowa) comma

These are defined by a yaza nowa or 7-limit-no-threes comma. See also subgroup temperaments.

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 M3 generator is ~5/4. The half-octave period is ~7/5 or ~10/7. 7/4 is equated to 1 period plus 1 generator.

Hemimean or Zozoquingu Nowa clan (P8, M2)

This clan tempers out the hemimean comma, [6 0 -5 2> = 3136/3125. The M2 generator is ~28/25 = ~194¢. Two generators equals ~5/4, and five of them equals ~7/4.

Quince or Lasepzo-agugu Nowa clan (P8, M2/2)

This clan tempers out the quince, [-15 0 -2 7> = 823543/819200. The generator is ~343/320 = ~116¢. Two generators equals ~8/7 (a M2), and seven generators equals ~8/5. An obvious 5-limit interpretation of the generator is 16/15, leading to the Magic temperament, which is in the Magic family.

Clans defined by a 3.5.7 (yaza noca) comma

These are defined by a yaza noca or 7-limit-no-twos comma. Any no-twos comma is a 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 (or some fraction of that). See also subgroup temperaments.

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 is ~9/7, and two generators equals ~5/3.

Mirkwai or Quinru-aquadyo Noca clan (P12, cM7/4)

This 3.5.7 clan tempers out the mirkwai comma, [0 3 4 -5> = 16875/16807. The generator is ~7/5, and four generators equals a compound major 7th = ~27/7.

Rank-3 temperaments

Even less familiar than rank-2 temperaments are the 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 generator in a rank-3 pergen is usually a comma, but sometimes it's some fraction of a 5-limit or 7-limit interval.

Families defined by a 2.3.5 (ya) comma

Every ya or 5-limit comma defines a rank-3 family, thus every comma in the list of rank-two 2.3.5 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a yaza or 7-limit temperament in which one of the generators is ~7/1. This generator can be reduced to ~7/4, which can be reduced further to ~64/63. Hence in all the pergens below, the ^1 or /1 generator is ~64/63. An additional 7-limit or 11-limit comma creates a yazala or 11-limit temperament, and so forth. All these examples are yaza:

Didymus or Gu rank three family (P8, P5, ^1)

These are the rank three temperaments tempering out the didymus or meantone comma, 81/80.

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. The half-octave period is ~45/32.

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. In the pergen, P4/3 is ~10/9.

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. In the pergen, P12/6 is ~6/5.

Families defined by a 2.3.7 (za) comma

Every za or 7-limit-no-fives comma defines a rank-3 family, thus every comma in the list of rank-two 2.3.7 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a yaza or 7-limit temperament 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. An additional 5-limit or 11-limit comma creates a yazala or 11-limit temperament, and so forth. All these examples are yaza:

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 or Sasaru family (P8, P5, ^1)

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

Semiphore or Zozo family (P8, P4/2, ^1)

Semiphore temperament tempers out 49/48 and thereby identifies the septimal minor third 7/6 with 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 and Slendro.

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. In the pergen, P5/3 is ~8/7.

Stearnsmic or Latribiru family (P8/2, P4/3, ^1)

Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6> = 118098/117649. In the pergen, P8/2 is 343/243 and P4/3 is ~54/49.

Families defined by a 2.3.5.7 (yaza) comma

Marvel or Ruyoyo family (P8, P5, ^1)

The head of the marvel family is marvel, which tempers out [-5 2 2 -1> = 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 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. In the pergen, ^1 = ~81/80.

Sensamagic or Zozoyo family (P8, P5, ^1)

These temper out [0 -5 1 2> = 245/243. In the pergen, ^1 = ~64/63.

Greenwoodmic or Ruruyo family (P8, P5, ^1)

These temper out the greenwoodma, [-3 4 1 -2> = 405/392. In the pergen, ^1 = ~64/63.

Avicennmic or Zoyoyo family (P8, P5, ^1)

These temper out the avicennma, [-9 1 2 1> = 525/512, also known as Avicenna's enharmonic diesis. In the pergen, ^1 = ~81/80.

Keemic or Zotriyo family (P8, P5, ^1)

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

Orwellismic or Triru-agu family (P8, P5, ^1)

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

Nuwell or Quadru-ayo family (P8, P5, ^1)

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

Ragisma or Zoquadyo family (P8, P5, ^1)

The 7-limit rank three microtemperament which tempers out the ragisma, [-1 -7 4 1> = 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. In the pergen, ^1 = ~81/80.

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. In the pergen, ^1 = ~81/80.

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. In the pergen, ^1 = ~81/80.

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. In the pergen, ^1 = ~64/63.

Tolermic or Sazoyoyo family (P8, P5, ^1)

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

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. In the pergen, ^1 = ~81/80 or ~64/63.

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. In the pergen, ^1 = ~81/80.

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 half-octave period. In the pergen, ^1 = ~81/80.

Cataharry or Labirugu family (P8, P4/2, ^1)

Cataharry temperaments temper out the cataharry comma, [-4 9 -2 -2> = 19683/19600. In the pergen, half a 4th is ~81/70, and ^1 = ~81/80.

Breed or Bizozogu family (P8, P5/2, /1)

Breed is a 7-limit microtemperament which tempers out [-5 -1 -2 4> = 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.

Mirwomo or Labizoyo family (P8, P5/2, ^1)

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

Landscape or Trizogugu family (P8/3, P5, ^1)

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

Dimcomp or Quadruyoyo family (P8/4, P5, ^1)

The dimcomp family of rank three temperaments tempers out the dimcomp comma, [-1 -4 8 -4> = 390625/388962. In the pergen, the quarter-octave period is ~25/21, and ^1 = ~81/80.

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 (the downminor 3rd in the pergen), and three = ~6/5.

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 the pergen's upminor 3rd of ~6/5.

Octagar or Rurutriyo family (P8, P5, ^m6/2)

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

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 the pergen's downmajor 3rd of ~5/4.

Wizmic or Quinzo-ayoyo family (P8, P5, vm7/2)

A wizmic temperament is one which tempers out the wizma, [ -6 -8 2 5 > = 420175/419904. Two ~324/245 generators equal the pergen's downminor 7th of ~7/4.

Canou or Saquadzo-atriyo family (P8, P5, vm6/3)

The canou family of rank three temperaments tempers out the canousma, [4 -14 3 4> = 4802000/4782969. Three ~81/70 generators equal the pergen's downminor 6th of ~14/9.

Mirkwai or Quinru-aquadyo family (P8, P5, c^M7/4)

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

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 hobbit scales can be constructed for them.

Valinorismic or Lorugugu temperaments

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

Rastmic or Lulu temperaments

These temper out the rastma, [-1 5 0 0 -2> = 243/242. As an ila (11-limit no-fives no-sevens) rank-2 temperament, it's (P8, P5/2).

Werckismic or Luzozogu temperaments

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

Swetismic or Lururuyo temperaments

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

Lehmerismic or Loloruyoyo temperaments

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

Kalismic or Bilorugu temperaments

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

Subgroup temperaments

A wide-open field. These are regular temperaments of various ranks which temper just intonation subgroups.

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 subgroups tempering out a given comma. For some commas, looking at the full commatic realm seems the best approach to discussing associated temperaments.

Orgonia or Satrilu-aruru

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

The Biosphere or Thozogu

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

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

26th-octave temperaments

These temperaments all have a period of 1/26 of an octave.

31-comma temperaments

These all have a period of 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 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

Microtemperaments which don't fit in elsewhere.