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Regular temperaments | Equal temperaments | Rank-2 (including linear) temperaments | Temperaments for a given comma | Rank-3 temperaments | Rank-4 temperaments | Subgroup temperaments | Commatic realms | Links

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. 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 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.

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.

Equal 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


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. 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 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-ET by tempering out the Pythagorean comma.

Regular temperaments of ranks two and three are cataloged here. Rank-2 temperaments are also listed here by their generator mappings and here by their generator size. There is also Graham Breed's giant list of regular temperaments.

Families


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

Meantone family

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

The schismatic family tempers out the schisma of 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.

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 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, 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 temperament, where the intervals 50/49 and 64/63 are tempered out, 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, 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.

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.

Würschmidt family

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 perfect 5th two octaves up); 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.

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" (3L 3s) in common 12-based music theory, as well as what is commonly called "Tcherepnin's scale" (3L 6s).

Dimipent family

The dimipent family tempers out the major diesis or diminished comma, 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.

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. Tunings include 8edo, 19edo, 46edo, and 65edo.

Orwell and the semicomma family

The semicomma (also known as Fokker's comma), 2109375/2097152 = |-21 3 7>, 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, 531441/524288 = |-19 12 0>. 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 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-equal, offset from one another justly tuned 5/4.

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>. 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>. 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.

Father family

This tempers out 16/15, the just diatonic semitone.

Sycamore family

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.

Escapade family

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

Amity family

This tempers out the amity comma, 1600000/1594323 = |9 -13 5>.

Vulture family

This tempers out the vulture comma, |24 -21 4>.

Vishnuzmic family

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

Luna Family

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

Laconic Family

This tempers out the laconic comma, 2187/2000, which is the difference between three 10/9's and one 3/2. Laconic is supported by 16-EDO, 21-EDO, and 37b-EDO, among others.

Immunity family

This tempers out the immunity comma, 1638400/1594323.

Ditonmic family

This tempers out the ditonma, 1220703125/1207959552.

Shibboleth family

This tempers out the shibboleth comma, 1953125/1889568.

Comic family

This tempers out the comic comma, 5120000/4782969.

Wesley family

This tempers out the wesley comma, 78125/73728.

Fifive family

This tempers out the fifive comma, 9765625/9565938.

Maja family

This tempers out the maja comma, 762939453125/753145430616.

Pental family

The pental family tempers out the pental comma, 847288609443/838860800000 = |-28 25 -5>.

Maquila family

This tempers out the maquila comma, 562949953421312/556182861328125 = |49 -6 -17>.

Mutt family

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

Clans


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.

Gamelismic clan

Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. 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 and 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.

Trienstonic clan

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

Slendro clan

This clan tempers out the slendro diesis, 49/48, a triprime comma with factors of 2, 3 and 7.

Jubilismic clan

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

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.

Sensamagic clan

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

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> = 16875/16807, a no-twos comma (ratio of odd numbers.)

Quince clan

This tempers out the quince, a no-threes comma |-15 0 -2 7> = 823543/819200.

Temperaments for a given comma


Septisemi temperaments

These are very low complexity temperaments tempering out the minor septimal semitone, 21/20 and hence equating 5/3 with 7/4.

Mint temperaments

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

Greenwoodmic temperaments

These temper out the greenwoodma, |-3 4 1 -2> = 405/392.

Avicennmic temperaments

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

Keemic temperaments

These temper out the keema, |-5 -3 3 1> = 875/864.

Starling temperaments

These temper out 126/125, the septimal semicomma or starling comma (the difference between three 6/5s plus one 7/6, and an octave), and include myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.

Marvel temperaments

These temper out |-5 2 2 -1> = 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.

Orwellismic temperaments

These temper out |6 3 -1 -3> = 1728/1715, the orwellisma.

Octagar temperaments

Octagar temperaments temper out the octagar comma, |5 -4 3 -2> = 4000/3969.

Hemifamity temperaments

Hemifamity temperaments temper out the hemifamity comma, |10 -6 1 -1> = 5120/5103.

Porwell temperaments

Porwell temperaments temper out the porwell comma, |11 1 -3 -2> = 6144/6125.

Stearnsmic temperaments

Stearnsmic temperaments temper out the stearnsma, |1 10 0 -6> = 118098/117649.

Hemimage temperaments

Hemimage temperaments temper out the hemimage comma, |5 -7 -1 3> = 10976/10935.

Cataharry temperaments

Cataharry temperaments temper out the cataharry comma, |-4 9 -2 -2> = 19683/19600.

Horwell temperaments

Horwell temperaments temper out the horwell comma, |-16 1 5 1> = 65625/65536.

Breedsmic temperaments

A breedsmic temperament is one which tempers out the breedsma, |-5 -1 -2 4> = 2401/2400. Some which do not get discussed elsewhere are collected on a page here.

Ragismic microtemperaments

A ragismic temperament is one which tempers out |-1 -7 4 1> = 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.

Landscape microtemperaments

A landscape temperament is one which tempers out |-4 6 -6 3> = 250047/250000.

Wizmic microtemperaments

A wizmic temperament is one which tempers out the wizma, | -6 -8 2 5 > = 420175/419904.

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 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.

High badness temperaments

High in badness, but worth cataloging for one reason or another.

11-limit comma temperaments


Rank-3 temperaments


Even less familiar than rank-2 temperaments are the 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, 2749edo will certainly do the trick. Breed has generators of 2/1, 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.

Landscape family

The 7-limit rank three microtemperament which tempers out the lanscape comma, 250047/250000, extends to various higher limit rank three temperaments such as tyr and odin.

Hemifamity family

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

Porwell family

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.

Sengic family

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.

Mirwomo family

The mirwomo family of rank three temperaments tempers out the mirwomo comma, 33075/32768.

Dimcomp family

The dimcomp family of rank three temperaments tempers out the dimcomp comma, 390625/388962.

Tolermic family

These temper out the tolerma, 179200/177147.

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.

Diaschismic rank three family

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.

Didymus rank three family

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 maximal 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.

Semiphore family

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".

Mint family

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

Valinorismic temperaments

These temper out the valinorsma, 176/175.

Rastmic temperaments

These temper out the rastma, 243/242.

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-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.

Subgroup temperaments


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

Commatic realms


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

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




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