Tour of regular temperaments

The following is a tour of many of the regular temperaments that contributors to this wiki have found notable. It is of course not meant to be comprehensive, and is not the result of a systematic search.

Rank-2 temperaments

A rank-2 temperament maps all JI intervals within its JI subgroup to a 2-dimensional lattice. This lattice has two generators. The larger generator is called the period, as the temperament will repeat periodically at that interval. The period is usually the octave, or some unit fraction of the octave. If the period is a full octave, the temperament is said to be a linear temperament. The smaller generator, referred to as "the" generator, is stacked repeatedly to generate a scale within that period.

A rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for the period and one for the generator. Each member of the JI subgroup is mapped to a period-generator pair.

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 81/80 out of 3-dimensional 5-limit JI), can be reduced to 1-dimensional 12-ET by tempering out the Pythagorean comma.

Families defined by a 2.3 (wa) comma

These are families defined by a wa or 3-limit comma. If only primes 2 and 3 are part of the subgroup, the comma creates a rank-1 temperament, an edo. But if another prime such as 5 is present, the comma creates a rank-2 temperament. Since edos are discussed elsewhere, this section assumes the presence of at least one additional prime. The rank-2 temperament created consists of multiple "copies" of an edo. The edo copies can be thought of as being offset from one another by a small comma. This small comma is represented in the pergen by ^1.

Limma or Sawati family (P8/5, ^1): This family tempers out the limma, [8 -5⟩ = 256/243. It equates 5 fifths with 3 octaves, which creates multiple copies of 5edo. The fifth is ~720¢, quite sharp. This family includes the Blackwood temperament, which is 5-limit. Blackwood's edo copies are offset from one another by 5/4, or alternatively by 81/80. 5/4 is usually tempered sharp, perhaps ~400¢, to match the sharp 5th.

Apotome or Lawati family (P8/7, ^1): This family tempers out the apotome, [-11 7⟩ = 2187/2048. It equates 7 fifths with 4 octaves, which creates multiple copies of 7edo. The fifth is ~685¢, which is very flat. This family includes the Whitewood temperament.

Compton or Lalawati family (P8/12, ^1): This tempers out the Pythagorean comma, [-19 12 0⟩ = 531441/524288. It equates 12 fifths with 7 octaves, which creates multiple copies of 12edo. Temperaments in this family include Compton and Catler. In the 5-limit compton temperament, the edo copies are offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.

Countercomp or Wa-41 family (P8/41, ^1): This family tempers out the Pythagorean countercomma, [65 -41⟩, which creates multiple copies of 41edo.

Mercator or Wa-53 family (P8/53, ^1): This family tempers out the Mercator's comma, [-84 53⟩, which creates multiple copies of 53edo.

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 Guti 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 12, 19, 31, 43, 50, 55 and 81 EDOs. 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 Layoti 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 12, 29, 41, 53, and 118 EDOs.

Pelogic or Layobiti 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 9, 16, 23, and 25 EDOs.

Father or Gubiti family (P8, P5): This tempers out 16/15, the just diatonic semitone, and equates 5/4 with 4/3.

Diaschismic or Saguguti family (P8/2, P5): The diaschismic family tempers out the diaschisma, [11 -4 -2⟩ = 2048/2025, such that two classic major thirds and a Pythagorean major third stack to an octave (i.e. 5/4 × 5/4 × 81/64 → 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 12, 22, 34, 46, 56, 58 and 80 EDOs. 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 Guguti 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 or Zozoti.

Immunity or Sasa-yoyoti 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 or Zozoti.

Dicot or Yoyoti 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 interpretation of the generator is ~11/9, which leads to Rastmic aka Neutral or Luluti.

Augmented or Triguti 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).

Misty or Sasa-triguti family (P8/3, P5): The misty family tempers out the misty comma of [26 -12 -3⟩, the difference between the Pythagorean comma and a stack of three schismas. The period is ~512/405 and the generator is ~3/2 (or alternatively ~135/128). 5/4 is equated to 8 periods minus 4 fifths, thus 5/4 is split into 4 equal parts, each 2 periods minus a fifth.

Porcupine or Triyoti 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 15, 22, 37, and 59 EDOs. An important 7-limit extension also tempers out 64/63.

Tricot or Quadsa-triyoti family (P8, P11/3): The tricot family tempers out the tricot comma, [39 -29 3⟩. The generator is ~59049/40960 = [-13 10 -1⟩ = 633¢, or its octave inverse ~81920/59049 = 567¢. Three of the latter generators equals a compound 4th of ~8/3. 5/4 is equated to 14 octaves minus 29 of these generators. An obvious 7-limit interpretation of the generator is 81/56 = 639¢, a much simpler ratio which leads to the Latriruti clan. An obvious 13-limit interpretation is 13/9 = 637¢, an even simpler ratio implying the Satrithoti clan.

Dimipent or Quadguti 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.

Undim or Trisa-quadguti family (P8/4, P5): The undim family tempers out the undim comma of [41 -20 -4⟩, the difference between the Pythagorean comma and a stack of four schismas.

Negri or Laquadyoti 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 Saquadyoti 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.

Smate or Saquadguti family (P8, P11/4): This tempers out the symbolic comma, [11 -1 -4⟩ = 2048/1875. Its generator is ~5/4 = ~421¢, four of which make ~8/3.

Vulture or Sasa-quadyoti 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 Saquadruti.

Pental or Trila-quinguti 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 Laquinzoti.

Ripple or Quinguti family (P8, P4/5): This tempers out the ripple comma, 6561/6250 = [-1 8 -5⟩, which equates a stack of four 10/9's with 8/5, and five of them with 16/9. The generator is 27/25, two of which equals 10/9, three of which equals 6/5, and five of which equals 4/3. 5/4 is equated to an octave minus 8 generators. As one might expect, 12EDO is about as accurate as it can be.

Passion or Saquinguti family (P8, P4/5): This tempers out the passion comma, 262144/253125 = [18 -4 -5⟩, which equates a stack of four 16/15's with 5/4, and five of them with 4/3.

Quintaleap or Trisa-quinguti family (P8, P4/5): This tempers out the quintaleap comma, [37 -16 -5⟩. The generator is ~135/128, five of them gives ~4/3, and sixteen of them gives 5/2. An obvious 17-limit interpretation of the generator is ~18/17, which makes Saquinsoti.

Quindromeda or Quinsa-quinguti family (P8, P4/5): This tempers out the quindromeda comma, [56 -28 -5⟩. The generator is ~4428675/4194304, five of them gives ~4/3, and 28 of them gives the fifth harmonic, 5/1. An obvious 17-limit interpretation of the generator is ~18/17, which makes Saquinsoti.

Amity or Saquinyoti 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 2 fifths minus 3 generators. An obvious 11-limit interpretation of the generator is 11/9, which makes Saquinloti. An obvious 13-limit interpretation of the generator is 39/32, which makes Lala-quinthoti.

Magic or Laquinyoti 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 16, 19, 22, 25, and 41 EDOs among its possible tunings, with the latter being near-optimal.

Fifive or Saquinbiyoti 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.

Quintosec or Quadsa-quinbiguti family (P8/5, P5/2): This tempers out the quintosec 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 Saquintriguti 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-tribiyoti 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 Tribiyoti 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 15, 19, 34, 49, 53, 72, 87 and 140 EDOs among its possible tunings.

Orson, semicomma or Lasepyoti 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 Lasepyobiti 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 Lasepruti temperament. An obvious 3-limit interpretation of the generator is 81/64, implying 29EDO.

Sensipent or Sepguti 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 Sasepzoti temperament.

Vishnuzmic or Sasepbiguti 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.

Unicorn or Laquadbiguti family (P8, P4/8): This tempers out the unicorn comma, 1594323/1562500 = [-2 13 -8⟩. The generator is ~250/243 = ~62¢ and eight of them equal ~4/3.

Würschmidt or Saquadbiguti 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 mos scales 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-tritriguti 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-tritriluti temperament.

Shibboleth or Tritriyoti 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.

Mabila or Sasa-quinbiguti family (P8, c4P4/10): The mabila family tempers out the mabila comma, [28 -3 -10⟩ = 268435456/263671875. The generator is ~512/375 = ~530¢, three generators equals ~5/2 and ten of them equals a quadruple-compound 4th of ~64/3. An obvious 11-limit interpretation of the generator is ~15/11.

Sycamore or Laleyoti 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.

Quartonic or Saleyoti family (P8, P4/11): The quartonic family tempers out the quartonic comma, [3 -18 11⟩ = 390625000/387420489. The generator is ~250/243 = ~45¢, seven generators equals ~6/5, and eleven generators equals ~4/3. An obvious 7-limit interpretation of the generator is ~36/35.

Lafa or Tribisa-quadtriguti family (P8, P12/12): This tempers out the lafa comma, [77 -31 -12⟩. The generator is ~4982259375/4294967296 = ~258.6¢. Twelve generators equals a twelfth (~3/1). 5/4 is equated to 7 octaves minus 31 generators.

Ditonmic or Lala-theyoti 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 53EDO, which is a good tuning for this high-accuracy family of temperaments.

Luna or Sasa-quintriguti 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.

Vavoom or Quinla-seyoti family (P8, P12/17): This tempers out the vavoom comma, [-68 18 17⟩. The generator is ~16/15 = ~111.9¢. Seventeen generators equals a twelfth (~3/1). 5/4 is equated to two octaves minus 18 generators.

Minortonic or Trila-seguti 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 Saseyoti 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-seguti 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-seloti temperament. However, Lala-seloti isn't nearly as accurate as Trisa-seguti.

Gammic or Laquinquadyoti family (P8, P5/20): The gammic family tempers out the gammic comma, [-29 -11 20⟩. Nine generators of about 35¢ equals ~6/5, eleven equals ~5/4 and twenty equals ~3/2. 34EDO 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 Ruti 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.

Trienstonic or Zoti 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.

Harrison or Laruti clan (P8, P5): This clan tempers out the Laru comma, [-13 10 0 -1⟩ = 59049/57344. It equates 7/4 to an augmented 6th. Its best downward extension is septimal meantone.

Garischismic or Sasaruti 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.

Leapfrog or Sasazoti clan (P8, P5): This clan tempers out the Sasazo comma, [21 -15 0 1⟩ = 14680064/14348907. It equates 7/6 to two apotomes and 7/4 to double augmented fifth. This clan includes leapday, leapweek and srutal.

Laruruti clan (P8/2, P5): This clan tempers out the Laruru comma, [-7 8 0 -2⟩ = 6561/6272. 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 Diaschismic or Saguguti temperament and the Jubilismic or Biruyoti temperament.

Slendro (Semaphore) or Zozoti 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.

Parahemif or Sasa-zozoti clan (P8, P5/2): This clan tempers out the parahemif comma, [15 -13 0 2⟩ = 1605632/1594323, and includes the hemif temperament and its strong extension hemififths. 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 Luluti temperament.

Triruti clan (P8/3, P5): This clan tempers out the Triru comma, [-1 6 0 -3⟩ = 729/686, 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 temperament.

Gamelismic or Latrizoti 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. Five generators is slightly flat of 2/1, making this a cluster temperament. See also Sawati and Lasepzoti.; 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 72EDO.

Trizoti clan (P8, P5/3): This clan tempers out the Trizo comma, [-2 -4 0 3⟩ = 343/324, 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 Latrizoti temperament.

Lee or Latriruti clan (P8, P11/3): This clan tempers out the Latriru comma, [-9 11 0 -3⟩ = 177147/175616. The generator is ~112/81 = ~566¢, and three such 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 Latribiruti clan (P8/2, P4/3): This clan 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 temperament, as does equating the period to ~7/5.

Skwares or Laquadruti clan (P8, P11/4): This clan tempers out the Laquadru comma, [-3 9 0 -4⟩ = 19683/19208. 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.

Buzzardismic or Saquadruti clan (P8, P12/4): This clan tempers out the buzzardisma, [16 -3 0 -4⟩ = 65536/64827. 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.

Cloudy or Laquinzoti clan (P8/5, P5): This clan tempers out the cloudy comma, [-14 0 0 5⟩ = 16807/16384. It has a period of 1/5 octave, which represents ~8/7. The generator is ~3/2. Unlike the Blackwood or Sawati family, ~3/2 is not equated with three-fifths of an octave, resulting in very small intervals.

Bleu or Quinruti clan (P8, P5/5): This clan tempers out the Quinru comma, [3 7 0 -5⟩ = 17496/16807. 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.

Saquinzoti clan (P8, P12/5): This clan tempers out the Saquinzo comma, [5 -12 0 5⟩ = 537824/531441. 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.

Lasepzoti clan (P8, P11/7): This clan tempers out the Lasepzo comma [-18 -1 0 7⟩ = 823543/786432. Its generator is ~8/7. Six generators equals ~7/3, and seven generators equals ~8/3. Three generators is ~30¢ sharp of 3/2, and five generators is ~15¢ sharp of 2/1, making this a cluster temperament. See also Sawati and Latrizoti.

Septiness or Sasasepruti clan (P8, P11/7): This clan tempers out the septiness comma [26 -4 0 -7⟩ = 67108864/66706983. Its generator is ~147/128, four of them gives ~7/4, and seven of them gives ~8/3. Five generators is ~12.5¢ sharp of 2/1, making this a cluster temperament.

Sepruti clan (P8, P12/7): This clan tempers out the sepru comma, [7 8 0 -7⟩ = 839808/823543. 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.

Tritrizoti clan (P8/9, P5): This clan tempers out the septiennealimma (tritrizo comma), [-11 -9 0 9⟩ = 40353607/40310784. It has a period of 1/9 octave, which represents ~7/6. The generator is ~3/2. This clan includes a number of regular temperaments including enneaportent, ennealimmal, and novemkleismic.

Clans defined by a 2.3.11 (ila) comma

Clans defined by a 2.3.13 (tha) comma

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. The pergen's multigen (the 2nd term, omitting any fraction) is always a 5-limit-no-threes ratio such as 5/4, 8/5, 25/8, etc.

Jubilismic or Biruyoti Nowa clan (P8/2, M3): This clan tempers out the jubilisma, 50/49, which is the difference between 10/7 and 7/5. The generator is the nowa major 3rd (~5/4). The half-octave period is ~7/5 or ~10/7. 7/4 is equated to 1 period plus 1 generator.

Bapbo or Ruruguti Nowa clan (P8, M3/2): This clan tempers out the bapbo comma, 256/245. The genarator is ~8/7 = ~202¢ and two of them equals ~5/4.

Hemimean or Zozoquinguti Nowa clan (P8, M3/2): This clan tempers out the hemimean comma, [6 0 -5 2⟩ = 3136/3125. The generator is ~28/25 = ~194¢. Two generators equals the nowa major 3rd (~5/4), three of them equals ~7/5, and five of them equals ~7/4.

Mabilismic or Latrizo-aquiniyoti Nowa clan (P8, cM3/3): This clan tempers out the mabilisma, [-20 0 5 3⟩ = 1071875/1048576. The generator is ~175/128 = ~527¢. Three generators equals ~5/2 and five of them equals ~32/7.

Vorwell or Sasatriru-aquadbiguti Nowa clan (P8, m6/3): This clan tempers out the vorwell comma (named for being tempered in septimal vulture and orwell), [27 0 -8 -3⟩ = 134217728/133984375. The rutrigu generator is ~1024/875 = ~272¢. Three generators equals ~8/5 and eight of them equals ~7/2.

Rainy or Quinzo-atriyoti Nowa clan (P8, M3/5): This clan tempers out the rainy comma, [-21 0 3 5⟩ = 2100875/2097152. The rurugu generator is ~256/245 = ~77¢. Three generators equals ~8/7 and five of them equals the classic major third (~5/4).

Llywelynsmic or Sasepru-aguti Nowa clan (P8, cM3/7): This clan tempers out the llywelynsma, [22 0 -1 -7⟩ = 4194304/4117715. The generator is ~8/7 = ~227¢ and seven of them equals ~5/2.

Quince or Lasepzo-aguguti Nowa clan (P8, m6/7): This clan tempers out the quince, [-15 0 -2 7⟩ = 823543/819200. The trizo-agu generator is ~343/320 = ~116¢. Two generators equals ~8/7, five generators equals ~7/5, and seven generators equals the nowa minor 6th ~8/5. An obvious 5-limit interpretation of the generator is 16/15, leading to the miracle temperament, which is in the gamelismic clan.

Slither or Satritriru-aquadyoti Nowa clan (P8, ccm6/9): This clan tempers out the slither comma, [16 0 4 -9⟩ = 40960000/40353607. The generator is ~49/40 = ~357¢. Four generators equals ~16/7, five of them equals ~14/5, and nine of them equals a double-compound minor 6th of ~32/5.

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. The pergen's multigen (the 2nd term, omitting any fraction) is always a 5-limit-no-twos ratio such as 5/3, 9/5, 25/9, etc. In any noca subgroup, "compound" means increased by 3/1 not 2/1.

Arcturus or Rutribiyoti Noca clan (P12, M6): This 3.5.7 clan tempers out the Arcturus comma [0 -7 6 -1⟩ = 15625/15309. The generator is the noca major 6th (~5/3), and six generators equals ~21/1.

Sensamagic or Zozoyoti Noca clan (P12, M6/2): This 3.5.7 clan tempers out the sensamagic comma [0 -5 1 2⟩ = 245/243. The generator is ~9/7, and two generators equals the classic major 6th (~5/3).

Gariboh or Triru-aquinyoti Noca clan (P12, M6/3): This 3.5.7 clan tempers out the gariboh comma [0 -2 5 -3⟩ = 3125/3087. The generator is ~25/21, two generators equals ~7/5, and three generators equals the noca major 6th (~5/3).

Mirkwai or Quinru-aquadyoti Noca clan (P12, cm7/5): This 3.5.7 clan tempers out the mirkwai comma, [0 3 4 -5⟩ = 16875/16807. The generator is ~7/5, four generators equals ~27/7, and five generators equals the noca compound minor 7th (~27/5).

Procyon or Sasepzo-atriguti Noca clan (P12, m7/7): This 3.5.7 clan tempers out the Procyon comma [0 -8 -3 7⟩ = 823543/820125. The generator is ~49/45, three generators equals ~9/7, four equals ~7/5, and seven equals the classic minor seventh (~9/5).

Betelgeuse or Satritrizo-aguguti Noca clan (P12, c³M6/9): This 3.5.7 clan tempers out the Betelgeuse comma [0 -13 -2 9⟩ = 40353607/39858075. The generator is ~3645/2401, two generators equals ~7/3, and nine generators equals the noca triple-compound major 6th (~45/1).

Izar or Saquadtrizo-asepguti Noca clan (P12, c⁵m7/12): This 3.5.7 clan tempers out the Izar comma (also known as bapbo schismina), [0 -11 -7 12⟩ = 13841287201/13839609375. The generator is ~16807/10125, five generators give ~63/5, seven give ~243/7, and twelve give ~2187/5.

Temperaments defined by a 2.3.5.7 (yaza) comma

These are defined by a full 7-limit (or yaza) comma.

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

Greenwoodmic or Ruruyoti temperaments: These temper out the greenwoodma, [-3 4 1 -2⟩ = 405/392.

Keegic or Trizoguti temperaments: Keegic rank-two temperaments temper out the keega, [-3 1 -3 3⟩ = 1029/1000.

Mint or Ruguti temperaments: Mint rank-two temperaments temper out the septimal quarter-tone 36/35, equating 7/6 with 6/5, and 5/4 with 9/7.

Avicennmic or Zoyoyoti temperaments: These temper out the avicennma, [-9 1 2 1⟩ = 525/512, also known as Avicenna's enharmonic diesis.

Sengic or Trizo-aguguti temperaments: Sengic rank-two temperaments temper out the senga, [1 -3 -2 3⟩ = 686/675.

Keemic or Zotriyoti temperaments: Keemic rank-two temperaments temper out the keema, [-5 -3 3 1⟩ = 875/864.

Secanticorn or Laruquinguti temperaments: Secanticorn rank-two temperaments temper out the secanticornisma, [-3 11 -5 -1⟩ = 177147/175000.

Nuwell or Quadru-ayoti temperaments: Nuwell rank-two temperaments temper out the nuwell comma, [1 5 1 -4⟩ = 2430/2401.

Mermismic or Sepruyoti temperaments: Mermismic rank-two temperaments temper out the mermisma, [5 -1 7 -7⟩ = 2500000/2470629.

Negricorn or Saquadzoguti temperaments: Negricorn rank-two temperaments temper out the negricorn comma, [6 -5 -4 4⟩ = 153664/151875.

Tolermic or Sazoyoyoti temperaments: These temper out the tolerma, [10 -11 2 1⟩ = 179200/177147.

Valenwuer or Sarutribiguti temperaments: Valenwuer rank-two temperaments temper out the valenwuer comma, [12 3 -6 -1⟩ = 110592/109375.

Mirwomo or Labizoyoti temperaments: Mirwomo rank-two temperaments temper out the mirwomo comma, [-15 3 2 2⟩ = 33075/32768.

Catasyc or Laruquadbiyoti temperaments: Catasyc rank-two temperaments temper out the catasyc comma, [-11 -3 8 -1⟩ = 390625/387072.

Compass or Quinruyoyoti temperaments: Compass rank-two temperaments temper out the compass comma, [-6 -2 10 -5⟩ = 9765625/9680832.

Trimyna or Quinzoguti temperaments: The trimyna rank-two temperaments temper out the trimyna comma, [-4 1 -5 5⟩ = 50421/50000.

Starling or Zotriguti temperaments: Starling rank-two temperaments temper 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. It includes myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.

Octagar or Rurutriyoti temperaments: Octagar rank-two temperaments temper out the octagar comma, [5 -4 3 -2⟩ = 4000/3969.

Orwellismic or Triru-aguti temperaments: Orwellismic rank-two temperaments temper out orwellisma, [6 3 -1 -3⟩ = 1728/1715.

Mynaslendric or Sepru-ayoti temperaments: Mynaslendric rank-two temperaments temper out the mynaslender comma, [11 4 1 -7⟩ = 829440/823543.

Mistismic or Sazoquadguti temperaments: Mistismic rank-two temperaments temper out the mistisma, [16 -6 -4 1⟩ = 458752/455625.

Varunismic or Labizoguguti temperaments: Varunismic rank-two temperaments temper out the varunisma, [-9 8 -4 2⟩ = 321489/320000.

Marvel or Ruyoyoti temperaments: Marvel rank-two temperaments temper out [-5 2 2 -1⟩ = 225/224. It includes negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.

Dimcomp or Quadruyoyoti temperaments: Dimcomp rank-two temperaments temper out the dimcomp comma, [-1 -4 8 -4⟩ = 390625/388962.

Cataharry or Labiruguti temperaments: Cataharry rank-two temperaments temper out the cataharry comma, [-4 9 -2 -2⟩ = 19683/19600.

Canousmic or Saquadzo-atriyoti temperaments: Canousmic rank-two temperaments temper out the canousma, [4 -14 3 4⟩ = 4802000/4782969.

Triwellismic or Tribizo-asepguti temperaments: Triwellismic rank-two temperaments temper out the triwellisma, [1 -1 -7 6⟩ = 235298/234375.

Hemimage or Satrizo-aguti temperaments: Hemimage rank-two temperaments temper out the hemimage comma, [5 -7 -1 3⟩ = 10976/10935.

Hemifamity or Saruyoti temperaments: Hemifamity rank-two temperaments temper out the hemifamity comma, [10 -6 1 -1⟩ = 5120/5103.

Parkleiness or Zotritriguti temperaments: Parkleiness rank-two temperaments temper out the parkleiness comma, [7 7 -9 1⟩ = 1959552/1953125.

Porwell or Sarurutriguti temperaments: Porwell rank-two temperaments temper out the porwell comma, [11 1 -3 -2⟩ = 6144/6125.

Cartoonismic or Satritrizo-asepbiguti temperaments: Cartoonismic temperaments temper out the cartoonisma, [12 -3 -14 9⟩ = 165288374272/164794921875.

Hemfiness or Saquinru-atriyoti temperaments: Hemfiness rank-two temperaments temper out the hemfiness comma, [15 -5 3 -5⟩ = 4096000/4084101.

Hewuermera or Satribiru-aguti temperaments: Hewuermera rank-two temperaments temper out the hewuermera comma, [16 2 -1 -6⟩ = 589824/588245.

Lokismic or Sasa-bizotriguti temperaments: Lokismic rank-two temperaments temper out the lokisma, [21 -8 -6 2⟩ = 102760448/102515625.

Decovulture or Sasabiruguguti temperaments: Decovulture rank-two temperaments temper out the decovulture comma, [26 -7 -4 -2⟩ = 67108864/66976875.

Pontiqak or Lazozotritriyoti temperaments: Pontiqak rank-two temperaments temper out the pontiqak comma, [-17 -6 9 2⟩ = 95703125/95551488.

Mitonismic or Laquadzo-aguti temperaments: Mitonismic rank-two temperaments temper out the mitonisma, [-20 7 -1 4⟩ = 5250987/5242880.

Horwell or Lazoquinyoti temperaments: Horwell rank-two temperaments temper out the horwell comma, [-16 1 5 1⟩ = 65625/65536.

Neptunismic or Laruruleyoti temperaments: Neptunismic rank-two temperaments temper out the neptunisma, [-12 -5 11 -2⟩ = 48828125/48771072.

Metric or Latriru-asepyoti temperaments: Metric rank-two temperaments temper out the meter comma, [-11 2 7 -3⟩ = 703125/702464.

Wizmic or Quinzo-ayoyoti temperaments: Wizmic rank-two temperaments temper out the wizma, [-6 -8 2 5⟩ = 420175/419904.

Supermatertismic or Lasepru-atritriyoti temperaments: Supermatertismic rank-two temperaments temper out the supermatertisma, [-6 3 9 -7⟩ = 52734375/52706752.

Breedsmic or Bizozoguti temperaments: Breedsmic rank-two temperaments temper out the breedsma, [-5 -1 -2 4⟩ = 2401/2400.

Supermasesquartismic or Laquadbiru-aquinyoti temperaments: Supermasesquartismic rank-two temperaments temper out the supermasesquartisma, [-5 10 5 -8⟩ = 184528125/184473632.

Ragismic or Zoquadyoti temperaments: Ragismic rank-two temperaments temper out the ragisma, [-1 -7 4 1⟩ = 4375/4374.

Akjaysmic or Trisa-sepruguti temperaments: Akjaysmic rank-two temperaments temper out the akjaysma, [47 -7 -7 -7⟩. These have a period of 1/7 octave, and five periods equals ~105/64. Unlike the Whitewood or Lawati family, ~3/2 is not equated with four-sevenths of an octave, resulting in small intervals.

Landscape or Trizoguguti temperaments: Landscape rank-two temperaments temper out the landscape comma, [-4 6 -6 3⟩ = 250047/250000. These have a period of 1/3 octave, but ~5/4 is not equated with a period, resulting in small intervals.

Rank-3 temperaments

Even less familiar than rank-2 temperaments are the rank-3 temperament, 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 Guti rank three family (P8, P5, ^1): These are the rank three temperaments tempering out the didymus or meantone comma, 81/80.

Diaschismic or Saguguti 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 Triyoti 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 Tribiyoti 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 Ruti 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 Sasaruti family (P8, P5, ^1): A garischismic temperament is one which tempers out the garischisma, [25 -14 0 -1⟩ = 33554432/33480783.

Laruruti clan (P8/2, P5): This clan tempers out the Laruru comma, [-7 8 0 -2⟩ = 6561/6272. 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 Diaschismic or Saguguti temperament and the Jubilismic or Biruyoti temperament.

Semiphore or Zozoti 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 Latrizoti 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 Latribiruti 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 Ruyoyoti family (P8, P5, ^1): The head of the marvel family is marvel, which tempers out [-5 2 2 -1⟩ = 225/224. It divides 8/7 into two 16/15s, or equivalently, two 15/14s. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.

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 Zotriguti 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. It divides 10/7 into two 6/5s. 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. In the pergen, ^1 = ~81/80.

Sensamagic or Zozoyoti family (P8, P5, ^1): These temper out [0 -5 1 2⟩ = 245/243, which divides 16/15 into two 28/27s. In the pergen, ^1 = ~64/63.

Greenwoodmic or Ruruyoti family (P8, P5, ^1): These temper out the greenwoodma, [-3 4 1 -2⟩ = 405/392, which divides 10/9 into two 15/14s. In the pergen, ^1 = ~64/63.

Avicennmic or Lazoyoyoti family (P8, P5, ^1): These temper out the avicennma, [-9 1 2 1⟩ = 525/512, which divides 7/6 into two 16/15s. In the pergen, ^1 = ~81/80.

Keemic or Zotriyoti family (P8, P5, ^1): These temper out the keema [-5 -3 3 1⟩ = 875/864, which divides 15/14 into two 25/24s. In the pergen, ^1 = ~81/80.

Orwellismic or Triru-aguti family (P8, P5, ^1): These temper out [6 3 -1 -3⟩ = 1728/1715. In the pergen, ^1 = ~64/63.

Nuwell or Quadru-ayoti family (P8, P5, ^1): These temper out the nuwell comma, [1 5 1 -4⟩ = 2430/2401. In the pergen, ^1 = ~64/63.

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

Hemifamity or Saruyoti family (P8, P5, ^1): The hemifamity family of rank three temperaments tempers out the hemifamity comma, [10 -6 1 -1⟩ = 5120/5103, which divides 10/7 into three 9/8s. In the pergen, ^1 = ~81/80.

Horwell or Lazoquinyoti 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-aguti family (P8, P5, ^1): The hemimage family of rank three temperaments tempers out the hemimage comma, [5 -7 -1 3⟩ = 10976/10935, which divides 10/9 into three 28/27s. In the pergen, ^1 = ~64/63.

Mint or Ruguti 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 Zoguti family (P8, P5, ^1): These are very low accuracy 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 Biruyoti 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 Labiruguti 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 Bizozoguti 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.

Sengic or Trizo-aguguti family (P8, P5, vm3/2): These temper out the senga, [1 -3 -2 3⟩ = 686/675. One generator is ~15/14, two give ~7/6 (the downminor 3rd in the pergen), and three give ~6/5.

Porwell or Sarurutriguti 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 Rurutriyoti 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 Zozoquinguti 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-ayoyoti 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.

Landscape or Trizoguguti 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.

Gariboh or Triru-aquinyoti family (P8, P5, vM6/3): The gariboh family of rank three temperaments tempers out the gariboh comma, [0 -2 5 -3⟩ = 3125/3087. Three ~25/21 generators equal the pergen's downmajor 6th of ~5/3.

Canou or Saquadzo-atriyoti 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.

Dimcomp or Quadruyoyoti 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.

Mirkwai or Quinru-aquadyoti 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.

Temperaments defined by an 11-limit comma

Ptolemismic or Luyoyoti clan (P8, P5, ^1): These temper out the ptolemisma, [2 -2 2 0 -1⟩ = 100/99. 11/8 is equated to 25/18, which is an octave minus two 6/5's. Since 25/18 is a 5-limit interval, every 2.3.5.11 interval is equated to a 5-limit interval, and both the pergen and the lattice are identical to that of 5-limit JI. In the pergen, ^1 = ~81/80.

Biyatismic or Lologuti clan (P8, P5, ^1): These temper out the biyatisma, [-3 -1 -1 0 2⟩ = 121/120. 5/4 is equated to 121/96, which is two 11/8's minus a 3/2 fifth. Since 121/96 is an ila (11-limit no-fives no-sevens) interval, every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI. In the pergen, ^1 is either ~33/32 or ~729/704.

Valinorsmic or Loruguguti clan: These temper out the valinorsma, [4 0 -2 -1 1⟩ = 176/175. To be a rank-3 temperament, either an additional comma must vanish or the prime subgroup must omit prime 3. Thus no assumptions can be made about the pergen.

Rastmic or Luluti rank-3 clan: These temper out the rastma, [1 5 0 0 -2⟩ = 243/242. In the corresponding 2.3.11 rank-2 temperament, the pergen is (P8, P5/2).

Pentacircle or Saluzoti clan (P8, P5, ^1): These temper out the pentacircle comma, [7 -4 0 1 -1⟩ = 896/891. The interval between 11/8 and 7/4 is equated to 81/64. Since that is a 3-limit interval, every 2.3.11 interval is equated to a 2.3.7 interval and vice versa, and both the pergen and the lattice are identical to that of either 2.3.7 JI or 2.3.11 JI. In the pergen, ^1 is either ~64/63 or ~33/32 or ~729/704.

Semicanousmic or Quadlo-aguti clan (P8, P5, ^1): These temper out the semicanousma, [-2 -6 -1 0 4⟩ = 14641/14580. 5/4 is equated to an ila (2.3.11) interval, thus every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI. In the pergen, ^1 is either ~33/32 or ~729/704.

Semiporwellismic or Saluluguti clan (P8, P5, ^1): These temper out the semiporwellisma, [14 -3 -1 0 -2⟩ = 16384/16335. 5/4 is equated to an ila (2.3.11) interval, thus every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI. In the pergen, ^1 is either ~33/32 or ~729/704.

Olympic or Salururuti clan (P8, P5, ^1): These temper out the olympia, [17 -5 0 -2 -1⟩ = 131072/130977. 11/8 is equated with a 2.3.7 interval, and thus every 2.3.7.11 interval is equated with a 2.3.7 interval. In the pergen, ^1 = ~64/63.

Alphaxenic or Laquadloti rank-3 clan: These temper out the Alpharabian comma, [-17 2 0 0 4⟩ = 131769/131072. In the corresponding 2.3.11 rank-2 temperament, the pergen is (P8/2, M2/4).

Rank-4 temperaments

Main article: Rank-4 temperament

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.

Keenanismic or Lozoyoti temperaments (P8, P5, ^1, /1): These temper out the keenanisma, [-7 -1 1 1 1⟩ = 385/384. In the pergen, ^1 could be either ~81/80 or ~64/63, and /1 could be either ~64/63 (if ^1 isn't), ~33/32 or ~729/704.

Werckismic or Luzozoguti temperaments (P8, P5, ^1, /1): These temper out the werckisma, [-3 2 -1 2 -1⟩ = 441/440. 11/8 is equated to [-6 2 -1 2⟩ and 5/4 is equated to [-5 2 0 2 -1⟩, thus the lattice can be thought of as either 7-limit JI or 11-limit no-fives JI. In the pergen, ^1 is ~64/63, and /1 is either ~81/80, ~33/32 or ~729/704.

Swetismic or Lururuyoti temperaments (P8, P5, ^1, /1): These temper out the swetisma, [2 3 1 -2 -1⟩ = 540/539. 11/8 is equated to -1 3 1 -2 = 135/98 and 5/4 is equated to [-4 -3 0 2 1⟩, thus the lattice can be thought of as either 7-limit JI or 11-limit no-fives JI. In the pergen, ^1 is ~64/63, and /1 can be either ~81/80, ~33/32 or ~729/704.

Lehmerismic or Loloruyoyoti temperaments (P8, P5, ^1, /1): These temper out the lehmerisma, [-4 -3 2 -1 2⟩ = 3025/3024. Since 7/4 is equated to a yala (11-limit no-sevens) interval, both the pergen and the lattice are identical to that of yala JI. In the pergen, ^1 = ~81/80 and/1 = either ~33/32 or ~729/704.

Kalismic or Biloruguti temperaments (P8/2, P5, ^1, /1): These temper out the kalisma, [-3 4 -2 -2 2⟩ = 9801/9800. The octave is split into two ~99/70 periods. In the pergen, ^1 could be either ~81/80 or ~64/63, and /1 could be either ~64/63 (if ^1 isn't), ~33/32 or ~729/704.

Subgroup temperaments

Main article: 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 subgroups (including full prime limits) tempering out a given comma. For some commas, looking at the full commatic realm seems the best approach to discussing associated temperaments.

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

Marveltwin or Thoyoyoti: This is the commatic realm of the 13-limit comma 325/324, the marveltwin comma.

The Archipelago or Bithoguti: The Archipelago is a name which has been given to the commatic realm of the 13-limit comma 676/675 = [2 -3 -2 0 0 2⟩, the island comma.

The Jacobins or Thotrilu-aguti: This is the commatic realm of the 13-limit comma 6656/6655, the jacobin comma.

Stellar remnants: This is the commatic realm of 7777/7776, the pulsar comma.

Orgonia or Satrilu-aruruti: This is the commatic realm of the 11-limit comma 65536/65219 = [16 0 0 -2 -3⟩, the orgonisma.

The Nexus or Tribiloti: This is the commatic realm of the 11-limit comma 1771561/1769472 = [-16 -3 0 0 6⟩, the nexus comma.

The Quartercache or Saquinlu-azoti: This is the commatic realm of the 11-limit comma 117440512/117406179 = [24 -6 0 1 -5⟩, the quartisma.

Miscellaneous other temperaments

Limmic temperaments: Various subgroup temperaments all tempering out the limma, 256/243.

Fractional-octave temperaments: These temperaments all have a fractional-octave period.

Miscellaneous 5-limit temperaments: High in badness, but worth cataloging for one reason or another.

Low harmonic entropy linear temperaments: Temperaments where the average harmonic entropy of their intervals is low in a particular scale size range.

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.

Middle Path tables

Tables of temperaments where complexity/7.65 + damage/10 < 1. Useful for beginners looking for a list of a manageable number of temperaments, which approximate harmonious intervals accurately with a manageable number of notes.

Middle Path table of five-limit rank two temperaments

Middle Path table of seven-limit rank two temperaments

Middle Path table of eleven-limit rank two temperaments

Maps of temperaments

Temperament nomenclature

Proposed names for rank 2 temperaments – a compact list of temperaments (out of date)
Temperament naming

External links

List of temperaments in Scala with ready to use values

Tour of regular temperaments

Contents

Rank-2 temperaments

Families defined by a 2.3 (wa) comma

Families defined by a 2.3.5 (ya) comma

Clans defined by a 2.3.7 (za) comma

Clans defined by a 2.3.11 (ila) comma

Clans defined by a 2.3.13 (tha) comma

Clans defined by a 2.5.7 (yaza nowa) comma

Clans defined by a 3.5.7 (yaza noca) comma

Temperaments defined by a 2.3.5.7 (yaza) comma

Rank-3 temperaments

Families defined by a 2.3.5 (ya) comma

Families defined by a 2.3.7 (za) comma

Families defined by a 2.3.5.7 (yaza) comma

Temperaments defined by an 11-limit comma

Rank-4 temperaments

Subgroup temperaments

Commatic realms

Miscellaneous other temperaments

Maps of temperaments

Temperament nomenclature

External links

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