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 comma

These are families defined by a 3-limit (color name: wa) 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.

Blackwood 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. The only member of this family is 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 fifth. Its color name is Sawati.

Whitewood 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. Its color name is Lawati.

Compton family (P8/12, ^1)

This tempers out the Pythagorean comma, [-19 12⟩ (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. Its color name is Lalawati.

Countercomp family (P8/41, ^1)

This family tempers out the Pythagorean countercomma, [65 -41⟩, which creates multiple copies of 41edo. Its color name is Wa-41.

Mercator family (P8/53, ^1)

This family tempers out the Mercator's comma, [-84 53⟩, which creates multiple copies of 53edo. Its color name is Wa-53.

Families defined by a 2.3.5 comma

These are families defined by a 5-limit (color name: ya) comma. As we go up in rank 2, 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 3, 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 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 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. Its color name is Guti.

Schismatic 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 118edo. Its color name is Layoti.

Mavila family (P8, P5)

This tempers out the mavila 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 stacked together and octave reduced leads you to 6/5 instead of 5/4, and one consequence of this is that it generates 2L 5s (antidiatonic) scales. 5/4 is equated to 3 fourths minus 1 octave. Septimal mavila and armodue are some of the most notable temperaments associated with the mavila comma. Tunings include 9-, 16-, 23-, and 25edo. Its color name is Layobiti.

Father family (P8, P5)

This tempers out 16/15, the just diatonic semitone, and equates 5/4 with 4/3. Its color name is Gubiti.

Diaschismic 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 second ~9/8 is divided in half, with each half equated to ~16/15. Diaschismic tunings include 12-, 22-, 34-, 46-, 56-, 58- and 80edo. Its color name is Saguguti. 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 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. Its color name is Guguti. An obvious 7-limit interpretation of the generator is 7/6~8/7, which leads to semaphore or Zozoti.

Immunity 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. Its color name is Sasa-yoyoti. An obvious 7-limit interpretation of the generator is 7/6~8/7, which leads to semaphore or Zozoti.

Dicot 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 between two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into a neutral-sized third 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 thirds span a 3/2. Tunings include 7edo, 10edo, and 17edo. Its color name is Yoyoti. An obvious 2.3.11 interpretation of the generator is ~11/9, which leads to neutral or Luluti.

Augmented 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 12edo-based music theory, as well as what is commonly called "Tcherepnin's scale" (3L 6s). Its color name is Triguti.

Misty 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. Its color name is Sasa-triguti.

Porcupine 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 59edo. Its color name is Triyoti. An important 7-limit extension also tempers out 64/63.

Alphatricot family (P8, P11/3)

The alphatricot family tempers out the alphatricot comma, [39 -29 3⟩. The generator is ~59049/40960 ([-13 10 -1⟩) = 633 ¢, or its octave inverse ~81920/59049 = 567 ¢. Three of the former generators equals the third harmonic, ~3/1. 5/4 is equated to 29 of these generators octave-reduced. Its color name is Quadsa-triyoti. 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.

Diminished family (P8/4, P5)

The 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. Its color name is Quadguti.

Undim 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. Its color name is Trisa-quadguti.

Negri family (P8, P4/4)

This tempers out the negri comma, [-14 3 4⟩. Its only member so far is negri. Its generator is ~16/15, four of which make ~4/3. 5/4 is equated to 3 generators. Its color name is Laquadyoti.

Tetracot 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. Its color name is Saquadyoti.

Smate 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. Its color name is Saquadguti.

Vulture 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. Its color name is Sasa-quadyoti. An obvious 7-limit interpretation of the generator is 21/16, which makes buzzard or Saquadruti.

Quintile family (P8/5, P5)

This tempers out the quintile comma, 847288609443/838860800000 ([-28 25 -5⟩). The period is ~59049/51200, and 5 periods make an octave. The generator is a fifth, or equivalently, 3/5 of an octave minus a fifth. 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. Its color name is Trila-quinguti. An obvious 7-limit interpretation of the generator is 8/7, which leads to Laquinzoti.

Ripple 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. Its color name is Quinguti.

Passion 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. Its color name is Saquinguti.

Quintaleap 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. Its color name is Trisa-quinguti. An obvious 17-limit interpretation of the generator is ~18/17, which makes Saquinsoti.

Quindromeda 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. Its color name is Quinsa-quinguti. An obvious 17-limit interpretation of the generator is ~18/17, which makes Saquinsoti.

Amity 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. Its color name is Saquinyoti. 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 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 41edo among its possible tunings, with the last being near-optimal. Its color name is Laquinyoti.

Fifive 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. Its color name is Saquinbiyoti.

Quintosec 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. Its color name is Quadsa-quinbiguti. An obvious 7-limit interpretation of the period is 8/7.

Trisedodge 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. Its color name is Saquintriguti. An obvious 7-limit interpretation of the period is 8/7.

Ampersand family (P8, P5/6)

This tempers out the ampersand comma, 34171875/33554432 ([-25 7 6⟩). Its only member is ampersand. The generator is ~16/15, of which six make ~3/2. 5/4 is equated to 1 octave minus 7 generators. Its color name is Lala-tribiyoti. If the generator is also equated to ~15/14, and three generators to ~11/9, one gets the miracle temperament.

Kleismic family (P8, P12/6)

The kleismic family of temperaments tempers out the kleisma, 15625/15552 ([-6 -5 6⟩), 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 140edo among its possible tunings. Its color name is Tribiyoti.

Orson or semicomma 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. Its color name is Lasepyoti. Its generator has a natural interpretation as ~7/6, leading to the orwell or Sepruti temperament.

Wesley family (P8, ccP4/7)

This tempers out the wesley comma, 78125/73728 ([-13 -2 7⟩). The generator is ~125/96 = ~412 ¢. Seven generators equals a double-compound fourth of ~16/3. 5/4 is equated to 1 octave minus 2 generators. Its color name is Lasepyobiti. 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 family (P8, ccP5/7)

The sensipent family tempers out the sensipent comma, 78732/78125 ([2 9 -7⟩), also known as the medium semicomma. Its generator is ~162/125 = ~443 ¢. Seven generators equals a double-compound fifth of ~6/1. 5/4 is equated to 9 generators minus 3 octaves. Tunings include 8edo, 19edo, 46edo, and 65edo. Its color name is Sepguti. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzoti temperament.

Vishnuzmic 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. Its color name is Sasepbiguti.

Unicorn 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. Its color name is Laquadbiguti.

Würschmidt family (P8, ccP5/8)

The würschmidt 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 fifth); that is, (5/4)⁸⋅(393216/390625) = 6. It tends to generate the same mos scales as the 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. Its color name is Saquadbiguti.

Escapade 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⟩ of ~55 ¢ and nine of them equal ~4/3. Seven of them equal ~5/4, thus two of them equal ~16/15. Its color name is Sasa-tritriguti. An obvious 11-limit interpretation of the generator is 33/32, leading to the Trisa-tritriluti temperament.

Mabila 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 fourth of ~64/3. Its color name is Sasa-quinbiguti. An obvious 11-limit interpretation of the generator is ~15/11.

Sycamore 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. Its color name is Laleyoti.

Quartonic 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. Its color name is Saleyoti. An obvious 7-limit interpretation of the generator is ~36/35.

Lafa 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. Its color name is Tribisa-quadtriguti.

Ditonmic 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 fourth. 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. Its color name is Lala-theyoti.

Luna family (P8, ccP4/15)

This tempers out the luna comma, [38 -2 -15⟩ (274877906944/274658203125). The generator is ~[18 -1 -7⟩ at ~193 ¢. Two generators equals ~5/4, and fifteen generators equals a double-compound fourth of ~16/3. Its color name is Sasa-quintriguti.

Vavoom 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. Its color name is Quinla-seyoti.

Minortonic 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 fifth (~6/1). 5/4 is equated to 35 generators minus 5 octaves. Its color name is Trila-seguti.

Maja 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 fourth. 5/4 is equated to 9 octaves minus 23 generators. Its color name is Saseyoti.

Maquila family (P8, c⁷P5/17)

This tempers out the maquila comma, [49 -6 -17⟩ (562949953421312/556182861328125). The generator is ~512/375 = ~535 ¢. Seventeen generators equals a septuple-compound fifth. 5/4 is equated to 3 octaves minus 6 generators. Its color name is Trisa-seguti. An obvious 11-limit interpretation of the generator is 11/8, leading to the Lala-seloti temperament. However, Lala-seloti is not nearly as accurate as Trisa-seguti.

Gammic 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 the neptune temperament. Its color name is Laquinquadyoti.

Clans defined by a 2.3.7 comma

These are defined by a no-5's 7-limit (color name: za) comma. See also subgroup temperaments.

If a 5-limit comma defines a family of rank-2 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 clan (P8, P5)

This clan tempers out Archytas' comma, 64/63. It equates 7/4 with 16/9. The clan consists of rank-2 temperaments, and should not be confused with the archytas family of rank-3 temperaments. Its color name is Ruti. Its best downward extension is superpyth.

Trienstonic 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. Its color name is Zoti.

Harrison clan (P8, P5)

This clan tempers out Harrison's comma, [-13 10 0 -1⟩ (59049/57344). It equates 7/4 to an augmented sixth. Its color name is Laruti. Its best downward extension is septimal meantone.

Garischismic 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 octave [23 -14⟩. This clan includes vulture, newt, garibaldi, sextile, and satin. Its color name is Sasaruti.

Sasazoti clan (P8, P5)

This clan tempers out the leapfrog 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 octave. The generator is ~3/2, and four generators minus three periods equals ~7/4. The major second ~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.

Semaphoresmic clan (P8, P4/2)

This clan tempers out the large septimal diesis, 49/48. Its generator is ~8/7 or ~7/6. Its color name is Zozoti. Its best downward extension is godzilla. See also semaphore.

Parahemif 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. Its color name is Sasa-zozoti. 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 octave. 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 clan (P8, P5/3)

This clan tempers out the gamelisma, [-10 1 0 3⟩ (1029/1024). Three ~8/7 generators equals a fifth. 7/4 is equated to an octave minus a generator. Five generators is slightly flat of 2/1, making this a cluster temperament. Its color name is Latrizoti. 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 divides the fifth into 6 equal steps, thus it is 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 fifth, 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.

Latriru clan (P8, P11/3)

This clan tempers out the lee 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. Its color name is Latriruti. An obvious full 7-limit interpretation of the generator is 7/5, leading to the liese temperament, which is a weak extension of meantone.

Stearnsmic 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 equal ~4/3. 7/4 is equated to five ~9/7 generators minus an octave. Its color name is Latribiruti. Equating the ~54/49 generator to ~10/9 creates a weak extension of the porcupine temperament, as does equating the period to ~7/5.

Skwaresmic clan (P8, P11/4)

This clan tempers out the skwaresma, [-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. Its color name is Laquadruti. This clan includes as a strong extension the squares temperament, which is a weak extension of meantone.

Buzzardsmic clan (P8, P12/4)

This clan tempers out the buzzardsma, [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. Its color name is Saquadruti. This clan includes as a strong extension the vulture temperament, which is in the vulture family.

Cloudy 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. Its color name is Laquinzoti.

Quinruti clan (P8, P5/5)

This clan tempers out the bleu 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 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. Its color name is Sasasepruti.

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.

Septiennealimmal clan (P8/9, P5)

This clan tempers out the septiennealimma, [-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. Its color name is Tritrizoti.

Clans defined by a 2.3.11 comma

Color name: ila. See also subgroup temperaments.

Lulubiti clan (P8/2, P5)

This low-accuracy 2.3.11 clan tempers out the Alpharabian limma, 128/121. Both 11/8 and 16/11 are equated to half-octave period. This clan includes as a strong extension the pajaric temperament, which is in the diaschismic family.

Rastmic 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. Its color name is Luluti.

Nexus clan (P8/3, P4/2)

This 2.3.11 clan tempers out the nexus comma [-16 -3 0 0 6⟩. Its 1/3-octave period is ~121/96 and its least-cents generator is ~12/11. A period plus a generator equals ~11/8. Six of these generators equals ~27/16. A period minus a generator equals ~1331/1152 or ~1536/1331. Two of these alternative generators equals ~4/3. Its color name is Tribiloti.

Alphaxenic or Laquadloti clan (P8/2, M2/4)

This 2.3.11 clan tempers out the Alpharabian comma [-17 2 0 0 4⟩. Its half-octave 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 a strong extension to the comic or Saquadyobiti temperament, which is in the jubilismic clan. Its color name is Laquadloti.

Clans defined by a 2.3.13 comma

Color name: tha. See also subgroup temperaments.

Thuthuti 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.

Satrithoti clan (P8, P11/3)

This 2.3.13 clan tempers out the threedie, 2197/2187 ([0 -7 0 0 0 3⟩). Its generator is ~18/13, and three generators equals ~8/3. 13/8 is equated to 7 generators minus three octaves. This clan is related to the Latriruti clan.

Clans defined by a 2.5.7 comma

These are defined by a no-3's 7-limit (color name: yaza nowa) comma. See also subgroup temperaments. The pergen's multigen (the second term, omitting any fraction) is always a no-3's 5-limit ratio such as 5/4, 8/5, 25/8, etc.

Jubilismic 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 classical major third (~5/4). The half-octave period is ~7/5 or ~10/7. 7/4 is equated to 1 period plus 1 generator. Its color name is Biruyoti.

Bapbo 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. Its color name is Ruruguti Nowa.

Hemimean 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 classical major third (~5/4), three of them equals ~7/5, and five of them equals ~7/4. Its color name is Zozoquinguti Nowa.

Mabilismic 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. Its color name is Latrizo-aquiniyoti Nowa.

Vorwell 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 generator is ~1024/875 = ~272 ¢. Three generators equals ~8/5 and eight of them equals ~7/2. Its color name is Sasatriru-aquadbiguti Nowa.

Quinzo-atriyoti Nowa clan (P8, M3/5)

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

Llywelynsmic 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. Its color name is Sasepru-aguti Nowa.

Quince clan (P8, m6/7)

This clan tempers out the quince comma, [-15 0 -2 7⟩ (823543/819200). The generator is ~343/320 = ~116 ¢. Two generators equals ~8/7, five generators equals ~7/5, and seven generators equals the classical minor sixth ~8/5. An obvious 5-limit interpretation of the generator is ~16/15, leading to the miracle temperament, which is in the gamelismic clan. Its color name is Lasepzo-aguguti Nowa.

Slither 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 sixth of ~32/5. Its color name is Satritriru-aquadyoti Nowa.

Clans defined by a 3.5.7 comma

These are defined by a no-2's 7-limit (color name: yaza noca) comma. Any no-2's 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 second term, omitting any fraction) is always a no-2's 5-limit ratio such as 5/3, 9/5, 25/9, etc. In any no-2's subgroup, "compound" means increased by 3/1 not 2/1.

Rutribiyoti Noca clan (P12, M6)

This 3.5.7 clan tempers out the arcturus comma [0 -7 6 -1⟩ (15625/15309). Its only member so far is arcturus. The generator is the classical major sixth (~5/3), and six generators equals ~21/1.

Sensamagic 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 sixth (~5/3). Its color name is Zozoyoti Noca.

Gariboh 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 classical major sixth (~5/3). Its color name is Triru-aquinyoti Noca.

Mirkwai 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 classical compound minor seventh (~27/5). Its color name is Quinru-aquadyoti Noca.

Sasepzo-atriguti Noca clan (P12, m7/7)

This 3.5.7 clan tempers out the procyon comma [0 -8 -3 7⟩ (823543/820125). Its only member so far is procyon. The generator is ~49/45, three generators equals ~9/7, four equals ~7/5, and seven equals the classic minor seventh (~9/5).

Satritrizo-aguguti Noca clan (P12, c³M6/9)

This 3.5.7 clan tempers out the betelgeuse comma [0 -13 -2 9⟩ (40353607/39858075). Its only member so far is betelgeuse. The generator is ~3645/2401, two generators equals ~7/3, and nine generators equals the classical triple-compound major sixth (~45/1).

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). Its only member so far is izar. 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 comma

These are defined by a full 7-limit (color name: yaza) 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. Its color name is Zoguti.

Greenwoodmic temperaments

These temper out the greenwoodma, [-3 4 1 -2⟩ (405/392). Its color name is Ruruyoti.

Keegic temperaments

Keegic rank-2 temperaments temper out the keega, [-3 1 -3 3⟩ (1029/1000). Its color name is Trizoguti.

Mint temperaments

Mint rank-2 temperaments temper out the septimal quartertone, 36/35, equating 7/6 with 6/5, and 5/4 with 9/7. Its color name is Ruguti.

Avicennmic temperaments

These temper out the avicennma, [-9 1 2 1⟩ (525/512), also known as Avicenna's enharmonic diesis. Its color name is Zoyoyoti.

Sengic temperaments

Sengic rank-2 temperaments temper out the senga, [1 -3 -2 3⟩ (686/675). Its color name is Trizo-aguguti.

Keemic temperaments

Keemic rank-2 temperaments temper out the keema, [-5 -3 3 1⟩ (875/864). Its color name is Zotriyoti.

Secanticorn temperaments

Secanticorn rank-2 temperaments temper out the secanticornisma, [-3 11 -5 -1⟩ (177147/175000). Its color name is Laruquinguti.

Nuwell temperaments

Nuwell rank-2 temperaments temper out the nuwell comma, [1 5 1 -4⟩ (2430/2401). Its color name is Quadru-ayoti.

Mermismic temperaments

Mermismic rank-2 temperaments temper out the mermisma, [5 -1 7 -7⟩ (2500000/2470629). Its color name is Sepruyoti.

Negricorn temperaments

Negricorn rank-2 temperaments temper out the negricorn comma, [6 -5 -4 4⟩ (153664/151875). Its color name is Saquadzoguti.

Tolermic temperaments

These temper out the tolerma, [10 -11 2 1⟩ (179200/177147). Its color name is Sazoyoyoti.

Valenwuer temperaments

Valenwuer rank-2 temperaments temper out the valenwuer comma, [12 3 -6 -1⟩ (110592/109375). Its color name is Sarutribiguti.

Mirwomo temperaments

Mirwomo rank-2 temperaments temper out the mirwomo comma, [-15 3 2 2⟩ (33075/32768). Its color name is Labizoyoti.

Catasyc temperaments

Catasyc rank-2 temperaments temper out the catasyc comma, [-11 -3 8 -1⟩ (390625/387072). Its color name is Laruquadbiyoti.

Compass temperaments

Compass rank-2 temperaments temper out the compass comma, [-6 -2 10 -5⟩ (9765625/9680832). Its color name is Quinruyoyoti.

Trimyna temperaments

Trimyna rank-2 temperaments temper out the trimyna comma, [-4 1 -5 5⟩ (50421/50000). Its color name is Quinzoguti.

Starling temperaments

Starling rank-2 temperaments temper out the starling comma a.k.a. septimal semicomma, [1 2 -3 1⟩ (126/125), the difference between three 6/5's plus one 7/6, and an octave. It includes myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum. Its color name is Zotriguti.

Octagar temperaments

Octagar rank-2 temperaments temper out the octagar comma, [5 -4 3 -2⟩ (4000/3969). Its color name is Rurutriyoti.

Orwellismic temperaments

Orwellismic rank-2 temperaments temper out orwellisma, [6 3 -1 -3⟩ (1728/1715). Its color name is Triru-aguti.

Mynaslendric temperaments

Mynaslendric rank-2 temperaments temper out the mynaslender comma, [11 4 1 -7⟩ (829440/823543). Its color name is Sepru-ayoti.

Mistismic temperaments

Mistismic rank-2 temperaments temper out the mistisma, [16 -6 -4 1⟩ (458752/455625). Its color name is Sazoquadguti.

Varunismic temperaments

Varunismic rank-2 temperaments temper out the varunisma, [-9 8 -4 2⟩ (321489/320000). Its color name is Labizoguguti.

Marvel temperaments

Marvel rank-2 temperaments temper out the marvel comma, [-5 2 2 -1⟩ (225/224). It includes wizard, tritonic, septimin, slender, triton, escapist and marv, not to mention negri, sharpie, pelogic, meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton. Its color name is Ruyoyoti.

Dimcomp temperaments

Dimcomp rank-2 temperaments temper out the dimcomp comma, [-1 -4 8 -4⟩ (390625/388962). Its color name is Quadruyoyoti.

Cataharry temperaments

Cataharry rank-2 temperaments temper out the cataharry comma, [-4 9 -2 -2⟩ (19683/19600). Its color name is Labiruguti.

Canousmic temperaments

Canousmic rank-2 temperaments temper out the canousma, [4 -14 3 4⟩ (4802000/4782969). Its color name is Saquadzo-atriyoti.

Triwellismic temperaments

Triwellismic rank-2 temperaments temper out the triwellisma, [1 -1 -7 6⟩ (235298/234375). Its color name is Tribizo-asepguti.

Hemimage temperaments

Hemimage rank-2 temperaments temper out the hemimage comma, [5 -7 -1 3⟩ (10976/10935). Its color name is Satrizo-aguti.

Hemifamity temperaments

Hemifamity rank-2 temperaments temper out the hemifamity comma, [10 -6 1 -1⟩ (5120/5103). Its color name is Saruyoti.

Parkleiness temperaments

Parkleiness rank-2 temperaments temper out the parkleiness comma, [7 7 -9 1⟩ (1959552/1953125). Its color name is Zotritriguti.

Porwell temperaments

Porwell rank-2 temperaments temper out the porwell comma, [11 1 -3 -2⟩ (6144/6125). Its color name is Sarurutriguti.

Cartoonismic temperaments

Cartoonismic rank-2 temperaments temper out the cartoonisma, [12 -3 -14 9⟩ (165288374272/164794921875). Its color name is Satritrizo-asepbiguti.

Hemfiness temperaments

Hemfiness rank-2 temperaments temper out the hemfiness comma, [15 -5 3 -5⟩ (4096000/4084101). Its color name is Saquinru-atriyoti.

Hewuermera temperaments

Hewuermera rank-2 temperaments temper out the hewuermera comma, [16 2 -1 -6⟩ (589824/588245). Its color name is Satribiru-aguti.

Lokismic temperaments

Lokismic rank-2 temperaments temper out the lokisma, [21 -8 -6 2⟩ (102760448/102515625). Its color name is Sasa-bizotriguti.

Decovulture temperaments

Decovulture rank-2 temperaments temper out the decovulture comma, [26 -7 -4 -2⟩ (67108864/66976875). Its color name is Sasabiruguguti.

Pontiqak temperaments

Pontiqak rank-2 temperaments temper out the pontiqak comma, [-17 -6 9 2⟩ (95703125/95551488). Its color name is Lazozotritriyoti.

Mitonismic temperaments

Mitonismic rank-2 temperaments temper out the mitonisma, [-20 7 -1 4⟩ (5250987/5242880). Its color name is Laquadzo-aguti.

Horwell temperaments

Horwell rank-2 temperaments temper out the horwell comma, [-16 1 5 1⟩ (65625/65536). Its color name is Lazoquinyoti.

Neptunismic temperaments

Neptunismic rank-2 temperaments temper out the neptunisma, [-12 -5 11 -2⟩ (48828125/48771072). Its color name is Laruruleyoti.

Metric microtemperaments

Metric rank-2 temperaments temper out the meter, [-11 2 7 -3⟩ (703125/702464). Its color name is Latriru-asepyoti.

Wizmic microtemperaments

Wizmic rank-2 temperaments temper out the wizma, [-6 -8 2 5⟩ (420175/419904). Its color name is Quinzo-ayoyoti.

Supermatertismic temperaments

Supermatertismic rank-2 temperaments temper out the supermatertisma, [-6 3 9 -7⟩ (52734375/52706752). Its color name is Lasepru-atritriyoti.

Breedsmic temperaments

Breedsmic rank-2 temperaments temper out the breedsma, [-5 -1 -2 4⟩ (2401/2400). Its color name is Bizozoguti.

Supermasesquartismic temperaments

Supermasesquartismic rank-2 temperaments temper out the supermasesquartisma, [-5 10 5 -8⟩ (184528125/184473632). Its color name is Laquadbiru-aquinyoti.

Ragismic microtemperaments

Ragismic rank-2 temperaments temper out the ragisma, [-1 -7 4 1⟩ (4375/4374). Its color name is Zoquadyoti.

Akjaysmic temperaments

Akjaysmic rank-2 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 4/7 of an octave, resulting in small intervals. Its color name is Trisa-sepruguti.

Landscape microtemperaments

Landscape rank-2 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. Its color name is Trizoguguti.

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 third generator in a rank-3 pergen is usually a comma, but sometimes it is some fraction of a 5-limit or 7-limit interval.

Families defined by a 2.3.5 comma

Every 5-limit (color name: ya) comma defines a rank-3 family, thus every comma in the list of rank-2 2.3.5 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a 7-limit (color name: yaza) 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 an 11-limit (color name: yazala) temperament, and so forth. All these examples are 7-limit:

Didymus rank-3 family (P8, P5, ^1)

These are the rank-3 temperaments tempering out the didymus or meantone comma, 81/80. Its color name is Guti.

Diaschismic rank-3 family (P8/2, P5, /1)

These are the rank-3 temperaments tempering out the diaschisma, [11 -4 -2⟩ (2048/2025). The half-octave period is ~45/32. Its color name is Saguguti.

Porcupine rank-3 family (P8, P4/3, /1)

These are the rank-3 temperaments tempering out the porcupine comma a.k.a. maximal diesis, [1 -5 3⟩ (250/243). In the pergen, P4/3 is ~10/9. Its color name is Triyoti.

Kleismic rank-3 family (P8, P12/6, /1)

These are the rank-3 temperaments tempering out the kleisma, [-6 -5 6⟩ (15625/15552). In the pergen, P12/6 is ~6/5. Its color name is Tribiyoti.

Families defined by a 2.3.7 comma

Every no-5's 7-limit (color name: za) comma defines a rank-3 family, thus every comma in the list of rank-2 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 7-limit (color name: yaza) 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 11-limit (color name: yazala) temperament, and so forth. All these examples are 7-limit:

Archytas 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. Its color name is Ruti.

Garischismic family (P8, P5, ^1)

A garischismic temperament is one which tempers out the garischisma, [25 -14 0 -1⟩ (33554432/33480783). Its color name is Sasaruti.

Laruruti clan (P8/2, P5)

This clan tempers out the Laruru comma, [-7 8 0 -2⟩ (6561/6272). Two ~81/56 periods equal an octave. The generator is ~3/2, and four generators minus three periods equals ~7/4. The major second ~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.

Semaphoresmic family (P8, P4/2, ^1)

Semaphoresmic 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. Its color name is Zozoti.

Gamelismic family (P8, P5/3, ^1)

Not to be confused with the gamelismic clan of rank-2 temperaments, the gamelismic family are those rank-3 temperaments which temper out the gamelisma, [-10 1 0 3⟩ (1029/1024). In the pergen, P5/3 is ~8/7. Its color name is Latrizoti.

Stearnsmic 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. Its color name is Latribiruti.

Families defined by a 2.3.5.7 comma

Color name: yaza.

Marvel family (P8, P5, ^1)

The head of the marvel family is marvel, which tempers out the marvel comma, [-5 2 2 -1⟩ (225/224). It divides 8/7 into two 16/15's, or equivalently, two 15/14's. 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. Its color name is Ruyoyoti.

Starling family (P8, P5, ^1)

Starling tempers out the starling comma a.k.a. septimal semicomma, [1 2 -3 1⟩ (126/125), the difference between three 6/5's plus one 7/6, and an octave. It divides 10/7 into two 6/5's. 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. Its color name is Zotriguti.

Sensamagic family (P8, P5, ^1)

These temper out [0 -5 1 2⟩ (245/243), which divides 16/15 into two 28/27's. In the pergen, ^1 = ~64/63. Its color name is Zozoyoti.

Greenwoodmic family (P8, P5, ^1)

These temper out the greenwoodma, [-3 4 1 -2⟩ (405/392), which divides 10/9 into two 15/14's. In the pergen, ^1 = ~64/63. Its color name is Ruruyoti.

Avicennmic family (P8, P5, ^1)

These temper out the avicennma, [-9 1 2 1⟩ (525/512), which divides 7/6 into two 16/15's. In the pergen, ^1 = ~81/80. Its color name is Lazoyoyoti.

Keemic family (P8, P5, ^1)

These temper out the keema, [-5 -3 3 1⟩ (875/864), which divides 15/14 into two 25/24's. In the pergen, ^1 = ~81/80. Its color name is Zotriyoti.

Orwellismic family (P8, P5, ^1)

These temper out the orwellisma, [6 3 -1 -3⟩ (1728/1715). In the pergen, ^1 = ~64/63. Its color name is Triru-aguti.

Nuwell family (P8, P5, ^1)

These temper out the nuwell comma, [1 5 1 -4⟩ (2430/2401). In the pergen, ^1 = ~64/63. Its color name is Quadru-ayoti.

Ragisma family (P8, P5, ^1)

The 7-limit rank-3 microtemperament which tempers out the ragisma, [-1 -7 4 1⟩ (4375/4374), extends to various higher-limit rank-3 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. Its color name is Zoquadyoti.

Hemifamity family (P8, P5, ^1)

The hemifamity family of rank-3 temperaments tempers out the hemifamity comma, [10 -6 1 -1⟩ (5120/5103), which divides 10/7 into three 9/8's. In the pergen, ^1 = ~81/80. Its color name is Saruyoti.

Horwell family (P8, P5, ^1)

The horwell family of rank-3 temperaments tempers out the horwell comma, [-16 1 5 1⟩ (65625/65536). In the pergen, ^1 = ~81/80. Its color name is Lazoquinyoti.

Hemimage family (P8, P5, ^1)

The hemimage family of rank-3 temperaments tempers out the hemimage comma, [5 -7 -1 3⟩ (10976/10935), which divides 10/9 into three 28/27's. In the pergen, ^1 = ~64/63. Its color name is Satrizo-aguti.

Mint family (P8, P5, ^1)

The mint temperament is a low-complexity, high-error temperament, tempering out the septimal quartertone 36/35, equating 7/6 with 6/5, and 5/4 with 9/7. In the pergen, ^1 = ~81/80 or ~64/63. Its color name is Ruguti.

Septisemi 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. Its color name is Zoguti.

Jubilismic 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. Its color name is Biruyoti.

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

Cataharry temperaments temper out the cataharry comma, [-4 9 -2 -2⟩ (19683/19600). In the pergen, half a fourth is ~81/70, and ^1 = ~81/80. Its color name is Labiruguti.

Breed 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. Its color name is Bizozoguti.

Sengic 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 third in the pergen), and three give ~6/5. Its color name is Trizo-aguguti.

Porwell family (P8, P5, ^m3/2)

The porwell family of rank-3 temperaments tempers out the porwell comma, [11 1 -3 -2⟩ (6144/6125). Two ~35/32 generators equal the pergen's upminor third of ~6/5. Its color name is Sarurutriguti.

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

The octagar family of rank-3 temperaments tempers out the octagar comma, [5 -4 3 -2⟩ (4000/3969). Two ~80/63 generators equal the pergen's upminor sixth of ~8/5. Its color name is Rurutriyoti.

Hemimean family (P8, P5, vM3/2)

The hemimean family of rank-3 temperaments tempers out the hemimean comma, [6 0 -5 2⟩ (3136/3125). Two ~28/25 generators equal the pergen's downmajor third of ~5/4. Its color name is Zozoquinguti.

Wizmic 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 seventh of ~7/4. Its color name is Quinzo-ayoyoti.

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

The 7-limit rank-3 microtemperament which tempers out the landscape comma, [-4 6 -6 3⟩ (250047/250000), extends to various higher-limit rank-3 temperaments such as tyr and odin. In the pergen, the 1/3-octave period is ~63/50, and ^1 = ~81/80. Its color name is Trizoguguti.

Gariboh family (P8, P5, vM6/3)

The gariboh family of rank-3 temperaments tempers out the gariboh comma, [0 -2 5 -3⟩ (3125/3087). Three ~25/21 generators equal the pergen's downmajor sixth of ~5/3. Its color name is Triru-aquinyoti.

Canou family (P8, P5, vm6/3)

The canou family of rank-3 temperaments tempers out the canousma, [4 -14 3 4⟩ (4802000/4782969). Three ~81/70 generators equal the pergen's downminor sixth of ~14/9. Its color name is Saquadzo-atriyoti.

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

The dimcomp family of rank-3 temperaments tempers out the dimcomp comma, [-1 -4 8 -4⟩ (390625/388962). In the pergen, the 1/4-octave period is ~25/21, and ^1 = ~81/80. Its color name is Quadruyoyoti.

Mirkwai family (P8, P5, c^M7/4)

The mirkwai family of rank-3 temperaments tempers out the mirkwai comma, [0 3 4 -5⟩ (16875/16807). Four ~7/5 generators equal the pergen's compound upmajor seventh of ~27/7. Its color name is Quinru-aquadyoti.

Temperaments defined by an 11-limit comma

Ptolemismic 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. Its color name is Luyoyoti.

Biyatismic 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. Its color name is Lologuti.

Valinorsmic 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. Its color name is Loruguguti.

Rastmic 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). Its color name is Luluti.

Pentacircle 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. Its color name is Saluzoti.

Semicanousmic clan (P8, P5, ^1)

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

Semiporwellismic clan (P8, P5, ^1)

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

Olympic 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. Its color name is Salururuti.

Alphaxenic 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). Its color name is Laquadloti.

Keenanismic temperaments

These temper out the keenanisma, [-7 -1 1 1 1⟩ (385/384). Its color name is Lozoyoti.

Werckismic temperaments

These temper out the werckisma, [-3 2 -1 2 -1⟩ (441/440). Its color name is Luzozoguti.

Swetismic temperaments

These temper out the swetisma, [2 3 1 -2 -1⟩ (540/539). Its color name is Lururuyoti.

Lehmerismic temperaments

These temper out the lehmerisma, [-4 -3 2 -1 2⟩ (3025/3024). Its color name is Loloruyoyoti.

Kalismic temperaments

These temper out the kalisma, [-3 4 -2 -2 2⟩ (9801/9800). Its color name is Biloruguti.

Rank-4 temperaments

Main article: Catalog of rank-4 temperaments

Even less explored than rank-3 temperaments are rank-4 temperaments. In fact, unless one counts 7-limit JI they do not seem to have been explored at all. However, they could be used; for example hobbit scales can be constructed for them.

Keenanismic family (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 is not), ~33/32 or ~729/704. Its color name is Lozoyoti.

Werckismic family (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 no-5's 11-limit JI. In the pergen, ^1 is ~64/63, and /1 is either ~81/80, ~33/32 or ~729/704. Its color name is Luzozoguti.

Swetismic family (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 no-5's 11-limit JI. In the pergen, ^1 is ~64/63, and /1 can be either ~81/80, ~33/32 or ~729/704. Its color name is Lururuyoti.

Lehmerismic family (P8, P5, ^1, /1)

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

Kalismic family (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 is not), ~33/32 or ~729/704. Its color name is Biloruguti.

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

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

Marveltwin

This is the commatic realm of the 13-limit comma 325/324, the marveltwin comma. Its color name is Thoyoyoti.

The Archipelago

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. Its color name is Bithoguti.

The Jacobins

This is the commatic realm of the 13-limit comma 6656/6655, the jacobin comma. Its color name is Thotrilu-aguti.

Orgonia

This is the commatic realm of the 11-limit comma 65536/65219 ([16 0 0 -2 -3⟩), the orgonisma. Its color name is Satrilu-aruruti.

The Nexus

This is the commatic realm of the 11-limit comma 1771561/1769472 ([-16 -3 0 0 6⟩), the nexus comma. Its color name is Tribiloti.

The Quartercache

This is the commatic realm of the 11-limit comma 117440512/117406179 ([24 -6 0 1 -5⟩), the quartisma. Its color name is Saquinlu-azoti.

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 do not 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

• Map of rank-2 temperaments, sorted by generator size
• Catalog of rank two temperaments
  • Catalog of seven-limit rank two temperaments
  • Catalog of eleven-limit rank two temperaments
  • Catalog of thirteen-limit rank two temperaments
• List of rank two temperaments by generator and period
• Rank-2 temperaments by mapping of 3
• Temperaments for MOS shapes
• Tree of rank two temperaments

Temperament nomenclature

• Temperament naming

External links

• List of temperaments in Scala with ready to use values