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 p-limit rank-2 temperament maps all intervals of p-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a linear temperament. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-ET by tempering out the Pythagorean comma.

Families defined by a 2.3 (wa) comma

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

Limma or Sawa family (P8/5, ^1): This family tempers out the limma, [8 -5 0⟩ = 256/243, which implies 5edo.

Apotome or Lawa family (P8/7, ^1): This family tempers out the apotome, [-11 7 0⟩ = 2187/2048, which implies 7edo.

Compton or Lalawa family (P8/12, ^1): This tempers out the Pythagorean comma, 531441/524288 = [-19 12 0⟩, which implies 12edo. This family includes the compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave The 5-limit compton temperament can be thought of as multiple rings of 12edo, 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 implies 41edo.

Mercator or Wa-53 family (P8/53, ^1): This family tempers out the Mercator's comma, [-84 53⟩, which implies 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 Gu family (P8, P5): The meantone family tempers out 81/80, also called the syntonic comma. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are 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 Layo family (P8, P5): The schismatic family tempers out the schisma of [-15 8 1⟩ = 32805/32768, which is the amount by which the Pythagorean comma exceeds the syntonic comma. The 5-limit version of the temperament is a microtemperament which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity; whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4. Tunings include 12, 29, 41, 53, and 118 EDOs.

Pelogic or Layobi family (P8, P5): This tempers out the pelogic comma, [-7 3 1⟩ = 135/128, also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates 2L 5s "anti-diatonic" scales. 5/4 is equated to 3 fourths minus 1 octave. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include 9, 16, 23, and 25 EDOs.

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

Diaschismic or Sagugu family (P8/2, P5): The diaschismic family tempers out the diaschisma, [11 -4 -2⟩ or 2048/2025, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period of an approximate 45/32 or 64/45, and its generator is an approximate 3/2. 5/4 is equated to 3 periods minus 2 fifths. The major 2nd ~9/8 is divided in half, with each half equated to ~16/15. Diaschismic tunings include 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 Gugu family (P8, P4/2): This low-accuracy family of temperaments tempers out 27/25, the large limma or bug comma. The generator is an approximate 6/5 or 10/9 = ~250¢, two of which make ~4/3. 5/4 is equated to 1 octave minus 3 generators. An obvious 7-limit interpretation of the generator is 7/6, which makes Slendro aka Semaphore or Zozo.

Immunity or Sasa-yoyo family (P8, P4/2): This tempers out the immunity comma, [16 -13 2⟩ (1638400/1594323). Its generator is ~729/640 = ~247¢, two of which make ~4/3. 5/4 is equated to 3 octaves minus 13 generators. An obvious 7-limit interpretation of the generator is 7/6, which leads to Slendro aka Semaphore or Zozo.

Dicot or Yoyo family (P8, P5/2): The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24 (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into a neutral-sized 3rd of ~350¢ that is taken to approximate both. 7EDO makes for a "good" dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the "neutral" dicot 3rds span a 3/2. Tunings include 7EDO, 10EDO, and 17EDO. An obvious 2.3.11 interpretation of the generator is ~11/9, which leads to Rastmic aka Neutral or Lulu.

Augmented or Trigu family (P8/3, P5): The augmented family tempers out the diesis of [7 0 -3⟩ = 128/125, the difference between three 5/4 major thirds and a 2/1 octave, and so identifies the major third with the third-octave. Hence it has the same 400-cent 5/4-approximations as 12EDO, which is an excellent tuning for augmented. It is the temperament that results in what is commonly called the "augmented scale" (3L 3s) in common 12-based music theory, as well as what is commonly called "Tcherepnin's scale" (3L 6s).

Misty or Sasa-trigu 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 Triyo family (P8, P4/3): The porcupine family tempers out [1 -5 3⟩ = 250/243, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. 5/4 is equated to 1 octave minus 5 generators. Some porcupine temperaments include 15, 22, 37, and 59 EDOs. An important 7-limit extension also tempers out 64/63.

Tricot or Quadsa-triyo 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 Latriru clan. An obvious 13-limit interpretation is 13/9 = 637¢, an even simpler ratio implying the Satritho clan.

Dimipent or Quadgu family (P8/4, P5): The dimipent (or diminished) family tempers out the major diesis or diminished comma, [3 4 -4⟩ or 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12EDO. 5/4 is equated to 1 fifth minus 1 period.

Undim or Trisa-quadgu 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 Laquadyo family (P8, P4/4): This tempers out the negri comma, [-14 3 4⟩. Its generator is ~16/15, four of which make ~4/3. 5/4 is equated to 3 generators.

Tetracot or Saquadyo family (P8, P5/4): The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by [5 -9 4⟩ (20000/19683), the minimal diesis or tetracot comma. 5/4 is equated to 9 generators minus an octave. 7EDO can also be considered a tetracot tuning, as can 20EDO, 27EDO, 34EDO, and 41EDO.

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

Vulture or Sasa-quadyo family (P8, P12/4): This tempers out the vulture comma, [24 -21 4⟩. Its generator is ~320/243 = ~475¢, four of which make ~3/1. 5/4 is equated to 21 generators minus 8 octaves. An obvious 7-limit interpretation of the generator is 21/16, which makes Saquadru.

Pental or Trila-quingu family (P8/5, P5): This tempers out the pental comma, 847288609443/838860800000 = [-28 25 -5⟩. The period is ~59049/51200, and 5 periods make an octave. The generator is a 5th, or equivalently, 3/5 of an 8ve minus a 5th. This alternate generator is only about 18¢, thus the scales have a very lopsided L/s ratio. 5/4 is equated to 2/5 of an octave minus 5 of these 18¢ generators. An obvious 7-limit interpretation of the generator is 8/7, which leads to Laquinzo.

Ripple or Quingu 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 Saquingu 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-quingu 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 saquinso.

Quindromeda or Quinsa-quingu 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 saquinso.

Amity or Saquinyo family (P8, P11/5): This tempers out the amity comma, 1600000/1594323 = [9 -13 5⟩. The generator is 243/200 = ~339.5¢, five of which make ~8/3. 5/4 is equated to 4 octaves minus 13 generators, or 2 fifths minus 3 generators. An obvious 11-limit interpretation of the generator is 11/9, which makes Saquinlo. An obvious 13-limit interpretation of the generator is 39/32, which makes Lala-quintho.

Magic or Laquinyo family (P8, P12/5): The magic family tempers out [-10 -1 5⟩ (3125/3072), known as the magic comma or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/1. The generator is itself an approximate 5/4. The magic family includes 16, 19, 22, 25, and 41 EDOs among its possible tunings, with the latter being near-optimal.

Fifive or Saquinbiyo family (P8/2, P5/5): This tempers out the fifive comma, [-1 -14 10⟩ = 9765625/9565938. The period is ~4374/3125 = [1 7 -5⟩, two of which make an octave. The generator is ~27/25, five of which make ~3/2. 5/4 is equated to 7 generators minus 1 period.

Quintosec or Quadsa-quinbigu 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 Saquintrigu family (P8/5, P4/3): This tempers out the trisedodge comma, 30958682112/30517578125 = [19 10 -15⟩. The period is ~144/125 = 240¢. The generator is ~6/5. Six periods minus three generators equals ~4/3. 5/4 is equated to 2 generators minus 1 period. An obvious 7-limit interpretation of the period is 8/7.

Ampersand or Lala-tribiyo family (P8, P5/6): This tempers out Ampersand's comma = 34171875/33554432 = [-25 7 6⟩. The generator is ~16/15, of which six make ~3/2. 5/4 is equated to 1 octave minus 7 generators. If the generator is also equated to ~15/14, and three generators to ~11/9, one gets the Miracle temperament.

Kleismic or Tribiyo family (P8, P12/6): The kleismic family of temperaments tempers out the kleisma [-6 -5 6⟩ = 15625/15552, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. 5/4 is equated to 5 generators minus 1 octave. The kleismic family includes 15, 19, 34, 49, 53, 72, 87 and 140 EDOs among its possible tunings.

Orson, semicomma or Lasepyo family (P8, P12/7): The semicomma (also known as Fokker's comma), 2109375/2097152 = [-21 3 7⟩, is tempered out by the members of the semicomma family. Its generator is ~75/64, seven of which equals ~3/1. 5/4 is equated to 1 octave minus 3 generators. This temperament doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as ~7/6, leading to Orwell or Sepru temperament.

Wesley or Lasepyobi family (P8, ccP4/7): This tempers out the wesley comma, [-13 -2 7⟩ = 78125/73728. The generator is ~125/96 = ~412¢. Seven generators equals a double-compound 4th of ~16/3. 5/4 is equated to 1 octave minus 2 generators. An obvious 7-limit interpretation of the generator is 9/7, leading to the Lasepru temperament. An obvious 3-limit interpretation of the generator is 81/64, implying 29EDO.

Sensipent or Sepgu family (P8, ccP5/7): The sensipent (sensi) family tempers out the sensipent comma, [2 9 -7⟩ (78732/78125), also known as the medium semicomma. Its generator is ~162/125 = ~443¢. Seven generators equals a double-compound 5th of ~6/1. 5/4 is equated to 9 generators minus 3 octaves. Tunings include 8EDO, 19EDO, 46EDO, and 65EDO. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzo temperament.

Vishnuzmic or Sasepbigu family (P8/2, P4/7): This tempers out the vishnuzma, [23 6 -14⟩, or the amount by which seven chromatic semitones (25/24) fall short of a perfect fourth (4/3), or (4/3)/(25/24)^7. The period is ~[-11 -3 7⟩ and the generator is ~25/24. 5/4 is equated to 1 period minus 3 generators.

Unicorn or Laquadbigu 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 Saquadbigu family (P8, ccP5/8): The würschmidt (or wuerschmidt) family tempers out the Würschmidt comma, 393216/390625 = [17 1 -8⟩. Würschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a double-compound perfect 5th); that is, (5/4)^8 * (393216/390625) = 6. It tends to generate the same MOSs as magic temperament, but is tuned slightly more accurately. Both 31EDO and 34EDO can be used as würschmidt tunings, as can 65EDO, which is quite accurate.

Escapade or Sasa-tritrigu family (P8, P4/9): This tempers out the escapade comma, [32 -7 -9⟩, which is the difference between nine just major thirds and seven just fourths. The generator is ~[-14 3 4⟩ = ~55¢ and nine of them equal ~4/3. Seven of them equal ~5/4, thus two of them equal ~16/15. An obvious 11-limit interpretation of the generator is 33/32, leading to the Trisa-tritrilu temperament.

Shibboleth or Tritriyo family (P8, ccP4/9): This tempers out the shibboleth comma, [-5 -10 9⟩ = 1953125/1889568. Nine generators of ~6/5 equal a double compound 4th of ~16/3. 5/4 is equated to 3 octaves minus 10 generators.

Mabila or Sasa-quinbigu family (P8, c⁴P4/10): The sycamore 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 Laleyo family (P8, P5/11): The sycamore family tempers out the sycamore comma, [-16 -6 11⟩ = 48828125/47775744, which is the amount by which five stacked chromatic semitones, 25/24, exceed 6/5, and hence also the amount six exceeds 5/4. Eleven of these generators equals ~3/2.

Quartonic or Saleyo 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.

Ditonmic or Lala-theyo family (P8, c⁴P4/13): This tempers out the ditonma, [-27 -2 13⟩ = 1220703125/1207959552. Thirteen ~[-12 -1 6⟩ generators of about 407¢ equals a quadruple-compound 4th. 5/4 is equated to 1 octave minus 2 generators. An obvious 3-limit interpretation of the generator is 81/64, which implies 53EDO, which is a good tuning for this high-accuracy family of temperaments.

Luna or Sasa-quintrigu family (P8, ccP4/15): This tempers out the luna comma, [38 -2 -15⟩ = 274877906944/274658203125. The generator is ~[18 -1 -7⟩ = ~193¢. Two generators equals ~5/4, and fifteen generators equals a double-compound 4th of ~16/3.

Minortonic or Trila-segu family (P8, ccP5/17): This tempers out the minortone comma, [-16 35 -17⟩. The head of the family is minortonic temperament, with a generator of a minor tone (~10/9). Seventeen generators equals a double-compound 5th = ~6/1. 5/4 is equated to 35 generators minus 5 octaves.

Maja or Saseyo family (P8, c⁶P4/17): This tempers out the maja comma, [-3 -23 17⟩ = 762939453125/753145430616. The generator is ~162/125 = ~453¢. Seventeen generators equals a sextuple-compound 4th. 5/4 is equated to 9 octaves minus 23 generators.

Maquila or Trisa-segu family (P8, c⁷P5/17): This tempers out the maquila comma, 562949953421312/556182861328125 = [49 -6 -17⟩. The generator is ~512/375 = ~535¢. Seventeen generators equals a septuple-compound 5th. 5/4 is equated to 3 octaves minus 6 generators. An obvious 11-limit interpretation of the generator is 11/8, leading to the Lala-selo temperament. However, Lala-selo isn't nearly as accurate as Trisa-segu.

Gammic or Laquinquadyo family (P8, P5/20): The gammic family tempers out the gammic comma, [-29 -11 20⟩. Nine generators of about 35¢ equals ~6/5, eleven 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 Ru clan (P8, P5): This clan tempers out the Archytas comma, 64/63. It equates 7/4 with 16/9. The clan consists of rank two temperaments, and should not be confused with the archytas family of rank three temperaments. Its best downward extension is superpyth.

Trienstonic or Zo clan (P8, P5): This clan tempers out the septimal third-tone 28/27, a low-accuracy temperament that equates 7/6 with 9/8, and 7/4 with 27/16.

Harrison or Laru 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 Sasaru clan (P8, P5): This clan tempers out the garischisma, [25 -14 0 -1⟩ = 33554432/33480783. It equates 8/7 to two apotomes ([-11 7⟩ = 2187/2048) and 7/4 to a double-diminished 8ve [23 -14⟩. This clan includes vulture, newt, garibaldi, sextile, and satin.

Leapfrog or Sasazo 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.

Slendro (Semaphore) or Zozo clan (P8, P4/2): This clan tempers out the slendro diesis, 49/48. Its generator is ~8/7 or ~7/6. Its best downward extension is godzilla. See also Semaphore.

Laruru 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 Sagugu temperament and the Jubilismic or Biruyo temperament.

Parahemif or Sasa-zozo 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 Lulu temperament.

Gamelismic or Latrizo clan (P8, P5/3): This clan tempers out the gamelisma, [-10 1 0 3⟩ = 1029/1024. Three ~8/7 generators equals a 5th. 7/4 is equated to an 8ve minus a generator. Five generators is slightly flat of 2/1, making this a cluster temperament. See also Sawa and Lasepzo.; 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.

Trizo 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 Latrizo temperament.

Triru 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.

Lee or Latriru clan (P8, P11/3): This clan tempers out the Latriru comma, [-9 11 0 -3⟩ = 177147/175616. Generator = ~112/81 = ~566¢. Three generators equals ~8/3. 7/4 is equated to 11 generators minus 5 octaves. An obvious 2.3.5.7 interpretation of the generator is 7/5, leading to the liese temperament, which is a weak extension of Meantone.

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

Buzzardismic or Saquadru 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.

Skwares or Laquadru 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.

Cloudy or Laquinzo 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 Sawa family, ~3/2 is not equated with three-fifths of an octave, resulting in very small intervals.

Bleu or Quinru 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.

Saquinzo 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.

Lasepzo 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 Sawa and Latrizo.

Septiness or Sasasepru 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.

Sepru 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.

Tritrizo 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 Biruyo Nowa clan (P8/2, M3): This clan tempers out the jubilisma, 50/49, which is the difference between 10/7 and 7/5. The 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 Rurugu 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 Zozoquingu 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-aquiniyo 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-aquadbigu Nowa clan (P8, m6/3): This clan tempers out the vorwell comma, [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-atriyo 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 nowa major 3rd = ~5/4.

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

Quince or Lasepzo-agugu 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-aquadyo 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 Rutribiyo 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 Zozoyo 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 noca major 6th = ~5/3.

Gariboh or Triru-aquinyo 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-aquadyo 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-atrigu 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 noca minor seventh = ~9/5.

Betelgeuse or Satritrizo-agugu 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-asepgu 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 equals ~63/5, seven equals ~243/7, and twelve equals ~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 Zogu 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 Ruruyo temperaments: These temper out the greenwoodma, [-3 4 1 -2⟩ = 405/392.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Akjaysmic or Trisa-seprugu 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 Lawa family, ~3/2 is not equated with four-sevenths of an octave, resulting in small intervals.

Landscape or Trizogugu 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 Gu rank three family (P8, P5, ^1): These are the rank three temperaments tempering out the didymus or meantone comma, 81/80.

Diaschismic or Sagugu rank three family (P8/2, P5, /1): These are the rank three temperaments tempering out the dischisma, [11 -4 -2⟩ = 2048/2025. The half-octave period is ~45/32.

Porcupine or Triyo rank three family (P8, P4/3, /1): These are the rank three temperaments tempering out the porcupine comma or maximal diesis, [1 -5 3⟩ = 250/243. In the pergen, P4/3 is ~10/9.

Kleismic or Tribiyo rank three family (P8, P12/6, /1): These are the rank three temperaments tempering out the kleisma, [-6 -5 6⟩ = 15625/15552. In the pergen, P12/6 is ~6/5.

Families defined by a 2.3.7 (za) comma

Every za or 7-limit-no-fives comma defines a rank-3 family, thus every comma in the list of rank-two 2.3.7 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a yaza or 7-limit temperament in which one of the generators is ~5/1. This generator can be reduced to ~5/4, which may be reduced further to ~81/80. Hence in all the pergens below, ^1 = ~81/80. An additional 5-limit or 11-limit comma creates a yazala or 11-limit temperament, and so forth. All these examples are yaza:

Archytas or Ru family (P8, P5, ^1): Archytas temperament tempers out 64/63. This comma equates every 7-limit interval to some 3-limit interval. If 81/80 were tempered out too, the otonal tetrad 4:5:6:7 would be identified with the dominant seventh chord.

Garischismic or Sasaru family (P8, P5, ^1): A garischismic temperament is one which tempers out the garischisma, [25 -14 0 -1⟩ = 33554432/33480783.

Semiphore or Zozo family (P8, P4/2, ^1): Semiphore temperament tempers out 49/48 and thereby identifies the septimal minor third 7/6 with the septimal whole tone 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like "semi-fourth". See also Semaphore and Slendro.

Gamelismic or Latrizo family (P8, P5/3, ^1): Not to be confused with the gamelismic clan of rank two temperaments, the gamelismic family are those rank three temperaments which temper out the gamelisma, [-10 1 0 3⟩ = 1029/1024. In the pergen, P5/3 is ~8/7.

Stearnsmic or Latribiru family (P8/2, P4/3, ^1): Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6⟩ = 118098/117649. In the pergen, P8/2 is ~343/243 and P4/3 is ~54/49.

Families defined by a 2.3.5.7 (yaza) comma

Marvel or Ruyoyo family (P8, P5, ^1): The head of the marvel family is marvel, which tempers out [-5 2 2 -1⟩ = 225/224. It 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 Zotrigu family (P8, P5, ^1): Starling tempers out the septimal semicomma or starling comma [1 2 -3 1⟩ = 126/125, the difference between three 6/5s plus one 7/6, and an octave. 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 Zozoyo 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 Ruruyo 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 Lazoyoyo 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 Zotriyo 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-agu family (P8, P5, ^1): These temper out [6 3 -1 -3⟩ = 1728/1715. In the pergen, ^1 = ~64/63.

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

Ragisma or Zoquadyo family (P8, P5, ^1): The 7-limit rank three microtemperament which tempers out the ragisma, [-1 -7 4 1⟩ = 4375/4374, extends to various higher limit rank three temperaments such as thor. These are not by any means all microtemperaments, but those which are not highly accurate are probably best discussed under another heading. In the pergen, ^1 = ~81/80.

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

Horwell or Lazoquinyo family (P8, P5, ^1): The horwell family of rank three temperaments tempers out the horwell comma, [-16 1 5 1⟩ = 65625/65536. In the pergen, ^1 = ~81/80.

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

Mint or Rugu family (P8, P5, ^1): The mint temperament is a low complexity, high error temperament, tempering out the septimal quarter-tone 36/35, equating 7/6 with 6/5, and 5/4 with 9/7. In the pergen, ^1 = ~81/80 or ~64/63.

Septisemi or Zogu family (P8, P5, ^1): These are very low 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 Biruyo family (P8/2, P5, ^1): Jubilismic temperament tempers out 50/49 and thereby equates the two septimal tritones, 7/5 and 10/7. This is the half-octave period. In the pergen, ^1 = ~81/80.

Cataharry or Labirugu family (P8, P4/2, ^1): Cataharry temperaments temper out the cataharry comma, [-4 9 -2 -2⟩ = 19683/19600. In the pergen, half a 4th is ~81/70, and ^1 = ~81/80.

Breed or Bizozogu family (P8, P5/2, ^1): Breed is a 7-limit microtemperament which tempers out [-5 -1 -2 4⟩ = 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749EDO will certainly do the trick. Breed has generators of 2/1, a 49/40-cum-60/49 neutral third, and ~64/63.

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

Porwell or Sarurutrigu family (P8, P5, ^m3/2): The porwell family of rank three temperaments tempers out the porwell comma, [11 1 -3 -2⟩ = 6144/6125. Two ~35/32 generators equal the pergen's upminor 3rd of ~6/5.

Octagar or Rurutriyo family (P8, P5, ^m6/2): The octagar family of rank three temperaments tempers out the octagar comma, [5 -4 3 -2⟩ = 4000/3969. Two ~80/63 generators equal the pergen's upminor 6th of ~8/5.

Hemimean or Zozoquingu family (P8, P5, vM3/2): The hemimean family of rank three temperaments tempers out the hemimean comma, [6 0 -5 2⟩ = 3136/3125. Two ~28/25 generators equal the pergen's downmajor 3rd of ~5/4.

Wizmic or Quinzo-ayoyo family (P8, P5, vm7/2): A wizmic temperament is one which tempers out the wizma, [-6 -8 2 5⟩ = 420175/419904. Two ~324/245 generators equal the pergen's downminor 7th of ~7/4.

Landscape or Trizogugu family (P8/3, P5, ^1): The 7-limit rank three microtemperament which tempers out the lanscape comma, [-4 6 -6 3⟩ = 250047/250000, extends to various higher limit rank three temperaments such as tyr and odin. In the pergen, the third-octave period is ~63/50, and ^1 = ~81/80.

Gariboh or Triru-aquinyo 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-atriyo family (P8, P5, vm6/3): The canou family of rank three temperaments tempers out the canousma, [4 -14 3 4⟩ = 4802000/4782969. Three ~81/70 generators equal the pergen's downminor 6th of ~14/9.

Dimcomp or Quadruyoyo family (P8/4, P5, ^1): The dimcomp family of rank three temperaments tempers out the dimcomp comma, [-1 -4 8 -4⟩ = 390625/388962. In the pergen, the quarter-octave period is ~25/21, and ^1 = ~81/80.

Mirkwai or Quinru-aquadyo family (P8, P5, c^M7/4): The mirkwai family of rank three temperaments tempers out the mirkwai comma, [0 3 4 -5⟩ = 16875/16807. Four ~7/5 generators equal the pergen's compound upmajor 7th of ~27/7.

Temperaments defined by an 11-limit comma

Ptolemismic or Luyoyo 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 Lologu 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 Lorugugu 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 Lulu 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 Saluzo 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-agu 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 Salulugu 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 Salururu 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 Laquadlo 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 Lozoyo 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 Luzozogu 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 Lururuyo 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 Loloruyoyo 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 Bilorugu 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 both full groups and subgroups tempering out a given comma. For some commas, looking at the full commatic realm seems the best approach to discussing associated temperaments.

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

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

The Archipelago or Bithogu: 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-agu: 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-aruru: This is the commatic realm of the 11-limit comma 65536/65219 = [16 0 0 -2 -3⟩, the orgonisma.

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

The Quartercache or Saquinlu-azo: 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.

High badness 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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