Tour of regular temperaments: Difference between revisions

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.

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]], {{monzo| 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, {{monzo| -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]], {{monzo| -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 [[41-comma|Pythagorean countercomma]], {{monzo| 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]], {{monzo| -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 lists|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 [[12edo|12-]], [[19edo|19-]], [[31edo|31-]], [[43edo|43-]], [[50edo|50-]], [[55edo|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 {{monzo| -15 8 1 }} ([[32805/32768]]), which is the amount by which the Pythagorean comma exceeds the syntonic comma. The 5-limit version of the temperament is a [[microtemperament]] which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity; whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4. Tunings include [[12edo|12-]], [[29edo|29-]], [[41edo|41-]], [[53edo|53-]], and [[118edo]]. Its color name is Layoti.  

; [[Mavila family]] (P8, P5)

: This tempers out the mavila comma, {{monzo| -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 [[9edo|9-]], [[16edo|16-]], [[23edo|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]], {{monzo| 11 -4 -2 }} (2048/2025), such that two classic major thirds and a [[81/64|Pythagorean major third]] stack to an octave (i.e. {{nowrap| (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 [[12edo|12-]], [[22edo|22-]], [[34edo|34-]], [[46edo|46-]], [[56edo|56-]], [[58edo|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 {{nowrap| 10/9 {{=}} ~250{{c}} }}, 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, {{monzo| 16 -13 2 }} (1638400/1594323). Its generator is {{nowrap| ~729/640 {{=}} ~247{{c}} }}, 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 {{monzo| 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 "[http://www.tcherepnin.com/alex/basic_elem1.htm#9step Tcherepnin's scale]" ([[3L 6s]]). Its color name is Triguti.  

; [[Misty family]] (P8/3, P5)

: The misty family tempers out the [[misty comma]] of {{monzo| 26 -12 -3 }}, the difference between the [[Pythagorean comma]] and a stack of three [[schisma]]s. 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 {{monzo| 1 -5 3 }} ([[250/243]]), the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. 5/4 is equated to 1 octave minus 5 generators. Some porcupine temperaments include [[15edo|15-]], [[22edo|22-]], [[37edo|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]], {{monzo| 39 -29 3 }}. The generator is {{nowrap| ~59049/40960 ({{monzo| -13 10 -1 }}) {{=}} 633{{c}} }}, or its octave inverse {{nowrap| ~81920/59049 {{=}} 567{{c}} }}. 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 {{nowrap| 81/56 {{=}} 639{{c}} }}, a much simpler ratio which leads to the [[Tour of Regular Temperaments #Latriruti clan (P8, P11/3)|Latriruti clan]]. An obvious 13-limit interpretation is {{nowrap| 13/9 {{=}} 637¢ }}, an even simpler ratio implying the [[Tour of Regular Temperaments #Satrithoti clan (P8, P11/3)|Satrithoti clan]].

; [[Diminished family]] (P8/4, P5)

: The diminished family tempers out the major diesis or diminished comma, {{monzo| 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 {{monzo| 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]], {{monzo| -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 {{monzo| 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]], {{monzo| 11 -1 -4 }} (2048/1875). Its generator is {{nowrap| ~5/4 {{=}} ~421{{c}} }}, four of which make ~8/3. Its color name is Saquadguti.  

; [[Vulture family]] (P8, P12/4)

: This tempers out the [[vulture comma]], {{monzo| 24 -21 4 }}. Its generator is {{nowrap| ~320/243 {{=}} ~475{{c}} }}, 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 ({{monzo| -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{{c}}, 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{{c}} 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 ({{monzo| -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 ({{monzo| 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]], {{monzo| 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]], {{monzo| 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 ({{monzo| 9 -13 5 }}). The generator is {{nowrap| 243/200 {{=}} ~339.5{{c}} }}, 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 {{monzo| -10 -1 5 }} (3125/3072), known as the [[magic comma]] or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/1. The generator is itself an approximate 5/4. The magic family includes [[16edo|16-]], [[19edo|19-]], [[22edo|22-]], [[25edo|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]], {{monzo| -1 -14 10 }} (9765625/9565938). The period is ~4374/3125 ({{monzo| 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 ({{monzo| 47 -15 -10 }}). The period is ~524288/455625 ({{monzo| 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 ({{monzo| 19 10 -15 }}). The period is {{nowrap| ~144/125 {{=}} 240{{c}} }}. 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 ({{monzo| -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 [[15625/15552|kleisma]], 15625/15552 ({{monzo| -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 [[15edo|15-]], [[19edo|19-]], [[34edo|34-]], [[49edo|49-]], [[53edo|53-]], [[72edo|72-]], [[87edo|87-]] and [[140edo]] among its possible tunings. Its color name is Tribiyoti.  

; [[Semicomma family|Orson or semicomma family]] (P8, P12/7)

: The [[semicomma]] (also known as Fokker's comma), 2109375/2097152 ({{monzo| -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|orwell or Sepruti]] temperament.

; [[Wesley family]] (P8, ccP4/7)

: This tempers out the [[wesley comma]], 78125/73728 ({{monzo| -13 -2 7 }}). The generator is {{nowrap| ~125/96 {{=}} ~412{{c}} }}. 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 ({{monzo| 2 9 -7 }}), also known as the medium semicomma. Its generator is {{nowrap| ~162/125 {{=}} ~443{{c}} }}. 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]], {{monzo| 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 ~{{monzo| -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 ({{monzo| -2 13 -8 }}). The generator is {{nowrap| ~250/243 {{=}} ~62{{c}} }} 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 ({{monzo| 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, {{nowrap| (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]], {{monzo| 32 -7 -9 }}, which is the difference between nine just major thirds and seven just fourths. The generator is {{monzo| -14 3 4 }} of ~55{{c}} 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]], {{monzo| 28 -3 -10 }} (268435456/263671875). The generator is {{nowrap| ~512/375 {{=}} ~530{{c}} }}, 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]], {{monzo| -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]], {{monzo| 3 -18 11 }} (390625000/387420489). The generator is {{nowrap| ~250/243 {{=}} ~45{{c}} }}, 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.

: This tempers out the [[lafa comma]], {{monzo| 77 -31 -12 }}. The generator is {{nowrap| ~4982259375/4294967296 {{=}} ~258.6{{c}} }}. Twelve generators equals a twelfth (~3/1). 5/4 is equated to 7 octaves minus 31 generators. Its color name is Tribisa-quadtriguti.

: This tempers out the [[ditonma]], {{monzo| -27 -2 13 }} (1220703125/1207959552). Thirteen ~{{monzo| -12 -1 6 }} generators of about 407{{c}} 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]], {{monzo| 38 -2 -15 }} (274877906944/274658203125). The generator is ~{{monzo| 18 -1 -7 }} at ~193{{c}}. 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]], {{monzo| -68 18 17 }}. The generator is {{nowrap| ~16/15 {{=}} ~111.9{{c}} }}. 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]], {{monzo| -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]], {{monzo| -3 -23 17 }} (762939453125/753145430616). The generator is {{nowrap| ~162/125 {{=}} ~453{{c}} }}. 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]], {{monzo| 49 -6 -17 }} (562949953421312/556182861328125). The generator is {{nowrap| ~512/375 {{=}} ~535{{c}} }}. 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]], {{monzo| -29 -11 20 }}. Nine generators of about 35{{c}} 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 lists|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]], {{monzo| -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]], {{monzo| 25 -14 0 -1 }} (33554432/33480783). It equates 8/7 to two apotomes ({{monzo| -11 7 }}, 2187/2048) and 7/4 to a double-diminished octave {{monzo| 23 -14 }}. This clan includes [[vulture family #Vulture|vulture]], [[breedsmic temperaments #Newt|newt]], [[schismatic family #Garibaldi|garibaldi]], [[landscape microtemperaments #Sextile|sextile]], and [[canousmic temperaments #Satin|satin]]. Its color name is Sasaruti.  

; Sasazoti clan (P8, P5)

: This clan tempers out the [[leapfrog comma]], {{monzo| 21 -15 0 1 }} (14680064/14348907). It equates 7/6 to two apotomes and 7/4 to double augmented fifth. This clan includes [[hemifamity temperaments #Leapday|leapday]], [[sensamagic clan #Leapweek|leapweek]] and [[diaschismic family #Srutal|srutal]].  

; Laruruti clan (P8/2, P5)

: This clan tempers out the Laruru comma, {{monzo| -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]], {{monzo| 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{{c}} generator is 11/9, leading to the Luluti temperament.

; Triruti clan (P8/3, P5)

: This clan tempers out the Triru comma, {{monzo| -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{{c}} period is 5/4, leading to the [[augmented]] temperament.

; [[Gamelismic clan]] (P8, P5/3)

: This clan tempers out the [[gamelisma]], {{monzo| -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, {{monzo| -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{{c}} 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]], {{monzo| -9 11 0 -3 }} (177147/175616). The generator is {{nowrap| ~112/81 {{=}} ~566{{c}} }}, 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]], {{monzo| 1 10 0 -6 }} (118098/117649). The period is {{nowrap| ~486/343 {{=}} ~600{{c}} }}. The generator is {{nowrap| ~9/7 {{=}} ~434{{c}} }}, or alternatively one period minus ~9/7, which equals {{nowrap| ~54/49 {{=}} ~166{{c}} }}. 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]], {{monzo| -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.  

: This clan tempers out the [[buzzardsma]], {{monzo| 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 family #Septimal vulture|vulture]] temperament, which is in the vulture family.  

; [[Cloudy clan]] (P8/5, P5)

: This clan tempers out the [[cloudy comma]], {{monzo| -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]], {{monzo| 3 7 0 -5 }} (17496/16807). The ~54/49 generator is about 139{{c}}. 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, {{monzo| 5 -12 0 5 }} (537824/531441). Its generator is {{nowrap| ~243/196 {{=}} ~380{{c}} }}. 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.

: This clan tempers out the Lasepzo comma {{monzo| -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{{c}} sharp of 3/2, and five generators is ~15{{c}} 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]] {{monzo| 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, {{monzo| 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 [[septimal ennealimma|septiennealimma]], {{monzo| -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]] ({{monzo| -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]] {{monzo| -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]] {{monzo| -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]] ({{monzo| 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]] ({{monzo| 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 {{nowrap| ~8/7 {{=}} ~202{{c}} }} 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]], {{monzo| 6 0 -5 2 }} (3136/3125). The generator is {{nowrap| ~28/25 {{=}} ~194{{c}} }}. 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]], {{monzo| -20 0 5 3 }} (1071875/1048576). The generator is {{nowrap| ~175/128 {{=}} ~527{{c}} }}. 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]]), {{monzo| 27 0 -8 -3 }} (134217728/133984375). The generator is {{nowrap| ~1024/875 {{=}} ~272{{c}} }}. 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]], {{monzo| -21 0 3 5 }} (2100875/2097152). The generator is {{nowrap| ~256/245 {{=}} ~77{{c}} }}. 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]], {{monzo| 22 0 -1 -7 }} (4194304/4117715). The generator is {{nowrap| ~8/7 {{=}} ~227{{c}} }} and seven of them equals ~5/2. Its color name is Sasepru-aguti Nowa.  

: This clan tempers out the [[quince comma]], {{monzo| -15 0 -2 7 }} (823543/819200). The generator is {{nowrap| ~343/320 {{=}} ~116{{c}} }}. 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]], {{monzo| 16 0 4 -9 }} (40960000/40353607). The generator is {{nowrap| ~49/40 {{=}} ~357{{c}} }}. 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.  

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]] {{monzo| 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]] {{monzo| 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]] {{monzo| 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]], {{monzo| 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]] {{monzo| 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]] {{monzo| 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), {{monzo| 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.

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]], {{monzo| -3 4 1 -2 }} (405/392). Its color name is Ruruyoti.  

; [[Keegic temperaments]]

: Keegic rank-2 temperaments temper out the [[keega]], {{monzo| -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]], {{monzo| -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]], {{monzo| 1 -3 -2 3 }} (686/675). Its color name is Trizo-aguguti.  

; [[Keemic temperaments]]

: Keemic rank-2 temperaments temper out the [[keema]], {{monzo| -5 -3 3 1 }} (875/864). Its color name is Zotriyoti.  

; Secanticorn temperaments

: Secanticorn rank-2 temperaments temper out the [[secanticornisma]], {{monzo| -3 11 -5 -1 }} (177147/175000). Its color name is Laruquinguti.  

; Nuwell temperaments

: Nuwell rank-2 temperaments temper out the [[nuwell comma]], {{monzo| 1 5 1 -4 }} (2430/2401). Its color name is Quadru-ayoti.  

; Mermismic temperaments

: Mermismic rank-2 temperaments temper out the [[mermisma]], {{monzo| 5 -1 7 -7 }} (2500000/2470629). Its color name is Sepruyoti.  

; Negricorn temperaments

: Negricorn rank-2 temperaments temper out the [[negricorn comma]], {{monzo| 6 -5 -4 4 }} (153664/151875). Its color name is Saquadzoguti.  

; Tolermic temperaments

: These temper out the [[tolerma]], {{monzo| 10 -11 2 1 }} (179200/177147). Its color name is Sazoyoyoti.  

; Valenwuer temperaments

: Valenwuer rank-2 temperaments temper out the [[valenwuer comma]], {{monzo| 12 3 -6 -1 }} (110592/109375). Its color name is Sarutribiguti.  

; [[Mirwomo temperaments]]

: Mirwomo rank-2 temperaments temper out the [[mirwomo comma]], {{monzo| -15 3 2 2 }} (33075/32768). Its color name is Labizoyoti.  

; Catasyc temperaments

: Catasyc rank-2 temperaments temper out the [[catasyc comma]], {{monzo| -11 -3 8 -1 }} (390625/387072). Its color name is Laruquadbiyoti.  

; Compass temperaments

: Compass rank-2 temperaments temper out the [[compass comma]], {{monzo| -6 -2 10 -5 }} (9765625/9680832). Its color name is Quinruyoyoti.  

; Trimyna temperaments

: Trimyna rank-2 temperaments temper out the [[trimyna comma]], {{monzo| -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, {{monzo| 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]], {{monzo| 5 -4 3 -2 }} (4000/3969). Its color name is Rurutriyoti.  

; [[Orwellismic temperaments]]

: Orwellismic rank-2 temperaments temper out [[orwellisma]], {{monzo| 6 3 -1 -3 }} (1728/1715). Its color name is Triru-aguti.  

; Mynaslendric temperaments

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

; [[Mistismic temperaments]]

: Mistismic rank-2 temperaments temper out the [[mistisma]], {{monzo| 16 -6 -4 1 }} (458752/455625). Its color name is Sazoquadguti.  

; [[Varunismic temperaments]]

: Varunismic rank-2 temperaments temper out the [[varunisma]], {{monzo| -9 8 -4 2 }} (321489/320000). Its color name is Labizoguguti.  

; [[Marvel temperaments]]

: Marvel rank-2 temperaments temper out the [[marvel comma]], {{monzo| -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]], {{monzo| -1 -4 8 -4 }} (390625/388962). Its color name is Quadruyoyoti.  

; [[Cataharry temperaments]]

: Cataharry rank-2 temperaments temper out the [[cataharry comma]], {{monzo| -4 9 -2 -2 }} (19683/19600). Its color name is Labiruguti.  

; [[Canousmic temperaments]]

: Canousmic rank-2 temperaments temper out the [[canousma]], {{monzo| 4 -14 3 4 }} (4802000/4782969). Its color name is Saquadzo-atriyoti.  

; [[Triwellismic temperaments]]

: Triwellismic rank-2 temperaments temper out the [[triwellisma]], {{monzo| 1 -1 -7 6 }} (235298/234375). Its color name is Tribizo-asepguti.  

; [[Hemimage temperaments]]

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

; [[Hemifamity temperaments]]

: Hemifamity rank-2 temperaments temper out the [[hemifamity comma]], {{monzo| 10 -6 1 -1 }} (5120/5103). Its color name is Saruyoti.  

; [[Parkleiness temperaments]]

: Parkleiness rank-2 temperaments temper out the [[parkleiness comma]], {{monzo| 7 7 -9 1 }} (1959552/1953125). Its color name is Zotritriguti.  

; [[Porwell temperaments]]

: Porwell rank-2 temperaments temper out the [[porwell comma]], {{monzo| 11 1 -3 -2 }} (6144/6125). Its color name is Sarurutriguti.  

; [[Cartoonismic temperaments]]

: Cartoonismic rank-2 temperaments temper out the [[cartoonisma]], {{monzo| 12 -3 -14 9 }} (165288374272/164794921875). Its color name is Satritrizo-asepbiguti.  

; [[Hemfiness temperaments]]

: Hemfiness rank-2 temperaments temper out the [[hemfiness comma]], {{monzo| 15 -5 3 -5 }} (4096000/4084101). Its color name is Saquinru-atriyoti.  

; [[Hewuermera temperaments]]

: Hewuermera rank-2 temperaments temper out the [[hewuermera comma]], {{monzo| 16 2 -1 -6 }} (589824/588245). Its color name is Satribiru-aguti.  

; [[Lokismic temperaments]]

: Lokismic rank-2 temperaments temper out the [[lokisma]], {{monzo| 21 -8 -6 2 }} (102760448/102515625). Its color name is Sasa-bizotriguti.  

; Decovulture temperaments

: Decovulture rank-2 temperaments temper out the [[decovulture comma]], {{monzo| 26 -7 -4 -2 }} (67108864/66976875). Its color name is Sasabiruguguti.  

 

 

 

 

; [[Metric microtemperaments]]

 

 

 

 

 

; [[Ragismic microtemperaments]]

: Ragismic rank-2 temperaments temper out the [[ragisma]], {{monzo| -1 -7 4 1 }} (4375/4374). Its color name is Zoquadyoti.

 

 

; [[Landscape microtemperaments]]

: Landscape rank-2 temperaments temper out the [[landscape comma]], {{monzo| -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.

Even less familiar than rank-2 temperaments are the [[rank-3 temperament]]s, 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.

 

 

 

 

 

 

; [[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, {{monzo| 25 -14 0 -1 }} (33554432/33480783). Its color name is Sasaruti.  

; Laruruti clan (P8/2, P5)

: This clan tempers out the Laruru comma, {{monzo| -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, {{monzo| -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, {{monzo| 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]], {{monzo| -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, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Ruyoyoti.  

; [[Starling family]] (P8, P5, ^1)

: Starling tempers out the [[starling comma]] a.k.a. septimal semicomma, {{monzo| 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, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Zotriguti.  

; [[Sensamagic family]] (P8, P5, ^1)

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

; Greenwoodmic family (P8, P5, ^1)

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