Tour of regular temperaments: Difference between revisions

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 family|Limma or Sawa family]] (P8/5, ^1)

: This family tempers out the limma, {{Monzo|8 -5 0}} = 256/243, which implies [[5edo]].

; [[Apotome family|Apotome or Lawa family]] (P8/7, ^1)

: This family tempers out the apotome, {{Monzo|-11 7 0}} = 2187/2048, which implies [[7edo]].

; [[Compton family|Compton or Lalawa family]] (P8/12, ^1)

: This tempers out the [[Pythagorean comma]], 531441/524288 = {{Monzo|-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 family|Countercomp or Wa-41 family]] (P8/41, ^1)

: This family tempers out the [[41-comma|Pythagorean countercomma]], {{Monzo| 65 -41 }}, which implies [[41edo]].

; [[Mercator family|Mercator or Wa-53 family]] (P8/53, ^1)

: This family tempers out the [[Mercator's comma]], {{Monzo| -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 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|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 [[12edo|12]], [[19edo|19]], [[31edo|31]], [[43edo|43]], [[50edo|50]], [[55edo|55]] and [[81edo|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.

: 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|118]] EDOs.

; [[Pelogic family|Pelogic or Layobi family]] (P8, P5)

: This tempers out the pelogic 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, 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 [[9edo|9]], [[16edo|16]], [[23edo|23]], and [[25edo|25]] EDOs.

; [[Father family|Father or Gubi family]] (P8, P5)

: This tempers out [[16/15]], the just diatonic semitone, and equates 5/4 with 4/3.

; [[Diaschismic family|Diaschismic or Sagugu family]] (P8/2, P5)

: The diaschismic family tempers out the [[diaschisma]], {{Monzo|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 [[12edo|12]], [[22edo|22]], [[34edo|34]], [[46edo|46]], [[56edo|56]], [[58edo|58]] and [[80edo|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|22EDO]] is an excellent pajara tuning.

; [[Bug family|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 family|Immunity or Sasa-yoyo family]] (P8, P4/2)

: This tempers out the immunity comma, {{Monzo|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 family|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|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|10EDO]], and [[17edo|17EDO]]. An obvious 2.3.11 interpretation of the generator is ~11/9, which leads to Rastmic aka Neutral or Lulu.

; [[Augmented family|Augmented or Trigu  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|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 "[http://www.tcherepnin.com/alex/basic_elem1.htm#9step Tcherepnin's scale]" ([[3L 6s]]).

; [[Misty family|Misty or Sasa-trigu 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|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 family|Porcupine or Triyo 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|59]] EDOs. An important 7-limit extension also tempers out 64/63.

; [[Tricot family|Tricot or Quadsa-triyo family]] (P8, P11/3)

: The tricot family tempers out the [[Tricot|tricot comma]], {{Monzo|39 -29 3}}. The generator is ~59049/40960 = {{monzo|-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 [[Tour of Regular Temperaments#Latriru clan (P8, P11/3)|Latriru clan]]. An obvious 13-limit interpretation is 13/9 = 637¢, an even simpler ratio implying the [[Tour of Regular Temperaments #Satritho clan (P8, P11/3)|Satritho clan]].

; [[Dimipent family|Dimipent or Quadgu family]] (P8/4, P5)

: The dimipent (or 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|12EDO]]. 5/4 is equated to 1 fifth minus 1 period.

; [[Undim family|Undim or Trisa-quadgu 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.  

; [[Negri|Negri or Laquadyo family]] (P8, P4/4)  

: This tempers out the [[negri comma]], {{Monzo|-14 3 4}}. Its generator is ~16/15, four of which make ~4/3. 5/4 is equated to 3 generators.  

; [[Tetracot family|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 {{Monzo|5 -9 4}} (20000/19683), the minimal diesis or [[tetracot comma]]. 5/4 is equated to 9 generators minus an octave. [[7edo|7EDO]] can also be considered a tetracot tuning, as can [[20edo|20EDO]], [[27edo|27EDO]], [[34edo|34EDO]], and [[41edo|41EDO]].

; [[Smate family|Smate or Saquadgu family]] (P8, P11/4)

: This tempers out the symbolic comma, 2048/1875 = {{Monzo|11 -1 -4}}. Its generator is ~5/4 = ~421¢, four of which make ~8/3.

; [[Vulture family|Vulture or Sasa-quadyo family]] (P8, P12/4)

: This tempers out the [[vulture comma]], {{Monzo|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 family|Pental or Trila-quingu family]] (P8/5, P5)

: This tempers out the pental comma, 847288609443/838860800000 = {{Monzo|-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 family|Ripple or Quingu 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|12EDO]] is about as accurate as it can be.

; [[Passion family|Passion or Saquingu 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]].

; [[Quintaleap family|Quintaleap or Trisa-quingu 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]]. An obvious 17-limit interpretation of the generator is ~18/17, which makes saquinso.

; [[Quindromeda family|Quindromeda or Quinsa-quingu 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]]. An obvious 17-limit interpretation of the generator is ~18/17, which makes saquinso.

; [[Amity family|Amity or Saquinyo family]] (P8, P11/5)

: This tempers out the [[amity comma]], 1600000/1594323 = {{Monzo|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 family|Magic or Laquinyo 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|41]] EDOs among its possible tunings, with the latter being near-optimal.

; [[Fifive family|Fifive or Saquinbiyo 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.  

; [[Quintosec family|Quintosec or Quadsa-quinbigu 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. An obvious 7-limit interpretation of the period is 8/7.

; [[Trisedodge family|Trisedodge or Saquintrigu family]] (P8/5, P4/3)

: This tempers out the trisedodge comma, 30958682112/30517578125 = {{Monzo|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|Ampersand or Lala-tribiyo family]] (P8, P5/6)  

: This tempers out Ampersand's comma = 34171875/33554432 = {{Monzo|-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 family|Kleismic or Tribiyo family]] (P8, P12/6)

: The kleismic family of temperaments tempers out the [[kleisma]] {{Monzo|-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 [[15edo|15]], [[19edo|19]], [[34edo|34]], [[49edo|49]], [[53edo|53]], [[72edo|72]], [[87edo|87]] and [[140edo|140]] EDOs among its possible tunings.

; [[Semicomma family|Orson, semicomma or Lasepyo 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. 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|Orwell or Sepru]] temperament.

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

: This tempers out the wesley comma, {{Monzo|-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|29EDO]].  

; [[Sensipent family|Sensipent or Sepgu family]] (P8, ccP5/7)

: The sensipent (sensi) family tempers out the [[sensipent comma]], {{Monzo|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|8EDO]], [[19edo|19EDO]], [[46edo|46EDO]], and [[65edo|65EDO]]. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzo temperament.  

; [[Vishnuzmic family|Vishnuzmic or Sasepbigu 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.

; [[Unicorn family|Unicorn or Laquadbigu family]] (P8, P4/8)

: This tempers out the unicorn comma, 1594323/1562500 = {{Monzo|-2 13 -8}}. The generator is ~250/243 = ~62¢ and eight of them equal ~4/3.

; [[Würschmidt family|Würschmidt or Saquadbigu family]] (P8, ccP5/8)

: The würschmidt (or wuerschmidt) 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 5th); that is, (5/4)^8 * (393216/390625) = 6. It tends to generate the same MOSs as [[Magic_family|magic temperament]], but is tuned slightly more accurately. Both [[31edo|31EDO]] and [[34edo|34EDO]] can be used as würschmidt tunings, as can [[65edo|65EDO]], which is quite accurate.

; [[Escapade family|Escapade or Sasa-tritrigu 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}} = ~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 family|Shibboleth or Tritriyo family]] (P8, ccP4/9)

: This tempers out the shibboleth comma, {{Monzo|-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 family|Mabila or Sasa-quinbigu family]] (P8, c⁴P4/10)

: The sycamore family tempers out the mabila comma, {{Monzo|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 family|Sycamore or Laleyo 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.

; [[Quartonic family|Quartonic or Saleyo family]] (P8, P4/11)

: The quartonic family tempers out the quartonic comma, {{Monzo|3 -18 11}} = 390625000/387420489. The generator is ~250/243 = ~45¢, seven generators equals ~6/5, and eleven generators equals ~4/3. An obvious 7-limit interpretation of the generator is ~36/35.

; [[Lafa family|Lafa or Tribisa-quadtrigu family]] (P8, P12/12)

: This tempers out the lafa comma, {{Monzo|77 -31 -12}}. The generator is ~4982259375/4294967296 = ~258.6¢. Twelve generators equals a twelfth = ~3/1. 5/4 is equated to 7 octaves minus 31 generators.

; [[Ditonmic family|Ditonmic or Lala-theyo family]] (P8, c⁴P4/13)

: This tempers out the ditonma, {{Monzo|-27 -2 13}} = 1220703125/1207959552. Thirteen ~{{Monzo|-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 family|Luna or Sasa-quintrigu family]] (P8, ccP4/15)

: This tempers out the luna comma, {{Monzo|38 -2 -15}} = 274877906944/274658203125. The generator is ~{{Monzo|18 -1 -7}} = ~193¢. Two generators equals ~5/4, and fifteen generators equals a double-compound 4th of ~16/3.  

; [[Vavoom family|Vavoom or Quinla-seyo family]] (P8, P12/17)

: This tempers out the vavoom comma, {{Monzo|-68 18 17}}. The generator is ~16/15 = ~111.9¢. Seventeen generators equals a twelfth = ~3/1. 5/4 is equated to two octaves minus 18 generators.

; [[Minortonic family|Minortonic or Trila-segu 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 5th = ~6/1. 5/4 is equated to 35 generators minus 5 octaves.

; [[Maja family|Maja or Saseyo family]] (P8, c⁶P4/17)

: This tempers out the maja comma, {{Monzo|-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 family|Maquila or Trisa-segu family]] (P8, c⁷P5/17)

: This tempers out the maquila comma, 562949953421312/556182861328125 = {{Monzo|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 family|Gammic or Laquinquadyo family]] (P8, P5/20)

: The gammic family tempers out the gammic comma, {{Monzo|-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 [[Just_intonation_subgroups|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.

: 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 clan|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's comma|Harrison or Laru clan]] (P8, P5)

: This clan tempers out the Laru comma, {{Monzo|-13 10 0 -1}} =  59049/57344. It equates 7/4 to an augmented 6th. Its best downward extension is [[Meantone family|septimal meantone]].

; [[Garischismic clan|Garischismic or Sasaru 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 8ve {{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]].

: This clan tempers out the Sasazo 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]].

; [[Slendro clan|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, {{Monzo|-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, {{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. An obvious 11-limit interpretation of the ~351¢ generator is 11/9, leading to the Lulu temperament.

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

: This clan tempers out the gamelisma, {{Monzo|-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, {{Monzo|-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, {{Monzo|-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, {{Monzo|-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 clan|Stearnsmic or Latribiru clan]] (P8/2, P4/3)

: This clan temper out the stearnsma, {{Monzo|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'', {{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. This clan includes as a strong extension the [[Vulture family|vulture]] temperament, which is in the vulture family.

; Skwares or Laquadru clan (P8, P11/4)

: This clan tempers out the Laquadru comma, {{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. This clan includes as a strong extension the [[squares]] temperament, which is a weak extension of meantone.

; [[Cloudy clan|Cloudy or Laquinzo 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 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, {{Monzo|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, {{Monzo|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 {{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¢ 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 {{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]].

; Sepru 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.

; [[Tritrizo clan]] (P8/9, P5)

: This clan tempers out the ''[[Septimal ennealimma|septiennealimma]]'' (tritrizo comma), {{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 [[Kleismic family #Novemkleismic|novemkleismic]].

=== Clans defined by a 2.3.11 (ila) comma ===

See also [[subgroup temperaments]].

; Lulubi clan (P8/2, P5)  

; [[Rastmic clan|Rastmic or Lulu 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.

; [[Nexus clan|Nexus or Tribilo Clan]] (P8/3, P4/2)

: This 2.3.11 clan tempers out the [[nexus comma]] {{monzo| -16 -3 0 0 6 }}. Its third-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.

; Alphaxenic or Laquadlo 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 saquadyobi temperament, which is in the comic family.

=== Clans defined by a 2.3.13 (tha) comma ===

See also [[subgroup temperaments]].

; Thuthu 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.

; Threedie or Satritho 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. Three generators equals ~8/3. 13/8 is equated to 7 generators minus three octaves. This clan is related to the Latriru clan.

=== 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 clan|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 clan|Hemimean or Zozoquingu Nowa clan]] (P8, M3/2)

: This clan tempers out the hemimean comma, {{Monzo|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 clan|Mabilismic or Latrizo-aquiniyo Nowa clan]] (P8, cM3/3)

: This clan tempers out the mabilisma, {{Monzo|-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, {{Monzo|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]], {{Monzo|-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 clan|Llywelyn or Sasepru-agu Nowa clan]] (P8, cM3/7)

: This clan tempers out the llywelyn comma, {{Monzo|22 0 -1 -7}} = 4194304/4117715. The generator is ~8/7 = ~227¢ and seven of them equals ~5/2.

; [[Quince clan|Quince or Lasepzo-agugu Nowa clan]] (P8, m6/7)

: This clan tempers out the quince, {{Monzo|-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, {{Monzo|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 clan|Arcturus or Rutribiyo Noca clan]] (P12, M6)  

: This 3.5.7 clan tempers out the Arcturus comma {{Monzo|0 -7 6 -1}} = 15625/15309. The generator is the noca major 6th = ~5/3, and six generators equals ~21/1.

; [[Sensamagic clan|Sensamagic or Zozoyo Noca 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 noca major 6th = ~5/3.

; [[Gariboh clan|Gariboh or Triru-aquinyo Noca 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 noca major 6th = ~5/3.

; [[Mirkwai clan|Mirkwai or Quinru-aquadyo Noca 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 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 {{Monzo|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 {{Monzo|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), {{Monzo|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 temperaments|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 temperaments|Greenwoodmic or Ruruyo temperaments]]

: These temper out the greenwoodma, {{Monzo|-3 4 1 -2}} = 405/392.

; [[Keegic temperaments|Keegic or Trizogu temperaments]]

: Keegic rank-two temperaments temper out the keega, {{Monzo|-3 1 -3 3}} = 1029/1000.

; [[Mint temperaments|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 temperaments|Avicennmic or Zoyoyo temperaments]]

: These temper out the avicennma, {{Monzo|-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, {{Monzo|1 -3 -2 3}} = 686/675.

; [[Keemic temperaments|Keemic or Zotriyo temperaments]]

: Keemic rank-two temperaments temper out the keema, {{Monzo|-5 -3 3 1}} = 875/864.

; Secanticorn or Laruquingu temperaments

: Secanticorn rank-two temperaments temper out the ''secanticornisma'', {{monzo|-3 11 -5 -1}} = 177147/175000.

; Nuwell or Quadru-ayo temperaments

: Nuwell rank-two temperaments temper out the nuwell comma, {{Monzo|1 5 1 -4}} = 2430/2401.

; Mermismic or Sepruyo temperaments

: Mermismic rank-two temperaments temper out the ''mermisma'', {{Monzo|5 -1 7 -7}} = 2500000/2470629.

; Negricorn or Saquadzogu temperaments

: Negricorn rank-two temperaments temper out the ''negricorn'' comma, {{Monzo|6 -5 -4 4}} = 153664/151875.

; Tolermic or Sazoyoyo temperaments

: These temper out the tolerma, {{Monzo|10 -11 2 1}} = 179200/177147.

; Valenwuer or Sarutribigu temperaments

: Valenwuer rank-two temperaments temper out the ''valenwuer'' comma, {{Monzo|12 3 -6 -1}} = 110592/109375.

; [[Mirwomo temperaments|Mirwomo or Labizoyo temperaments]]

: Mirwomo rank-two temperaments temper out the mirwomo comma, {{Monzo|-15 3 2 2}} = 33075/32768.

; Catasyc or Laruquadbiyo temperaments

: Catasyc rank-two temperaments temper out the ''catasyc'' comma, {{Monzo|-11 -3 8 -1}} = 390625/387072.

; Compass or Quinruyoyo temperaments

: Compass rank-two temperaments temper out the compass comma, {{Monzo|-6 -2 10 -5}} = 9765625/9680832.

; Trimyna or Quinzogu temperaments

: The trimyna rank-two temperaments temper out the trimyna comma, {{Monzo|-4 1 -5 5}} = 50421/50000.

; [[Starling temperaments|Starling or Zotrigu temperaments]]

: Starling rank-two temperaments temper out the septimal semicomma or starling comma {{Monzo|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 temperaments|Octagar or Rurutriyo temperaments]]

: Octagar rank-two temperaments temper out the octagar comma, {{Monzo|5 -4 3 -2}} = 4000/3969.

; [[Orwellismic temperaments|Orwellismic or Triru-agu temperaments]]

: Orwellismic rank-two temperaments temper out orwellisma, {{Monzo|6 3 -1 -3}} = 1728/1715.

; Mynaslendric or Sepru-ayo temperaments

: Mynaslendric rank-two temperaments temper out the ''mynaslender'' comma, {{Monzo|11 4 1 -7}} = 829440/823543.

; [[Mistismic temperaments|Mistismic or Sazoquadgu temperaments]]

: Mistismic rank-two temperaments temper out the ''mistisma'', {{Monzo|16 -6 -4 1}} = 458752/455625.

; [[Varunismic temperaments|Varunismic or Labizogugu temperaments]]

: Varunismic rank-two temperaments temper out the varunisma, {{monzo|-9 8 -4 2}} = 321489/320000.

; [[Marvel temperaments|Marvel or Ruyoyo temperaments]]

: Marvel rank-two temperaments temper out {{Monzo|-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, {{Monzo|-1 -4 8 -4}} = 390625/388962.

; [[Cataharry temperaments|Cataharry or Labirugu temperaments]]

: Cataharry rank-two temperaments temper out the cataharry comma, {{Monzo|-4 9 -2 -2}} = 19683/19600.

; [[Canousmic temperaments|Canousmic or Saquadzo-atriyo temperaments]]

: Canousmic rank-two temperaments temper out the canousma, {{Monzo|4 -14 3 4}} = 4802000/4782969.

; [[Triwellismic temperaments|Triwellismic or Tribizo-asepgu temperaments]]

: Triwellismic rank-two temperaments temper out the ''triwellisma'', {{Monzo|1 -1 -7 6}} = 235298/234375.

; [[Hemimage temperaments|Hemimage or Satrizo-agu temperaments]]

: Hemimage rank-two temperaments temper out the hemimage comma, {{Monzo|5 -7 -1 3}} = 10976/10935.

; [[Hemifamity temperaments|Hemifamity or Saruyo temperaments]]

: Hemifamity rank-two temperaments temper out the hemifamity comma, {{Monzo|10 -6 1 -1}} = 5120/5103.

; [[Parkleiness temperaments|Parkleiness or Zotritrigu temperaments]]

: Parkleiness rank-two temperaments temper out the ''parkleiness'' comma, {{Monzo|7 7 -9 1}} = 1959552/1953125.

; [[Porwell temperaments|Porwell or Sarurutrigu temperaments]]

: Porwell rank-two temperaments temper out the porwell comma, {{Monzo|11 1 -3 -2}} = 6144/6125.

; [[Cartoonismic temperaments|Cartoonismic or Satritrizo-asepbigu temperaments]]

: Cartoonismic temperaments temper out the cartoonisma, {{monzo| 12 -3 -14 9 }} = 165288374272/164794921875.

; [[Hemfiness temperaments|Hemfiness or Saquinru-atriyo temperaments]]

: Hemfiness rank-two temperaments temper out the ''hemfiness'' comma, {{Monzo|15 -5 3 -5}} = 4096000/4084101.

; [[Hewuermera temperaments|Hewuermera or Satribiru-agu temperaments]]

: Hewuermera rank-two temperaments temper out the ''hewuermera'' comma, {{Monzo|16 2 -1 -6}} = 589824/588245.

; [[Lokismic temperaments|Lokismic or Sasa-bizotrigu temperaments]]

: Lokismic rank-two temperaments temper out the ''lokisma'', {{Monzo|21 -8 -6 2}} = 102760448/102515625.

; Decovulture or Sasabirugugu temperaments

: Decovulture rank-two temperaments temper out the ''decovulture'' comma, {{Monzo|26 -7 -4 -2}} = 67108864/66976875.

; Pontiqak or Lazozotritriyo temperaments

: Pontiqak rank-two temperaments temper out the ''pontiqak'' comma, {{Monzo|-17 -6 9 2}} = 95703125/95551488.

; [[Mitonismic temperaments|Mitonismic or Laquadzo-agu temperaments]]

: Mitonismic rank-two temperaments temper out the ''mitonisma'', {{Monzo|-20 7 -1 4}} = 5250987/5242880.

; [[Horwell temperaments|Horwell or Lazoquinyo temperaments]]

: Horwell rank-two temperaments temper out the horwell comma, {{Monzo|-16 1 5 1}} = 65625/65536.

; Neptunismic or Laruruleyo temperaments

: Neptunismic rank-two temperaments temper out the ''neptunisma'', {{Monzo|-12 -5 11 -2}} = 48828125/48771072.

; [[Metric microtemperaments|Metric or Latriru-asepyo temperaments]]

: Metric rank-two temperaments temper out the meter comma, {{Monzo|-11 2 7 -3}} = 703125/702464.

; [[Wizmic microtemperaments|Wizmic or Quinzo-ayoyo temperaments]]

: Wizmic rank-two temperaments temper out the wizma, {{Monzo|-6 -8 2 5}} = 420175/419904.

; [[Supermatertismic temperaments|Supermatertismic or Lasepru-atritriyo temperaments]]

: Supermatertismic rank-two temperaments temper out the ''supermatertisma'', {{Monzo|-6 3 9 -7}} = 52734375/52706752.

; [[Breedsmic temperaments|Breedsmic or Bizozogu temperaments]]

: Breedsmic rank-two temperaments temper out the breedsma, {{Monzo|-5 -1 -2 4}} = 2401/2400.

; Supermasesquartismic or Laquadbiru-aquinyo temperaments

: Supermasesquartismic rank-two temperaments temper out the ''supermasesquartisma'', {{Monzo|-5 10 5 -8}} = 184528125/184473632.

; [[Ragismic microtemperaments|Ragismic or Zoquadyo temperaments]]

: Ragismic rank-two temperaments temper out the ragisma, {{Monzo|-1 -7 4 1}} = 4375/4374.

; Akjaysmic or Trisa-seprugu temperaments

: Akjaysmic rank-two temperaments temper out the akjaysma, {{Monzo|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 microtemperaments|Landscape or Trizogugu temperaments]]

: Landscape rank-two 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.

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 rank three family|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 rank three family|Diaschismic or Sagugu rank three family]] (P8/2, P5, /1)

: These are the rank three temperaments tempering out the dischisma, {{Monzo|11 -4 -2}} = 2048/2025. The half-octave period is ~45/32.

; [[Porcupine rank three family|Porcupine or Triyo rank three family]] (P8, P4/3, /1)

: These are the rank three temperaments tempering out the porcupine comma or maximal diesis, {{Monzo|1 -5 3}} = 250/243. In the pergen, P4/3 is ~10/9.

; [[Kleismic rank three family|Kleismic or Tribiyo rank three family]] (P8, P12/6, /1)

: These are the rank three temperaments tempering out the kleisma, {{Monzo|-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 family|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 family|Garischismic or Sasaru family]] (P8, P5, ^1)

: A garischismic temperament is one which tempers out the garischisma, {{Monzo|25 -14 0 -1}} = 33554432/33480783.

; [[Semiphore family|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|Sem'''<u>a</u>'''phore]] and [[Slendro clan|Slendro]].

; [[Gamelismic family|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,  {{Monzo|-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, {{Monzo|1 10 0 -6}} = 118098/117649. In the pergen, P8/2 is ~343/243 and P4/3 is ~54/49.

; [[Marvel family|Marvel or Ruyoyo family]] (P8, P5, ^1)

: The head of the marvel family is marvel, which tempers out {{Monzo|-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 family|Starling or Zotrigu family]] (P8, P5, ^1)

: Starling tempers out the septimal semicomma or starling comma {{Monzo|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|77EDO]], but 31, 46 or 58 also work nicely. In the pergen, ^1 = ~81/80.

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

: These temper out {{Monzo|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, {{Monzo|-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, {{Monzo|-9 1 2 1}} = 525/512, which divides 7/6 into two 16/15s. In the pergen, ^1 = ~81/80.

; [[Keemic family|Keemic or Zotriyo family]] (P8, P5, ^1)

: These temper out the keema {{Monzo|-5 -3 3 1}} = 875/864, which divides 15/14 into two 25/24s. In the pergen, ^1 = ~81/80.

; [[Orwellismic family|Orwellismic or Triru-agu family]] (P8, P5, ^1)

: These temper out {{Monzo|6 3 -1 -3}} = 1728/1715. In the pergen, ^1 = ~64/63.

; [[Nuwell family|Nuwell or Quadru-ayo family]] (P8, P5, ^1)

: These temper out the nuwell comma, {{Monzo|1 5 1 -4}} = 2430/2401. In the pergen, ^1 = ~64/63.

; [[Ragisma family|Ragisma or Zoquadyo family]] (P8, P5, ^1)

: The 7-limit rank three microtemperament which tempers out the ragisma, {{Monzo|-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 family|Hemifamity or Saruyo family]] (P8, P5, ^1)

: The hemifamity family of rank three temperaments tempers out the hemifamity comma, {{Monzo|10 -6 1 -1}} = 5120/5103, which divides 10/7 into three 9/8s. In the pergen, ^1 = ~81/80.

; [[Horwell family|Horwell or Lazoquinyo family]] (P8, P5, ^1)

: The horwell family of rank three temperaments tempers out the horwell comma, {{Monzo|-16 1 5 1}} = 65625/65536. In the pergen, ^1 = ~81/80.

; [[Hemimage family|Hemimage or Satrizo-agu family]] (P8, P5, ^1)

: The hemimage family of rank three temperaments tempers out the hemimage comma, {{Monzo|5 -7 -1 3}} = 10976/10935, which divides 10/9 into three 28/27s. In the pergen, ^1 = ~64/63.

; [[Mint family|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.