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


=== Families defined by a 2.3 (wa) comma ===
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.
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.


; [[Blackwood|Blackwood or Sawa family]] (P8/5, ^1)
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.
: This family tempers out the limma, {{Monzo|8 -5 0}} = 256/243, which implies [[5edo|5EDO]].


; [[Apotome family|Apotome or Lawa family]] (P8/7, ^1)
=== Families defined by a 2.3 comma ===
: This family tempers out the apotome, {{Monzo|-11 7 0}} = 2187/2048, which implies [[7edo|7EDO]].
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.  


; [[Compton family|Pythagorean or Lalawa family]] (P8/12, ^1)
; Blackwood family (P8/5, ^1)
: The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = {{Monzo|-19 12 0}}, which implies [[12edo|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 12-edo, offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12-edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.
: 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.  


; [[Counterpyth family|Counterpyth or Tribisawa family]] (P8/41, ^1)
; [[Whitewood family]] (P8/7, ^1)
: The Counterpyth family tempers out the [[41-comma|counterpyth comma]], {{Monzo| 65 -41}}, which implies [[41edo|41EDO]].
: 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.  


; [[Mercator family|Mercator or Quadbilawa family]] (P8/53, ^1)
; [[Compton family]] (P8/12, ^1)
: The Mercator family tempers out the [[Mercator's comma]], {{Monzo| -84 53 }}, which implies [[53edo|53EDO]].  
: 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.  


=== Families defined by a 2.3.5 (ya) comma ===
; [[Countercomp family]] (P8/41, ^1)
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.
: This family tempers out the [[41-comma|Pythagorean countercomma]], {{monzo| 65 -41 }}, which creates multiple copies of [[41edo]]. Its color name is Wa-41.  


; [[Meantone family|Meantone or Gu family]] (P8, P5)  
; [[Mercator family]] (P8/53, ^1)
: 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]], [[19edo]], [[31edo]], [[43edo]], [[50edo]], [[55edo]] 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.
: This family tempers out the [[Mercator's comma]], {{monzo| -84 53 }}, which creates multiple copies of [[53edo]]. Its color name is Wa-53.  


; [[Schismatic family|Schismatic or Layo family]] (P8, P5)
=== Families defined by a 2.3.5 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]], [[29edo]], [[41edo]], [[53edo]], and [[118edo]].
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.


; [[Pelogic family|Pelogic or Layobi family]] (P8, P5)
; [[Meantone 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]], [[16edo]], [[23edo]], and [[25edo]].
: 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)<sup>4</sup>) 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.  


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


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


; [[Bug family|Bug or Gugu family]] (P8, P4/2)
; [[Father family]] (P8, P5)
: 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 aka Zozo.
: This tempers out [[16/15]], the just diatonic semitone, and equates 5/4 with 4/3. Its color name is Gubiti.  


; [[Immunity family|Immunity or Sasa-yoyo family]] (P8, P4/2)
; [[Diaschismic family]] (P8/2, P5)
: 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 aka Zozo.
: 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.


; [[Dicot family|Dicot or Yoyo family]] (P8, P5/2)
; [[Bug family]] (P8, P4/2)
: 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 nterpretation of the generator is ~11/9, which leads to Rastmic aka Neutral aka Lulu.
: 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.


; [[Augmented_family|Augmented or Trigu  family]] (P8/3, P5)
; [[Immunity family]] (P8, P4/2)
: 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]]).
: 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.


; [[Porcupine family|Porcupine or Triyo family]] (P8, P4/3)
; [[Dicot family]] (P8, P5/2)
: 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.
: 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.


; [[Tricot family|Tricot or Quadsatriyo family]] (P8, P11/3)
; [[Augmented family]] (P8/3, P5)
: 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]].
: 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.  


; [[Dimipent family|Dimipent or Quadgu family]] (P8/4, P5)
; [[Misty family]] (P8/3, 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.
: 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.  


; [[Negri|Negri or Laquadyo family]] (P8, P4/4)  
; [[Porcupine family]] (P8, P4/3)
: 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.  
: 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.


; [[Tetracot family|Tetracot or Saquadyo family]] (P8, P5/4)
; [[Alphatricot family]] (P8, P11/3)
: 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]].
: 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]].


; [[Vulture family|Vulture or Sasa-quadyo family]] (P8, P12/4)
; [[Diminished family]] (P8/4, P5)
: 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.
: 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.  


; [[Pental family|Pental or Trila-quingu family]] (P8/5, P5)
; [[Undim family]] (P8/4, 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.
: 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.  


; [[Ripple family|Ripple or Quingu family]] (P8, P4/5)
; Negri family (P8, P4/4)  
: This tempers out the ripple comma, 6561/6250 = {{Monzo| -1 8 -5 }}, which equates a stack of four [[10/9]]'s with [[8/5]]. As one might expect, [[12edo|12EDO]] is about as accurate as it can be.  
: 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.  


; [[Amity family|Amity or Saquinyo family]] (P8, P11/5)
; [[Tetracot family]] (P8, P5/4)
: 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  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.
: 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.  


; [[Magic family|Magic or Laquinyo family]] (P8, P12/5)
; [[Smate family]] (P8, P11/4)
: 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.
: 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.  


; [[Fifive family|Fifive or Saquinbiyo family]] (P8/2, P5/5)
; [[Vulture family]] (P8, P12/4)
: 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.  
: 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.


; [[Qintosec family|Qintosec or Quadsa-quinbigu family]] (P8/5, P5/2)
; [[Quintile family]] (P8/5, P5)
: This tempers out the qintosec 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.
: 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.


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


; [[Ampersand|Ampersand or Lala-tribiyo family]] (P8, P5/6)  
; [[Passion family]] (P8, P4/5)
: 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.
: 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.  


; [[Kleismic family|Kleismic or Tribiyo family]] (P8, P12/6)
; [[Quintaleap family]] (P8, P4/5)
: 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.
: 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.


; [[Semicomma family|Orwell or Sepru, and the semicomma or Lasepyo family]] (P8, P12/7)
; [[Quindromeda family]] (P8, P4/5)
: 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.
: 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.


; [[Wesley family|Wesley or Lasepyobi family]] (P8, ccP4/7)
; [[Amity family]] (P8, P11/5)
: 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]].  
: 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.


; [[Sensipent family|Sensipent or Sepgu family]] (P8, ccP5/7)
; [[Magic family]] (P8, P12/5)
: 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.  
: 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.  


; [[Vishnuzmic family|Vishnuzmic or Sasepbigu family]] (P8/2, P4/7)
; [[Fifive family]] (P8/2, P5/5)
: 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.
: 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.  


; [[Mutt temperament|Mutt or Trila-septriyo family]] (P8/3, ccP4/7)
; [[Quintosec family]] (P8/5, P5/2)
: This tempers out the [[mutt comma]], {{Monzo|-44 -3 21}}, leading to some strange properties. Seven ~5/4 generators equals a double-compound 4th = ~16/3. The third-octave period is <u>not</u> 5/4, thus the generator is equivalently a period minus ~5/4, only about 14¢. The L/s ratio is very lopsided, and scales resemble a "fuzzy" augmented chord.
: 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.  


; [[Würschmidt family|Würschmidt or Saquadbigu family]] (P8, ccP5/8)
; [[Trisedodge family]] (P8/5, P4/3)
: 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.
: 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.  


; [[Escapade family|Escapade or Sasa-tritrigu family]] (P8, P4/9)
; Ampersand family (P8, P5/6)  
: 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.
: 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.


; [[Shibboleth family|Shibboleth or Tritriyo family]] (P8, ccP4/9)
; [[Kleismic family]] (P8, P12/6)
: 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.
: 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.  


; [[Sycamore family|Sycamore or Laleyo family]] (P8, P5/11)
; [[Semicomma family|Orson or semicomma family]] (P8, P12/7)
: 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.
: 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.


; [[Ditonmic family|Ditonmic or Lala-theyo family]] (P8, c<sup>4</sup>P4/13)
; [[Wesley family]] (P8, ccP4/7)
: 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.
: 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]].  


; [[Luna family|Luna or Sasa-quintrigu family]] (P8, ccP4/15)
; [[Sensipent family]] (P8, ccP5/7)
: 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.  
: 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.  


; [[Minortonic family|Minortonic or Trila-segu family]] (P8, ccP5/17)
; [[Vishnuzmic family]] (P8/2, P4/7)
: 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.
: 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)<sup>7</sup>. 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.  


; [[Maja family|Maja or Saseyo family]] (P8, c<sup>6</sup>P4/17)
; [[Unicorn family]] (P8, P4/8)
: 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.
: 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.  


; [[Maquila family|Maquila or Trisa-segu family]] (P8, c<sup>7</sup>P5/17)
; [[Würschmidt family]] (P8, ccP5/8)
: 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.
: 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)<sup>8</sup>⋅(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.  


; [[Gammic family|Gammic or Laquinquadyo family]] (P8, P5/20)
; [[Escapade family]] (P8, P4/9)
: The gammic family tempers out the gammic comma, {{Monzo|-29 -11 20}};. Nine generators of about 35¢ equals ~6/5, eleven equal ~5/4 and twenty equal ~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.
: 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.


=== Clans defined by a 2.3.7 (za) comma ===
; [[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.


These are defined by a za or 7-limit-no-fives comma. See also [[subgroup temperaments]].  
; [[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.  


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.
; [[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.


; [[Archytas clan|Archytas or Ru clan]] (P8, P5)
; [[Lafa family]] (P8, P12/12)
: 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]].
: 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.  


; [[Trienstonic clan|Trienstonic or Zo clan]] (P8, P5)  
; [[Ditonmic family]] (P8, c4P4/13)
: 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.
: 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.  


; [[Harrison's comma|Harrison or Laru clan]] (P8, P5)  
; [[Luna family]] (P8, ccP4/15)
: 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]].
: 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.  


; [[Garischismic clan|Garischismic or Sasaru clan]] (P8, P5)
; [[Vavoom family]] (P8, P12/17)
: 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 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.  


; [[Slendro clan|Slendro (Semaphore) or Zozo clan]] (P8, P4/2)
; [[Minortonic family]] (P8, ccP5/17)
: 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]].
: 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.  


; Laruru clan (P8/2, P5)  
; [[Maja family]] (P8, c<sup>6</sup>P4/17)
: 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 Diaschismatic or Sagugu temperament and the Jubalismic or Biruyo temperament.
: 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.  


; Sasa-zozo clan (P8, P5/2)  
; [[Maquila family]] (P8, c<sup>7</sup>P5/17)
: This clan tempers out the Sasa-zozo comma, {{Monzo|15 -13 0 2}} = 1605632/1594323, and includes as a strong extension the [[Hemififths]] temperament. 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.
: 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.


; [[Gamelismic clan|Gamelismic or Latrizo clan]] (P8, P5/3)
; [[Gammic family]] (P8, P5/20)
: 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.
: 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.  
: 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)
=== Clans defined by a 2.3.7 comma ===
: 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.
These are defined by a no-5's 7-limit (color name: za) comma. See also [[subgroup temperaments]].


; Triru clan (P8/3, P5)
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.
: 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.


; Latriru clan (P8, P11/3)  
; [[Archytas clan]] (P8, P5)
: 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.
: 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]].


; Saquadru clan (P8, P12/4)  
; [[Trienstonic clan]] (P8, P5)
: This clan tempers out the Saquadru comma, {{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.
: 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.  


; Laquadru clan (P8, P11/4)  
; Harrison clan (P8, P5)
: 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.
: 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]].  


; [[Cloudy clan|Cloudy or Laquinzo clan]] (P8/5, P5)  
; [[Garischismic clan]] (P8, 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.
: 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.  


; Quinru clan (P8, P5/5)  
; Sasazoti clan (P8, P5)
: 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.  
: 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]].  


; Saquinzo clan (P8, P12/5)  
; Laruruti clan (P8/2, P5)
: 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.
: 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.


; [[Stearnsmic clan|Stearnsmic or Latribiru clan]] (P8/2, P4/3)
; [[Semaphoresmic clan]] (P8, P4/2)
: 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.
: 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]].  


; Lasepzo clan (P8, P11/7)  
; Parahemif clan (P8, P5/2)
: 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.
: 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.


; Septiness or Sasasepru clan (P8, P11/7)  
; Triruti clan (P8/3, P5)
: 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]].
: 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.


; Sepru clan (P8, P12/7)  
; [[Gamelismic clan]] (P8, P5/3)
: 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.
: 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.


; [[Tritrizo clan]] (P8/9, P5)
; Trizoti clan (P8, P5/3)
: This clan tempers out the ''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]].
: 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.


=== Clans defined by a 2.3.11 (ila) comma ===
; Latriru clan (P8, P11/3)
See also [[subgroup temperaments]].
: 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.


; Lulu clan (P8/2, P5)  
; [[Stearnsmic clan]] (P8/2, P4/3)
: This 2.3.11 clan tempers out alpharabian limma, 128/121. Both 11/8 and 16/11 is equated to half-octave period. This clan includes as a strong extension the pajaric temperament, which is in the diaschismic family.
: 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.


; [[Rastmic temperaments|Rastmic or Neutral or Lulu clan]] (P8, P5/2)  
; Skwaresmic clan (P8, P11/4)
: 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.
: 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.  


; Laquadlo clan (P8/2, M2/4)  
; [[Buzzardsmic clan]] (P8, P12/4)
: This 2.3.11 clan tempers out the Laquadlo 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 as a strong extension the comic or saquadyobi temperament, which is in the comic family.
: 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.  


=== Clans defined by a 2.3.13 (tha) comma ===
; [[Cloudy clan]] (P8/5, P5)
See also [[subgroup temperaments]].
: 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.  


; [[Hemif|Hemif or Thuthu clan]] (P8, P5/2)  
; Quinruti clan (P8, P5/5)
: 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.
: 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.  


; Satritho clan (P8, P11/3)  
; Saquinzoti clan (P8, P12/5)
: This 2.3.13 clan tempers out the Satritho comma 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.
: 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.


=== Clans defined by a 2.5.7 (yaza nowa) comma ===
; Lasepzoti clan (P8, P11/7)
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.
: 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.


; Bapbo or Rurugu Nowa clan (P8, M3/2)  
; Septiness clan (P8, P11/7)
: This clan tempers out the bapbo comma, [[256/245]]. The genarator is ~8/7 = ~202¢ and two of them equals ~5/4.
: 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.  


; [[Jubilismic clan|Jubilismic or Biruyo Nowa clan]] (P8/2, M3)  
; Sepruti clan (P8, P12/7)
: 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.  
: 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.


; Slither or Satritriru-aquadyo Nowa clan (P8, ccm6/9)  
; [[Septiennealimmal clan]] (P8/9, P5)
: 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.
: 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.  


; [[Hemimean clan|Hemimean or Zozoquingu Nowa clan]] (P8, M3/2)
=== Clans defined by a 2.3.11 comma ===
: 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.
Color name: ila. See also [[subgroup temperaments]].


; [[Quince clan|Quince or Lasepzo-agugu Nowa clan]] (P8, m6/7)
; Lulubiti clan (P8/2, P5)  
: 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.
: 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.


; Rainy or Quinzo-atriyo Nowa clan (P8, M3/5)
; [[Rastmic clan]] (P8, P5/2)  
: 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.
: 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.  


; Vorwell or Sasatriru-aquadbigu Nowa clan (P8, m6/3)
; [[Nexus clan]] (P8/3, P4/2)
: 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.
: 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.  


=== Clans defined by a 3.5.7 (yaza noca) comma ===
; Alphaxenic or Laquadloti clan (P8/2, M2/4)  
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 a noca subgroup, "compound" means increased by 3/1 not 2/1.
: 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.  


; Arcturus or Rutribiyo Noca clan (P12, M6)
=== Clans defined by a 2.3.13 comma ===
: 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.
Color name: tha. See also [[subgroup temperaments]].


; [[Sensamagic clan|Sensamagic or Zozoyo Noca clan]] (P12, M6/2)
; Thuthuti clan (P8, P5/2)  
: 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.
: 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.


; Betelgeuse or Satritrizo-agugu Noca clan (P12, c<sup>3</sup>M6)  
; Satrithoti clan (P8, P11/3)  
: 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.
: 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.


; [[Gariboh clan|Gariboh or Triru-aquinyo Noca clan]] (P12, M6/3)
=== Clans defined by a 2.5.7 comma ===
: 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.
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.


; [[Mirkwai clan|Mirkwai or Quinru-aquadyo Noca clan]] (P12, cm7/5)
; [[Jubilismic clan]] (P8/2, M3)
: 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.
: 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.  


; Procyon or Sasepzo-atrigu Noca clan (P12, m7/7)
; [[Bapbo clan]] (P8, M3/2)  
: 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.
: 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.  


; Izar or Saquadtrizo-asepgu Noca clan (P12, c<sup>5</sup>m7/12)
; [[Hemimean clan]] (P8, M3/2)
: 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.
: 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.  


=== Temperaments defined by a 2.3.5.7 (yaza) comma ===
; Mabilismic clan (P8, cM3/3)
These are defined by a full 7-limit (or yaza) comma.
: 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.  


; [[Septisemi temperaments|Septisemi or Zogu temperaments]]
; Vorwell clan (P8, m6/3)
: These are very low complexity temperaments tempering out the minor septimal semitone, [[21/20]] and hence equating 5/3 with 7/4.
: 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.  


; [[Greenwoodmic temperaments|Greenwoodmic or Ruruyo temperaments]]
; Quinzo-atriyoti Nowa clan (P8, M3/5)
: These temper out the greenwoodma, {{Monzo|-3 4 1 -2}} = 405/392.
: 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).


; Keegic or Trizogu temperaments
; [[Llywelynsmic clan]] (P8, cM3/7)
: Keegic rank-two temperaments temper out the keega, {{Monzo|-3 1 -3 3}} = 1029/1000.
: 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.  


; [[Mint temperaments|Mint or Rugu temperaments]]
; [[Quince clan]] (P8, m6/7)
: Mint rank-two temperaments temper out the septimal quarter-tone 36/35, equating 7/6 with 6/5, and 5/4 with 9/7.
: 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.  


; [[Avicennmic temperaments|Avicennmic or Zoyoyo temperaments]]
; Slither clan (P8, ccm6/9)
: These temper out the avicennma, {{Monzo|-9 1 2 1}} = 525/512, also known as Avicenna's enharmonic diesis.
: 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.  


; Sengic or Trizo-agugu temperaments
=== Clans defined by a 3.5.7 comma ===
: Sengic rank-two temperaments temper out the senga, {{Monzo|1 -3 -2 3}} = 686/675.
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.


; [[Keemic temperaments|Keemic or Zotriyo temperaments]]
; Rutribiyoti Noca clan (P12, M6)
: Keemic rank-two temperaments temper out the keema, {{Monzo|-5 -3 3 1}} = 875/864.
: 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.


; Secanticorn or Laruquingu temperaments
; [[Sensamagic clan]] (P12, M6/2)
: Secanticorn rank-two temperaments temper out the ''secanticornisma'', {{monzo|-3 11 -5 -1}} = 177147/175000.
: 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.  


; Nuwell or Quadru-ayo temperaments
; [[Gariboh clan]] (P12, M6/3)
: Nuwell rank-two temperaments temper out the nuwell comma, {{Monzo|1 5 1 -4}} = 2430/2401.
: 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.  


; Mermismic or Sepruyo temperaments
; [[Mirkwai clan]] (P12, cm7/5)
: Mermismic rank-two temperaments temper out the ''mermisma'', {{Monzo|5 -1 7 -7}} = 2500000/2470629.
: 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.  


; Negricorn or Saquadzogu temperaments
; Sasepzo-atriguti Noca clan (P12, m7/7)
: Negricorn rank-two temperaments temper out the ''negricorn'' comma, {{Monzo|6 -5 -4 4}} = 153664/151875.
: 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).


; Tolermic or Sazoyoyo temperaments
; Satritrizo-aguguti Noca clan (P12, c<sup>3</sup>M6/9)
: These temper out the tolerma, {{Monzo|10 -11 2 1}} = 179200/177147.
: 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).


; Valenwuer or Sarutribigu temperaments
; Saquadtrizo-asepguti Noca clan (P12, c<sup>5</sup>m7/12)
: Valenwuer rank-two temperaments temper out the ''valenwuer'' comma, {{Monzo|12 3 -6 -1}} = 110592/109375.
: 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.


; [[Mirwomo temperaments|Mirwomo or Labizoyo temperaments]]
=== Temperaments defined by a 2.3.5.7 comma ===
: Mirwomo rank-two temperaments temper out the mirwomo comma, {{Monzo|-15 3 2 2}} = 33075/32768.
These are defined by a full 7-limit (color name: yaza) comma.


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


; Compass or Quinruyoyo temperaments
; [[Greenwoodmic temperaments]]
: Compass rank-two temperaments temper out the compass comma, {{Monzo|-6 -2 10 -5}} = 9765625/9680832.
: These temper out the [[greenwoodma]], {{monzo| -3 4 1 -2 }} (405/392). Its color name is Ruruyoti.  


; Trimyna or Quinzogu temperaments
; [[Keegic temperaments]]
: The trimyna rank-two temperaments temper out the trimyna comma, {{Monzo|-4 1 -5 5}} = 50421/50000.
: Keegic rank-2 temperaments temper out the [[keega]], {{monzo| -3 1 -3 3 }} (1029/1000). Its color name is Trizoguti.  


; [[Starling temperaments|Starling or Zotrigu temperaments]]
; [[Mint 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.
: 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.  


; [[Octagar temperaments|Octagar or Rurutriyo temperaments]]
; [[Avicennmic temperaments]]
: Octagar rank-two temperaments temper out the octagar comma, {{Monzo|5 -4 3 -2}} = 4000/3969.
: These temper out the [[avicennma]], {{monzo| -9 1 2 1 }} (525/512), also known as Avicenna's enharmonic diesis. Its color name is Zoyoyoti.  


; [[Orwellismic temperaments|Orwellismic or Triru-agu temperaments]]
; Sengic temperaments
: Orwellismic rank-two temperaments temper out orwellisma, {{Monzo|6 3 -1 -3}} = 1728/1715.
: Sengic rank-2 temperaments temper out the [[senga]], {{monzo| 1 -3 -2 3 }} (686/675). Its color name is Trizo-aguguti.  


; Mynaslendric or Sepru-ayo temperaments
; [[Keemic temperaments]]
: Mynaslendric rank-two temperaments temper out the ''mynaslender'' comma, {{Monzo|11 4 1 -7}} = 829440/823543.
: Keemic rank-2 temperaments temper out the [[keema]], {{monzo| -5 -3 3 1 }} (875/864). Its color name is Zotriyoti.  


; [[Mistismic temperaments|Mistismic or Sazoquadgu temperaments]]
; Secanticorn temperaments
: Mistismic rank-two temperaments temper out the ''mistisma'', {{Monzo|16 -6 -4 1}} = 458752/455625.
: Secanticorn rank-2 temperaments temper out the [[secanticornisma]], {{monzo| -3 11 -5 -1 }} (177147/175000). Its color name is Laruquinguti.  


; [[Varunismic temperaments|Varunismic or Labizogugu temperaments]]
; Nuwell temperaments
: Varunismic rank-two temperaments temper out the varunisma, {{monzo|-9 8 -4 2}} = 321489/320000.
: Nuwell rank-2 temperaments temper out the [[nuwell comma]], {{monzo| 1 5 1 -4 }} (2430/2401). Its color name is Quadru-ayoti.  


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


; Dimcomp or Quadruyoyo temperaments
; Negricorn temperaments
: Dimcomp rank-two temperaments temper out the dimcomp comma, {{Monzo|-1 -4 8 -4}} = 390625/388962.
: Negricorn rank-2 temperaments temper out the [[negricorn comma]], {{monzo| 6 -5 -4 4 }} (153664/151875). Its color name is Saquadzoguti.  


; [[Cataharry temperaments|Cataharry or Labirugu temperaments]]
; Tolermic temperaments
: Cataharry rank-two temperaments temper out the cataharry comma, {{Monzo|-4 9 -2 -2}} = 19683/19600.
: These temper out the [[tolerma]], {{monzo| 10 -11 2 1 }} (179200/177147). Its color name is Sazoyoyoti.  


; [[Canousmic temperaments|Canousmic or Saquadzo-atriyo temperaments]]
; Valenwuer temperaments
: Canousmic rank-two temperaments temper out the canousma, {{Monzo|4 -14 3 4}} = 4802000/4782969.
: Valenwuer rank-2 temperaments temper out the [[valenwuer comma]], {{monzo| 12 3 -6 -1 }} (110592/109375). Its color name is Sarutribiguti.  


; Triwellismic or Tribizo-asepgu temperaments
; [[Mirwomo temperaments]]
: Triwellismic rank-two temperaments temper out the ''triwellisma'', {{Monzo|1 -1 -7 6}} = 235298/234375.
: Mirwomo rank-2 temperaments temper out the [[mirwomo comma]], {{monzo| -15 3 2 2 }} (33075/32768). Its color name is Labizoyoti.  


; [[Hemimage temperaments|Hemimage or Satrizo-agu temperaments]]
; Catasyc temperaments
: Hemimage rank-two temperaments temper out the hemimage comma, {{Monzo|5 -7 -1 3}} = 10976/10935.
: Catasyc rank-2 temperaments temper out the [[catasyc comma]], {{monzo| -11 -3 8 -1 }} (390625/387072). Its color name is Laruquadbiyoti.  


; [[Hemifamity temperaments|Hemifamity or Saruyo temperaments]]
; Compass temperaments
: Hemifamity rank-two temperaments temper out the hemifamity comma, {{Monzo|10 -6 1 -1}} = 5120/5103.
: Compass rank-2 temperaments temper out the [[compass comma]], {{monzo| -6 -2 10 -5 }} (9765625/9680832). Its color name is Quinruyoyoti.  


; Parkleiness or Zotritrigu temperaments
; Trimyna temperaments
: Parkleiness rank-two temperaments temper out the ''parkleiness'' comma, {{Monzo|7 7 -9 1}} = 1959552/1953125.
: Trimyna rank-2 temperaments temper out the [[trimyna comma]], {{monzo| -4 1 -5 5 }} (50421/50000). Its color name is Quinzoguti.  


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


; [[Hemfiness temperaments|Hemfiness or Saquinru-atriyo temperaments]]
; [[Octagar temperaments]]
: Hemfiness rank-two temperaments temper out the ''hemfiness'' comma, {{Monzo|15 -5 3 -5}} = 4096000/4084101.
: Octagar rank-2 temperaments temper out the [[octagar comma]], {{monzo| 5 -4 3 -2 }} (4000/3969). Its color name is Rurutriyoti.  


; [[Hewuermera temperaments|Hewuermera or Satribiru-agu temperaments]]
; [[Orwellismic temperaments]]
: Hewuermera rank-two temperaments temper out the ''hewuermera'' comma, {{Monzo|16 2 -1 -6}} = 589824/588245.
: Orwellismic rank-2 temperaments temper out [[orwellisma]], {{monzo| 6 3 -1 -3 }} (1728/1715). Its color name is Triru-aguti.  


; Decovulture or Sasabirugugu temperaments
; Mynaslendric temperaments
: Decovulture rank-two temperaments temper out the ''decovulture'' comma, {{Monzo|26 -7 -4 -2}} = 67108864/66976875.
: Mynaslendric rank-2 temperaments temper out the [[mynaslender comma]], {{monzo| 11 4 1 -7 }} (829440/823543). Its color name is Sepru-ayoti.  


; Pontiqak or Lazozotritriyo temperaments
; [[Mistismic temperaments]]
: Pontiqak rank-two temperaments temper out the ''pontiqak'' comma, {{Monzo|-17 -6 9 2}} = 95703125/95551488.
: Mistismic rank-2 temperaments temper out the [[mistisma]], {{monzo| 16 -6 -4 1 }} (458752/455625). Its color name is Sazoquadguti.  


; [[Mitonismic temperaments|Mitonismic or Laquadzo-agu temperaments]]
; [[Varunismic temperaments]]
: Mitonismic rank-two temperaments temper out the ''mitonisma'', {{Monzo|-20 7 -1 4}} = 5250987/5242880.
: Varunismic rank-2 temperaments temper out the [[varunisma]], {{monzo| -9 8 -4 2 }} (321489/320000). Its color name is Labizoguguti.  


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


; [[Metric microtemperaments|Metric or Latriru-asepyo temperaments]]
; Dimcomp temperaments
: Metric rank-two temperaments temper out the meter comma, {{Monzo|-11 2 7 -3}} = 703125/702464.
: Dimcomp rank-2 temperaments temper out the [[dimcomp comma]], {{monzo| -1 -4 8 -4 }} (390625/388962). Its color name is Quadruyoyoti.  


; [[Wizmic microtemperaments|Wizmic or Quinzo-ayoyo temperaments]]
; [[Cataharry temperaments]]
: Wizmic rank-two temperaments temper out the wizma, {{Monzo|-6 -8 2 5}} = 420175/419904.
: Cataharry rank-2 temperaments temper out the [[cataharry comma]], {{monzo| -4 9 -2 -2 }} (19683/19600). Its color name is Labiruguti.  


; Supermatertismic or Lasepru-atritriyo temperaments
; [[Canousmic temperaments]]
: Supermatertismic rank-two temperaments temper out the ''supermatertisma'', {{Monzo|-6 3 9 -7}} = 52734375/52706752.
: Canousmic rank-2 temperaments temper out the [[canousma]], {{monzo| 4 -14 3 4 }} (4802000/4782969). Its color name is Saquadzo-atriyoti.  


; [[Breedsmic temperaments|Breedsmic or Bizozogu temperaments]]
; [[Triwellismic temperaments]]
: Breedsmic rank-two temperaments temper out the breedsma, {{Monzo|-5 -1 -2 4}} = 2401/2400.
: Triwellismic rank-2 temperaments temper out the [[triwellisma]], {{monzo| 1 -1 -7 6 }} (235298/234375). Its color name is Tribizo-asepguti.  


; Supermasesquartismic or Laquadbiru-aquinyo temperaments
; [[Hemimage temperaments]]
: Supermasesquartismic rank-two temperaments temper out the ''supermasesquartisma'', {{Monzo|-5 10 5 -8}} = 184528125/184473632.
: Hemimage rank-2 temperaments temper out the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} (10976/10935). Its color name is Satrizo-aguti.  


; [[Ragismic microtemperaments|Ragismic or Zoquadyo temperaments]]
; [[Hemifamity temperaments]]
: Ragismic rank-two temperaments temper out the ragisma, {{Monzo|-1 -7 4 1}} = 4375/4374.
: Hemifamity rank-2 temperaments temper out the [[hemifamity comma]], {{monzo| 10 -6 1 -1 }} (5120/5103). Its color name is Saruyoti.  


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


; [[Landscape microtemperaments|Landscape or Trizogugu temperaments]]
; [[Porwell temperaments]]
: Lanscape rank-two temperaments temper out the lanscape 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.
: 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.
 
; Pontiqak temperaments
: Pontiqak rank-2 temperaments temper out the [[pontiqak comma]], {{monzo| -17 -6 9 2 }} (95703125/95551488). Its color name is Lazozotritriyoti.
 
; [[Mitonismic temperaments]]
: Mitonismic rank-2 temperaments temper out the [[mitonisma]], {{monzo| -20 7 -1 4 }} (5250987/5242880). Its color name is Laquadzo-aguti.
 
; [[Horwell temperaments]]
: Horwell rank-2 temperaments temper out the [[horwell comma]], {{monzo| -16 1 5 1 }} (65625/65536). Its color name is Lazoquinyoti.
 
; Neptunismic temperaments
: Neptunismic rank-2 temperaments temper out the [[neptunisma]], {{monzo| -12 -5 11 -2 }} (48828125/48771072). Its color name is Laruruleyoti.
 
; [[Metric microtemperaments]]
: Metric rank-2 temperaments temper out the [[meter]], {{monzo| -11 2 7 -3 }} (703125/702464). Its color name is Latriru-asepyoti.
 
; [[Wizmic microtemperaments]]
: Wizmic rank-2 temperaments temper out the [[wizma]], {{monzo| -6 -8 2 5 }} (420175/419904). Its color name is Quinzo-ayoyoti.
 
; [[Supermatertismic temperaments]]
: Supermatertismic rank-2 temperaments temper out the [[supermatertisma]], {{monzo| -6 3 9 -7 }} (52734375/52706752). Its color name is Lasepru-atritriyoti.
 
; [[Breedsmic temperaments]]
: Breedsmic rank-2 temperaments temper out the [[breedsma]], {{monzo| -5 -1 -2 4 }} (2401/2400). Its color name is Bizozoguti.
 
; Supermasesquartismic temperaments
: Supermasesquartismic rank-2 temperaments temper out the [[supermasesquartisma]], {{monzo| -5 10 5 -8 }} (184528125/184473632). Its color name is Laquadbiru-aquinyoti.
 
; [[Ragismic microtemperaments]]
: Ragismic rank-2 temperaments temper out the [[ragisma]], {{monzo| -1 -7 4 1 }} (4375/4374). Its color name is Zoquadyoti.
 
; Akjaysmic temperaments
: Akjaysmic rank-2 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 Lawati family, ~3/2 is not equated with 4/7 of an octave, resulting in small intervals. Its color name is Trisa-sepruguti.
 
; [[Landscape microtemperaments]]
: Landscape rank-2 temperaments temper out the [[landscape comma]], {{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.


== Rank-3 temperaments ==
== Rank-3 temperaments ==
Even less familiar than rank-2 temperaments are the [[Planar temperament|rank-3 temperaments]], generated by a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank-3 temperaments by the commas they temper out. They can also be identified by pergen. The 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.
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.
 
=== Families defined by a 2.3.5 comma ===
Every 5-limit (color name: ya) comma defines a rank-3 family, thus every comma in the list of rank-2 2.3.5 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a 7-limit (color name: yaza) temperament in which one of the generators is ~7/1. This generator can be reduced to ~7/4, which can be reduced further to ~64/63. Hence in all the pergens below, the ^1 or /1 generator is ~64/63. An additional 7-limit or 11-limit comma creates an 11-limit (color name: yazala) temperament, and so forth. All these examples are 7-limit:


=== Families defined by a 2.3.5 (ya) comma ===
; [[Didymus rank three family|Didymus rank-3 family]] (P8, P5, ^1)
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:
: These are the rank-3 temperaments tempering out the didymus or meantone comma, 81/80. Its color name is Guti.  


; [[Didymus rank three family|Didymus or Gu rank three family]] (P8, P5, ^1)
; [[Diaschismic rank three family|Diaschismic rank-3 family]] (P8/2, P5, /1)
: These are the rank three temperaments tempering out the didymus or meantone comma, 81/80.  
: These are the rank-3 temperaments tempering out the diaschisma, {{monzo| 11 -4 -2 }} (2048/2025). The half-octave period is ~45/32. Its color name is Saguguti.  


; [[Diaschismic rank three family|Diaschismic or Sagugu rank three family]] (P8/2, P5, /1)
; [[Porcupine rank three family|Porcupine rank-3 family]] (P8, P4/3, /1)
: These are the rank three temperaments tempering out the dischisma, {{Monzo|11 -4 -2}} = 2048/2025. The half-octave period is ~45/32.
: These are the rank-3 temperaments tempering out the porcupine comma a.k.a. maximal diesis, {{monzo| 1 -5 3 }} (250/243). In the pergen, P4/3 is ~10/9. Its color name is Triyoti.  


; [[Porcupine rank three family|Porcupine or Triyo rank three family]] (P8, P4/3, /1)
; [[Kleismic rank three family|Kleismic rank-3 family]] (P8, P12/6, /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.
: These are the rank-3 temperaments tempering out the kleisma, {{monzo| -6 -5 6 }} (15625/15552). In the pergen, P12/6 is ~6/5. Its color name is Tribiyoti.  


; [[Kleismic rank three family|Kleismic or Tribiyo rank three family]] (P8, P12/6, /1)
=== Families defined by a 2.3.7 comma ===
: These are the rank three temperaments tempering out the kleisma, {{Monzo|-6 -5 6}} = 15625/15552. In the pergen, P12/6 is ~6/5.
Every no-5's 7-limit (color name: za) comma defines a rank-3 family, thus every comma in the list of rank-2 2.3.7 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a 7-limit (color name: yaza) temperament in which one of the generators is ~5/1. This generator can be reduced to ~5/4, which may be reduced further to ~81/80. Hence in all the pergens below, {{nowrap| ^1 {{=}} ~81/80 }}. An additional 5-limit or 11-limit comma creates a 11-limit (color name: yazala) temperament, and so forth. All these examples are 7-limit:


=== Families defined by a 2.3.7 (za) comma ===
; [[Archytas family]] (P8, P5, ^1)
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 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.  


; [[Archytas family|Archytas or Ru family]] (P8, P5, ^1)
; [[Garischismic 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.
: A garischismic temperament is one which tempers out the garischisma, {{monzo| 25 -14 0 -1 }} (33554432/33480783). Its color name is Sasaruti.  


; Garischismic or Sasaru family (P8, P5, ^1)
; Laruruti clan (P8/2, P5)
: A garischismic temperament is one which tempers out the garischisma, {{Monzo|25 -14 0 -1}} = 33554432/33480783.
: 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.


; [[Semiphore family|Semiphore or Zozo family]] (P8, P4/2, ^1)
; [[Semaphoresmic 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]].
: 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|Gamelismic or Latrizo family]] (P8, P5/3, ^1)
; [[Gamelismic 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.  
: 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 or Latribiru family (P8/2, P4/3, ^1)
; 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.
: 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 (yaza) comma ===
=== Families defined by a 2.3.5.7 comma ===
; [[Marvel family|Marvel or Ruyoyo family]] (P8, P5, ^1)
Color name: yaza.  
: The head of the marvel family is marvel, which tempers out {{Monzo|-5 2 2 -1}} = [[225/224]]. 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.  
; [[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|Starling or Zotrigu family]] (P8, P5, ^1)
; [[Starling 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. 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.
: 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|Sensamagic or Zozoyo family]] (P8, P5, ^1)
; [[Sensamagic family]] (P8, P5, ^1)
: These temper out {{Monzo|0 -5 1 2}} = 245/243. In the pergen, ^1 = ~64/63.
: 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 or Ruruyo family (P8, P5, ^1)
; Greenwoodmic family (P8, P5, ^1)
: These temper out the greenwoodma, {{Monzo|-3 4 1 -2}} = 405/392. In the pergen, ^1 = ~64/63.
: 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.  


; Avicennmic or Zoyoyo family (P8, P5, ^1)
; Avicennmic family (P8, P5, ^1)
: These temper out the avicennma, {{Monzo|-9 1 2 1}} = 525/512. In the pergen, ^1 = ~81/80.
: These temper out the avicennma, {{monzo| -9 1 2 1 }} (525/512), which divides 7/6 into two 16/15's. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Lazoyoyoti.  


; [[Keemic family|Keemic or Zotriyo family]] (P8, P5, ^1)
; [[Keemic family]] (P8, P5, ^1)
: These temper out the keema {{Monzo|-5 -3 3 1}} = 875/864. In the pergen, ^1 = ~81/80.
: These temper out the keema, {{monzo| -5 -3 3 1 }} (875/864), which divides 15/14 into two 25/24's. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Zotriyoti.  


; [[Orwellismic family|Orwellismic or Triru-agu family]] (P8, P5, ^1)
; [[Orwellismic family]] (P8, P5, ^1)
: These temper out {{Monzo|6 3 -1 -3}} = 1728/1715. In the pergen, ^1 = ~64/63.
: These temper out the orwellisma, {{monzo| 6 3 -1 -3 }} (1728/1715). In the pergen, {{nowrap| ^1 {{=}} ~64/63 }}. Its color name is Triru-aguti.  


; [[Nuwell family|Nuwell or Quadru-ayo family]] (P8, P5, ^1)
; [[Nuwell family]] (P8, P5, ^1)
: These temper out the nuwell comma, {{Monzo|1 5 1 -4}} = 2430/2401. In the pergen, ^1 = ~64/63.
: These temper out the nuwell comma, {{monzo| 1 5 1 -4 }} (2430/2401). In the pergen, {{nowrap| ^1 {{=}} ~64/63 }}. Its color name is Quadru-ayoti.  


; [[Ragisma family|Ragisma or Zoquadyo family]] (P8, P5, ^1)
; [[Ragisma 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.
: The 7-limit rank-3 microtemperament which tempers out the ragisma, {{monzo| -1 -7 4 1 }} (4375/4374), extends to various higher-limit rank-3 temperaments such as thor. These are not by any means all microtemperaments, but those which are not highly accurate are probably best discussed under another heading. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Zoquadyoti.  


; [[Hemifamity family|Hemifamity or Saruyo family]] (P8, P5, ^1)
; [[Hemifamity family]] (P8, P5, ^1)
: The hemifamity family of rank three temperaments tempers out the hemifamity comma, {{Monzo|10 -6 1 -1}} = 5120/5103. In the pergen, ^1 = ~81/80.
: The hemifamity family of rank-3 temperaments tempers out the hemifamity comma, {{monzo| 10 -6 1 -1 }} (5120/5103), which divides 10/7 into three 9/8's. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Saruyoti.  


; [[Horwell family|Horwell or Lazoquinyo family]] (P8, P5, ^1)
; [[Horwell 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.
: The horwell family of rank-3 temperaments tempers out the horwell comma, {{monzo| -16 1 5 1 }} (65625/65536). In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Lazoquinyoti.  


; [[Hemimage family|Hemimage or Satrizo-agu family]] (P8, P5, ^1)
; [[Hemimage family]] (P8, P5, ^1)
: The hemimage family of rank three temperaments tempers out the hemimage comma, {{Monzo|5 -7 -1 3}} = 10976/10935. In the pergen, ^1 = ~64/63.
: The hemimage family of rank-3 temperaments tempers out the hemimage comma, {{monzo| 5 -7 -1 3 }} (10976/10935), which divides 10/9 into three 28/27's. In the pergen, {{nowrap| ^1 {{=}} ~64/63 }}. Its color name is Satrizo-aguti.  


; [[Tolermic family|Tolermic or Sazoyoyo family]] (P8, P5, ^1)
; [[Mint family]] (P8, P5, ^1)
: These temper out the tolerma, {{Monzo|10 -11 2 1}} = 179200/177147. In the pergen, ^1 = ~81/80.
: The mint temperament is a low-complexity, high-error temperament, tempering out the septimal quartertone 36/35, equating 7/6 with 6/5, and 5/4 with 9/7. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }} or ~64/63. Its color name is Ruguti.  


; [[Mint family|Mint or Rugu family]] (P8, P5, ^1)
; Septisemi 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.
: 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, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Zoguti.  


; Septisemi or Zogu family (P8, P5, ^1)
; [[Jubilismic family]] (P8/2, P5, ^1)
: These are very low complexity temperaments tempering out the minor septimal semitone, [[21/20]] and hence equating 5/3 with 7/4. In the pergen, ^1 = ~81/80.
: Jubilismic temperament tempers out 50/49 and thereby equates the two septimal tritones, 7/5 and 10/7. This is the half-octave period. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Biruyoti.  


; [[Jubilismic family|Jubilismic or Biruyo family]] (P8/2, P5, ^1)
; [[Cataharry family]] (P8, P4/2, ^1)
: Jubilismic temperament tempers out 50/49 and thereby equates the two septimal tritones, 7/5 and 10/7. This is the half-octave period. In the pergen, ^1 = ~81/80.
: Cataharry temperaments temper out the cataharry comma, {{monzo| -4 9 -2 -2 }} (19683/19600). In the pergen, half a fourth is ~81/70, and {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Labiruguti.  


; [[Cataharry family|Cataharry or Labirugu family]] (P8, P4/2, ^1)
; [[Breed family]] (P8, P5/2, ^1)
: Cataharry temperaments temper out the cataharry comma, {{Monzo|-4 9 -2 -2}} = 19683/19600. In the pergen, half a 4th is ~81/70, and ^1 = ~81/80.
: Breed is a 7-limit microtemperament which tempers out {{monzo| -5 -1 -2 4 }} (2401/2400). While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749edo will certainly do the trick. Breed has generators of 2/1, a 49/40-cum-60/49 neutral third, and ~64/63. Its color name is Bizozoguti.  


; [[Breed family|Breed or Bizozogu family]] (P8, P5/2, ^1)
; [[Sengic family]] (P8, P5, vm3/2)
: Breed is a 7-limit microtemperament which tempers out {{Monzo|-5 -1 -2 4}} = 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749EDO will certainly do the trick. Breed has generators of 2/1, a 49/40-cum-60/49 neutral third, and ~64/63.
: These temper out the senga, {{monzo| 1 -3 -2 3 }} (686/675). One generator is ~15/14, two give ~7/6 (the downminor third in the pergen), and three give ~6/5. Its color name is Trizo-aguguti.  


; [[Mirwomo family|Mirwomo or Labizoyo family]] (P8, P5/2, ^1)
; [[Porwell family]] (P8, P5, ^m3/2)
: The mirwomo family of rank three temperaments tempers out the mirwomo comma, {{Monzo|-15 3 2 2}} = 33075/32768. In the pergen, half a fifth is ~128/105, and ^1 = ~81/80.
: The porwell family of rank-3 temperaments tempers out the porwell comma, {{monzo| 11 1 -3 -2 }} (6144/6125). Two ~35/32 generators equal the pergen's upminor third of ~6/5. Its color name is Sarurutriguti.  


; [[Sengic family|Sengic or Trizo-agugu family]] (P8, P5, vm3/2)
; [[Octagar family]] (P8, P5, ^m6/2)
: These temper out the senga, {{Monzo|1 -3 -2 3}} = 686/675. One generator = ~15/14, two = ~7/6 (the downminor 3rd in the pergen), and three = ~6/5.
: The octagar family of rank-3 temperaments tempers out the octagar comma, {{monzo| 5 -4 3 -2 }} (4000/3969). Two ~80/63 generators equal the pergen's upminor sixth of ~8/5. Its color name is Rurutriyoti.  


; [[Porwell family|Porwell or Sarurutrigu family]] (P8, P5, ^m3/2)
; [[Hemimean family]] (P8, P5, vM3/2)
: The porwell family of rank three temperaments tempers out the porwell comma, {{Monzo|11 1 -3 -2}} = 6144/6125. Two ~35/32 generators equal the pergen's upminor 3rd of ~6/5.
: The hemimean family of rank-3 temperaments tempers out the hemimean comma, {{monzo| 6 0 -5 2 }} (3136/3125). Two ~28/25 generators equal the pergen's downmajor third of ~5/4. Its color name is Zozoquinguti.  


; [[Octagar family|Octagar or Rurutriyo family]] (P8, P5, ^m6/2)
; Wizmic family (P8, P5, vm7/2)
: The octagar family of rank three temperaments tempers out the octagar comma, {{Monzo|5 -4 3 -2}} = 4000/3969. Two ~80/63 generators equal the pergen's upminor 6th of ~8/5.
: A wizmic temperament is one which tempers out the wizma, {{monzo| -6 -8 2 5 }}, 420175/419904. Two ~324/245 generators equal the pergen's downminor seventh of ~7/4. Its color name is Quinzo-ayoyoti.  


; [[Hemimean family|Hemimean or Zozoquingu family]] (P8, P5, vM3/2)
; [[Landscape family]] (P8/3, P5, ^1)  
: The hemimean family of rank three temperaments tempers out the hemimean comma, {{Monzo|6 0 -5 2}} = 3136/3125. Two ~28/25 generators equal the pergen's downmajor 3rd of ~5/4.
: The 7-limit rank-3 microtemperament which tempers out the landscape comma, {{monzo| -4 6 -6 3 }} (250047/250000), extends to various higher-limit rank-3 temperaments such as tyr and odin. In the pergen, the 1/3-octave period is ~63/50, and {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Trizoguguti.  


; Wizmic or Quinzo-ayoyo family (P8, P5, vm7/2)
; [[Gariboh family]] (P8, P5, vM6/3)
: A wizmic temperament is one which tempers out the wizma, {{Monzo|-6 -8 2 5}} = 420175/419904. Two ~324/245 generators equal the pergen's downminor 7th of ~7/4.
: The gariboh family of rank-3 temperaments tempers out the gariboh comma, {{monzo| 0 -2 5 -3 }} (3125/3087). Three ~25/21 generators equal the pergen's downmajor sixth of ~5/3. Its color name is Triru-aquinyoti.  


; [[Landscape family|Landscape or Trizogugu family]] (P8/3, P5, ^1)  
; [[Canou family]] (P8, P5, vm6/3)
: The 7-limit rank three microtemperament which tempers out the lanscape comma, {{Monzo|-4 6 -6 3}} = 250047/250000, extends to various higher limit rank three temperaments such as tyr and odin. In the pergen, the third-octave period is ~63/50, and ^1 = ~81/80.
: The canou family of rank-3 temperaments tempers out the canousma, {{monzo| 4 -14 3 4 }} (4802000/4782969). Three ~81/70 generators equal the pergen's downminor sixth of ~14/9. Its color name is Saquadzo-atriyoti.  


; [[Canou family|Canou or Saquadzo-atriyo family]] (P8, P5, vm6/3)
; [[Dimcomp family]] (P8/4, P5, ^1)  
: The canou family of rank three temperaments tempers out the canousma, {{Monzo|4 -14 3 4}} = 4802000/4782969. Three ~81/70 generators equal the pergen's downminor 6th of ~14/9.  
: The dimcomp family of rank-3 temperaments tempers out the dimcomp comma, {{monzo| -1 -4 8 -4 }} (390625/388962). In the pergen, the 1/4-octave period is ~25/21, and {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Quadruyoyoti.  


; [[Dimcomp family|Dimcomp or Quadruyoyo family]] (P8/4, P5, ^1)  
; [[Mirkwai family]] (P8, P5, c^M7/4)
: The dimcomp family of rank three temperaments tempers out the dimcomp comma, {{Monzo|-1 -4 8 -4}} = 390625/388962. In the pergen, the quarter-octave period is ~25/21, and ^1 = ~81/80.
: The mirkwai family of rank-3 temperaments tempers out the mirkwai comma, {{monzo| 0 3 4 -5 }} (16875/16807). Four ~7/5 generators equal the pergen's compound upmajor seventh of  ~27/7. Its color name is Quinru-aquadyoti.  


; [[Mirkwai family|Mirkwai or Quinru-aquadyo family]] (P8, P5, c^M7/4)
=== Temperaments defined by an 11-limit comma ===
: The mirkwai family of rank three temperaments tempers out the mirkwai comma, {{Monzo|0 3 4 -5}} = 16875/16807. Four ~7/5 generators equal the pergen's compound upmajor 7th of ~27/7.
; [[Ptolemismic clan]] (P8, P5, ^1)
: These temper out the [[ptolemisma]], {{monzo| 2 -2 2 0 -1 }} (100/99). 11/8 is equated to 25/18, which is an octave minus two 6/5's. Since 25/18 is a 5-limit interval, every 2.3.5.11 interval is equated to a 5-limit interval, and both the pergen and the lattice are identical to that of 5-limit JI. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Luyoyoti.  


== [[Rank_four_temperaments|Rank-4 temperaments]] ==
; [[Biyatismic clan]] (P8, P5, ^1)
Even less explored than rank three temperaments are rank four temperaments. In fact, unless one counts 7-limit JI they don't seem to have been explored at all. However, they could be used; for example [[Hobbits|hobbit scales]] can be constructed for them.
: These temper out the [[biyatisma]], {{monzo| -3 -1 -1 0 2 }} (121/120). 5/4 is equated to 121/96, which is two 11/8's minus a 3/2 fifth. Since 121/96 is an ila (11-limit no-fives no-sevens) interval, every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Lologuti.  


; [[Valinorismic temperaments|Valinorismic or Lorugugu temperaments]]  
; [[Valinorsmic clan]]  
: These temper out the valinorsma, {{Monzo|4 0 -2 -1 1}} = 176/175.
: These temper out the [[valinorsma]], {{monzo| 4 0 -2 -1 1 }} (176/175). To be a rank-3 temperament, either an additional comma must vanish or the prime subgroup must omit prime 3. Thus no assumptions can be made about the pergen. Its color name is Loruguguti.  


; [[Rastmic temperaments|Rastmic or Lulu temperaments]]
; [[Rastmic rank three clan|Rastmic rank-3 clan]]
: These temper out the rastma, {{Monzo|1 5 0 0 -2}} = 243/242. As an ila (11-limit no-fives no-sevens) rank-2 temperament, it's (P8, P5/2).
: These temper out the [[rastma]], {{monzo| 1 5 0 0 -2 }} (243/242). In the corresponding [[#Clans defined by a 2.3.11 comma|2.3.11 rank-2 temperament]], the pergen is (P8, P5/2). Its color name is Luluti.  


; [[Werckismic temperaments|Werckismic or Luzozogu temperaments]]
; [[Pentacircle clan]] (P8, P5, ^1)
: These temper out the werckisma, {{Monzo|-3 2 -1 2 -1}} = 441/440.
: These temper out the [[pentacircle comma]], {{monzo| 7 -4 0 1 -1 }} (896/891). The interval between 11/8 and 7/4 is equated to 81/64. Since that is a 3-limit interval, every 2.3.11 interval is equated to a 2.3.7 interval and vice versa, and both the pergen and the lattice are identical to that of either 2.3.7 JI or 2.3.11 JI. In the pergen, ^1 is either ~64/63 or ~33/32 or ~729/704. Its color name is Saluzoti.  


; [[Swetismic temperaments|Swetismic or Lururuyo temperaments]]
; [[Semicanousmic clan]] (P8, P5, ^1)
: These temper out the swetisma, {{Monzo|2 3 1 -2 -1}} = 540/539.
: These temper out the [[semicanousma]], {{monzo| -2 -6 -1 0 4 }} (14641/14580). 5/4 is equated to a 2.3.11 (color name: ila) interval, thus every 2.3.5.11 interval is equated to a 2.3.11 interval, and both the pergen and the lattice are identical to that of 2.3.11-subgroup JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Quadlo-aguti.  


; [[Lehmerismic temperaments|Lehmerismic or Loloruyoyo temperaments]]
; [[Semiporwellismic clan]] (P8, P5, ^1)
: These temper out the lehmerisma, {{Monzo|-4 -3 2 -1 2}} = 3025/3024.
: These temper out the [[semiporwellisma]], {{monzo| 14 -3 -1 0 -2 }} (16384/16335). 5/4 is equated to an 2.3.11 (color name: ila) interval, thus every 2.3.5.11 interval is equated to a 2.3.11 interval, and both the pergen and the lattice are identical to that of 2.3.11-subgroup JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Saluluguti.  


; [[Kalismic temperaments|Kalismic or Bilorugu temperaments]]
; [[Olympic clan]] (P8, P5, ^1)
: These temper out the kalisma, {{Monzo|-3 4 -2 -2 2}} = 9801/9800.
: These temper out the [[olympia]], {{monzo| 17 -5 0 -2 -1 }} (131072/130977). 11/8 is equated with a 2.3.7 interval, and thus every 2.3.7.11 interval is equated with a 2.3.7 interval. In the pergen, {{nowrap| ^1 {{=}} ~64/63 }}. Its color name is Salururuti.  


== [[Subgroup temperaments]] ==
; [[Alphaxenic rank three clan|Alphaxenic rank-3 clan]]
: These temper out the [[Alpharabian comma]], {{monzo| -17 2 0 0 4 }} (131769/131072). In the corresponding [[#Clans defined by a 2.3.11 comma|2.3.11 rank-2 temperament]], the pergen is (P8/2, M2/4). Its color name is Laquadloti.
 
; [[Keenanismic temperaments]]
: These temper out the [[keenanisma]], {{monzo| -7 -1 1 1 1 }} (385/384). Its color name is Lozoyoti.
 
; [[Werckismic temperaments]]
: These temper out the [[werckisma]], {{monzo| -3 2 -1 2 -1 }} (441/440). Its color name is Luzozoguti.
 
; [[Swetismic temperaments]]
: These temper out the [[swetisma]], {{monzo| 2 3 1 -2 -1 }} (540/539). Its color name is Lururuyoti.
 
; [[Lehmerismic temperaments]]
: These temper out the [[lehmerisma]], {{monzo| -4 -3 2 -1 2 }} (3025/3024). Its color name is Loloruyoyoti.
 
; [[Kalismic temperaments]]
: These temper out the [[kalisma]], {{monzo| -3 4 -2 -2 2 }} (9801/9800). Its color name is Biloruguti.
 
== Rank-4 temperaments ==
{{Main| Catalog of rank-4 temperaments }}
 
Even less explored than rank-3 temperaments are rank-4 temperaments. In fact, unless one counts 7-limit JI they do not seem to have been explored at all. However, they could be used; for example [[hobbit]] scales can be constructed for them.
 
; [[Keenanismic family]] (P8, P5, ^1, /1)
: These temper out the keenanisma, {{monzo| -7 -1 1 1 1 }} (385/384). In the pergen, ^1 could be either ~81/80 or ~64/63, and /1 could be either ~64/63 (if ^1 is not), ~33/32 or ~729/704. Its color name is Lozoyoti.
 
; Werckismic family (P8, P5, ^1, /1)
: These temper out the werckisma, {{monzo| -3 2 -1 2 -1 }} (441/440). 11/8 is equated to {{monzo| -6 2 -1 2 }} and 5/4 is equated to {{monzo| -5 2 0 2 -1 }}, thus the lattice can be thought of as either 7-limit JI or no-5's 11-limit JI. In the pergen, ^1 is ~64/63, and /1 is either ~81/80, ~33/32 or ~729/704. Its color name is Luzozoguti.
 
; Swetismic family (P8, P5, ^1, /1)
: These temper out the swetisma, {{monzo| 2 3 1 -2 -1 }} (540/539). 11/8 is equated to {{monzo| -1 3 1 -2 }} (135/98) and 5/4 is equated to {{monzo| -4 -3 0 2 1 }}, thus the lattice can be thought of as either 7-limit JI or no-5's 11-limit JI. In the pergen, ^1 is ~64/63, and /1 can be either ~81/80, ~33/32 or ~729/704. Its color name is Lururuyoti.
 
; Lehmerismic family (P8, P5, ^1, /1)
: These temper out the lehmerisma, {{monzo| -4 -3 2 -1 2 }} (3025/3024). Since 7/4 is equated to a no-7's 11-limit (color name: yala) interval, both the pergen and the lattice are identical to that of no-7's 11-limit JI. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }} and /1 is either ~33/32 or ~729/704. Its color name is Loloruyoyoti.
 
; Kalismic family (P8/2, P5, ^1, /1)
: These temper out the kalisma, {{monzo| -3 4 -2 -2 2 }} (9801/9800). The octave is split into two ~99/70 periods. In the pergen, ^1 could be either ~81/80 or ~64/63, and /1 could be either ~64/63 (if ^1 is not), ~33/32 or ~729/704. Its color name is Biloruguti.
 
== Subgroup temperaments ==
{{Main| Subgroup temperaments }}


A wide-open field. These are regular temperaments of various ranks which temper [[just intonation subgroups]].
A wide-open field. These are regular temperaments of various ranks which temper [[just intonation subgroups]].


== Commatic realms of 11-limit and 13-limit commas ==
== Commatic realms ==
By a ''commatic realm'' is meant the whole collection of regular temperaments of various ranks and for [[subgroup]]s (including full [[prime limit]]s) tempering out a given comma. For some commas, looking at the full commatic realm seems the best approach to discussing associated temperaments.


By a ''commatic realm'' is meant the whole collection of regular temperaments of various ranks and for both full groups and [[Just_intonation_subgroups|subgroups]] tempering out a given comma. For some commas, looking at the full commatic realm seems the best approach to discussing associated temperaments.
; [[The Biosphere]]
: The Biosphere is the name given to the commatic realm of the [[13-limit]] comma 91/90. Its color name is Thozoguti.  


; [[Orgonia|Orgonia or Satrilu-aruru]]
; [[Marveltwin]]
: Orgonia is the commatic realm of the [[11-limit]] comma 65536/65219 = {{Monzo|16 0 0 -2 -3}}, the orgonisma.
: This is the commatic realm of the 13-limit comma 325/324, the [[marveltwin comma]]. Its color name is Thoyoyoti.  


; [[The Biosphere|The Biosphere or Thozogu]]
; [[The Archipelago]]
: The Biosphere is the name given to the commatic realm of the 13-limit comma 91/90.
: The Archipelago is a name which has been given to the commatic realm of the 13-limit comma 676/675 ({{monzo| 2 -3 -2 0 0 2 }}), the [[island comma]]. Its color name is Bithoguti.  


; [[The Archipelago|The Archipelago or Bithogu]]
; [[The Jacobins]]
The Archipelago is a name which has been given to the commatic realm of the [[13-limit]] comma {{Monzo|2 -3 -2 0 0 2}} = 676/675.
: This is the commatic realm of the 13-limit comma 6656/6655, the [[jacobin comma]]. Its color name is Thotrilu-aguti.  


; [[Marveltwin|Marveltwin or Thoyoyo]]
; [[Orgonia]]
: This is the commatic realm of the 13-limit comma 325/324.
: This is the commatic realm of the 11-limit comma 65536/65219 ({{monzo| 16 0 0 -2 -3 }}), the [[orgonisma]]. Its color name is Satrilu-aruruti.
 
; [[The Nexus]]
: This is the commatic realm of the 11-limit comma 1771561/1769472 ({{monzo| -16 -3 0 0 6 }}), the [[nexus comma]]. Its color name is Tribiloti.
 
; [[The Quartercache]]
: This is the commatic realm of the 11-limit comma 117440512/117406179 ({{monzo| 24 -6 0 1 -5 }}), the [[quartisma]]. Its color name is Saquinlu-azoti.  


== Miscellaneous other temperaments ==
== Miscellaneous other temperaments ==
; [[Limmic temperaments]]
: Various subgroup temperaments all tempering out the limma, 256/243.


; [[26th-octave temperaments]]
; [[Fractional-octave temperaments]]
: These temperaments all have a period of 1/26 of an octave.
: These temperaments all have a fractional-octave period.


; [[31 comma temperaments|31-comma temperaments]]
; [[Miscellaneous 5-limit temperaments]]
: These all have a period of 1/31 of an octave.
: High in badness, but worth cataloging for one reason or another.
 
; [[Low harmonic entropy linear temperaments]]
: Temperaments where the average [[harmonic entropy]] of their intervals is low in a particular scale size range.


; [[Turkish maqam music temperaments]]
; [[Turkish maqam music temperaments]]
: Various theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish [[Arabic,_Turkish,_Persian|makam (maqam) music]] in a systematic way. This includes, in effect, certain linear temperaments.
: Various theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish [[Arabic, Turkish, Persian music|makam (maqam) music]] in a systematic way. This includes, in effect, certain linear temperaments.


; [[Very low accuracy temperaments]]
; [[Very low accuracy temperaments]]
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; [[Very high accuracy temperaments]]
; [[Very high accuracy temperaments]]
: Microtemperaments which don't fit in elsewhere.
: Microtemperaments which do not fit in elsewhere.


; [[High badness temperaments]]
; Middle Path tables
: High in badness, but worth cataloging for one reason or another.
: Tables of temperaments where {{nowrap| complexity/7.65 + damage/10 < 1 }}. Useful for beginners looking for a list of a manageable number of temperaments, which approximate harmonious intervals accurately with a manageable number of notes.
:: [[Middle Path table of five-limit rank two temperaments]]
:: [[Middle Path table of seven-limit rank two temperaments]]
:: [[Middle Path table of eleven-limit rank two temperaments]]


== See also ==
== Maps of temperaments ==
* [[Map of rank-2 temperaments]], sorted by generator size
* [[Map of rank-2 temperaments]], sorted by generator size
* [[Catalog of rank two temperaments]]
** [[Catalog of seven-limit rank two temperaments]]
** [[Catalog of eleven-limit rank two temperaments]]
** [[Catalog of thirteen-limit rank two temperaments]]
* [[List of rank two temperaments by generator and period]]
* [[Rank-2 temperaments by mapping of 3]]
* [[Temperaments for MOS shapes]]
* [[Tree of rank two temperaments]]
== Temperament nomenclature ==
* [[Temperament naming]]


== External links ==
== External links ==
* [http://www.huygens-fokker.org/docs/lintemps.html List of temperaments] in [http://www.huygens-fokker.org/scala Scala] with ready to use values
* [http://www.huygens-fokker.org/docs/lintemps.html List of temperaments] in [http://www.huygens-fokker.org/scala Scala] with ready to use values


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[[Category:Overview]]
[[Category:Temperament]]