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The following is a tour of many of the [[regular temperament]]s that contributors to this wiki have found notable. It is of course not meant to be comprehensive, and is not the result of a systematic search.
The following is a tour of many of the [[regular temperament]]s that contributors to this wiki have found notable. It is of course not meant to be comprehensive, and is not the result of a systematic search.


= Rank-2 temperaments =
== 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.


=== [[Suprapyth|Suprapyth or Sayo family]] (P8, P5) ===
; [[Meantone family]] (P8, P5)  
The Sup'''<u>ra</u>'''pyth or Sayo family tempers out {{Monzo|12 -9 1}} = 20480/19683, which equates 5/4 to a Pythagorean augmented 2nd. Being a fourthward comma, it tends to sharpen the 5th, hence it's "super-pythagorean". The best 7-limit extension adds the Archy or Ru comma to make the [[Superpyth|Sup'''<u>e</u>'''rpyth]] temperament.
: 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.  


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


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


===[[Diaschismic family|Diaschismic or Sagugu family]] (P8/2, P5)===
; [[Father 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]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. 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.
: This tempers out [[16/15]], the just diatonic semitone, and equates 5/4 with 4/3. Its color name is Gubiti.  


===[[Bug family|Bug or Gugu family]] (P8, P4/2)===
; [[Diaschismic family]] (P8/2, 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.
: 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.


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


===[[Dicot family|Dicot or Yoyo family]] (P8, P5/2)===
; [[Immunity 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]] makes for a "good" dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the "neutral" dicot 3rds span a 3/2. Tunings include [[7edo]], [[10edo]], and [[17edo]]. An obvious 2.3.11 nterpretation of the generator is ~11/9, which leads to Rastmic aka Neutral aka Lulu.
: 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.


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


===[[Porcupine family|Porcupine or Triyo family]] (P8, P4/3)===
; [[Augmented family]] (P8/3, P5)
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]], [[22edo]], [[37edo]], and [[59edo]]. An important 7-limit extension also tempers out 64/63.
: 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.  


===[[Tricot family|Tricot or Quadsatriyo family]] (P8, P11/3)===
; [[Misty family]] (P8/3, P5)
The tricot family tempers out the [[Tricot|tricot comma]], {{Monzo|39 -29 3}}. The generator is ~59049/40960 = [-13 10 -1> = 633¢, or its octave inverse ~81920/59049 = 567¢. Three of the latter generators equals a compound 4th of ~8/3. 5/4 is equated to 14 octaves minus 29 of these generators. An obvious 7-limit interpretation of the generator is 81/56 = 639¢, a much simpler ratio which leads to the [[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 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.  


===[[Dimipent family|Dimipent or Quadgu family]] (P8/4, P5)===
; [[Porcupine family]] (P8, P4/3)
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]]. 5/4 is equated to 1 fifth minus 1 period.
: 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.


=== [[Negri|Negri or Laquadyo family]] (P8, P4/4) ===
; [[Alphatricot family]] (P8, P11/3)
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 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]].


===[[Tetracot family|Tetracot or Saquadyo family]] (P8, P5/4)===
; [[Diminished family]] (P8/4, P5)
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]].
: 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.  


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


===[[Pental family|Pental or Trila-quingu family]] (P8/5, P5)===
; Negri family (P8, P4/4)  
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.
: 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.  


===[[Ripple family|Ripple or Quingu family]] (P8, P4/5)===
; [[Tetracot family]] (P8, P5/4)
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]] is about as accurate as it can be.  
: 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.  


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


===[[Magic family|Magic or Laquinyo family]] (P8, P12/5)===
; [[Vulture family]] (P8, P12/4)
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]], [[19edo]], [[22edo]], [[25edo]], and [[41edo]] among its possible tunings, with the latter being near-optimal.
: 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.


===[[Fifive family|Fifive or Saquinbiyo family]] (P8/2, P5/5)===
; [[Quintile family]] (P8/5, P5)
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 [[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.


===[[Qintosec family|Qintosec or Quadsa-quinbigu family]] (P8/5, P5/2)===
; [[Ripple family]] (P8, P4/5)
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 [[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.  


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


===[[Ampersand|Ampersand or Lala-tribiyo family]] (P8, P5/6) ===
; [[Quintaleap 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 [[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.


===[[Kleismic family|Kleismic or Tribiyo family]] (P8, P12/6)===
; [[Quindromeda 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]], [[19edo]], [[34edo]], [[49edo]], [[53edo]], [[72edo]], [[87edo]] and [[140edo]] among its possible tunings.
: 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.


===[[Semicomma_family|Orwell or Sepru, and the semicomma or Lasepyo family]] (P8, P12/7)===
; [[Amity family]] (P8, P11/5)
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 [[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.


===[[Wesley family|Wesley or Lasepyobi family]] (P8, ccP4/7)===
; [[Magic family]] (P8, P12/5)
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 [[29-edo]].  
: 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.  


===[[Sensipent family|Sensipent or Sepgu family]] (P8, ccP5/7)===
; [[Fifive family]] (P8/2, P5/5)
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]], [[19edo]], [[46edo]], and [[65edo]]. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzo temperament.  
: 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.  


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


===[[Mutt temperament|Mutt or Trila-septriyo family]] (P8/3, ccP4/7)===
; [[Trisedodge family]] (P8/5, P4/3)
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 [[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.  


===[[Würschmidt family|Würschmidt or Saquadbigu family]] (P8, ccP5/8)===
; Ampersand family (P8, P5/6)  
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]] and [[34edo]] can be used as würschmidt tunings, as can [[65edo]], which is quite accurate.
: 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.


===[[Escapade family|Escapade or Sasa-tritrigu family]] (P8, P4/9)===
; [[Kleismic family]] (P8, P12/6)
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.
: 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.  


===[[Shibboleth family|Shibboleth or Tritriyo family]] (P8, ccP4/9)===
; [[Semicomma family|Orson or semicomma family]] (P8, P12/7)
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 [[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.


===[[Sycamore family|Sycamore or Laleyo family]] (P8, P5/11)===
; [[Wesley family]] (P8, ccP4/7)
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.
: 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]].  


===[[Ditonmic family|Ditonmic or Lala-theyo family]] (P8, c<sup>4</sup>P4/13)===
; [[Sensipent family]] (P8, ccP5/7)
This tempers out the ditonma, {{Monzo|-27 -2 13}} = 1220703125/1207959552. Thirteen ~[-12 -1 6> generators of about 407¢ equals a quadruple-compound 4th. 5/4 is equated to 1 octave minus 2 generators. An obvious 3-limit interpretation of the generator is 81/64, which implies 53-edo, which is a good tuning for this high-accuracy family of temperaments.
: 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.  


===[[Luna family|Luna or Sasa-quintrigu family]] (P8, ccP4/15)===
; [[Vishnuzmic family]] (P8/2, P4/7)
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.  
: 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.  


===[[Minortonic family|Minortonic or Trila-segu family]] (P8, ccP5/17)===
; [[Unicorn family]] (P8, P4/8)
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 [[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.  


===[[Maja family|Maja or Saseyo family]] (P8, c<sup>6</sup>P4/17)===
; [[Würschmidt family]] (P8, ccP5/8)
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.
: 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.  


===[[Maquila family|Maquila or Trisa-segu family]] (P8, c<sup>7</sup>P5/17)===
; [[Escapade family]] (P8, P4/9)
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.
: 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.


===[[Gammic family|Gammic or Laquinquadyo family]] (P8, P5/20)===
; [[Mabila family]] (P8, c4P4/10)
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. 34-edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is Neptune temperament.
: 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.


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


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


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.
; [[Lafa family]] (P8, P12/12)
: 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.  


===[[Archytas clan|Archytas or Ru clan]] (P8, P5)===
; [[Ditonmic family]] (P8, c4P4/13)
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 [[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.  


===[[Trienstonic clan|Trienstonic or Zo clan]] (P8, P5) ===
; [[Minortonic family]] (P8, ccP5/17)
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 [[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.  


===[[Slendro clan|Slendro (Semaphore) or Zozo clan]] (P8, P4/2)===
; [[Maquila family]] (P8, c<sup>7</sup>P5/17)
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 [[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.


=== Sasa-zozo clan (P8, P5/2) ===
; [[Gammic family]] (P8, P5/20)
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.
: 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.  


=== Triru clan (P8/3, P5) ===
=== Clans defined by a 2.3.7 comma ===
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.
These are defined by a no-5's 7-limit (color name: za) comma. See also [[subgroup temperaments]].


=== Trizo clan (P8, P5/3) ===
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 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.


===[[Gamelismic clan|Gamelismic or Latrizo clan]] (P8, P5/3)===
; [[Archytas clan]] (P8, P5)
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.
: 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]].


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 72-EDO.
; [[Trienstonic clan]] (P8, P5)
: This clan tempers out the septimal third-tone, [[28/27]], a low-accuracy temperament that equates 7/6 with 9/8, and 7/4 with 27/16. Its color name is Zoti.  


=== Latriru clan (P8, P11/3) ===
; Harrison 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 [[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]].  


===[[Stearnsmic clan|Stearnsmic or Latribiru clan]] (P8/2, P4/3)===
; [[Garischismic clan]] (P8, P5)
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 [[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.  


=== Laquadru clan (P8, P11/4) ===
; Sasazoti 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 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]].  


=== Saquadru clan (P8, P12/4) ===
; Laruruti clan (P8/2, 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 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.


=== [[Cloudy clan|Cloudy or Laquinzo clan]] (P8/5, P5) ===
; [[Semaphoresmic clan]] (P8, P4/2)
This clan tempers out the [[cloudy comma]], {{Monzo|-14 0 0 5}} = 16807/16384. Five ~8/7 periods equals an 8ve, and four periods equals ~7/4. 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 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]].  


=== Quinru clan (P8, P5/5) ===
; Parahemif clan (P8, P5/2)
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 [[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.


=== Saquinzo clan (P8, P12/5) ===
; Triruti clan (P8/3, 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 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.


=== Lasepzo clan (P8, P11/7) ===
; [[Gamelismic clan]] (P8, P5/3)
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 [[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.


=== Sepru clan (P8, P12/7) ===
; Trizoti 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 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.


=== [[Rastmic temperaments|Rastmic or Neutral or Lulu clan]] (P8, P5/2) ===
; [[Stearnsmic clan]] (P8/2, P4/3)
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 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.


=== Laquadlo clan (P8/2, M2/4) ===
; Skwaresmic clan (P8, P11/4)
This 2.3.11 clan tempers out the Laquadlo comma {{Monzo|-17 2 0 0 4}}. Its half-ocave 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 aka Saquadyobi temperament, which is in the Comic 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.  


== Clans defined by a 2.3.13 (tha) comma ==
; [[Buzzardsmic clan]] (P8, P12/4)
See also [[subgroup temperaments]].
: 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.  


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


=== Satritho clan (P8, P11/3) ===
; Quinruti clan (P8, P5/5)
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 [[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.  


== Clans defined by a 2.5.7 (yaza nowa) comma ==
; Saquinzoti clan (P8, P12/5)
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 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.


=== [[Jubilismic clan|Jubilismic or Biruyo Nowa clan]] (P8/2, M3) ===
; Lasepzoti clan (P8, P11/7)
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 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.


===[[Hemimean clan|Hemimean or Zozoquingu Nowa clan]] (P8, M3/2)===
; Septiness clan (P8, P11/7)
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.
: 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.  


===[[Rainy clan|Rainy or Quinzo-atriyo Nowa clan]] (P8, M3/5)===
; Sepruti clan (P8, P12/7)
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 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.


===[[Quince clan|Quince or Lasepzo-agugu Nowa clan]] (P8, m6/7)===
; [[Septiennealimmal clan]] (P8/9, 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 clan tempers out the [[septimal ennealimma|septiennealimma]], {{monzo| -11 -9 0 9 }} (40353607/40310784). It has a period of 1/9 octave, which represents ~7/6. The generator is ~3/2. This clan includes a number of regular temperaments including [[enneaportent]], [[ennealimmal]], and [[novemkleismic]]. Its color name is Tritrizoti.  


== Clans defined by a 3.5.7 (yaza noca) comma ==
=== Clans defined by a 2.3.11 comma ===
These are defined by a yaza noca or 7-limit-no-twos comma. Any no-twos comma is a ratio of odd numbers. If no other commas are tempered out, It is a non-octave tuning, with a period of a perfect 12th or tritave, 3/1 (or some fraction of that). See also [[subgroup temperaments]]. The pergen's multigen (the 2nd term, omitting any fraction) is always a 5-limit-no-twos ratio such as 5/3, 9/5, 25/9, etc.
Color name: ila. See also [[subgroup temperaments]].


=== [[Arcturus clan|Arcturus or Rutribiyo Noca clan]] (P12, M6) ===
; Lulubiti clan (P8/2, P5)  
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.
: 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.


===[[Sensamagic clan|Sensamagic or Zozoyo Noca clan]] (P12, M6/2)===
; [[Rastmic 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 2.3.11 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.  


===[[Gariboh clan|Gariboh or Triru-aquinyo Noca clan]] (P12, M6/3)===
; [[Nexus clan]] (P8/3, P4/2)
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.
: 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.  


===[[Mirkwai clan|Mirkwai or Quinru-aquadyo Noca clan]] (P12, cm7/5)===
; Alphaxenic or Laquadloti clan (P8/2, M2/4)  
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. 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.  


===[[Procyon clan|Procyon or Sasepzo-atrigu Noca clan]] (P12, m7/7)===
=== Clans defined by a 2.3.13 comma ===
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.
Color name: tha. See also [[subgroup temperaments]].


== Rank-two temperaments defined by a 2.3.5.7 (yaza) comma ==
; Thuthuti clan (P8, P5/2)
These are defined by a full 7-limit (or yaza) comma.
: 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.


===[[Marvel temperaments|Marvel or Ruyoyo temperaments]]===
; Satrithoti clan (P8, P11/3)
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.
: 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.


===[[Starling temperaments|Starling or Zotrigu temperaments]]===
=== Clans defined by a 2.5.7 comma ===
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.
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.


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


===[[Avicennmic temperaments|Avicennmic or Zoyoyo temperaments]]===
; [[Bapbo clan]] (P8, M3/2)
These temper out the avicennma, {{Monzo|-9 1 2 1}} = 525/512, also known as Avicenna's enharmonic diesis.
: 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.  


===[[Keemic temperaments|Keemic or Zotriyo temperaments]]===
; [[Hemimean clan]] (P8, M3/2)
Keemic rank-two temperaments temper out the keema, {{Monzo|-5 -3 3 1}} = 875/864.
: 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.  


===[[Orwellismic temperaments|Orwellismic or Triru-agu temperaments]]===
; Mabilismic clan (P8, cM3/3)
Orwellismic rank-two temperaments temper out orwellisma, {{Monzo|6 3 -1 -3}} = 1728/1715.
: 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.  


===[[Nuwell temperaments|Nuwell or Quadru-ayo temperaments]]===
; Vorwell clan (P8, m6/3)
Nuwell rank-two temperaments temper out the nuwell comma, {{Monzo|1 5 1 -4}} = 2430/2401.
: 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.  


===[[Ragismic microtemperaments|Ragismic or Zoquadyo microtemperaments]]===
; Quinzo-atriyoti Nowa clan (P8, M3/5)
Ragismic rank-two microtemperaments temper out the ragisma, {{Monzo|-1 -7 4 1}} = 4375/4374.
: 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).


===[[Hemifamity temperaments|Hemifamity or Saruyo temperaments]]===
; [[Llywelynsmic clan]] (P8, cM3/7)
Hemifamity rank-two temperaments temper out the hemifamity comma, {{Monzo|10 -6 1 -1}} = 5120/5103.
: 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.  


===[[Horwell temperaments|Horwell or Lazoquinyo temperaments]]===
; [[Quince clan]] (P8, m6/7)
Horwell rank-two temperaments temper out the horwell comma, {{Monzo|-16 1 5 1}} = 65625/65536.
: 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.  


===[[Hemimage temperaments|Hemimage or Satrizo-agu temperaments]]===
; Slither clan (P8, ccm6/9)
Hemimage rank-two temperaments temper out the hemimage comma, {{Monzo|5 -7 -1 3}} = 10976/10935.
: 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.  


===[[Tolermic temperaments|Tolermic or Sazoyoyo temperaments]]===
=== Clans defined by a 3.5.7 comma ===
These temper out the tolerma, {{Monzo|10 -11 2 1}} = 179200/177147.
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.


===[[Mint temperaments|Mint or Rugu temperaments]]===
; Rutribiyoti Noca clan (P12, M6)
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 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.


===[[Septisemi temperaments|Septisemi or Zogu temperaments]]===
; [[Sensamagic clan]] (P12, M6/2)
These are very low complexity temperaments tempering out the minor septimal semitone, [[21/20]] and hence equating 5/3 with 7/4.
: 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.  


===[[Cataharry temperaments|Cataharry or Labirugu temperaments]]===
; [[Gariboh clan]] (P12, M6/3)
Cataharry rank-two temperaments temper out the cataharry comma, {{Monzo|-4 9 -2 -2}} = 19683/19600.
: 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.  


===[[Breedsmic temperaments|Breedsmic or Bizozogu temperaments]]===
; [[Mirkwai clan]] (P12, cm7/5)
Breedsmic rank-two temperaments temper out the breedsma, {{Monzo|-5 -1 -2 4}} = 2401/2400.
: 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.  


===[[Mirwomo temperaments|Mirwomo or Labizoyo temperaments]]===
; Sasepzo-atriguti Noca clan (P12, m7/7)
Mirwomo rank-two temperaments temper out the mirwomo comma, {{Monzo|-15 3 2 2}} = 33075/32768.
: 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).


===[[Landscape microtemperaments|Landscape or Trizogugu temperaments]]===
; Satritrizo-aguguti Noca clan (P12, c<sup>3</sup>M6/9)
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.
: 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).


===[[Dimcomp temperaments|Dimcomp or Quadruyoyo temperaments]]===
; Saquadtrizo-asepguti Noca clan (P12, c<sup>5</sup>m7/12)
Dimcomp rank-two temperaments temper out the dimcomp comma, {{Monzo|-1 -4 8 -4}} = 390625/388962.
: 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.


===[[Sengic temperaments|Sengic or Trizo-agugu temperaments]]===
=== Temperaments defined by a 2.3.5.7 comma ===
Sengic rank-two temperaments temper out the senga, {{Monzo|1 -3 -2 3}} = 686/675.
These are defined by a full 7-limit (color name: yaza) comma.


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


===[[Octagar temperaments|Octagar or Rurutriyo temperaments]]===
; [[Greenwoodmic temperaments]]
Octagar rank-two temperaments temper out the octagar comma, {{Monzo|5 -4 3 -2}} = 4000/3969.
: These temper out the [[greenwoodma]], {{monzo| -3 4 1 -2 }} (405/392). Its color name is Ruruyoti.  


===[[Wizmic microtemperaments|Wizmic or Quinzo-ayoyo temperaments]]===
; [[Keegic temperaments]]
Wizmic rank-two temperaments temper out the wizma, {{Monzo|-6 -8 2 5}} = 420175/419904.
: Keegic rank-2 temperaments temper out the [[keega]], {{monzo| -3 1 -3 3 }} (1029/1000). Its color name is Trizoguti.  


===[[Canousmic temperaments|Canousmic or Saquadzo-atriyo temperaments]]===
; [[Mint temperaments]]
Canousmic rank-two temperaments temper out the canousma, {{Monzo|4 -14 3 4}} = 4802000/4782969.
: 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.  


===[[Triwellismic temperaments|Triwellismic or Tribizo-asepgu temperaments]]===
; [[Avicennmic temperaments]]
Triwellismic rank-two temperaments temper out the ''triwellisma'' (named by [[User:Xenllium|Xenllium]]), {{Monzo|1 -1 -7 6}} = 235298/234375.
: These temper out the [[avicennma]], {{monzo| -9 1 2 1 }} (525/512), also known as Avicenna's enharmonic diesis. Its color name is Zoyoyoti.  


===[[Hewuermera temperaments|Hewuermera or Satribiru-agu temperaments]]===
; Sengic temperaments
Hewuermera rank-two temperaments temper out the ''hewuermera'' comma (named by [[User:Xenllium|Xenllium]]), {{Monzo|16 2 -1 -6}} = 589824/588245.
: Sengic rank-2 temperaments temper out the [[senga]], {{monzo| 1 -3 -2 3 }} (686/675). Its color name is Trizo-aguguti.  


===[[Metric microtemperaments|Metric or Latriru-asepyo temperaments]]===
; [[Keemic temperaments]]
Metric rank-two temperaments temper out the meter comma, {{Monzo|-11 2 7 -3}} = 703125/702464.
: Keemic rank-2 temperaments temper out the [[keema]], {{monzo| -5 -3 3 1 }} (875/864). Its color name is Zotriyoti.  


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


= Rank-3 temperaments =
; Nuwell 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.
: Nuwell rank-2 temperaments temper out the [[nuwell comma]], {{monzo| 1 5 1 -4 }} (2430/2401). Its color name is Quadru-ayoti.  


== Families defined by a 2.3.5 (ya) comma ==
; Mermismic temperaments
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:
: Mermismic rank-2 temperaments temper out the [[mermisma]], {{monzo| 5 -1 7 -7 }} (2500000/2470629). Its color name is Sepruyoti.  


===[[Didymus rank three family|Didymus or Gu rank three family]] (P8, P5, ^1)===
; Negricorn temperaments
These are the rank three temperaments tempering out the didymus or meantone comma, 81/80.  
: Negricorn rank-2 temperaments temper out the [[negricorn comma]], {{monzo| 6 -5 -4 4 }} (153664/151875). Its color name is Saquadzoguti.  


===[[Diaschismic rank three family|Diaschismic or Sagugu rank three family]] (P8/2, P5, /1)===
; Tolermic temperaments
These are the rank three temperaments tempering out the dischisma, {{Monzo|11 -4 -2}} = 2048/2025. The half-octave period is ~45/32.
: These temper out the [[tolerma]], {{monzo| 10 -11 2 1 }} (179200/177147). Its color name is Sazoyoyoti.  


===[[Porcupine rank three family|Porcupine or Triyo rank three family]] (P8, P4/3, /1)===
; Valenwuer temperaments
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.
: Valenwuer rank-2 temperaments temper out the [[valenwuer comma]], {{monzo| 12 3 -6 -1 }} (110592/109375). Its color name is Sarutribiguti.  


===[[Kleismic rank three family|Kleismic or Tribiyo rank three family]] (P8, P12/6, /1)===
; [[Mirwomo temperaments]]
These are the rank three temperaments tempering out the kleisma, {{Monzo|-6 -5 6}} = 15625/15552. In the pergen, P12/6 is ~6/5.
: Mirwomo rank-2 temperaments temper out the [[mirwomo comma]], {{monzo| -15 3 2 2 }} (33075/32768). Its color name is Labizoyoti.  


== Families defined by a 2.3.7 (za) comma ==
; Catasyc temperaments
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:
: Catasyc rank-2 temperaments temper out the [[catasyc comma]], {{monzo| -11 -3 8 -1 }} (390625/387072). Its color name is Laruquadbiyoti.  


===[[Archytas family|Archytas or Ru family]] (P8, P5, ^1)===
; Compass temperaments
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.
: Compass rank-2 temperaments temper out the [[compass comma]], {{monzo| -6 -2 10 -5 }} (9765625/9680832). Its color name is Quinruyoyoti.  


===[[Garischismic family|Garischismic or Sasaru family]] (P8, P5, ^1)===
; Trimyna temperaments
A garischismic temperament is one which tempers out the garischisma, {{Monzo|25 -14 0 -1}} = 33554432/33480783.
: Trimyna rank-2 temperaments temper out the [[trimyna comma]], {{monzo| -4 1 -5 5 }} (50421/50000). Its color name is Quinzoguti.  


===[[Semiphore family|Semiphore or Zozo family]] (P8, P4/2, ^1)===
; [[Starling temperaments]]
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]].
: 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.  


===[[Gamelismic family|Gamelismic or Latrizo family]] (P8, P5/3, ^1)===
; [[Octagar temperaments]]
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.  
: Octagar rank-2 temperaments temper out the [[octagar comma]], {{monzo| 5 -4 3 -2 }} (4000/3969). Its color name is Rurutriyoti.  


===[[Stearnsmic family|Stearnsmic or Latribiru family]] (P8/2, P4/3, ^1)===
; [[Orwellismic temperaments]]
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.
: Orwellismic rank-2 temperaments temper out [[orwellisma]], {{monzo| 6 3 -1 -3 }} (1728/1715). Its color name is Triru-aguti.  


== Families defined by a 2.3.5.7 (yaza) comma ==
; Mynaslendric temperaments
===[[Marvel family|Marvel or Ruyoyo family]] (P8, P5, ^1)===
: Mynaslendric rank-2 temperaments temper out the [[mynaslender comma]], {{monzo| 11 4 1 -7 }} (829440/823543). Its color name is Sepru-ayoti.  
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.  
; [[Mistismic temperaments]]
: Mistismic rank-2 temperaments temper out the [[mistisma]], {{monzo| 16 -6 -4 1 }} (458752/455625). Its color name is Sazoquadguti.  


===[[Starling family|Starling or Zotrigu family]] (P8, P5, ^1)===
; [[Varunismic temperaments]]
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]], but 31, 46 or 58 also work nicely. In the pergen, ^1 = ~81/80.
: Varunismic rank-2 temperaments temper out the [[varunisma]], {{monzo| -9 8 -4 2 }} (321489/320000). Its color name is Labizoguguti.  


===[[Sensamagic family|Sensamagic or Zozoyo family]] (P8, P5, ^1)===
; [[Marvel temperaments]]
These temper out {{Monzo|0 -5 1 2}} = 245/243. In the pergen, ^1 = ~64/63.
: 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.  


===[[Greenwoodmic family|Greenwoodmic or Ruruyo family]] (P8, P5, ^1)===
; Dimcomp temperaments
These temper out the greenwoodma, {{Monzo|-3 4 1 -2}} = 405/392. In the pergen, ^1 = ~64/63.
: Dimcomp rank-2 temperaments temper out the [[dimcomp comma]], {{monzo| -1 -4 8 -4 }} (390625/388962). Its color name is Quadruyoyoti.  


===[[Avicennmic family|Avicennmic or Zoyoyo family]] (P8, P5, ^1)===
; [[Cataharry temperaments]]
These temper out the avicennma, {{Monzo|-9 1 2 1}} = 525/512. In the pergen, ^1 = ~81/80.
: Cataharry rank-2 temperaments temper out the [[cataharry comma]], {{monzo| -4 9 -2 -2 }} (19683/19600). Its color name is Labiruguti.  


===[[Keemic family|Keemic or Zotriyo family]] (P8, P5, ^1)===
; [[Canousmic temperaments]]
These temper out the keema {{Monzo|-5 -3 3 1}} = 875/864. In the pergen, ^1 = ~81/80.
: Canousmic rank-2 temperaments temper out the [[canousma]], {{monzo| 4 -14 3 4 }} (4802000/4782969). Its color name is Saquadzo-atriyoti.  


===[[Orwellismic family|Orwellismic or Triru-agu family]] (P8, P5, ^1)===
; [[Triwellismic temperaments]]
These temper out {{Monzo|6 3 -1 -3}} = 1728/1715. In the pergen, ^1 = ~64/63.
: Triwellismic rank-2 temperaments temper out the [[triwellisma]], {{monzo| 1 -1 -7 6 }} (235298/234375). Its color name is Tribizo-asepguti.  


===[[Nuwell family|Nuwell or Quadru-ayo family]] (P8, P5, ^1)===
; [[Hemimage temperaments]]
These temper out the nuwell comma, {{Monzo|1 5 1 -4}} = 2430/2401. In the pergen, ^1 = ~64/63.
: Hemimage rank-2 temperaments temper out the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} (10976/10935). Its color name is Satrizo-aguti.  


===[[Ragisma family|Ragisma or Zoquadyo family]] (P8, P5, ^1)===
; [[Hemifamity temperaments]]
The 7-limit rank three microtemperament which tempers out the ragisma, {{Monzo|-1 -7 4 1}} = 4375/4374, extends to various higher limit rank three temperaments such as thor. These are not by any means all microtemperaments, but those which are not highly accurate are probably best discussed under another heading. In the pergen, ^1 = ~81/80.
: Hemifamity rank-2 temperaments temper out the [[hemifamity comma]], {{monzo| 10 -6 1 -1 }} (5120/5103). Its color name is Saruyoti.  


===[[Hemifamity family|Hemifamity or Saruyo family]] (P8, P5, ^1)===
; [[Parkleiness temperaments]]
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.
: Parkleiness rank-2 temperaments temper out the [[parkleiness comma]], {{monzo| 7 7 -9 1 }} (1959552/1953125). Its color name is Zotritriguti.  


===[[Horwell family|Horwell or Lazoquinyo family]] (P8, P5, ^1)===
; [[Porwell temperaments]]
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.
: Porwell rank-2 temperaments temper out the [[porwell comma]], {{monzo| 11 1 -3 -2 }} (6144/6125). Its color name is Sarurutriguti.  


===[[Hemimage family|Hemimage or Satrizo-agu family]] (P8, P5, ^1)===
; [[Cartoonismic temperaments]]
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.
: Cartoonismic rank-2 temperaments temper out the [[cartoonisma]], {{monzo| 12 -3 -14 9 }} (165288374272/164794921875). Its color name is Satritrizo-asepbiguti.  


===[[Tolermic family|Tolermic or Sazoyoyo family]] (P8, P5, ^1)===
; [[Hemfiness temperaments]]
These temper out the tolerma, {{Monzo|10 -11 2 1}} = 179200/177147. In the pergen, ^1 = ~81/80.
: Hemfiness rank-2 temperaments temper out the [[hemfiness comma]], {{monzo| 15 -5 3 -5 }} (4096000/4084101). Its color name is Saquinru-atriyoti.  


===[[Mint family|Mint or Rugu family]] (P8, P5, ^1)===
; [[Hewuermera temperaments]]
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.
: Hewuermera rank-2 temperaments temper out the [[hewuermera comma]], {{monzo| 16 2 -1 -6 }} (589824/588245). Its color name is Satribiru-aguti.  


===[[Septisemi family|Septisemi or Zogu family]] (P8, P5, ^1)===
; [[Lokismic temperaments]]
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.
: Lokismic rank-2 temperaments temper out the [[lokisma]], {{monzo| 21 -8 -6 2 }} (102760448/102515625). Its color name is Sasa-bizotriguti.  


===[[Jubilismic family|Jubilismic or Biruyo family]] (P8/2, P5, ^1)===
; Decovulture temperaments
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.
: Decovulture rank-2 temperaments temper out the [[decovulture comma]], {{monzo| 26 -7 -4 -2 }} (67108864/66976875). Its color name is Sasabiruguguti.  


===[[Cataharry family|Cataharry or Labirugu family]] (P8, P4/2, ^1)===
; Pontiqak temperaments
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.
: Pontiqak rank-2 temperaments temper out the [[pontiqak comma]], {{monzo| -17 -6 9 2 }} (95703125/95551488). Its color name is Lazozotritriyoti.  


===[[Breed family|Breed or Bizozogu family]] (P8, P5/2, ^1)===
; [[Mitonismic temperaments]]
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.
: Mitonismic rank-2 temperaments temper out the [[mitonisma]], {{monzo| -20 7 -1 4 }} (5250987/5242880). Its color name is Laquadzo-aguti.  


===[[Mirwomo family|Mirwomo or Labizoyo family]] (P8, P5/2, ^1)===
; [[Horwell temperaments]]
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.
: Horwell rank-2 temperaments temper out the [[horwell comma]], {{monzo| -16 1 5 1 }} (65625/65536). Its color name is Lazoquinyoti.  


===[[Trimyna family|Trimyna or Quinzogu family]] (P8, ccP4/5, ^1)===
; Neptunismic temperaments
The trimyna family of rank three temperaments tempers out the trimyna comma, {{Monzo|-4 1 -5 5}} = 50421/50000. In the pergen, 1/5 of double-compound fourth is ~7/5, and ^1 = ~81/80.
: Neptunismic rank-2 temperaments temper out the [[neptunisma]], {{monzo| -12 -5 11 -2 }} (48828125/48771072). Its color name is Laruruleyoti.  


===[[Sengic family|Sengic or Trizo-agugu family]] (P8, P5, vm3/2)===
; [[Metric microtemperaments]]
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.
: Metric rank-2 temperaments temper out the [[meter]], {{monzo| -11 2 7 -3 }} (703125/702464). Its color name is Latriru-asepyoti.  


===[[Porwell family|Porwell or Sarurutrigu family]] (P8, P5, ^m3/2)===
; [[Wizmic microtemperaments]]
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.
: Wizmic rank-2 temperaments temper out the [[wizma]], {{monzo| -6 -8 2 5 }} (420175/419904). Its color name is Quinzo-ayoyoti.  


===[[Octagar family|Octagar or Rurutriyo family]] (P8, P5, ^m6/2)===
; [[Supermatertismic temperaments]]
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.
: Supermatertismic rank-2 temperaments temper out the [[supermatertisma]], {{monzo| -6 3 9 -7 }} (52734375/52706752). Its color name is Lasepru-atritriyoti.  


===[[Hemimean family|Hemimean or Zozoquingu family]] (P8, P5, vM3/2)===
; [[Breedsmic temperaments]]
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.
: Breedsmic rank-2 temperaments temper out the [[breedsma]], {{monzo| -5 -1 -2 4 }} (2401/2400). Its color name is Bizozoguti.  


===[[Wizmic family|Wizmic or Quinzo-ayoyo family]] (P8, P5, vm7/2)===
; Supermasesquartismic temperaments
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.
: Supermasesquartismic rank-2 temperaments temper out the [[supermasesquartisma]], {{monzo| -5 10 5 -8 }} (184528125/184473632). Its color name is Laquadbiru-aquinyoti.  


=== [[Landscape family|Landscape or Trizogugu family]] (P8/3, P5, ^1) ===
; [[Ragismic microtemperaments]]
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.
: Ragismic rank-2 temperaments temper out the [[ragisma]], {{monzo| -1 -7 4 1 }} (4375/4374). Its color name is Zoquadyoti.  


===[[Canou family|Canou or Saquadzo-atriyo family]] (P8, P5, vm6/3)===
; Akjaysmic temperaments
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.  
: 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.  


=== [[Dimcomp family|Dimcomp or Quadruyoyo family]] (P8/4, P5, ^1) ===
; [[Landscape microtemperaments]]
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.
: 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.


===[[Mirkwai family|Mirkwai or Quinru-aquadyo family]] (P8, P5, c^M7/4)===
== Rank-3 temperaments ==
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.
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.


=[[Rank_four_temperaments|Rank-4 temperaments]]=
=== Families defined by a 2.3.5 comma ===
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.
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:


===[[Valinorismic temperaments|Valinorismic or Lorugugu temperaments]] ===
; [[Didymus rank three family|Didymus rank-3 family]] (P8, P5, ^1)
These temper out the valinorsma, {{Monzo|4 0 -2 -1 1}} = 176/175.
: These are the rank-3 temperaments tempering out the didymus or meantone comma, 81/80. Its color name is Guti.  


===[[Rastmic temperaments|Rastmic or Lulu temperaments]]===
; [[Diaschismic rank three family|Diaschismic rank-3 family]] (P8/2, P5, /1)
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 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.  


===[[Werckismic temperaments|Werckismic or Luzozogu temperaments]]===
; [[Porcupine rank three family|Porcupine rank-3 family]] (P8, P4/3, /1)
These temper out the werckisma, {{Monzo|-3 2 -1 2 -1}} = 441/440.
: 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.  


===[[Swetismic temperaments|Swetismic or Lururuyo temperaments]]===
; [[Kleismic rank three family|Kleismic rank-3 family]] (P8, P12/6, /1)
These temper out the swetisma, {{Monzo|2 3 1 -2 -1}} = 540/539.
: 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.  


===[[Lehmerismic temperaments|Lehmerismic or Loloruyoyo temperaments]]===
=== Families defined by a 2.3.7 comma ===
These temper out the lehmerisma, {{Monzo|-4 -3 2 -1 2}} = 3025/3024.
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:


===[[Kalismic temperaments|Kalismic or Bilorugu temperaments]]===
; [[Archytas family]] (P8, P5, ^1)
These temper out the kalisma, {{Monzo|-3 4 -2 -2 2}} = 9801/9800.
: 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.  


=[[Subgroup temperaments]]=
; [[Garischismic family]] (P8, P5, ^1)
: A garischismic temperament is one which tempers out the garischisma, {{monzo| 25 -14 0 -1 }} (33554432/33480783). Its color name is Sasaruti.
 
; Laruruti clan (P8/2, P5)
: This clan tempers out the Laruru comma, {{monzo| -7 8 0 -2 }} (6561/6272). Two ~81/56 periods equal an octave. The generator is ~3/2, and four generators minus three periods equals ~7/4. The major second ~9/8 is divided in half, with each half equated to ~28/27. See also the diaschismic or Saguguti temperament and the jubilismic or Biruyoti temperament.
 
; [[Semaphoresmic family]] (P8, P4/2, ^1)
: Semaphoresmic temperament tempers out 49/48 and thereby identifies the septimal minor third 7/6 with the septimal whole tone 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like ''semi-fourth''. See also [[semaphore]]. Its color name is Zozoti.
 
; [[Gamelismic family]] (P8, P5/3, ^1)
: Not to be confused with the gamelismic clan of rank-2 temperaments, the gamelismic family are those rank-3 temperaments which temper out the gamelisma, {{monzo| -10 1 0 3 }} (1029/1024). In the pergen, P5/3 is ~8/7. Its color name is Latrizoti.
 
; Stearnsmic family (P8/2, P4/3, ^1)
: Stearnsmic temperaments temper out the stearnsma, {{monzo| 1 10 0 -6 }} (118098/117649). In the pergen, P8/2 is ~343/243 and P4/3 is ~54/49. Its color name is Latribiruti.
 
=== Families defined by a 2.3.5.7 comma ===
Color name: yaza.
 
; [[Marvel family]] (P8, P5, ^1)
: The head of the marvel family is marvel, which tempers out the [[marvel comma]], {{monzo| -5 2 2 -1 }} (225/224). It divides 8/7 into two 16/15's, or equivalently, two 15/14's. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.
: The marvel comma equates every 7-limit interval to some 5-limit interval, therefore the generators are the same as for 5-limit JI: 2/1, 3/1 and 5/1. These may be reduced to 2/1, 3/2 and 5/4, and 5/4 may be reduced further to 81/80. Hence in the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Ruyoyoti.
 
; [[Starling family]] (P8, P5, ^1)
: Starling tempers out the [[starling comma]] a.k.a. septimal semicomma, {{monzo| 1 2 -3 1 }} (126/125), the difference between three 6/5's plus one 7/6, and an octave. It divides 10/7 into two 6/5's. Like marvel, it has the same generators as 5-limit JI. An excellent tuning for starling is [[77edo]], but 31, 46 or 58 also work nicely. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Zotriguti.
 
; [[Sensamagic family]] (P8, P5, ^1)
: These temper out {{monzo| 0 -5 1 2 }} (245/243), which divides 16/15 into two 28/27's. In the pergen, {{nowrap| ^1 {{=}} ~64/63 }}. Its color name is Zozoyoti.
 
; Greenwoodmic family (P8, P5, ^1)
: These temper out the greenwoodma, {{monzo| -3 4 1 -2 }} (405/392), which divides 10/9 into two 15/14's. In the pergen, {{nowrap| ^1 {{=}} ~64/63 }}. Its color name is Ruruyoti.
 
; Avicennmic family (P8, P5, ^1)
: 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]] (P8, P5, ^1)
: 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]] (P8, P5, ^1)
: 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]] (P8, P5, ^1)
: 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]] (P8, P5, ^1)
: 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]] (P8, P5, ^1)
: 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]] (P8, P5, ^1)
: 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]] (P8, P5, ^1)
: 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.
 
; [[Mint family]] (P8, P5, ^1)
: The mint temperament is a low-complexity, high-error temperament, tempering out the septimal quartertone 36/35, equating 7/6 with 6/5, and 5/4 with 9/7. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }} or ~64/63. Its color name is Ruguti.
 
; Septisemi family (P8, P5, ^1)
: These are very low-accuracy temperaments tempering out the minor septimal semitone, [[21/20]], and hence equating 5/3 with 7/4. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Zoguti.
 
; [[Jubilismic family]] (P8/2, P5, ^1)
: Jubilismic temperament tempers out 50/49 and thereby equates the two septimal tritones, 7/5 and 10/7. This is the half-octave period. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Biruyoti.
 
; [[Cataharry family]] (P8, P4/2, ^1)
: 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.
 
; [[Breed family]] (P8, P5/2, ^1)
: 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.
 
; [[Sengic family]] (P8, P5, vm3/2)
: 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.
 
; [[Porwell family]] (P8, P5, ^m3/2)
: 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.
 
; [[Octagar family]] (P8, P5, ^m6/2)
: 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.
 
; [[Hemimean family]] (P8, P5, vM3/2)
: 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.
 
; Wizmic family (P8, P5, vm7/2)
: 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.
 
; [[Landscape family]] (P8/3, P5, ^1)
: 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.
 
; [[Gariboh family]] (P8, P5, vM6/3)
: 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.
 
; [[Canou family]] (P8, P5, vm6/3)
: 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.
 
; [[Dimcomp family]] (P8/4, P5, ^1)
: 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.
 
; [[Mirkwai family]] (P8, P5, c^M7/4)
: 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.
 
=== Temperaments defined by an 11-limit comma ===
; [[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.
 
; [[Biyatismic clan]] (P8, P5, ^1)
: 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.
 
; [[Valinorsmic clan]]
: 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 rank three clan|Rastmic rank-3 clan]]
: 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.
 
; [[Pentacircle clan]] (P8, P5, ^1)
: 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.
 
; [[Semicanousmic clan]] (P8, P5, ^1)
: 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.
 
; [[Semiporwellismic clan]] (P8, P5, ^1)
: 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.
 
; [[Olympic clan]] (P8, P5, ^1)
: 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.
 
; [[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.
 
; [[The Biosphere]]
: The Biosphere is the name given to the commatic realm of the [[13-limit]] comma 91/90. Its color name is Thozoguti.
 
; [[Marveltwin]]
: This is the commatic realm of the 13-limit comma 325/324, the [[marveltwin comma]]. Its color name is Thoyoyoti.
 
; [[The Archipelago]]
: The Archipelago is a name which has been given to the commatic realm of the 13-limit comma 676/675 ({{monzo| 2 -3 -2 0 0 2 }}), the [[island comma]]. Its color name is Bithoguti.
 
; [[The Jacobins]]
: This is the commatic realm of the 13-limit comma 6656/6655, the [[jacobin comma]]. Its color name is Thotrilu-aguti.
 
; [[Orgonia]]
: This is the commatic realm of the 11-limit comma 65536/65219 ({{monzo| 16 0 0 -2 -3 }}), the [[orgonisma]]. Its color name is Satrilu-aruruti.


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 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.  


==[[Orgonia|Orgonia or Satrilu-aruru]]==
; [[The Quartercache]]
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 11-limit comma 117440512/117406179 ({{monzo| 24 -6 0 1 -5 }}), the [[quartisma]]. Its color name is Saquinlu-azoti.  


==[[The Biosphere|The Biosphere or Thozogu]] ==
== Miscellaneous other temperaments ==
The Biosphere is the name given to the commatic realm of the 13-limit comma 91/90.
; [[Limmic temperaments]]
: Various subgroup temperaments all tempering out the limma, 256/243.


==[[The Archipelago|The Archipelago or Bithogu]]==
; [[Fractional-octave temperaments]]
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.
: These temperaments all have a fractional-octave period.


==[[Marveltwin|Marveltwin or Thoyoyo]] ==
; [[Miscellaneous 5-limit temperaments]]
This is the commatic realm of the 13-limit comma 325/324.
: High in badness, but worth cataloging for one reason or another.


= Miscellaneous other temperaments =
; [[Low harmonic entropy linear temperaments]]
: Temperaments where the average [[harmonic entropy]] of their intervals is low in a particular scale size range.


===[[26th-octave temperaments]]===
; [[Turkish maqam music temperaments]]
These temperaments all have a period of 1/26 of an octave.
: 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.


===[[31 comma temperaments|31-comma temperaments]]===
; [[Very low accuracy temperaments]]
These all have a period of 1/31 of an octave.
: All hope abandon ye who enter here.


===[[Turkish maqam music temperaments]]===
; [[Very high accuracy 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.
: Microtemperaments which do not fit in elsewhere.


===[[Very low accuracy temperaments]]===
; Middle Path tables
All hope abandon ye who enter here.
: 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]]


===[[Very high accuracy temperaments]]===
== Maps of temperaments ==
Microtemperaments which don't fit in elsewhere.
* [[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]]


===[[High badness temperaments]]===
== Temperament nomenclature ==
High in badness, but worth cataloging for one reason or another.
* [[Temperament naming]]


= 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
* [[Map_of_rank-2_temperaments|Map of rank-2 temperaments]], sorted by generator size


[[Category:Regular temperament theory| ]] <!-- main article -->
[[Category:Lists of temperaments]] <!-- main article -->
[[Category:Overview]]
[[Category:Temperament]]