Tour of regular temperaments: Difference between revisions

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Deleted duplicate temperaments, merged "Temperaments for a given comma" section into "Rank-3 temperaments"
→Rank-2 temperaments: dimipent -> diminished
 
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<span style="display: block; text-align: right;">[[de:Reguläre_Temperaturen]]  
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.
[[レギュラーテンペラメントとランクrテンペラメント|日本語]]</span>
=Regular temperaments=


Regular temperaments are non-Just tunings in which the infinite number of intervals in [[Harmonic_Limit|p-limit]] [[Just_intonation|Just intonation]], or any [[Just_intonation_subgroups|subgroup]] thereof, are mapped to a smaller, though still infinite, set of [[tempering_out|tempered]] intervals. This is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the "dimensionality" of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.
== Rank-2 temperaments ==
A rank-2 temperament maps all JI intervals within its [[JI subgroup]] to a 2-dimensional lattice. This lattice has two generators. The larger generator is called the period, as the temperament will repeat periodically at that interval. The period is usually the octave, or some unit fraction of the octave. If the period is a full octave, the temperament is said to be a '''linear temperament'''. The smaller generator, referred to as "the" generator, is stacked repeatedly to generate a scale within that period.


A rank r [http://en.wikipedia.org/wiki/Regular_temperament regular temperament] in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r [[vals]]. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of r independent vals defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[Comma_pump_examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.
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.


==Why would I want to use a regular 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 81/80 out of 3-dimensional 5-limit JI), can be reduced to 1-dimensional 12-ET by tempering out the Pythagorean comma.


Regular temperaments are of most use to musicians who want their music to sound as much as possible like Just intonation, but without the difficulties normally associated with JI, such as wolf intervals, commas, and comma pumps. They are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals.
=== Families defined by a 2.3 comma ===
These are families defined by a 3-limit (color name: wa) comma. If only primes 2 and 3 are part of the [[subgroup]], the comma creates a rank-1 temperament, an [[edo]]. But if another prime such as 5 is present, the comma creates a rank-2 temperament. Since edos are discussed elsewhere, this section assumes the presence of at least one additional prime. The rank-2 temperament created consists of multiple "copies" of an edo. The edo copies can be thought of as being offset from one another by a small comma. This small comma is represented in the [[pergen]] by ^1.  


==What do I need to know to understand all the numbers on the pages for individual regular temperaments?==
; Blackwood family (P8/5, ^1)
: This family tempers out the [[limma]], {{monzo| 8 -5 }} (256/243). It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~720¢, quite sharp. The only member of this family is the [[blackwood]] temperament, which is 5-limit. Blackwood's edo copies are offset from one another by 5/4, or alternatively by 81/80. 5/4 is usually tempered sharp, perhaps ~400¢, to match the sharp fifth. Its color name is Sawati.


Although the concept of regular temperament is centuries old and predates much of modern mathematics, members of the Yahoo! Alternative Tuning List have developed a particular form of numerical shorthand for describing the properties of temperaments. The most important of these are vals and commas, which any student of the modern regular temperament paradigm should become familiar with. These concepts are rather straightforward and require little math to understand.
; [[Whitewood family]] (P8/7, ^1)
: This family tempers out the apotome, {{monzo| -11 7 }} (2187/2048). It equates 7 fifths with 4 octaves, which creates multiple copies of [[7edo]]. The fifth is ~685¢, which is very flat. This family includes the [[whitewood]] temperament. Its color name is Lawati.  


Another recent contribution to the field of temperament is the concept of optimization, which can take many forms. The point of optimization is to minimize the difference between a temperament and JI by finding an optimal tuning for the generator. The two most frequently used forms of optimization are [[POTE_tuning|POTE]] ("Pure-Octave Tenney-Euclidean") and [[TOP_tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please"). Optimization is rather intensive mathematically, but it is seldom left as an exercise to the reader; most temperaments are presented here in their optimal forms.
; [[Compton family]] (P8/12, ^1)
: This tempers out the [[Pythagorean comma]], {{monzo| -19 12 }} (531441/524288). It equates 12 fifths with 7 octaves, which creates multiple copies of [[12edo]]. Temperaments in this family include [[compton]] and [[catler]]. In the 5-limit compton temperament, the edo copies are offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12edo instruments slightly detuned from each other provide an easy way to make music with these temperaments. Its color name is Lalawati.  


Yet another recent development is the concept of a [[pergen]], appearing here as (P8, P5/2) or somesuch. Every rank-2, rank-3, rank-4, etc. temperament has a pergen, which specifies the period and the generator. Assuming the prime subgroup includes both 2 and 3, the period is either an octave or some fraction of it, and the generator is either some 3-limit interval or some fraction of one. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every strong extension of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but don't uniquely identify them. "W" in a pergen means "widened by one 8ve", e.g. WWP5 is a 5th plus two 8ves, or 6/1.
; [[Countercomp family]] (P8/41, ^1)
: This family tempers out the [[41-comma|Pythagorean countercomma]], {{monzo| 65 -41 }}, which creates multiple copies of [[41edo]]. Its color name is Wa-41.  


Each temperament has two names: a traditional name and a [[Color notation|color name]]. The traditional names are [[Temperament Names|arbitrary and ideosyncratic]], but the color names are systematic and rigorous, and the comma can be deduced from the color name. Wa = 3-limit, yo = 5-over, gu = 5-under, zo = 7-over, and ru = 7-under. See also [[Color notation/Temperament Names|Color Notation/Temperament Names]].
; [[Mercator family]] (P8/53, ^1)
: This family tempers out the [[Mercator's comma]], {{monzo| -84 53 }}, which creates multiple copies of [[53edo]]. Its color name is Wa-53.  


==Equal temperaments (Rank-1 temperaments)==
=== Families defined by a 2.3.5 comma ===
These are families defined by a 5-limit (color name: ya) comma. As we go up in rank 2, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) The same comment applies to 7-limit temperaments and rank 3, etc. Members of families and their relationships can be classified by the [[normal lists|normal comma list]] of the various temperaments. Families include weak extensions as well as strong, in other words, the [[pergen]] shown here may change.


[[Equal-step tuning|Equal temperaments]] (abbreviated ET or tET) and [[EDO|equal divisions of the octave]] (abbreviated EDO or ED2) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator, making it a rank-1 temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval, let alone the octave, and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO, although one can also use unsupported vals or poorly-supported vals to achieve "fun" results. The familiar 12-note equal temperament, or 12edo, reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-ET.
; [[Meantone family]] (P8, P5)  
: The meantone family tempers out [[81/80]], also called the syntonic comma. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)<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.  


=Rank-2 (including linear) temperaments=
; [[Schismatic family]] (P8, P5)
: The schismatic family tempers out the schisma of {{monzo| -15 8 1 }} ([[32805/32768]]), which is the amount by which the Pythagorean comma exceeds the syntonic comma. The 5-limit version of the temperament is a [[microtemperament]] which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity; whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4. Tunings include [[12edo|12-]], [[29edo|29-]], [[41edo|41-]], [[53edo|53-]], and [[118edo]]. Its color name is Layoti.


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 that is not already tempered out. 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.
; [[Mavila family]] (P8, P5)
: This tempers out the mavila comma, {{monzo| -7 3 1 }} ([[135/128]]), also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them stacked together and octave reduced leads you to 6/5 instead of 5/4, and one consequence of this is that it generates [[2L 5s]] (antidiatonic) scales. 5/4 is equated to 3 fourths minus 1 octave. Septimal mavila and armodue are some of the most notable temperaments associated with the mavila comma. Tunings include [[9edo|9-]], [[16edo|16-]], [[23edo|23-]], and [[25edo]]. Its color name is Layobiti.  


Regular temperaments of ranks two and three are cataloged on the [[Optimal patent val]] page. Rank-2 temperaments are also listed at [[Proposed names for rank 2 temperaments]] by their generator mappings, and at [[Map of rank-2 temperaments]] by their generator size. See also the [[pergen]]s page. There is also [[Graham Breed]]'s [http://x31eq.com/catalog2.html giant list of regular temperaments].
; [[Father family]] (P8, P5)
: This tempers out [[16/15]], the just diatonic semitone, and equates 5/4 with 4/3. Its color name is Gubiti.  


== Families ==
; [[Diaschismic family]] (P8/2, P5)
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.
: 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.


===[[Meantone family|Meantone or Gu family]] (P8, P5) ===
; [[Bug family]] (P8, P4/2)
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 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.


===[[Schismatic family|Schismatic or Layo family]] (P8, P5)===
; [[Immunity family]] (P8, P4/2)
The schismatic family tempers out the schisma of [-15 8 1> = [[32805/32768]], which is the amount by which the Pythagorean comma exceeds the syntonic comma. The 5-limit version of the temperament is a [[Microtempering|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]].
: 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.


===[[Kleismic family|Kleismic or Tribiyo family]] (P8, P12/6)===
; [[Dicot family]] (P8, P5/2)
The kleismic family of temperaments tempers out the [[kleisma]] [-6 -5 6> = 15625/15552, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. The kleismic family includes [[15edo]], [[19edo]], [[34edo]], [[49edo]], [[53edo]], [[72edo]], [[87edo]] and [[140edo]] among its possible tunings.
: 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.


===[[Magic family|Magic or Laquinyo family]] (P8, P12/5)===
; [[Augmented family]] (P8/3, P5)
The magic family tempers out [-10 -1 5> (3125/3072), known as the [[magic comma]] or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/1. The generator is itself an approximate 5/4. The magic family includes [[16edo]], [[19edo]], [[22edo]], [[25edo]], and [[41edo]] among its possible tunings, with the latter being near-optimal.
: 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.  


===[[Diaschismic family|Diaschismic or Sagugu family]] (P8/2, P5)===
; [[Misty family]] (P8/3, P5)
The diaschismic family tempers out the [[diaschisma]], [11 -4 -2> or 2048/2025, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. A noted 7-limit extension to diaschismic is [[pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out, of which [[22edo]] is an excellent tuning.
: 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.  


===[[Pelogic family|Pelogic or Layobi family]] (P8, P5)===
; [[Porcupine family]] (P8, P4/3)
This tempers out the pelogic comma, [-7 3 1> = [[135/128]], also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates [[2L 5s]] "anti-diatonic" scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo]], [[16edo]], [[23edo]], and [[25edo]].
: 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.


===[[Porcupine family|Porcupine or Triyo family]] (P8, P4/3)===
; [[Alphatricot family]] (P8, P11/3)
The porcupine family tempers out [1 -5 3> = [[250/243]], the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include [[15edo]], [[22edo]], [[37edo]], and [[59edo]].
: 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]].


===[[Würschmidt family|Würschmidt or Saquadbigu family]] (P8, WWP5/8)===
; [[Diminished family]] (P8/4, P5)
The würschmidt (or wuerschmidt) family tempers out the [[Würschmidt comma]], 393216/390625 = [17 1 -8&gt;. Würschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a perfect 5th two octaves up); 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.
: 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.  


===[[Augmented_family|Augmented or Trigu  family]] (P8/3, P5)===
; [[Undim family]] (P8/4, P5)
The augmented family tempers out the diesis of [7 0 -3> = [[128/125]], the difference between three 5/4 major thirds and a 2/1 octave, and so identifies the major third with the third-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]], which is an excellent tuning for augmented. It is the temperament that results in what is commonly called the "augmented scale" ([[3L 3s]]) in common 12-based music theory, as well as what is commonly called "[http://www.tcherepnin.com/alex/basic_elem1.htm#9step Tcherepnin's scale]" ([[3L 6s]]).
: 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.  


===[[Dimipent family|Dimipent or Quadgu family]] (P8/4, P5)===
; Negri family (P8, P4/4)  
The dimipent (or diminished) family tempers out the major diesis or diminished comma, [3 4 -4> or [[648/625]], the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as [[12edo]].
: 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.  


===[[Dicot family|Dicot or Yoyo family]] (P8, P5/2)===
; [[Tetracot family]] (P8, P5/4)
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 two neutral-sized intervals that are 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]].
: 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.  


===[[Tetracot family|Tetracot or Saquadyo family]] (P8, P5/4)===
; [[Smate family]] (P8, P11/4)
The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by [5 -9 4> (20000/19683), the minimal diesis or [[tetracot comma]]. [[7edo]] can also be considered a tetracot tuning, as can [[20edo]], [[27edo]], [[34edo]], and [[41edo]].
: 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.  


===[[Sensipent family|Sensipent or Sepgu family]] (P8, WWP5/7)===
; [[Vulture family]] (P8, P12/4)
The sensipent (sensi) family tempers out the [[sensipent comma]], [2 9 -7> (78732/78125), also known as the medium semicomma. Tunings include [[8edo]], [[19edo]], [[46edo]], and [[65edo]].
: 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.


===[[Semicomma_family|Orwell or Sepru, and the semicomma or Lasepyo family]] (P8, P12/7)===
; [[Quintile family]] (P8/5, P5)
The [[semicomma]] (also known as Fokker's comma), 2109375/2097152 = [-21 3 7&gt;, is tempered out by the members of the semicomma family. It doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as 7/6, leading to [[orwell]] temperament.
: 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.


===[[Pythagorean family|Pythagorean or Lalawa family]] (P8/12, ^1)===
; [[Ripple family]] (P8, P4/5)
The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = [-19 12 0&gt;. Since this is a 3-limit comma, it is also a 5-limit comma and can stand as parent to a 7-limit or higher family, in this case containing compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate chains of 12-equal, offset from one another justly tuned 5/4.
: This tempers out the [[ripple comma]], 6561/6250 ({{monzo| -1 8 -5 }}), which equates a stack of four [[10/9]]'s with [[8/5]], 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.  


===[[Apotome family|Apotome or Lawa family]] (P8/7, ^1)===
; [[Passion family]] (P8, P4/5)
This family tempers out the apotome, [-11 7 0> = 2187/2048, which is a 3-limit comma.
: 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.  


===[[Gammic family|Gammic or Laquinquadyo family]] (P8, P5/20)===
; [[Quintaleap family]] (P8, P4/5)
The gammic family tempers out the gammic comma, [-29 -11 20&gt;. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is Neptune 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.


===[[Minortonic family|Minortonic or Trila-segu family]] (P8, WWP5/17)===
; [[Quindromeda family]] (P8, P4/5)
This tempers out the minortone comma, [-16 35 -17&gt;. The head of the family is minortonic temperament, with a generator of a minor tone (~10/9).
: 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.


===[[Bug family|Bug or Gugu family]] (P8, P4/2)===
; [[Amity family]] (P8, P11/5)
This tempers out [[27/25]], the large limma or bug comma.
: 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.


===[[Father family|Father or Gubi family]] (P8, P5)===
; [[Magic family]] (P8, P12/5)
This tempers out [[16/15]], the just diatonic semitone.
: 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.  


===[[Sycamore family|Sycamore or Laleyo family]] (P8, P5/11)===
; [[Fifive family]] (P8/2, P5/5)
The sycamore family tempers out the sycamore comma, [-16 -6 11&gt; = 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.
: 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.  


===[[Escapade family|Escapade or Sasa-tritrigu family]] (P8, P4/9)===
; [[Quintosec family]] (P8/5, P5/2)
This tempers out the [[escapade comma]], [32 -7 -9&gt;, which is the difference between nine just major thirds and seven just fourths.
: 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.  


===[[Amity family|Amity or Saquinyo family]] (P8, P11/5)===
; [[Trisedodge family]] (P8/5, P4/3)
This tempers out the [[amity comma]], 1600000/1594323 = [9 -13 5&gt;.
: 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.  


===[[Vulture family|Vulture or Sasa-quadyo family]] (P8, P12/4)===
; Ampersand family (P8, P5/6)  
This tempers out the [[vulture comma]], [24 -21 4&gt;.
: 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.


===[[Vishnuzmic family|Vishnuzmic or Sasepbigu family]] (P8/2, P4/7)===
; [[Kleismic family]] (P8, P12/6)
This tempers out the vishnuzma, [23 6 -14&gt;, or the amount by which seven chromatic semitones (25/24) fall short of a perfect fourth (4/), or (4/3)/(25/24)^7.
: 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.  


===[[Luna family|Luna or Sasa-quintrigu family]] (P8, WWP4/15)===
; [[Semicomma family|Orson or semicomma family]] (P8, P12/7)
This tempers out the luna comma, [38 -2 -15&gt; (274877906944/274658203125)
: 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.


===[[Laconic family|Laconic or Latrigubi family]] (P8, P5/3)===
; [[Wesley family]] (P8, ccP4/7)
This tempers out the laconic comma, [-4 7 -3> (2187/2000), which is the difference between three 10/9's and one 3/2. Laconic is supported by [[16edo]], [[21edo]], and [[37edo]] (using the 37b mapping), among others.
: 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]].  


===[[Immunity family|Immunity or Sasa-yoyo family]] (P8, P4/2)===
; [[Sensipent family]] (P8, ccP5/7)
This tempers out the immunity comma, [16 -13 2> (1638400/1594323).
: 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.  


===[[Ditonmic family|Ditonmic or Lala-theyo family]] (P8, W<sup>4</sup>P4/13)===
; [[Vishnuzmic family]] (P8/2, P4/7)
This tempers out the ditonma, [-27 -2 13> = 1220703125/1207959552.
: 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.  


===[[Shibboleth family|Shibboleth or Tritriyo family]] (P8, WWP4/9)===
; [[Unicorn family]] (P8, P4/8)
This tempers out the shibboleth comma, [-5 -10 9> = 1953125/1889568.
: 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.  


===[[Comic family|Comic or Saquadyobi family]] (P8/2, M2/4)===
; [[Würschmidt family]] (P8, ccP5/8)
This tempers out the comic comma, [13 -14 4> = 5120000/4782969.
: 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.  


===[[Wesley family|Wesley or Lasepyobi family]] (P8, WWP4/7)===
; [[Escapade family]] (P8, P4/9)
This tempers out the wesley comma, [-13 -2 7> = 78125/73728.
: 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.


===[[Fifive family|Fifive or Saquinbiyo family]] (P8/2, P5/5)===
; [[Mabila family]] (P8, c4P4/10)
This tempers out the fifive comma, [-1 -14 10> = 9765625/9565938.
: 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.


===[[Maja family|Maja or Saseyo family]] (P8, W<sup>6</sup>P4/17)===
; [[Sycamore family]] (P8, P5/11)
This tempers out the maja comma, [-3 -23 17> = 762939453125/753145430616.
: 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.  


===[[Pental family|Pental or Trila-quingu family]] (P8/5, P5)===
; [[Quartonic family]] (P8, P4/11)
This tempers out the pental comma, 847288609443/838860800000 = [-28 25 -5&gt;.
: 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.


===[[Qintosec family|Qintosec or Quadsa-quinbigu family]] (P8/5, P5/2)===
; [[Lafa family]] (P8, P12/12)
This tempers out the qintosec comma, 140737488355328/140126044921875 = [47 -15 -10&gt;.
: 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.  


===[[Trisedodge family|Trisedodge or Saquintrigu family]] (P8/5, P4/3)===
; [[Ditonmic family]] (P8, c4P4/13)
This tempers out the trisedodge comma, 30958682112/30517578125 = [19 10 -15&gt;.
: 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.  


===[[Maquila family|Maquila or Trisa-segu family]] (P8, W<sup>7</sup>P5/17)===
; [[Luna family]] (P8, ccP4/15)
This tempers out the maquila comma, 562949953421312/556182861328125 = [49 -6 -17&gt;.
: 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.  


===[[Mutt family|Mutt or Trila-septriyo family]] (P8/3, WWP4/7)===
; [[Vavoom family]] (P8, P12/17)
This tempers out the [[mutt_comma|mutt comma]], [-44 -3 21&gt;, leading to some strange properties.
: 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.  


==Clans==
; [[Minortonic family]] (P8, ccP5/17)
: This tempers out the [[minortone comma]], {{monzo| -16 35 -17 }}. The head of the family is minortonic temperament, with a generator of a minor tone (~10/9). Seventeen generators equals a double-compound fifth (~6/1). 5/4 is equated to 35 generators minus 5 octaves. Its color name is Trila-seguti.


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.
; [[Maja family]] (P8, c<sup>6</sup>P4/17)
: This tempers out the [[maja comma]], {{monzo| -3 -23 17 }} (762939453125/753145430616). The generator is {{nowrap| ~162/125 {{=}} ~453{{c}} }}. Seventeen generators equals a sextuple-compound fourth. 5/4 is equated to 9 octaves minus 23 generators. Its color name is Saseyoti.  


===[[Trienstonic clan|Trienstonic or Zo clan]] (P8, P5) ===
; [[Maquila family]] (P8, c<sup>7</sup>P5/17)
This 2.3.7 clan tempers out the septimal third-tone, [[28/27]], a no-fives comma.
: 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.


===[[Slendro clan|Slendro or Zozo clan]] (P8, P4/2)===
; [[Gammic family]] (P8, P5/20)
This 2.3.7 clan tempers out the slendro diesis, [[49/48]], a no-fives comma.
: 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.  


===[[Archytas clan|Archytas or Ru clan]] (P8, P5)===
=== Clans defined by a 2.3.7 comma ===
This 2.3.7 clan tempers out the Archytas comma, [[64/63]], a no-fives comma. The clan consists of rank two temperaments, and should not be confused with the [[Archytas family]] of rank three temperaments.
These are defined by a no-5's 7-limit (color name: za) comma. See also [[subgroup temperaments]].


===[[Gamelismic clan|Gamelismic or Latrizo 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 2.3.7 clan tempers out the gamelisma, [-10 1 0 3> = 1029/1024, a no-fives comma. Three 8/7 generators equals a 5th. 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.


===[[Jubilismic clan|Jubilismic or Biruyo Nowa clan]] (P8/2, M3)===
; [[Archytas clan]] (P8, P5)
This 2.5.7 clan tempers out the jubilisma, [[50/49]], a no-threes comma which is the difference between 10/7 and 7/5. The pergen's M3 generator equals 5/4. The half-octave period equals 7/5.
: 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]].


===[[Hemimean clan|Hemimean or Zozoquingu Nowa clan]] (P8, M2)===
; [[Trienstonic clan]] (P8, P5)
This 2.5.7 clan tempers out the hemimean comma, [6 0 -5 2> = 3136/3125, a no-threes comma. Two M2 generators equals 5/4, and five of them equals 7/4.
: 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.  


===[[Quince clan|Quince or Lasepzo-agugu Nowa clan]] (P8, M2/2)===
; Harrison clan (P8, P5)
This 2.5.7 clan tempers out the quince, [-15 0 -2 7&gt; = 823543/819200, a no-threes comma. Two generators equals 8/7 (a M2), and seven generators equals 8/5.
: 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]].  


===[[Sensamagic clan|Sensamagic or Zozoyo Noca clan]] (P12, M3)===
; [[Garischismic clan]] (P8, P5)
This 3.5.7 clan tempers out the sensamagic comma [0 -5 1 2> = 245/243, a no-twos comma (ratio of odd numbers). The M3 generator = 9/7, and two generators equals 5/3.
: 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.  


===[[Mirkwai clan|Mirkwai or Quinru-aquadyo Noca clan]] (P12, WM7/4)===
; Sasazoti clan (P8, P5)
This 3.5.7 clan tempers out the mirkwai comma, [0 3 4 -5&gt; = 16875/16807, a no-twos comma. Four generators = a major 14th = 27/7.
: 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]].  


=Rank-3 temperaments=
; 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.


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 element of a rank-3 pergen is usually a comma, but sometimes it's some fraction of a 5-limit or 7-limit interval.
; [[Semaphoresmic clan]] (P8, P4/2)
: This clan tempers out the large septimal diesis, [[49/48]]. Its generator is ~8/7 or ~7/6. Its color name is Zozoti. Its best downward extension is [[godzilla]]. See also [[semaphore]].  


===[[Didymus rank three family|Didymus or Gu rank three family]] (P8, P5, ^1)===
; Parahemif clan (P8, P5/2)
These are the rank three temperaments tempering out the didymus or meantone comma, 81/80. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth. The meantone comma equates every 5-limit interval to some 3-limit interval, therefore the generators are the same as for 2.3.7 JI: 2/1, 3/1 and 7/1. These may be reduced to 2/1, 3/2 and 7/4, and 7/4 may be reduced further to 64/63. Hence in the pergen,^1 = 64/63.
: 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.


===[[Porcupine rank three family|Porcupine or Triyo rank three family]] (P8, P4/3, /1)===
; Triruti clan (P8/3, P5)
These are the rank three temperaments tempering out the porcupine comma or maximal diesis, [1 -5 3> = 250/243. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth. /1 = 64/63.
: 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.


===[[Diaschismic rank three family|Diaschismic or Sagugu rank three family]] (P8/2, P5, /1)===
; [[Gamelismic clan]] (P8, P5/3)
These are the rank three temperaments tempering out the dischisma, [11 -4 -2> = 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth. /1 = 64/63.
: 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.


===[[Kleismic rank three family|Kleismic or Tribiyo rank three family]] (P8, P12/6, /1)===
; Trizoti clan (P8, P5/3)
These are the rank three temperaments tempering out the kleisma, [-6 -5 6> = 15625/15552. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth. /1 = 64/63.
: 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.


===[[Archytas family|Archytas or Ru family]] (P8, P5, ^1)===
; Latriru clan (P8, P11/3)
Archytas temperament tempers out 64/63. This comma equates every 7-limit interval to some 3-limit interval, therefore the generators are the same as for 2.3.5 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. If 81/80 were tempered out too, the otonal tetrad 4:5:6:7 would be identified with the 3-dominant seventh chord.
: 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.


===[[Semiphore family|Semiphore or Zozo family]] (P8, P4/2, ^1)===
; [[Stearnsmic clan]] (P8/2, P4/3)
Semiphore temperament tempers out 49/48 and thereby identifies the septimal minor third, 7/6 and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like "semi-fourth". ^1 = 81/80.
: 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.


===[[Gamelismic family|Gamelismic or Latrizo family]] (P8, P5/3, ^1)===
; Skwaresmic clan (P8, P11/4)
Not to be confused with the gamelismic clan of rank two temperaments, the gamelismic family are those rank three temperaments which temper out the gamelisma, [-10 1 0 3> = 1029/1024. ^1 = 81/80.
: 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.  


===[[Marvel family|Marvel or Ruyoyo family]] (P8, P5, ^1)===
; [[Buzzardsmic clan]] (P8, P12/4)
The head of the marvel family is marvel, which tempers out [-5 2 2 -1&gt; = [[225/224]]. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle. Other family members include 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 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.  


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.  
; [[Cloudy clan]] (P8/5, P5)
: This clan tempers out the [[cloudy comma]], {{monzo| -14 0 0 5 }} (16807/16384). It has a period of 1/5 octave, which represents ~8/7. The generator is ~3/2. Unlike the blackwood or Sawati family, ~3/2 is not equated with three-fifths of an octave, resulting in very small intervals. Its color name is Laquinzoti.  


===[[Starling family|Starling or Zotrigu family]] (P8, P5, ^1)===
; Quinruti clan (P8, P5/5)
Starling tempers out the septimal semicomma or starling comma [1 2 -3 1> = [[126/125]], the difference between three 6/5s plus one 7/6, and an octave. 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. Its family includes myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum. ^1 = 81/80.
: 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.  


===[[Breed family|Breed or Bizozogu family]] (P8, P5/2, /1)===
; Saquinzoti clan (P8, P12/5)
Breed is a 7-limit microtemperament which tempers out [-5 -1 -2 4&gt; = 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.
: 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.


===[[Ragisma family|Ragisma or Zoquadyo family]] (P8, P5, ^1)===
; Lasepzoti clan (P8, P11/7)
The 7-limit rank three microtemperament which tempers out the ragisma, [-1 -7 4 1&gt; = 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. Accurate ones include ennealimmal, supermajor, enneadecal, amity, mitonic, parakleismic, gamera and vishnu. Pontiac belongs on the list but falls under the schismatic family rubric. ^1 = 81/80.
: 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.


===[[Landscape family|Landscape or Trizogugu family]] (P8/3, P5, ^1)===
; Septiness clan (P8, P11/7)
The 7-limit rank three microtemperament which tempers out the lanscape comma, [-4 6 -6 3&gt; = 250047/250000, extends to various higher limit rank three temperaments such as tyr and odin. The period is 63/50 and ^1 = 81/80.
: 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.  


===[[Hemifamity family|Hemifamity or Saruyo family]] (P8, P5, ^1)===
; Sepruti clan (P8, P12/7)
The hemifamity family of rank three temperaments tempers out the hemifamity comma, [10 -6 1 -1> = 5120/5103. ^1 = 81/80.
: 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.


===[[Porwell family|Porwell or Sarurutrigu family]] (P8, P5, vM6/2)===
; [[Septiennealimmal clan]] (P8/9, P5)
The porwell family of rank three temperaments tempers out the porwell comma, [11 1 -3 -2> = 6144/6125. vM6 = a downmajor 6th = 5/3.
: 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.  


===[[Horwell family|Horwell or Lazoquinyo family]] (P8, P5, ^1)===
=== Clans defined by a 2.3.11 comma ===
The horwell family of rank three temperaments tempers out the horwell comma, [-16 1 5 1> = 65625/65536. ^1 = 81/80.
Color name: ila. See also [[subgroup temperaments]].


===[[Hemimage family|Hemimage or Satrizo-agu family]] (P8, P5, ^1)===
; Lulubiti clan (P8/2, P5)  
The hemimage family of rank three temperaments tempers out the hemimage comma, [5 -7 -1 3> = 10976/10935. ^1 = 64/63.
: 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 family|Sensamagic or Zozoyo family]] (P8, P5, ^1)===
; [[Rastmic clan]] (P8, P5/2)  
These temper out [0 -5 1 2> = 245/243. ^1 = 64/63.
: 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.  


===[[Keemic family|Keemic or Zotriyo family]] (P8, P5, ^1)===
; [[Nexus clan]] (P8/3, P4/2)
These temper out the keema [-5 -3 3 1> = 875/864. ^1 = 81/80.
: 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.  


===[[Sengic family|Sengic or Trizo-agugu family]] (P8, P5, vm3/2)===
; Alphaxenic or Laquadloti clan (P8/2, M2/4)  
These temper out the senga, [1 -3 -2 3> = 686/675. One generator =15/14, two = 7/6 (a downminor 3rd), and three = 6/5.
: 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.  


===[[Orwellismic family|Orwellismic or Triru-agu family]] (P8, P5, ^1)===
=== Clans defined by a 2.3.13 comma ===
These temper out [6, 3, -1, -3> = 1728/1715. ^1 = 64/63.
Color name: tha. See also [[subgroup temperaments]].


===[[Nuwell family|Nuwell or Quadru-ayo family]] (P8, P5, ^1)===
; Thuthuti clan (P8, P5/2)  
These temper out the nuwell comma, [1, 5, 1, -4> = 2430/2401. ^1 = 64/63.
: 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.


===[[Octagar family|Octagar or Rurutriyo family]] (P8, P5, ^m6/2)===
; Satrithoti clan (P8, P11/3)  
The octagar family of rank three temperaments tempers out the octagar comma, [5 -4 3 -2> = 4000/3969. ^m6 = an upminor 6th = 8/5.
: 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.


===[[Mirkwai family|Mirkwai or Quinru-aquadyo family]] (P8, P5, W^M7/4)===
=== Clans defined by a 2.5.7 comma ===
The mirkwai family of rank three temperaments tempers out the mirkwai comma, [0 3 4 -5> = 16875/16807. W^M7 = a wide upmajor 7th =  27/7.
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.


===[[Hemimean family|Hemimean or Zozoquingu family]] (P8, P5, vM3/2)===
; [[Jubilismic clan]] (P8/2, M3)
The hemimean family of rank three temperaments tempers out the hemimean comma, [6 0 -5 2> = 3136/3125. vM3 = a downmajor 3rd = 5/4.
: 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.  


===[[Mirwomo family|Mirwomo or Labizoyo family]] (P8, P5/2, ^1)===
; [[Bapbo clan]] (P8, M3/2)  
The mirwomo family of rank three temperaments tempers out the mirwomo comma, [-15 3 2 2> = 33075/32768. P5/2 = 128/105 and ^1 = 81/80.
: 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.  


===[[Dimcomp family|Dimcomp or Quadruyoyo family]] (P8/4, P5, ^1)===
; [[Hemimean clan]] (P8, M3/2)
The dimcomp family of rank three temperaments tempers out the dimcomp comma, [-1 -4 8 -4> = 390625/388962. The period = 25/21.
: 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.  


===[[Tolermic family|Tolermic or Sazoyoyo family]] (P8, P5, ^1)===
; Mabilismic clan (P8, cM3/3)
These temper out the tolerma, [10 -11 2 1> = 179200/177147. ^1 = 81/80.
: 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.  


===[[Jubilismic family|Jubilismic or Biruyo family]] (P8/2, P5, ^1)===
; Vorwell clan (P8, m6/3)
Jubilismic temperament tempers out 50/49 and thereby identifies the two septimal tritones, 7/5 and 10/7. The period is 7/5, and ^1 = 81/80.
: 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.  


===[[Mint family|Mint or Rugu family]] (P8, P5, ^1)===
; Quinzo-atriyoti Nowa clan (P8, M3/5)
The mint temperament is a low complexity, high error temperament, tempering out the septimal quarter-tone 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7. ^1 = 81/80 or 64/63.
: 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).


===[[Septisemi temperaments|Septisemi or Zogu temperaments]] (P8, P5, ^1)===
; [[Llywelynsmic clan]] (P8, cM3/7)
These are very low complexity temperaments tempering out the minor septimal semitone, [[21/20]] and hence equating 5/3 with 7/4. ^1 = 81/80.
: 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.  


===[[Wizmic microtemperaments|Wizmic or Quinzo-ayoyo microtemperaments]] (P8, P5, vm7/2)===
; [[Quince clan]] (P8, m6/7)
A wizmic temperament is one which tempers out the wizma, [ -6 -8 2 5 &gt; = 420175/419904. vm7 = 7/4.
: 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.  


===[[Greenwoodmic temperaments|Greenwoodmic or Ruruyo temperaments]] (P8, P5, ^1)===
; Slither clan (P8, ccm6/9)
These temper out the greenwoodma, [-3 4 1 -2&gt; = 405/392. ^1 = 64/63.
: 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.  


===[[Avicennmic temperaments|Avicennmic or Zoyoyo temperaments]] (P8, P5, ^1)===
=== Clans defined by a 3.5.7 comma ===
These temper out the avicennma, [-9 1 2 1&gt; = 525/512, also known as Avicenna's enharmonic diesis. ^1 = 81/80.
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.


===[[Garischismic temperaments|Garischismic or Sasaru temperaments]] (P8, P5, ^1)===
; Rutribiyoti Noca clan (P12, M6)  
A garischismic temperament is one which tempers out the garischisma, [25 -14 0 -1&gt; = 33554432/33480783. ^1 = 81/80.
: 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.


===[[Stearnsmic temperaments|Stearnsmic or Latribiru temperaments]] (P8/2, P4/3, ^1)===
; [[Sensamagic clan]] (P12, M6/2)
Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6&gt; = 118098/117649. ^1 = 81/80.
: 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]] (P8, P12/2, ^1)===
; [[Gariboh clan]] (P12, M6/3)
Cataharry temperaments temper out the cataharry comma, [-4 9 -2 -2&gt; = 19683/19600. ^1 = 81/80.
: 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.  


=[[Rank_four_temperaments|Rank-4 temperaments]]=
; [[Mirkwai clan]] (P12, cm7/5)
: This 3.5.7 clan tempers out the [[mirkwai comma]], {{monzo| 0 3 4 -5 }} (16875/16807). The generator is ~7/5, four generators equals ~27/7, and five generators equals the classical compound minor seventh (~27/5). Its color name is Quinru-aquadyoti Noca.


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.
; Sasepzo-atriguti Noca clan (P12, m7/7)
: This 3.5.7 clan tempers out the [[procyon comma]] {{monzo| 0 -8 -3 7 }} (823543/820125). Its only member so far is [[procyon]]. The generator is ~49/45, three generators equals ~9/7, four equals ~7/5, and seven equals the classic minor seventh (~9/5).


===[[Valinorismic temperaments|Valinorismic or Lorugugu temperaments]] ===
; Satritrizo-aguguti Noca clan (P12, c<sup>3</sup>M6/9)
These temper out the valinorsma, [4 0 -2 -1 1> = 176/175.
: 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).


===[[Rastmic temperaments|Rastmic or Lulu temperaments]]===
; Saquadtrizo-asepguti Noca clan (P12, c<sup>5</sup>m7/12)
These temper out the rastma, [-1 5 0 0 -2> = 243/242. As a no-fives no-sevens rank-2 temperament, it's (P8, P5/2).
: 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.


===[[Werckismic temperaments|Werckismic or Luzozogu temperaments]]===
=== Temperaments defined by a 2.3.5.7 comma ===
These temper out the werckisma, [-3 2 -1 2 -1> = 441/440.
These are defined by a full 7-limit (color name: yaza) comma.


===[[Swetismic temperaments|Swetismic or Lururuyo temperaments]]===
; [[Septisemi temperaments]]
These temper out the swetisma, [2 3 1 -2 -1> = 540/539.
: 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.  


===[[Lehmerismic temperaments|Lehmerismic or Loloruyoyo temperaments]]===
; [[Greenwoodmic temperaments]]
These temper out the lehmerisma, [-4 -3 2 -1 2> = 3025/3024.
: These temper out the [[greenwoodma]], {{monzo| -3 4 1 -2 }} (405/392). Its color name is Ruruyoti.  


===[[Kalismic temperaments|Kalismic or Bilorugu temperaments]]===
; [[Keegic temperaments]]
These temper out the kalisma, [-3 4 -2 -2 2> = 9801/9800.
: Keegic rank-2 temperaments temper out the [[keega]], {{monzo| -3 1 -3 3 }} (1029/1000). Its color name is Trizoguti.  


=[[Subgroup temperaments]]=
; [[Mint temperaments]]
: Mint rank-2 temperaments temper out the septimal quartertone, [[36/35]], equating 7/6 with 6/5, and 5/4 with 9/7. Its color name is Ruguti.
 
; [[Avicennmic temperaments]]
: These temper out the [[avicennma]], {{monzo| -9 1 2 1 }} (525/512), also known as Avicenna's enharmonic diesis. Its color name is Zoyoyoti.
 
; Sengic temperaments
: Sengic rank-2 temperaments temper out the [[senga]], {{monzo| 1 -3 -2 3 }} (686/675). Its color name is Trizo-aguguti.
 
; [[Keemic temperaments]]
: Keemic rank-2 temperaments temper out the [[keema]], {{monzo| -5 -3 3 1 }} (875/864). Its color name is Zotriyoti.
 
; Secanticorn temperaments
: Secanticorn rank-2 temperaments temper out the [[secanticornisma]], {{monzo| -3 11 -5 -1 }} (177147/175000). Its color name is Laruquinguti.
 
; Nuwell temperaments
: Nuwell rank-2 temperaments temper out the [[nuwell comma]], {{monzo| 1 5 1 -4 }} (2430/2401). Its color name is Quadru-ayoti.
 
; Mermismic temperaments
: Mermismic rank-2 temperaments temper out the [[mermisma]], {{monzo| 5 -1 7 -7 }} (2500000/2470629). Its color name is Sepruyoti.
 
; Negricorn temperaments
: Negricorn rank-2 temperaments temper out the [[negricorn comma]], {{monzo| 6 -5 -4 4 }} (153664/151875). Its color name is Saquadzoguti.
 
; Tolermic temperaments
: These temper out the [[tolerma]], {{monzo| 10 -11 2 1 }} (179200/177147). Its color name is Sazoyoyoti.
 
; Valenwuer temperaments
: Valenwuer rank-2 temperaments temper out the [[valenwuer comma]], {{monzo| 12 3 -6 -1 }} (110592/109375). Its color name is Sarutribiguti.
 
; [[Mirwomo temperaments]]
: Mirwomo rank-2 temperaments temper out the [[mirwomo comma]], {{monzo| -15 3 2 2 }} (33075/32768). Its color name is Labizoyoti.
 
; Catasyc temperaments
: Catasyc rank-2 temperaments temper out the [[catasyc comma]], {{monzo| -11 -3 8 -1 }} (390625/387072). Its color name is Laruquadbiyoti.
 
; Compass temperaments
: Compass rank-2 temperaments temper out the [[compass comma]], {{monzo| -6 -2 10 -5 }} (9765625/9680832). Its color name is Quinruyoyoti.
 
; Trimyna temperaments
: Trimyna rank-2 temperaments temper out the [[trimyna comma]], {{monzo| -4 1 -5 5 }} (50421/50000). Its color name is Quinzoguti.
 
; [[Starling temperaments]]
: Starling rank-2 temperaments temper out the [[starling comma]] a.k.a. septimal semicomma, {{monzo| 1 2 -3 1 }} ([[126/125]]), the difference between three 6/5's plus one 7/6, and an octave. It includes myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum. Its color name is Zotriguti.
 
; [[Octagar temperaments]]
: Octagar rank-2 temperaments temper out the [[octagar comma]], {{monzo| 5 -4 3 -2 }} (4000/3969). Its color name is Rurutriyoti.
 
; [[Orwellismic temperaments]]
: Orwellismic rank-2 temperaments temper out [[orwellisma]], {{monzo| 6 3 -1 -3 }} (1728/1715). Its color name is Triru-aguti.
 
; Mynaslendric temperaments
: Mynaslendric rank-2 temperaments temper out the [[mynaslender comma]], {{monzo| 11 4 1 -7 }} (829440/823543). Its color name is Sepru-ayoti.
 
; [[Mistismic temperaments]]
: Mistismic rank-2 temperaments temper out the [[mistisma]], {{monzo| 16 -6 -4 1 }} (458752/455625). Its color name is Sazoquadguti.
 
; [[Varunismic temperaments]]
: Varunismic rank-2 temperaments temper out the [[varunisma]], {{monzo| -9 8 -4 2 }} (321489/320000). Its color name is Labizoguguti.
 
; [[Marvel temperaments]]
: Marvel rank-2 temperaments temper out the [[marvel comma]], {{monzo| -5 2 2 -1 }} (225/224). It includes wizard, tritonic, septimin, slender, triton, escapist and marv, not to mention negri, sharpie, pelogic, meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton. Its color name is Ruyoyoti.
 
; Dimcomp temperaments
: Dimcomp rank-2 temperaments temper out the [[dimcomp comma]], {{monzo| -1 -4 8 -4 }} (390625/388962). Its color name is Quadruyoyoti.
 
; [[Cataharry temperaments]]
: Cataharry rank-2 temperaments temper out the [[cataharry comma]], {{monzo| -4 9 -2 -2 }} (19683/19600). Its color name is Labiruguti.
 
; [[Canousmic temperaments]]
: Canousmic rank-2 temperaments temper out the [[canousma]], {{monzo| 4 -14 3 4 }} (4802000/4782969). Its color name is Saquadzo-atriyoti.
 
; [[Triwellismic temperaments]]
: Triwellismic rank-2 temperaments temper out the [[triwellisma]], {{monzo| 1 -1 -7 6 }} (235298/234375). Its color name is Tribizo-asepguti.
 
; [[Hemimage temperaments]]
: Hemimage rank-2 temperaments temper out the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} (10976/10935). Its color name is Satrizo-aguti.
 
; [[Hemifamity temperaments]]
: Hemifamity rank-2 temperaments temper out the [[hemifamity comma]], {{monzo| 10 -6 1 -1 }} (5120/5103). Its color name is Saruyoti.
 
; [[Parkleiness temperaments]]
: Parkleiness rank-2 temperaments temper out the [[parkleiness comma]], {{monzo| 7 7 -9 1 }} (1959552/1953125). Its color name is Zotritriguti.
 
; [[Porwell temperaments]]
: Porwell rank-2 temperaments temper out the [[porwell comma]], {{monzo| 11 1 -3 -2 }} (6144/6125). Its color name is Sarurutriguti.
 
; [[Cartoonismic temperaments]]
: Cartoonismic rank-2 temperaments temper out the [[cartoonisma]], {{monzo| 12 -3 -14 9 }} (165288374272/164794921875). Its color name is Satritrizo-asepbiguti.
 
; [[Hemfiness temperaments]]
: Hemfiness rank-2 temperaments temper out the [[hemfiness comma]], {{monzo| 15 -5 3 -5 }} (4096000/4084101). Its color name is Saquinru-atriyoti.
 
; [[Hewuermera temperaments]]
: Hewuermera rank-2 temperaments temper out the [[hewuermera comma]], {{monzo| 16 2 -1 -6 }} (589824/588245). Its color name is Satribiru-aguti.
 
; [[Lokismic temperaments]]
: Lokismic rank-2 temperaments temper out the [[lokisma]], {{monzo| 21 -8 -6 2 }} (102760448/102515625). Its color name is Sasa-bizotriguti.
 
; Decovulture temperaments
: Decovulture rank-2 temperaments temper out the [[decovulture comma]], {{monzo| 26 -7 -4 -2 }} (67108864/66976875). Its color name is Sasabiruguguti.
 
; Pontiqak temperaments
: Pontiqak rank-2 temperaments temper out the [[pontiqak comma]], {{monzo| -17 -6 9 2 }} (95703125/95551488). Its color name is Lazozotritriyoti.
 
; [[Mitonismic temperaments]]
: Mitonismic rank-2 temperaments temper out the [[mitonisma]], {{monzo| -20 7 -1 4 }} (5250987/5242880). Its color name is Laquadzo-aguti.
 
; [[Horwell temperaments]]
: Horwell rank-2 temperaments temper out the [[horwell comma]], {{monzo| -16 1 5 1 }} (65625/65536). Its color name is Lazoquinyoti.
 
; Neptunismic temperaments
: Neptunismic rank-2 temperaments temper out the [[neptunisma]], {{monzo| -12 -5 11 -2 }} (48828125/48771072). Its color name is Laruruleyoti.
 
; [[Metric microtemperaments]]
: Metric rank-2 temperaments temper out the [[meter]], {{monzo| -11 2 7 -3 }} (703125/702464). Its color name is Latriru-asepyoti.
 
; [[Wizmic microtemperaments]]
: Wizmic rank-2 temperaments temper out the [[wizma]], {{monzo| -6 -8 2 5 }} (420175/419904). Its color name is Quinzo-ayoyoti.
 
; [[Supermatertismic temperaments]]
: Supermatertismic rank-2 temperaments temper out the [[supermatertisma]], {{monzo| -6 3 9 -7 }} (52734375/52706752). Its color name is Lasepru-atritriyoti.
 
; [[Breedsmic temperaments]]
: Breedsmic rank-2 temperaments temper out the [[breedsma]], {{monzo| -5 -1 -2 4 }} (2401/2400). Its color name is Bizozoguti.
 
; Supermasesquartismic temperaments
: Supermasesquartismic rank-2 temperaments temper out the [[supermasesquartisma]], {{monzo| -5 10 5 -8 }} (184528125/184473632). Its color name is Laquadbiru-aquinyoti.
 
; [[Ragismic microtemperaments]]
: Ragismic rank-2 temperaments temper out the [[ragisma]], {{monzo| -1 -7 4 1 }} (4375/4374). Its color name is Zoquadyoti.
 
; Akjaysmic temperaments
: Akjaysmic rank-2 temperaments temper out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }}. These have a period of 1/7 octave, and five periods equals ~105/64. Unlike the whitewood or Lawati family, ~3/2 is not equated with 4/7 of an octave, resulting in small intervals. Its color name is Trisa-sepruguti.
 
; [[Landscape microtemperaments]]
: Landscape rank-2 temperaments temper out the [[landscape comma]], {{monzo| -4 6 -6 3 }} (250047/250000). These have a period of 1/3 octave, but ~5/4 is not equated with a period, resulting in small intervals. Its color name is Trizoguguti.
 
== Rank-3 temperaments ==
Even less familiar than rank-2 temperaments are the [[rank-3 temperament]]s, generated by a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank-3 temperaments by the commas they temper out. They can also be identified by pergen. The third generator in a rank-3 pergen is usually a comma, but sometimes it is some fraction of a 5-limit or 7-limit interval.
 
=== Families defined by a 2.3.5 comma ===
Every 5-limit (color name: ya) comma defines a rank-3 family, thus every comma in the list of rank-2 2.3.5 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a 7-limit (color name: yaza) temperament in which one of the generators is ~7/1. This generator can be reduced to ~7/4, which can be reduced further to ~64/63. Hence in all the pergens below, the ^1 or /1 generator is ~64/63. An additional 7-limit or 11-limit comma creates an 11-limit (color name: yazala) temperament, and so forth. All these examples are 7-limit:
 
; [[Didymus rank three family|Didymus rank-3 family]] (P8, P5, ^1)
: These are the rank-3 temperaments tempering out the didymus or meantone comma, 81/80. Its color name is Guti.
 
; [[Diaschismic rank three family|Diaschismic rank-3 family]] (P8/2, P5, /1)
: 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.
 
; [[Porcupine rank three family|Porcupine rank-3 family]] (P8, P4/3, /1)
: 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.
 
; [[Kleismic rank three family|Kleismic rank-3 family]] (P8, P12/6, /1)
: 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.
 
=== Families defined by a 2.3.7 comma ===
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:
 
; [[Archytas family]] (P8, P5, ^1)
: Archytas temperament tempers out 64/63. This comma equates every 7-limit interval to some 3-limit interval. If 81/80 were tempered out too, the otonal tetrad 4:5:6:7 would be identified with the dominant seventh chord. Its color name is Ruti.
 
; [[Garischismic family]] (P8, P5, ^1)
: A garischismic temperament is one which tempers out the garischisma, {{monzo| 25 -14 0 -1 }} (33554432/33480783). Its color name is Sasaruti.
 
; Laruruti clan (P8/2, P5)
: This clan tempers out the Laruru comma, {{monzo| -7 8 0 -2 }} (6561/6272). Two ~81/56 periods equal an octave. The generator is ~3/2, and four generators minus three periods equals ~7/4. The major second ~9/8 is divided in half, with each half equated to ~28/27. See also the diaschismic or Saguguti temperament and the jubilismic or Biruyoti temperament.
 
; [[Semaphoresmic family]] (P8, P4/2, ^1)
: Semaphoresmic temperament tempers out 49/48 and thereby identifies the septimal minor third 7/6 with the septimal whole tone 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like ''semi-fourth''. See also [[semaphore]]. Its color name is Zozoti.
 
; [[Gamelismic family]] (P8, P5/3, ^1)
: Not to be confused with the gamelismic clan of rank-2 temperaments, the gamelismic family are those rank-3 temperaments which temper out the gamelisma, {{monzo| -10 1 0 3 }} (1029/1024). In the pergen, P5/3 is ~8/7. Its color name is Latrizoti.
 
; Stearnsmic family (P8/2, P4/3, ^1)
: Stearnsmic temperaments temper out the stearnsma, {{monzo| 1 10 0 -6 }} (118098/117649). In the pergen, P8/2 is ~343/243 and P4/3 is ~54/49. Its color name is Latribiruti.
 
=== Families defined by a 2.3.5.7 comma ===
Color name: yaza.
 
; [[Marvel family]] (P8, P5, ^1)
: The head of the marvel family is marvel, which tempers out the [[marvel comma]], {{monzo| -5 2 2 -1 }} (225/224). It divides 8/7 into two 16/15's, or equivalently, two 15/14's. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.
: The marvel comma equates every 7-limit interval to some 5-limit interval, therefore the generators are the same as for 5-limit JI: 2/1, 3/1 and 5/1. These may be reduced to 2/1, 3/2 and 5/4, and 5/4 may be reduced further to 81/80. Hence in the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Ruyoyoti.
 
; [[Starling family]] (P8, P5, ^1)
: Starling tempers out the [[starling comma]] a.k.a. septimal semicomma, {{monzo| 1 2 -3 1 }} (126/125), the difference between three 6/5's plus one 7/6, and an octave. It divides 10/7 into two 6/5's. Like marvel, it has the same generators as 5-limit JI. An excellent tuning for starling is [[77edo]], but 31, 46 or 58 also work nicely. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Zotriguti.
 
; [[Sensamagic family]] (P8, P5, ^1)
: These temper out {{monzo| 0 -5 1 2 }} (245/243), which divides 16/15 into two 28/27's. In the pergen, {{nowrap| ^1 {{=}} ~64/63 }}. Its color name is Zozoyoti.
 
; Greenwoodmic family (P8, P5, ^1)
: These temper out the greenwoodma, {{monzo| -3 4 1 -2 }} (405/392), which divides 10/9 into two 15/14's. In the pergen, {{nowrap| ^1 {{=}} ~64/63 }}. Its color name is Ruruyoti.
 
; 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=
== 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.
 
; [[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.


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 Quartercache]]
: This is the commatic realm of the 11-limit comma 117440512/117406179 ({{monzo| 24 -6 0 1 -5 }}), the [[quartisma]]. Its color name is Saquinlu-azoti.  


==[[Orgonia|Orgonia or Satrilu-aruru]]==
== Miscellaneous other temperaments ==
Orgonia is the commatic realm of the [[11-limit]] comma 65536/65219 = [16 0 0 -2 -3&gt;, the orgonisma.
; [[Limmic temperaments]]
: Various subgroup temperaments all tempering out the limma, 256/243.


==[[The Biosphere|The Biosphere or Thozogu]] ==
; [[Fractional-octave temperaments]]
The Biosphere is the name given to the commatic realm of the 13-limit comma 91/90.
: These temperaments all have a fractional-octave period.


==[[The Archipelago|The Archipelago or Bithogu]]==
; [[Miscellaneous 5-limit temperaments]]
The Archipelago is a name which has been given to the commatic realm of the [[13-limit]] comma [2 -3 -2 0 0 2> = 676/675.
: 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.


===[[31 comma temperaments]]===
; [[Turkish maqam music temperaments]]
These all have period 1/31 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.


===[[Turkish maqam music temperaments]]===
; [[Very low 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.
: All hope abandon ye who enter here.


===[[Very low accuracy temperaments]]===
; [[Very high accuracy temperaments]]
All hope abandon ye who enter here.
: Microtemperaments which do not fit in elsewhere.


===[[Very high accuracy temperaments]]===
; Middle Path tables
Microtemperaments which don't fit in elsewhere.
: 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]]


===[[High badness temperaments]]===
== Maps of temperaments ==
High in badness, but worth cataloging for one reason or another.
* [[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]]


===[[11-limit comma temperaments]]===
== Temperament nomenclature ==
These temperaments go to 11...
* [[Temperament naming]]


=Links=
== External links ==
* [http://en.wikipedia.org/wiki/Regular_temperament Regular temperaments - Wikipedia]
* [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


[[Category:overview]]
[[Category:Lists of temperaments]] <!-- main article -->
[[Category:temperament]]

Latest revision as of 12:35, 21 August 2025

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

Rank-2 temperaments

A rank-2 temperament maps all JI intervals within its JI subgroup to a 2-dimensional lattice. This lattice has two generators. The larger generator is called the period, as the temperament will repeat periodically at that interval. The period is usually the octave, or some unit fraction of the octave. If the period is a full octave, the temperament is said to be a linear temperament. The smaller generator, referred to as "the" generator, is stacked repeatedly to generate a scale within that period.

A rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for the period and one for the generator. Each member of the JI subgroup is mapped to a period-generator pair.

Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma 81/80 out of 3-dimensional 5-limit JI), can be reduced to 1-dimensional 12-ET by tempering out the Pythagorean comma.

Families defined by a 2.3 comma

These are families defined by a 3-limit (color name: wa) comma. If only primes 2 and 3 are part of the subgroup, the comma creates a rank-1 temperament, an edo. But if another prime such as 5 is present, the comma creates a rank-2 temperament. Since edos are discussed elsewhere, this section assumes the presence of at least one additional prime. The rank-2 temperament created consists of multiple "copies" of an edo. The edo copies can be thought of as being offset from one another by a small comma. This small comma is represented in the pergen by ^1.

Blackwood family (P8/5, ^1)
This family tempers out the limma, [8 -5⟩ (256/243). It equates 5 fifths with 3 octaves, which creates multiple copies of 5edo. The fifth is ~720¢, quite sharp. The only member of this family is the blackwood temperament, which is 5-limit. Blackwood's edo copies are offset from one another by 5/4, or alternatively by 81/80. 5/4 is usually tempered sharp, perhaps ~400¢, to match the sharp fifth. Its color name is Sawati.
Whitewood family (P8/7, ^1)
This family tempers out the apotome, [-11 7⟩ (2187/2048). It equates 7 fifths with 4 octaves, which creates multiple copies of 7edo. The fifth is ~685¢, which is very flat. This family includes the whitewood temperament. Its color name is Lawati.
Compton family (P8/12, ^1)
This tempers out the Pythagorean comma, [-19 12⟩ (531441/524288). It equates 12 fifths with 7 octaves, which creates multiple copies of 12edo. Temperaments in this family include compton and catler. In the 5-limit compton temperament, the edo copies are offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12edo instruments slightly detuned from each other provide an easy way to make music with these temperaments. Its color name is Lalawati.
Countercomp family (P8/41, ^1)
This family tempers out the Pythagorean countercomma, [65 -41⟩, which creates multiple copies of 41edo. Its color name is Wa-41.
Mercator family (P8/53, ^1)
This family tempers out the Mercator's comma, [-84 53⟩, which creates multiple copies of 53edo. Its color name is Wa-53.

Families defined by a 2.3.5 comma

These are families defined by a 5-limit (color name: ya) comma. As we go up in rank 2, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) The same comment applies to 7-limit temperaments and rank 3, etc. Members of families and their relationships can be classified by the normal comma list of the various temperaments. Families include weak extensions as well as strong, in other words, the pergen shown here may change.

Meantone family (P8, P5)
The meantone family tempers out 81/80, also called the syntonic comma. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are 12-, 19-, 31-, 43-, 50-, 55- and 81edo. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma. Its color name is Guti.
Schismatic family (P8, P5)
The schismatic family tempers out the schisma of [-15 8 1⟩ (32805/32768), which is the amount by which the Pythagorean comma exceeds the syntonic comma. The 5-limit version of the temperament is a microtemperament which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity; whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4. Tunings include 12-, 29-, 41-, 53-, and 118edo. Its color name is Layoti.
Mavila family (P8, P5)
This tempers out the mavila comma, [-7 3 1⟩ (135/128), also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them 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 9-, 16-, 23-, and 25edo. Its color name is Layobiti.
Father family (P8, P5)
This tempers out 16/15, the just diatonic semitone, and equates 5/4 with 4/3. Its color name is Gubiti.
Diaschismic family (P8/2, P5)
The diaschismic family tempers out the diaschisma, [11 -4 -2⟩ (2048/2025), such that two classic major thirds and a Pythagorean major third stack to an octave (i.e. (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 12-, 22-, 34-, 46-, 56-, 58- and 80edo. Its color name is Saguguti. An obvious 7-limit interpretation of the period is 7/5, which makes pajara temperament, where the intervals 50/49 and 64/63 are tempered out. 22edo is an excellent pajara tuning.
Bug family (P8, P4/2)
This low-accuracy family of temperaments tempers out 27/25, the large limma or bug comma. The generator is an approximate 6/5 or 10/9 = ~250 ¢, two of which make ~4/3. 5/4 is equated to 1 octave minus 3 generators. Its color name is Guguti. An obvious 7-limit interpretation of the generator is 7/6~8/7, which leads to semaphore or Zozoti.
Immunity family (P8, P4/2)
This tempers out the immunity comma, [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. Its color name is Sasa-yoyoti. An obvious 7-limit interpretation of the generator is 7/6~8/7, which leads to semaphore or Zozoti.
Dicot family (P8, P5/2)
The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24 (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 or between two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into a neutral-sized third of ~350¢ that is taken to approximate both. 7edo makes for a "good" dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the neutral dicot thirds span a 3/2. Tunings include 7edo, 10edo, and 17edo. Its color name is Yoyoti. An obvious 2.3.11 interpretation of the generator is ~11/9, which leads to neutral or Luluti.
Augmented family (P8/3, P5)
The augmented family tempers out the diesis of [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 "Tcherepnin's scale" (3L 6s). Its color name is Triguti.
Misty family (P8/3, P5)
The misty family tempers out the misty comma of [26 -12 -3⟩, the difference between the Pythagorean comma and a stack of three schismas. The period is ~512/405 and the generator is ~3/2 (or alternatively ~135/128). 5/4 is equated to 8 periods minus 4 fifths, thus 5/4 is split into 4 equal parts, each 2 periods minus a fifth. Its color name is Sasa-triguti.
Porcupine family (P8, P4/3)
The porcupine family tempers out [1 -5 3⟩ (250/243), the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. 5/4 is equated to 1 octave minus 5 generators. Some porcupine temperaments include 15-, 22-, 37-, and 59edo. Its color name is Triyoti. An important 7-limit extension also tempers out 64/63.
Alphatricot family (P8, P11/3)
The alphatricot family tempers out the alphatricot comma, [39 -29 3⟩. The generator is ~59049/40960 ([-13 10 -1⟩) = 633 ¢, or its octave inverse ~81920/59049 = 567 ¢. 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 81/56 = 639 ¢, a much simpler ratio which leads to the Latriruti clan. An obvious 13-limit interpretation is 13/9 = 637¢, an even simpler ratio implying the Satrithoti clan.
Diminished family (P8/4, P5)
The diminished family tempers out the major diesis or diminished comma, [3 4 -4⟩ or 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12edo. 5/4 is equated to 1 fifth minus 1 period. Its color name is Quadguti.
Undim family (P8/4, P5)
The undim family tempers out the undim comma of [41 -20 -4⟩, the difference between the Pythagorean comma and a stack of four schismas. Its color name is Trisa-quadguti.
Negri family (P8, P4/4)
This tempers out the negri comma, [-14 3 4⟩. Its only member so far is negri. Its generator is ~16/15, four of which make ~4/3. 5/4 is equated to 3 generators. Its color name is Laquadyoti.
Tetracot family (P8, P5/4)
The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by [5 -9 4⟩ (20000/19683), the minimal diesis or tetracot comma. 5/4 is equated to 9 generators minus an octave. 7edo can also be considered a tetracot tuning, as can 20edo, 27edo, 34edo, and 41edo. Its color name is Saquadyoti.
Smate family (P8, P11/4)
This tempers out the symbolic comma, [11 -1 -4⟩ (2048/1875). Its generator is ~5/4 = ~421 ¢, four of which make ~8/3. Its color name is Saquadguti.
Vulture family (P8, P12/4)
This tempers out the vulture comma, [24 -21 4⟩. Its generator is ~320/243 = ~475 ¢, four of which make ~3/1. 5/4 is equated to 21 generators minus 8 octaves. Its color name is Sasa-quadyoti. An obvious 7-limit interpretation of the generator is 21/16, which makes buzzard or Saquadruti.
Quintile family (P8/5, P5)
This tempers out the quintile comma, 847288609443/838860800000 ([-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 ¢, 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. Its color name is Trila-quinguti. An obvious 7-limit interpretation of the generator is 8/7, which leads to Laquinzoti.
Ripple family (P8, P4/5)
This tempers out the ripple comma, 6561/6250 ([-1 8 -5⟩), which equates a stack of four 10/9's with 8/5, and five of them with 16/9. The generator is 27/25, two of which equals 10/9, three of which equals 6/5, and five of which equals 4/3. 5/4 is equated to an octave minus 8 generators. As one might expect, 12edo is about as accurate as it can be. Its color name is Quinguti.
Passion family (P8, P4/5)
This tempers out the passion comma, 262144/253125 ([18 -4 -5⟩), which equates a stack of four 16/15's with 5/4, and five of them with 4/3. Its color name is Saquinguti.
Quintaleap family (P8, P4/5)
This tempers out the quintaleap comma, [37 -16 -5⟩. The generator is ~135/128, five of them gives ~4/3, and sixteen of them gives 5/2. Its color name is Trisa-quinguti. An obvious 17-limit interpretation of the generator is ~18/17, which makes Saquinsoti.
Quindromeda family (P8, P4/5)
This tempers out the quindromeda comma, [56 -28 -5⟩. The generator is ~4428675/4194304, five of them gives ~4/3, and 28 of them gives the fifth harmonic, 5/1. Its color name is Quinsa-quinguti. An obvious 17-limit interpretation of the generator is ~18/17, which makes Saquinsoti.
Amity family (P8, P11/5)
This tempers out the amity comma, 1600000/1594323 ([9 -13 5⟩). The generator is 243/200 = ~339.5 ¢, five of which make ~8/3. 5/4 is equated to 4 octaves minus 13 generators, or 2 fifths minus 3 generators. Its color name is Saquinyoti. An obvious 11-limit interpretation of the generator is 11/9, which makes Saquinloti. An obvious 13-limit interpretation of the generator is 39/32, which makes Lala-quinthoti.
Magic family (P8, P12/5)
The magic family tempers out [-10 -1 5⟩ (3125/3072), known as the magic comma or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/1. The generator is itself an approximate 5/4. The magic family includes 16-, 19-, 22-, 25-, and 41edo among its possible tunings, with the last being near-optimal. Its color name is Laquinyoti.
Fifive family (P8/2, P5/5)
This tempers out the fifive comma, [-1 -14 10⟩ (9765625/9565938). The period is ~4374/3125 ([1 7 -5⟩), two of which make an octave. The generator is ~27/25, five of which make ~3/2. 5/4 is equated to 7 generators minus 1 period. Its color name is Saquinbiyoti.
Quintosec family (P8/5, P5/2)
This tempers out the quintosec comma, 140737488355328/140126044921875 ([47 -15 -10⟩). The period is ~524288/455625 ([19 -6 -4⟩), five of which equals an octave. The generator is ~16/15. A period plus a generator makes half a fifth. 5/4 is equated to 3 periods minus 3 generators. Its color name is Quadsa-quinbiguti. An obvious 7-limit interpretation of the period is 8/7.
Trisedodge family (P8/5, P4/3)
This tempers out the trisedodge comma, 30958682112/30517578125 ([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. Its color name is Saquintriguti. An obvious 7-limit interpretation of the period is 8/7.
Ampersand family (P8, P5/6)
This tempers out the ampersand comma, 34171875/33554432 ([-25 7 6⟩). Its only member is ampersand. The generator is ~16/15, of which six make ~3/2. 5/4 is equated to 1 octave minus 7 generators. Its color name is Lala-tribiyoti. If the generator is also equated to ~15/14, and three generators to ~11/9, one gets the miracle temperament.
Kleismic family (P8, P12/6)
The kleismic family of temperaments tempers out the kleisma, 15625/15552 ([-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 15-, 19-, 34-, 49-, 53-, 72-, 87- and 140edo among its possible tunings. Its color name is Tribiyoti.
Orson or semicomma family (P8, P12/7)
The semicomma (also known as Fokker's comma), 2109375/2097152 ([-21 3 7⟩), is tempered out by the members of the semicomma family. Its generator is ~75/64, seven of which equals ~3/1. 5/4 is equated to 1 octave minus 3 generators. Its color name is Lasepyoti. Its generator has a natural interpretation as ~7/6, leading to the orwell or Sepruti temperament.
Wesley family (P8, ccP4/7)
This tempers out the wesley comma, 78125/73728 ([-13 -2 7⟩). The generator is ~125/96 = ~412 ¢. Seven generators equals a double-compound fourth of ~16/3. 5/4 is equated to 1 octave minus 2 generators. Its color name is Lasepyobiti. An obvious 7-limit interpretation of the generator is 9/7, leading to the Lasepruti temperament. An obvious 3-limit interpretation of the generator is 81/64, implying 29edo.
Sensipent family (P8, ccP5/7)
The sensipent family tempers out the sensipent comma, 78732/78125 ([2 9 -7⟩), also known as the medium semicomma. Its generator is ~162/125 = ~443 ¢. Seven generators equals a double-compound fifth of ~6/1. 5/4 is equated to 9 generators minus 3 octaves. Tunings include 8edo, 19edo, 46edo, and 65edo. Its color name is Sepguti. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzoti temperament.
Vishnuzmic family (P8/2, P4/7)
This tempers out the vishnuzma, [23 6 -14⟩, or the amount by which seven chromatic semitones (25/24) fall short of a perfect fourth (4/3), or (4/3)/(25/24)7. The period is ~[-11 -3 7⟩ and the generator is ~25/24. 5/4 is equated to 1 period minus 3 generators. Its color name is Sasepbiguti.
Unicorn family (P8, P4/8)
This tempers out the unicorn comma, 1594323/1562500 ([-2 13 -8⟩). The generator is ~250/243 = ~62 ¢ and eight of them equal ~4/3. Its color name is Laquadbiguti.
Würschmidt family (P8, ccP5/8)
The würschmidt family tempers out the würschmidt comma, 393216/390625 ([17 1 -8⟩). Würschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a double-compound perfect fifth); that is, (5/4)8⋅(393216/390625) = 6. It tends to generate the same mos scales as the magic temperament, but is tuned slightly more accurately. Both 31edo and 34edo can be used as würschmidt tunings, as can 65edo, which is quite accurate. Its color name is Saquadbiguti.
Escapade family (P8, P4/9)
This tempers out the escapade comma, [32 -7 -9⟩, which is the difference between nine just major thirds and seven just fourths. The generator is [-14 3 4⟩ of ~55 ¢ and nine of them equal ~4/3. Seven of them equal ~5/4, thus two of them equal ~16/15. Its color name is Sasa-tritriguti. An obvious 11-limit interpretation of the generator is 33/32, leading to the Trisa-tritriluti temperament.
Mabila family (P8, c4P4/10)
The mabila family tempers out the mabila comma, [28 -3 -10⟩ (268435456/263671875). The generator is ~512/375 = ~530 ¢, three generators equals ~5/2 and ten of them equals a quadruple-compound fourth of ~64/3. Its color name is Sasa-quinbiguti. An obvious 11-limit interpretation of the generator is ~15/11.
Sycamore family (P8, P5/11)
The sycamore family tempers out the sycamore comma, [-16 -6 11⟩ (48828125/47775744), which is the amount by which five stacked chromatic semitones, 25/24, exceed 6/5, and hence also the amount six exceeds 5/4. Eleven of these generators equals ~3/2. Its color name is Laleyoti.
Quartonic family (P8, P4/11)
The quartonic family tempers out the quartonic comma, [3 -18 11⟩ (390625000/387420489). The generator is ~250/243 = ~45 ¢, 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.
Lafa family (P8, P12/12)
This tempers out the lafa comma, [77 -31 -12⟩. The generator is ~4982259375/4294967296 = ~258.6 ¢. Twelve generators equals a twelfth (~3/1). 5/4 is equated to 7 octaves minus 31 generators. Its color name is Tribisa-quadtriguti.
Ditonmic family (P8, c4P4/13)
This tempers out the ditonma, [-27 -2 13⟩ (1220703125/1207959552). Thirteen ~[-12 -1 6⟩ generators of about 407 ¢ equals a quadruple-compound fourth. 5/4 is equated to 1 octave minus 2 generators. An obvious 3-limit interpretation of the generator is 81/64, which implies 53edo, which is a good tuning for this high-accuracy family of temperaments. Its color name is Lala-theyoti.
Luna family (P8, ccP4/15)
This tempers out the luna comma, [38 -2 -15⟩ (274877906944/274658203125). The generator is ~[18 -1 -7⟩ at ~193 ¢. Two generators equals ~5/4, and fifteen generators equals a double-compound fourth of ~16/3. Its color name is Sasa-quintriguti.
Vavoom family (P8, P12/17)
This tempers out the vavoom comma, [-68 18 17⟩. The generator is ~16/15 = ~111.9 ¢. Seventeen generators equals a twelfth (~3/1). 5/4 is equated to two octaves minus 18 generators. Its color name is Quinla-seyoti.
Minortonic family (P8, ccP5/17)
This tempers out the minortone comma, [-16 35 -17⟩. The head of the family is minortonic temperament, with a generator of a minor tone (~10/9). Seventeen generators equals a double-compound fifth (~6/1). 5/4 is equated to 35 generators minus 5 octaves. Its color name is Trila-seguti.
Maja family (P8, c6P4/17)
This tempers out the maja comma, [-3 -23 17⟩ (762939453125/753145430616). The generator is ~162/125 = ~453 ¢. Seventeen generators equals a sextuple-compound fourth. 5/4 is equated to 9 octaves minus 23 generators. Its color name is Saseyoti.
Maquila family (P8, c7P5/17)
This tempers out the maquila comma, [49 -6 -17⟩ (562949953421312/556182861328125). The generator is ~512/375 = ~535 ¢. Seventeen generators equals a septuple-compound fifth. 5/4 is equated to 3 octaves minus 6 generators. Its color name is Trisa-seguti. An obvious 11-limit interpretation of the generator is 11/8, leading to the Lala-seloti temperament. However, Lala-seloti is not nearly as accurate as Trisa-seguti.
Gammic family (P8, P5/20)
The gammic family tempers out the gammic comma, [-29 -11 20⟩. Nine generators of about 35 ¢ equals ~6/5, eleven equals ~5/4 and twenty equals ~3/2. 34edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is Carlos Gamma. Another member is the neptune temperament. Its color name is Laquinquadyoti.

Clans defined by a 2.3.7 comma

These are defined by a no-5's 7-limit (color name: za) comma. See also subgroup temperaments.

If a 5-limit comma defines a family of rank-2 temperaments, then we might say a comma belonging to another subgroup of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. We can modify the definition of normal comma list for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.

Archytas clan (P8, P5)
This clan tempers out Archytas' comma, 64/63. It equates 7/4 with 16/9. The clan consists of rank-2 temperaments, and should not be confused with the archytas family of rank-3 temperaments. Its color name is Ruti. Its best downward extension is superpyth.
Trienstonic clan (P8, P5)
This clan tempers out the septimal third-tone, 28/27, a low-accuracy temperament that equates 7/6 with 9/8, and 7/4 with 27/16. Its color name is Zoti.
Harrison clan (P8, P5)
This clan tempers out Harrison's comma, [-13 10 0 -1⟩ (59049/57344). It equates 7/4 to an augmented sixth. Its color name is Laruti. Its best downward extension is septimal meantone.
Garischismic clan (P8, P5)
This clan tempers out the garischisma, [25 -14 0 -1⟩ (33554432/33480783). It equates 8/7 to two apotomes ([-11 7⟩, 2187/2048) and 7/4 to a double-diminished octave [23 -14⟩. This clan includes vulture, newt, garibaldi, sextile, and satin. Its color name is Sasaruti.
Sasazoti clan (P8, P5)
This clan tempers out the leapfrog comma, [21 -15 0 1⟩ (14680064/14348907). It equates 7/6 to two apotomes and 7/4 to double augmented fifth. This clan includes leapday, leapweek and srutal.
Laruruti clan (P8/2, P5)
This clan tempers out the Laruru comma, [-7 8 0 -2⟩ (6561/6272). Two ~81/56 periods equal an octave. The generator is ~3/2, and four generators minus three periods equals ~7/4. The major second ~9/8 is divided in half, with each half equated to ~28/27. See also the Diaschismic or Saguguti temperament and the Jubilismic or Biruyoti temperament.
Semaphoresmic clan (P8, P4/2)
This clan tempers out the large septimal diesis, 49/48. Its generator is ~8/7 or ~7/6. Its color name is Zozoti. Its best downward extension is godzilla. See also semaphore.
Parahemif clan (P8, P5/2)
This clan tempers out the parahemif comma, [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 ¢ generator is 11/9, leading to the Luluti temperament.
Triruti clan (P8/3, P5)
This clan tempers out the Triru comma, [-1 6 0 -3⟩ (729/686), a low-accuracy temperament. Three ~9/7 periods equals an octave. 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.
Gamelismic clan (P8, P5/3)
This clan tempers out the gamelisma, [-10 1 0 3⟩ (1029/1024). Three ~8/7 generators equals a fifth. 7/4 is equated to an octave minus a generator. Five generators is slightly flat of 2/1, making this a cluster temperament. Its color name is Latrizoti. See also Sawati and Lasepzoti.
A particularly noteworthy member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds. Miracle divides the fifth into 6 equal steps, thus it is a weak extension. Its 21-note scale called Blackjack and 31-note scale called Canasta have some useful properties. It is the most efficient 11-limit temperament for many purposes, with a tuning close to 72edo.
Trizoti clan (P8, P5/3)
This clan tempers out the Trizo comma, [-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 ¢ generator is 8/7, leading to the much more accurate gamelismic or Latrizoti temperament.
Latriru clan (P8, P11/3)
This clan tempers out the lee comma, [-9 11 0 -3⟩ (177147/175616). The generator is ~112/81 = ~566 ¢, and three such generators equals ~8/3. 7/4 is equated to 11 generators minus 5 octaves. Its color name is Latriruti. An obvious full 7-limit interpretation of the generator is 7/5, leading to the liese temperament, which is a weak extension of meantone.
Stearnsmic clan (P8/2, P4/3)
This clan temper out the stearnsma, [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 equal ~4/3. 7/4 is equated to five ~9/7 generators minus an octave. Its color name is Latribiruti. Equating the ~54/49 generator to ~10/9 creates a weak extension of the porcupine temperament, as does equating the period to ~7/5.
Skwaresmic clan (P8, P11/4)
This clan tempers out the skwaresma, [-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.
Buzzardsmic clan (P8, P12/4)
This clan tempers out the buzzardsma, [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 temperament, which is in the vulture family.
Cloudy clan (P8/5, P5)
This clan tempers out the cloudy comma, [-14 0 0 5⟩ (16807/16384). It has a period of 1/5 octave, which represents ~8/7. The generator is ~3/2. Unlike the blackwood or Sawati family, ~3/2 is not equated with three-fifths of an octave, resulting in very small intervals. Its color name is Laquinzoti.
Quinruti clan (P8, P5/5)
This clan tempers out the bleu comma, [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.
Saquinzoti clan (P8, P12/5)
This clan tempers out the Saquinzo comma, [5 -12 0 5⟩ (537824/531441). Its generator is ~243/196 = ~380 ¢. Five generators makes ~3/1. 7/4 is equated to 12 generators minus 3 octaves. An obvious 5-limit interpretation of the generator is 5/4, leading to the magic temperament, which is in the magic family.
Lasepzoti clan (P8, P11/7)
This clan tempers out the Lasepzo comma [-18 -1 0 7⟩ (823543/786432). Its generator is ~8/7. Six generators equals ~7/3, and seven generators equals ~8/3. Three generators is ~30 ¢ sharp of 3/2, and five generators is ~15 ¢ sharp of 2/1, making this a cluster temperament. See also Sawati and Latrizoti.
Septiness clan (P8, P11/7)
This clan tempers out the septiness comma [26 -4 0 -7⟩ (67108864/66706983). Its generator is ~147/128, four of them gives ~7/4, and seven of them gives ~8/3. Five generators is ~12.5¢ sharp of 2/1, making this a cluster temperament. Its color name is Sasasepruti.
Sepruti clan (P8, P12/7)
This clan tempers out the Sepru comma, [7 8 0 -7⟩ (839808/823543). Its generator is ~7/6. Seven generators equals ~3/1. 7/4 is equated to 8 generators minus 1 octave. This clan includes as a strong extension the orwell temperament, which is in the semicomma family.
Septiennealimmal clan (P8/9, P5)
This clan tempers out the septiennealimma, [-11 -9 0 9⟩ (40353607/40310784). It has a period of 1/9 octave, which represents ~7/6. The generator is ~3/2. This clan includes a number of regular temperaments including enneaportent, ennealimmal, and novemkleismic. Its color name is Tritrizoti.

Clans defined by a 2.3.11 comma

Color name: ila. See also subgroup temperaments.

Lulubiti clan (P8/2, P5)
This low-accuracy 2.3.11 clan tempers out the Alpharabian limma, 128/121. Both 11/8 and 16/11 are equated to half-octave period. This clan includes as a strong extension the pajaric temperament, which is in the diaschismic family.
Rastmic clan (P8, P5/2)
This 2.3.11 clan tempers out 243/242 ([-1 5 0 0 -2⟩). Its generator is ~11/9. Two generators equals ~3/2. 11/8 is equated to 5 generators minus an octave. This clan includes as a strong extension the dicot temperament, which is in the dicot family. Its color name is Luluti.
Nexus clan (P8/3, P4/2)
This 2.3.11 clan tempers out the nexus comma [-16 -3 0 0 6⟩. Its 1/3-octave period is ~121/96 and its least-cents generator is ~12/11. A period plus a generator equals ~11/8. Six of these generators equals ~27/16. A period minus a generator equals ~1331/1152 or ~1536/1331. Two of these alternative generators equals ~4/3. Its color name is Tribiloti.
Alphaxenic or Laquadloti clan (P8/2, M2/4)
This 2.3.11 clan tempers out the Alpharabian comma [-17 2 0 0 4⟩. Its half-octave period is ~363/256, and its generator is ~33/32. Four generators equals ~9/8. 3/2 is equated to a period plus 2 generators, and 11/8 is equated to a period minus a generator. This clan includes a strong extension to the comic or Saquadyobiti temperament, which is in the jubilismic clan. Its color name is Laquadloti.

Clans defined by a 2.3.13 comma

Color name: tha. See also subgroup temperaments.

Thuthuti clan (P8, P5/2)
This 2.3.13 clan tempers out 512/507 ([9 -1 0 0 0 -2⟩). Its generator is ~16/13. Two generators equals ~3/2. 13/8 is equated to 1 octave minus 1 generator. This clan includes as a strong extension the dicot temperament, which is in the dicot family.
Satrithoti clan (P8, P11/3)
This 2.3.13 clan tempers out the threedie, 2197/2187 ([0 -7 0 0 0 3⟩). Its generator is ~18/13, and three generators equals ~8/3. 13/8 is equated to 7 generators minus three octaves. This clan is related to the Latriruti clan.

Clans defined by a 2.5.7 comma

These are defined by a no-3's 7-limit (color name: yaza nowa) comma. See also subgroup temperaments. The pergen's multigen (the second term, omitting any fraction) is always a no-3's 5-limit ratio such as 5/4, 8/5, 25/8, etc.

Jubilismic clan (P8/2, M3)
This clan tempers out the jubilisma, 50/49, which is the difference between 10/7 and 7/5. The generator is the classical major third (~5/4). The half-octave period is ~7/5 or ~10/7. 7/4 is equated to 1 period plus 1 generator. Its color name is Biruyoti.
Bapbo clan (P8, M3/2)
This clan tempers out the bapbo comma, 256/245. The genarator is ~8/7 = ~202 ¢ and two of them equals ~5/4. Its color name is Ruruguti Nowa.
Hemimean clan (P8, M3/2)
This clan tempers out the hemimean comma, [6 0 -5 2⟩ (3136/3125). The generator is ~28/25 = ~194 ¢. Two generators equals the classical major third (~5/4), three of them equals ~7/5, and five of them equals ~7/4. Its color name is Zozoquinguti Nowa.
Mabilismic clan (P8, cM3/3)
This clan tempers out the mabilisma, [-20 0 5 3⟩ (1071875/1048576). The generator is ~175/128 = ~527 ¢. Three generators equals ~5/2 and five of them equals ~32/7. Its color name is Latrizo-aquiniyoti Nowa.
Vorwell clan (P8, m6/3)
This clan tempers out the vorwell comma (named for being tempered in septimal vulture and orwell), [27 0 -8 -3⟩ (134217728/133984375). The generator is ~1024/875 = ~272 ¢. Three generators equals ~8/5 and eight of them equals ~7/2. Its color name is Sasatriru-aquadbiguti Nowa.
Quinzo-atriyoti Nowa clan (P8, M3/5)
This clan tempers out the rainy comma, [-21 0 3 5⟩ (2100875/2097152). The generator is ~256/245 = ~77 ¢. Three generators equals ~8/7 and five of them equals the classic major third (~5/4).
Llywelynsmic clan (P8, cM3/7)
This clan tempers out the llywelynsma, [22 0 -1 -7⟩ (4194304/4117715). The generator is ~8/7 = ~227 ¢ and seven of them equals ~5/2. Its color name is Sasepru-aguti Nowa.
Quince clan (P8, m6/7)
This clan tempers out the quince comma, [-15 0 -2 7⟩ (823543/819200). The generator is ~343/320 = ~116 ¢. Two generators equals ~8/7, five generators equals ~7/5, and seven generators equals the classical minor sixth ~8/5. An obvious 5-limit interpretation of the generator is ~16/15, leading to the miracle temperament, which is in the gamelismic clan. Its color name is Lasepzo-aguguti Nowa.
Slither clan (P8, ccm6/9)
This clan tempers out the slither comma, [16 0 4 -9⟩ (40960000/40353607). The generator is ~49/40 = ~357 ¢. Four generators equals ~16/7, five of them equals ~14/5, and nine of them equals a double-compound minor sixth of ~32/5. Its color name is Satritriru-aquadyoti Nowa.

Clans defined by a 3.5.7 comma

These are defined by a no-2's 7-limit (color name: yaza noca) comma. Any no-2's comma is a ratio of odd numbers. If no other commas are tempered out, It is a non-octave tuning, with a period of a perfect 12th or tritave, 3/1 (or some fraction of that). See also subgroup temperaments. The pergen's multigen (the second term, omitting any fraction) is always a no-2's 5-limit ratio such as 5/3, 9/5, 25/9, etc. In any no-2's subgroup, "compound" means increased by 3/1 not 2/1.

Rutribiyoti Noca clan (P12, M6)
This 3.5.7 clan tempers out the arcturus comma [0 -7 6 -1⟩ (15625/15309). Its only member so far is arcturus. The generator is the classical major sixth (~5/3), and six generators equals ~21/1.
Sensamagic clan (P12, M6/2)
This 3.5.7 clan tempers out the sensamagic comma [0 -5 1 2⟩ (245/243). The generator is ~9/7, and two generators equals the classic major sixth (~5/3). Its color name is Zozoyoti Noca.
Gariboh clan (P12, M6/3)
This 3.5.7 clan tempers out the gariboh comma [0 -2 5 -3⟩ (3125/3087). The generator is ~25/21, two generators equals ~7/5, and three generators equals the classical major sixth (~5/3). Its color name is Triru-aquinyoti Noca.
Mirkwai clan (P12, cm7/5)
This 3.5.7 clan tempers out the mirkwai comma, [0 3 4 -5⟩ (16875/16807). The generator is ~7/5, four generators equals ~27/7, and five generators equals the classical compound minor seventh (~27/5). Its color name is Quinru-aquadyoti Noca.
Sasepzo-atriguti Noca clan (P12, m7/7)
This 3.5.7 clan tempers out the procyon comma [0 -8 -3 7⟩ (823543/820125). Its only member so far is procyon. The generator is ~49/45, three generators equals ~9/7, four equals ~7/5, and seven equals the classic minor seventh (~9/5).
Satritrizo-aguguti Noca clan (P12, c3M6/9)
This 3.5.7 clan tempers out the betelgeuse comma [0 -13 -2 9⟩ (40353607/39858075). Its only member so far is betelgeuse. The generator is ~3645/2401, two generators equals ~7/3, and nine generators equals the classical triple-compound major sixth (~45/1).
Saquadtrizo-asepguti Noca clan (P12, c5m7/12)
This 3.5.7 clan tempers out the izar comma (also known as bapbo schismina), [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.

Temperaments defined by a 2.3.5.7 comma

These are defined by a full 7-limit (color name: yaza) comma.

Septisemi temperaments
These are very low complexity temperaments tempering out the minor septimal semitone, 21/20, and hence equating 5/3 with 7/4. Its color name is Zoguti.
Greenwoodmic temperaments
These temper out the greenwoodma, [-3 4 1 -2⟩ (405/392). Its color name is Ruruyoti.
Keegic temperaments
Keegic rank-2 temperaments temper out the keega, [-3 1 -3 3⟩ (1029/1000). Its color name is Trizoguti.
Mint temperaments
Mint rank-2 temperaments temper out the septimal quartertone, 36/35, equating 7/6 with 6/5, and 5/4 with 9/7. Its color name is Ruguti.
Avicennmic temperaments
These temper out the avicennma, [-9 1 2 1⟩ (525/512), also known as Avicenna's enharmonic diesis. Its color name is Zoyoyoti.
Sengic temperaments
Sengic rank-2 temperaments temper out the senga, [1 -3 -2 3⟩ (686/675). Its color name is Trizo-aguguti.
Keemic temperaments
Keemic rank-2 temperaments temper out the keema, [-5 -3 3 1⟩ (875/864). Its color name is Zotriyoti.
Secanticorn temperaments
Secanticorn rank-2 temperaments temper out the secanticornisma, [-3 11 -5 -1⟩ (177147/175000). Its color name is Laruquinguti.
Nuwell temperaments
Nuwell rank-2 temperaments temper out the nuwell comma, [1 5 1 -4⟩ (2430/2401). Its color name is Quadru-ayoti.
Mermismic temperaments
Mermismic rank-2 temperaments temper out the mermisma, [5 -1 7 -7⟩ (2500000/2470629). Its color name is Sepruyoti.
Negricorn temperaments
Negricorn rank-2 temperaments temper out the negricorn comma, [6 -5 -4 4⟩ (153664/151875). Its color name is Saquadzoguti.
Tolermic temperaments
These temper out the tolerma, [10 -11 2 1⟩ (179200/177147). Its color name is Sazoyoyoti.
Valenwuer temperaments
Valenwuer rank-2 temperaments temper out the valenwuer comma, [12 3 -6 -1⟩ (110592/109375). Its color name is Sarutribiguti.
Mirwomo temperaments
Mirwomo rank-2 temperaments temper out the mirwomo comma, [-15 3 2 2⟩ (33075/32768). Its color name is Labizoyoti.
Catasyc temperaments
Catasyc rank-2 temperaments temper out the catasyc comma, [-11 -3 8 -1⟩ (390625/387072). Its color name is Laruquadbiyoti.
Compass temperaments
Compass rank-2 temperaments temper out the compass comma, [-6 -2 10 -5⟩ (9765625/9680832). Its color name is Quinruyoyoti.
Trimyna temperaments
Trimyna rank-2 temperaments temper out the trimyna comma, [-4 1 -5 5⟩ (50421/50000). Its color name is Quinzoguti.
Starling temperaments
Starling rank-2 temperaments temper out the starling comma a.k.a. septimal semicomma, [1 2 -3 1⟩ (126/125), the difference between three 6/5's plus one 7/6, and an octave. It includes myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum. Its color name is Zotriguti.
Octagar temperaments
Octagar rank-2 temperaments temper out the octagar comma, [5 -4 3 -2⟩ (4000/3969). Its color name is Rurutriyoti.
Orwellismic temperaments
Orwellismic rank-2 temperaments temper out orwellisma, [6 3 -1 -3⟩ (1728/1715). Its color name is Triru-aguti.
Mynaslendric temperaments
Mynaslendric rank-2 temperaments temper out the mynaslender comma, [11 4 1 -7⟩ (829440/823543). Its color name is Sepru-ayoti.
Mistismic temperaments
Mistismic rank-2 temperaments temper out the mistisma, [16 -6 -4 1⟩ (458752/455625). Its color name is Sazoquadguti.
Varunismic temperaments
Varunismic rank-2 temperaments temper out the varunisma, [-9 8 -4 2⟩ (321489/320000). Its color name is Labizoguguti.
Marvel temperaments
Marvel rank-2 temperaments temper out the marvel comma, [-5 2 2 -1⟩ (225/224). It includes wizard, tritonic, septimin, slender, triton, escapist and marv, not to mention negri, sharpie, pelogic, meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton. Its color name is Ruyoyoti.
Dimcomp temperaments
Dimcomp rank-2 temperaments temper out the dimcomp comma, [-1 -4 8 -4⟩ (390625/388962). Its color name is Quadruyoyoti.
Cataharry temperaments
Cataharry rank-2 temperaments temper out the cataharry comma, [-4 9 -2 -2⟩ (19683/19600). Its color name is Labiruguti.
Canousmic temperaments
Canousmic rank-2 temperaments temper out the canousma, [4 -14 3 4⟩ (4802000/4782969). Its color name is Saquadzo-atriyoti.
Triwellismic temperaments
Triwellismic rank-2 temperaments temper out the triwellisma, [1 -1 -7 6⟩ (235298/234375). Its color name is Tribizo-asepguti.
Hemimage temperaments
Hemimage rank-2 temperaments temper out the hemimage comma, [5 -7 -1 3⟩ (10976/10935). Its color name is Satrizo-aguti.
Hemifamity temperaments
Hemifamity rank-2 temperaments temper out the hemifamity comma, [10 -6 1 -1⟩ (5120/5103). Its color name is Saruyoti.
Parkleiness temperaments
Parkleiness rank-2 temperaments temper out the parkleiness comma, [7 7 -9 1⟩ (1959552/1953125). Its color name is Zotritriguti.
Porwell temperaments
Porwell rank-2 temperaments temper out the porwell comma, [11 1 -3 -2⟩ (6144/6125). Its color name is Sarurutriguti.
Cartoonismic temperaments
Cartoonismic rank-2 temperaments temper out the cartoonisma, [12 -3 -14 9⟩ (165288374272/164794921875). Its color name is Satritrizo-asepbiguti.
Hemfiness temperaments
Hemfiness rank-2 temperaments temper out the hemfiness comma, [15 -5 3 -5⟩ (4096000/4084101). Its color name is Saquinru-atriyoti.
Hewuermera temperaments
Hewuermera rank-2 temperaments temper out the hewuermera comma, [16 2 -1 -6⟩ (589824/588245). Its color name is Satribiru-aguti.
Lokismic temperaments
Lokismic rank-2 temperaments temper out the lokisma, [21 -8 -6 2⟩ (102760448/102515625). Its color name is Sasa-bizotriguti.
Decovulture temperaments
Decovulture rank-2 temperaments temper out the decovulture comma, [26 -7 -4 -2⟩ (67108864/66976875). Its color name is Sasabiruguguti.
Pontiqak temperaments
Pontiqak rank-2 temperaments temper out the pontiqak comma, [-17 -6 9 2⟩ (95703125/95551488). Its color name is Lazozotritriyoti.
Mitonismic temperaments
Mitonismic rank-2 temperaments temper out the mitonisma, [-20 7 -1 4⟩ (5250987/5242880). Its color name is Laquadzo-aguti.
Horwell temperaments
Horwell rank-2 temperaments temper out the horwell comma, [-16 1 5 1⟩ (65625/65536). Its color name is Lazoquinyoti.
Neptunismic temperaments
Neptunismic rank-2 temperaments temper out the neptunisma, [-12 -5 11 -2⟩ (48828125/48771072). Its color name is Laruruleyoti.
Metric microtemperaments
Metric rank-2 temperaments temper out the meter, [-11 2 7 -3⟩ (703125/702464). Its color name is Latriru-asepyoti.
Wizmic microtemperaments
Wizmic rank-2 temperaments temper out the wizma, [-6 -8 2 5⟩ (420175/419904). Its color name is Quinzo-ayoyoti.
Supermatertismic temperaments
Supermatertismic rank-2 temperaments temper out the supermatertisma, [-6 3 9 -7⟩ (52734375/52706752). Its color name is Lasepru-atritriyoti.
Breedsmic temperaments
Breedsmic rank-2 temperaments temper out the breedsma, [-5 -1 -2 4⟩ (2401/2400). Its color name is Bizozoguti.
Supermasesquartismic temperaments
Supermasesquartismic rank-2 temperaments temper out the supermasesquartisma, [-5 10 5 -8⟩ (184528125/184473632). Its color name is Laquadbiru-aquinyoti.
Ragismic microtemperaments
Ragismic rank-2 temperaments temper out the ragisma, [-1 -7 4 1⟩ (4375/4374). Its color name is Zoquadyoti.
Akjaysmic temperaments
Akjaysmic rank-2 temperaments temper out the akjaysma, [47 -7 -7 -7⟩. These have a period of 1/7 octave, and five periods equals ~105/64. Unlike the whitewood or Lawati family, ~3/2 is not equated with 4/7 of an octave, resulting in small intervals. Its color name is Trisa-sepruguti.
Landscape microtemperaments
Landscape rank-2 temperaments temper out the landscape comma, [-4 6 -6 3⟩ (250047/250000). These have a period of 1/3 octave, but ~5/4 is not equated with a period, resulting in small intervals. Its color name is Trizoguguti.

Rank-3 temperaments

Even less familiar than rank-2 temperaments are the 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 third generator in a rank-3 pergen is usually a comma, but sometimes it is some fraction of a 5-limit or 7-limit interval.

Families defined by a 2.3.5 comma

Every 5-limit (color name: ya) comma defines a rank-3 family, thus every comma in the list of rank-2 2.3.5 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a 7-limit (color name: yaza) temperament in which one of the generators is ~7/1. This generator can be reduced to ~7/4, which can be reduced further to ~64/63. Hence in all the pergens below, the ^1 or /1 generator is ~64/63. An additional 7-limit or 11-limit comma creates an 11-limit (color name: yazala) temperament, and so forth. All these examples are 7-limit:

Didymus rank-3 family (P8, P5, ^1)
These are the rank-3 temperaments tempering out the didymus or meantone comma, 81/80. Its color name is Guti.
Diaschismic rank-3 family (P8/2, P5, /1)
These are the rank-3 temperaments tempering out the diaschisma, [11 -4 -2⟩ (2048/2025). The half-octave period is ~45/32. Its color name is Saguguti.
Porcupine rank-3 family (P8, P4/3, /1)
These are the rank-3 temperaments tempering out the porcupine comma a.k.a. maximal diesis, [1 -5 3⟩ (250/243). In the pergen, P4/3 is ~10/9. Its color name is Triyoti.
Kleismic rank-3 family (P8, P12/6, /1)
These are the rank-3 temperaments tempering out the kleisma, [-6 -5 6⟩ (15625/15552). In the pergen, P12/6 is ~6/5. Its color name is Tribiyoti.

Families defined by a 2.3.7 comma

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, ^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:

Archytas family (P8, P5, ^1)
Archytas temperament tempers out 64/63. This comma equates every 7-limit interval to some 3-limit interval. If 81/80 were tempered out too, the otonal tetrad 4:5:6:7 would be identified with the dominant seventh chord. Its color name is Ruti.
Garischismic family (P8, P5, ^1)
A garischismic temperament is one which tempers out the garischisma, [25 -14 0 -1⟩ (33554432/33480783). Its color name is Sasaruti.
Laruruti clan (P8/2, P5)
This clan tempers out the Laruru comma, [-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, [-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, [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, [-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, ^1 = ~81/80. Its color name is Ruyoyoti.
Starling family (P8, P5, ^1)
Starling tempers out the starling comma a.k.a. septimal semicomma, [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, ^1 = ~81/80. Its color name is Zotriguti.
Sensamagic family (P8, P5, ^1)
These temper out [0 -5 1 2⟩ (245/243), which divides 16/15 into two 28/27's. In the pergen, ^1 = ~64/63. Its color name is Zozoyoti.
Greenwoodmic family (P8, P5, ^1)
These temper out the greenwoodma, [-3 4 1 -2⟩ (405/392), which divides 10/9 into two 15/14's. In the pergen, ^1 = ~64/63. Its color name is Ruruyoti.
Avicennmic family (P8, P5, ^1)
These temper out the avicennma, [-9 1 2 1⟩ (525/512), which divides 7/6 into two 16/15's. In the pergen, ^1 = ~81/80. Its color name is Lazoyoyoti.
Keemic family (P8, P5, ^1)
These temper out the keema, [-5 -3 3 1⟩ (875/864), which divides 15/14 into two 25/24's. In the pergen, ^1 = ~81/80. Its color name is Zotriyoti.
Orwellismic family (P8, P5, ^1)
These temper out the orwellisma, [6 3 -1 -3⟩ (1728/1715). In the pergen, ^1 = ~64/63. Its color name is Triru-aguti.
Nuwell family (P8, P5, ^1)
These temper out the nuwell comma, [1 5 1 -4⟩ (2430/2401). In the pergen, ^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, [-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, ^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, [10 -6 1 -1⟩ (5120/5103), which divides 10/7 into three 9/8's. In the pergen, ^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, [-16 1 5 1⟩ (65625/65536). In the pergen, ^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, [5 -7 -1 3⟩ (10976/10935), which divides 10/9 into three 28/27's. In the pergen, ^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, ^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, ^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, ^1 = ~81/80. Its color name is Biruyoti.
Cataharry family (P8, P4/2, ^1)
Cataharry temperaments temper out the cataharry comma, [-4 9 -2 -2⟩ (19683/19600). In the pergen, half a fourth is ~81/70, and ^1 = ~81/80. Its color name is Labiruguti.
Breed family (P8, P5/2, ^1)
Breed is a 7-limit microtemperament which tempers out [-5 -1 -2 4⟩ (2401/2400). While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749edo will certainly do the trick. Breed has generators of 2/1, a 49/40-cum-60/49 neutral third, and ~64/63. Its color name is Bizozoguti.
Sengic family (P8, P5, vm3/2)
These temper out the senga, [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, [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, [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, [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, [-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, [-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 ^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, [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, [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, [-1 -4 8 -4⟩ (390625/388962). In the pergen, the 1/4-octave period is ~25/21, and ^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, [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, [2 -2 2 0 -1⟩ (100/99). 11/8 is equated to 25/18, which is an octave minus two 6/5's. Since 25/18 is a 5-limit interval, every 2.3.5.11 interval is equated to a 5-limit interval, and both the pergen and the lattice are identical to that of 5-limit JI. In the pergen, ^1 = ~81/80. Its color name is Luyoyoti.
Biyatismic clan (P8, P5, ^1)
These temper out the biyatisma, [-3 -1 -1 0 2⟩ (121/120). 5/4 is equated to 121/96, which is two 11/8's minus a 3/2 fifth. Since 121/96 is an ila (11-limit no-fives no-sevens) interval, every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Lologuti.
Valinorsmic clan
These temper out the valinorsma, [4 0 -2 -1 1⟩ (176/175). To be a rank-3 temperament, either an additional comma must vanish or the prime subgroup must omit prime 3. Thus no assumptions can be made about the pergen. Its color name is Loruguguti.
Rastmic rank-3 clan
These temper out the rastma, [1 5 0 0 -2⟩ (243/242). In the corresponding 2.3.11 rank-2 temperament, the pergen is (P8, P5/2). Its color name is Luluti.
Pentacircle clan (P8, P5, ^1)
These temper out the pentacircle comma, [7 -4 0 1 -1⟩ (896/891). The interval between 11/8 and 7/4 is equated to 81/64. Since that is a 3-limit interval, every 2.3.11 interval is equated to a 2.3.7 interval and vice versa, and both the pergen and the lattice are identical to that of either 2.3.7 JI or 2.3.11 JI. In the pergen, ^1 is either ~64/63 or ~33/32 or ~729/704. Its color name is Saluzoti.
Semicanousmic clan (P8, P5, ^1)
These temper out the semicanousma, [-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, [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, [17 -5 0 -2 -1⟩ (131072/130977). 11/8 is equated with a 2.3.7 interval, and thus every 2.3.7.11 interval is equated with a 2.3.7 interval. In the pergen, ^1 = ~64/63. Its color name is Salururuti.
Alphaxenic rank-3 clan
These temper out the Alpharabian comma, [-17 2 0 0 4⟩ (131769/131072). In the corresponding 2.3.11 rank-2 temperament, the pergen is (P8/2, M2/4). Its color name is Laquadloti.
Keenanismic temperaments
These temper out the keenanisma, [-7 -1 1 1 1⟩ (385/384). Its color name is Lozoyoti.
Werckismic temperaments
These temper out the werckisma, [-3 2 -1 2 -1⟩ (441/440). Its color name is Luzozoguti.
Swetismic temperaments
These temper out the swetisma, [2 3 1 -2 -1⟩ (540/539). Its color name is Lururuyoti.
Lehmerismic temperaments
These temper out the lehmerisma, [-4 -3 2 -1 2⟩ (3025/3024). Its color name is Loloruyoyoti.
Kalismic temperaments
These temper out the kalisma, [-3 4 -2 -2 2⟩ (9801/9800). Its color name is Biloruguti.

Rank-4 temperaments

Main article: 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, [-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, [-3 2 -1 2 -1⟩ (441/440). 11/8 is equated to [-6 2 -1 2⟩ and 5/4 is equated to [-5 2 0 2 -1⟩, thus the lattice can be thought of as either 7-limit JI or 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, [2 3 1 -2 -1⟩ (540/539). 11/8 is equated to [-1 3 1 -2⟩ (135/98) and 5/4 is equated to [-4 -3 0 2 1⟩, thus the lattice can be thought of as either 7-limit JI or 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, [-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, ^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, [-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 article: Subgroup temperaments

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

Commatic realms

By a commatic realm is meant the whole collection of regular temperaments of various ranks and for subgroups (including full prime limits) 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 ([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 ([16 0 0 -2 -3⟩), the orgonisma. Its color name is Satrilu-aruruti.
The Nexus
This is the commatic realm of the 11-limit comma 1771561/1769472 ([-16 -3 0 0 6⟩), the nexus comma. Its color name is Tribiloti.
The Quartercache
This is the commatic realm of the 11-limit comma 117440512/117406179 ([24 -6 0 1 -5⟩), the quartisma. Its color name is Saquinlu-azoti.

Miscellaneous other temperaments

Limmic temperaments
Various subgroup temperaments all tempering out the limma, 256/243.
Fractional-octave temperaments
These temperaments all have a fractional-octave period.
Miscellaneous 5-limit temperaments
High in badness, but worth cataloging for one reason or another.
Low harmonic entropy linear temperaments
Temperaments where the average harmonic entropy of their intervals is low in a particular scale size range.
Turkish maqam music temperaments
Various theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish makam (maqam) music in a systematic way. This includes, in effect, certain linear temperaments.
Very low accuracy temperaments
All hope abandon ye who enter here.
Very high accuracy temperaments
Microtemperaments which do not fit in elsewhere.
Middle Path tables
Tables of temperaments where complexity/7.65 + damage/10 < 1. Useful for beginners looking for a list of a manageable number of temperaments, which approximate harmonious intervals accurately with a manageable number of notes.
Middle Path table of five-limit rank two temperaments
Middle Path table of seven-limit rank two temperaments
Middle Path table of eleven-limit rank two temperaments

Maps of temperaments

Temperament nomenclature

External links

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