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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | 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. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-04 16:49:21 UTC</tt>.<br>
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| : The original revision id was <tt>506390598</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"><span style="display: block; text-align: right;">Other languages: [[xenharmonie/Reguläre Temperaturen|Deutsch]]
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| </span>
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| [[toc|flat]]
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| ----
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| =Regular temperaments=
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| Regular temperaments are non-Just tunings wherein the infinite number of intervals in p-limit Just intonation (or any subgroup thereof) are mapped to a smaller (though still infinite) set of tempered intervals, by "tempering" (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 would be located by a four-dimensional set of coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which commas (and how many) are tempered out, and 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, or in other words 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. |
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| 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 [[comma|commas]] tempered out by the temperament, or a set of r independent [[Vals and Tuning Space|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. |
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| ==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. |
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| 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, which 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. |
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| ==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. |
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| 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 as a first order of business. These concepts are rather straightforward and require very 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. |
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| 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 ("Pure-Octave Tenney-Euclidean") and [[Top tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please"). Optimization is rather
| | ; [[Compton family]] (P8/12, ^1) |
| intensive mathematically, but it is seldom (if ever) left as an exercise to the reader; most temperaments are presented here in their optimal forms.
| | : 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. |
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| =[[edo|Equal temperaments]]=
| | ; [[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. |
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| [[Equal Temperaments|Equal temperaments]] (abbreviated ET or tET) and 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 (12-ET) 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-tET. | | ; [[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. |
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| =Rank 2 (including "linear") temperaments[[#lineartemperaments]]= | | === 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. |
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| Regular temperaments of ranks two and three are cataloged [[Optimal patent val|here]]. Other pages listing them are [[Paul Erlich]]'s [[Catalog of five-limit rank two temperaments|catalog of 5-limit rank two temperaments]], and a [[Proposed names for rank 2 temperaments|page]] listing higher limit rank two temperaments. There is also [[Graham Breed]]'s [[http://x31eq.com/catalog2.html|giant list of regular temperaments]].
| | ; [[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. |
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| P-limit Rank 2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators (though they may have any number of step-sizes). This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. Rank-2 temperaments can be reduced to a related rank-1 temperaments 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-TET by tempering out the Pythagorean comma.
| | ; [[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. |
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| ==Families==
| | ; [[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. |
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| 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, where a **comma** is a small interval, not a square or cube or other power, which is tempered out by the temperament.
| | ; [[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. |
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| Meantone is a familiar historical temperament based on a chain of fifths (or fourths), but it is only one of many possibilities for temperaments based on a chain of generating intervals. These are referred to as "rank 2" temperaments, since they are based on a set of two linearly independent intervals. One of these intervals (typically an octave or fraction of an octave) can be selected as the "period", and another interval, usually chosen to be smaller than the period, is referred to as the "generator".
| | ; [[Diaschismic family]] (P8/2, P5) |
| | : The diaschismic family tempers out the [[diaschisma]], {{monzo| 11 -4 -2 }} (2048/2025), such that two classic major thirds and a [[81/64|Pythagorean major third]] stack to an octave (i.e. {{nowrap| (5/4)⋅(5/4)⋅(81/64) → 2/1 }}). It has a half-octave period of an approximate 45/32 or 64/45, and its generator is an approximate 3/2. 5/4 is equated to 3 periods minus 2 fifths. The major second ~9/8 is divided in half, with each half equated to ~16/15. Diaschismic tunings include [[12edo|12-]], [[22edo|22-]], [[34edo|34-]], [[46edo|46-]], [[56edo|56-]], [[58edo|58-]] and [[80edo]]. Its color name is Saguguti. An obvious 7-limit interpretation of the period is 7/5, which makes [[pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out. [[22edo]] is an excellent pajara tuning. |
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| ===[[Meantone family]]===
| | ; [[Bug family]] (P8, P4/2) |
| The meantone family tempers out 81/80, also called the syntonic comma. 81/80 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 some sort of average or "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 syntonic comma (the 81/80 interval.)
| | : 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. |
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| ===[[Schismatic family]]===
| | ; [[Immunity family]] (P8, P4/2) |
| The schismatic family tempers out the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the syntonic comma (81/80). 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 [[xenharmonic/12edo|12edo]], [[xenharmonic/29edo|29edo]], [[xenharmonic/41edo|41edo]], [[xenharmonic/53edo|53edo]], and [[xenharmonic/118edo|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. |
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| ===[[Kleismic family]]===
| | ; [[Dicot family]] (P8, P5/2) |
| The kleismic family of temperaments tempers out the kleisma of 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. |
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| ===[[Magic family]]===
| | ; [[Augmented family]] (P8/3, P5) |
| The magic family tempers out 3125/3072, known as the magic comma or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/2. The magic 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. |
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| ===[[Diaschismic family]]===
| | ; [[Misty family]] (P8/3, P5) |
| The diaschismic family tempers out 2048/2025, the [[diaschisma]], which tempers things 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| pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out, and 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. |
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| ===[[Pelogic family]]===
| | ; [[Porcupine family]] (P8, P4/3) |
| This tempers out the pelogic comma, 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, the accumulated flatness leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates 2L5s "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. |
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| ===[[Porcupine family]]===
| | ; [[Alphatricot family]] (P8, P11/3) |
| The porcupine family tempers out 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]]. |
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| ===[[Würschmidt family]]===
| | ; [[Diminished family]] (P8/4, P5) |
| The würschmidt (or wuerschmidt) family tempers out Würschmidt's 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 perfect 5th two octaves up); that is, (5/4)^8 * (393216/390625) = 6. It tends to generate the same MOS's 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. |
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| ===[[Augmented family]]===
| | ; [[Undim family]] (P8/4, P5) |
| The augmented family tempers out the diesis of 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" (3L3s) 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]]" (3L6s). | | : 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. |
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| ===[[Dimipent family]]===
| | ; Negri family (P8, P4/4) |
| The dimipent family tempers out the major diesis aka diminished comma, 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. |
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| ===[[Dicot family]]===
| | ; [[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. |
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| ===[[Tetracot family]]===
| | ; [[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 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. |
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| ===[[Sensipent family]]===
| | ; [[Vulture family]] (P8, P12/4) |
| This tempers out the sensipent comma, 78732/78125, also known as the medium semicomma. Tunings include [[xenharmonic/8edo|8edo]], [[xenharmonic/19edo|19edo]], [[xenharmonic/46edo|46edo]], and [[xenharmonic/65edo|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. |
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| ===[[Semicomma family|Orwell and the semicomma family]]===
| | ; [[Quintile family]] (P8/5, P5) |
| The semicomma (also known as **Fokker's comma)** 2109375/2097152 = |-21 3 7> 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. |
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| ===[[Pythagorean family]]===
| | ; [[Ripple family]] (P8, P4/5) |
| The Pythagorean family tempers out the Pythagorean comma, |-19 12 0>. 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 temperament and catler temperament. 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. |
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| ===[[Apotome family]]===
| | ; [[Passion family]] (P8, P4/5) |
| This family tempers out the apotome, 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. |
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| ===[[Gammic family]]===
| | ; [[Quintaleap family]] (P8, P4/5) |
| The gammic family tempers out the gammic comma, |-29 -11 20>. 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. |
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| ===[[Minortonic family]]===
| | ; [[Quindromeda family]] (P8, P4/5) |
| 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). | | : 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. |
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| ===[[Bug family]]===
| | ; [[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. |
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| ===[[Father family]]===
| | ; [[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. |
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| ===[[Sycamore family]]===
| | ; [[Fifive family]] (P8/2, P5/5) |
| 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.
| | : 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. |
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| ===[[Escapade family]]===
| | ; [[Quintosec family]] (P8/5, P5/2) |
| This tempers out the escapade comma, |32 -7 -9>, 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. |
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| ===[[Amity family]]===
| | ; [[Trisedodge family]] (P8/5, P4/3) |
| This tempers out the amity comma, 1600000/1594323 = |9 -13 5>. | | : 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. |
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| ===[[Vulture family]]===
| | ; Ampersand family (P8, P5/6) |
| This tempers out the vulture comma, |24 -21 4>. | | : 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. |
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| ===[[Vishnuzmic family]]===
| | ; [[Kleismic family]] (P8, P12/6) |
| 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/), 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. |
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| ===[[Luna Family]]===
| | ; [[Semicomma family|Orson or semicomma family]] (P8, P12/7) |
| This tempers out the Luna comma; |38 -2 -15> (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. |
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| ===[[Laconic Family]]===
| | ; [[Wesley family]] (P8, ccP4/7) |
| This tempers out the laconic comma, 2187/2000, which is the difference between three 10/9's and one 3/2. Laconic is supported by 16-EDO, 21-EDO, and 37b-EDO, 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]]. |
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| ===[[Immunity family]]===
| | ; [[Sensipent family]] (P8, ccP5/7) |
| This tempers out the immunity comma, 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. |
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| ===[[Ditonmic family]]===
| | ; [[Vishnuzmic family]] (P8/2, P4/7) |
| This tempers out the ditonma, 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. |
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| ===[[Shibboleth family]]===
| | ; [[Unicorn family]] (P8, P4/8) |
| This tempers out the shibboleth comma, 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. |
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| ===[[Comic family]]===
| | ; [[Würschmidt family]] (P8, ccP5/8) |
| This tempers out the comic comma, 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. |
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| ===[[Wesley family]]===
| | ; [[Escapade family]] (P8, P4/9) |
| This tempers out the wesley comma, 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. |
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| ===[[Fifive family]]===
| | ; [[Mabila family]] (P8, c4P4/10) |
| This tempers out the fifive comma, 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. |
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| ===[[Maja family]]===
| | ; [[Sycamore family]] (P8, P5/11) |
| This tempers out the maja comma, 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. |
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| ===[[Mutt family]]===
| | ; [[Quartonic family]] (P8, P4/11) |
| This tempers out the mutt comma, |-44 -3 21>, leading to some strange properties.
| | : 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. |
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| ==Clans== | | ; [[Lafa family]] (P8, P12/12) |
| | : This tempers out the [[lafa comma]], {{monzo| 77 -31 -12 }}. The generator is {{nowrap| ~4982259375/4294967296 {{=}} ~258.6{{c}} }}. Twelve generators equals a twelfth (~3/1). 5/4 is equated to 7 octaves minus 31 generators. Its color name is Tribisa-quadtriguti. |
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| ===[[Gamelismic clan]]===
| | ; [[Ditonmic family]] (P8, c4P4/13) |
| 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. Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. 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 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. |
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| Particularly noteworthy as member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds.
| | ; [[Luna family]] (P8, ccP4/15) |
| | : This tempers out the [[luna comma]], {{monzo| 38 -2 -15 }} (274877906944/274658203125). The generator is ~{{monzo| 18 -1 -7 }} at ~193{{c}}. Two generators equals ~5/4, and fifteen generators equals a double-compound fourth of ~16/3. Its color name is Sasa-quintriguti. |
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| Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called "blackjack" and a 31-note scale called "canasta" have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.
| | ; [[Vavoom family]] (P8, P12/17) |
| | : This tempers out the [[vavoom comma]], {{monzo| -68 18 17 }}. The generator is {{nowrap| ~16/15 {{=}} ~111.9{{c}} }}. Seventeen generators equals a twelfth (~3/1). 5/4 is equated to two octaves minus 18 generators. Its color name is Quinla-seyoti. |
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| ===[[Trienstonic clan]]===
| | ; [[Minortonic family]] (P8, ccP5/17) |
| This clan tempers out the septimal third-tone, [[28_27|28/27]], a triprime comma with factors of 2, 3 and 7. | | : This tempers out the [[minortone comma]], {{monzo| -16 35 -17 }}. The head of the family is minortonic temperament, with a generator of a minor tone (~10/9). Seventeen generators equals a double-compound fifth (~6/1). 5/4 is equated to 35 generators minus 5 octaves. Its color name is Trila-seguti. |
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| ===[[Slendro clan]]===
| | ; [[Maja family]] (P8, c<sup>6</sup>P4/17) |
| This clan tempers out the slendro diesis, [[49_48|49/48]], a triprime comma with factors of 2, 3 and 7. | | : 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. |
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| ===[[Jubilismic clan]]===
| | ; [[Maquila family]] (P8, c<sup>7</sup>P5/17) |
| This tempers out the jubilisma, [[50_49|50/49]], E.A. the difference between 10/7 and 7/5. | | : 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. |
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| ===[[Archytas clan]]===
| | ; [[Gammic family]] (P8, P5/20) |
| This clan tempers out the Archytas comma, [[64_63|64/63]], which is a triprime comma with factors of 2, 3 and 7. The clan consists of rank two temperaments, and should not be confused with the [[Archytas family]] of rank three temperaments.
| | : 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. |
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| ===[[Sensamagic clan]]=== | | === Clans defined by a 2.3.7 comma === |
| This clan tempers out 245/243, the sensamagic comma.
| | These are defined by a no-5's 7-limit (color name: za) comma. See also [[subgroup temperaments]]. |
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| ===[[Hemimean clan]]===
| | 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 tempers out the hemimean comma, 3136/3125, a no-threes comma.
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| ===[[Mirkwai clan]]===
| | ; [[Archytas clan]] (P8, P5) |
| This tempers out the mirkwai comma, |0 3 4 -5> = 16875/16807, a no-twos comma (ratio of odd numbers.) | | : 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]]. |
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| ===[[Quince clan]]===
| | ; [[Trienstonic clan]] (P8, P5) |
| This tempers out quince, the no-threes comma |-15 0 -2 7> = 823543/819200. | | : 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. |
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| =Temperaments for a given comma=
| | ; Harrison clan (P8, P5) |
| | : This clan tempers out [[Harrison's comma]], {{monzo| -13 10 0 -1 }} (59049/57344). It equates 7/4 to an augmented sixth. Its color name is Laruti. Its best downward extension is [[septimal meantone]]. |
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| ===[[Septisemi temperaments]]===
| | ; [[Garischismic clan]] (P8, P5) |
| These are very low complexity temperaments tempering out the minor septimal semitone, 21/20 and hence equating 5/3 with 7/4.
| | : This clan tempers out the [[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. |
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| ===[[Mint temperaments]]===
| | ; Sasazoti clan (P8, P5) |
| These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35.
| | : 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]]. |
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| ===[[Greenwoodmic temperaments]]===
| | ; Laruruti clan (P8/2, P5) |
| These temper out the greenwoodma, |-3 4 1 -2> = 405/392.
| | : 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. |
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| ===[[Avicennmic temperaments]]===
| | ; [[Semaphoresmic clan]] (P8, P4/2) |
| These temper out the avicennma, |-9 1 2 1> = 525/512, also known as Avicenna's enharmonic diesis.
| | : 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]]. |
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| ===[[Keemic temperaments]]===
| | ; Parahemif clan (P8, P5/2) |
| These temper out the keema, |-5 -3 3 1> = 875/864.
| | : 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. |
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| ===[[Starling temperaments]]===
| | ; Triruti clan (P8/3, P5) |
| Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma (<span class="commentBody">the difference between three 6/5s plus one 7/6, and an octave) </span>is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.
| | : 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. |
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| ===[[Marvel temperaments]]===
| | ; [[Gamelismic clan]] (P8, P5/3) |
| These temper out |-5 2 2 -1> = 225/224, the marvel comma, and 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 [[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. |
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| ===[[Orwellismic temperaments]]===
| | ; Trizoti clan (P8, P5/3) |
| These temper out |6 3 -1 -3> = 1728/1715, the orwellisma.
| | : 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. |
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| ===[[Octagar temperaments]]===
| | ; Latriru clan (P8, P11/3) |
| Octagar temperaments temper out the octagar comma, |5 -4 3 -2> = 4000/3969.
| | : 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. |
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| ===[[Hemifamity temperaments]]===
| | ; [[Stearnsmic clan]] (P8/2, P4/3) |
| Hemifamity temperaments temper out the hemifamity comma, |10 -6 1 -1> = 5120/5103.
| | : 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. |
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| ===[[Porwell temperaments]]===
| | ; Skwaresmic clan (P8, P11/4) |
| Porwell temperaments temper out the porwell comma, |11 1 -3 -2> = 6144/6125.
| | : 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. |
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| ===[[Stearnsmic temperaments]]===
| | ; [[Buzzardsmic clan]] (P8, P12/4) |
| Stearnsmic temperaments temper out the stearnsma, |1 10 0 -6> = 118098/117649.
| | : 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. |
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| ===[[Hemimage temperaments]]===
| | ; [[Cloudy clan]] (P8/5, P5) |
| Hemimage temperaments temper out the hemimage comma, |5 -7 -1 3> = 10976/10935.
| | : 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. |
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| ===[[Horwell temperaments]]===
| | ; Quinruti clan (P8, P5/5) |
| Horwell temperaments temper out the horwell comma, |-16 1 5 1> = 65625/65536.
| | : 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. |
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| ===[[Breedsmic temperaments]]===
| | ; Saquinzoti clan (P8, P12/5) |
| A breedsmic temperament is one which tempers out the breedsma, |-5 -1 -2 4> = 2401/2400. Some which do not get discussed elsewhere are collected on a page [[Breedsmic temperaments|here]].
| | : 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. |
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| ===[[Ragismic microtemperaments]]===
| | ; Lasepzoti clan (P8, P11/7) |
| A ragismic temperament is one which tempers out |-1 -7 4 1> = 4375/4374. 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.
| | : 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. |
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| ===[[Landscape microtemperaments]]===
| | ; Septiness clan (P8, P11/7) |
| A landscape temperament is one which tempers out |-4 6 -6 3> = 250047/250000.
| | : 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. |
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| ==[[Wizmic microtemperaments]]==
| | ; Sepruti clan (P8, P12/7) |
| A wizmic temperament is one which tempers out the wizma, | -6 -8 2 5 > = 420175/419904.
| | : 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. |
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| ===[[31 comma temperaments]]===
| | ; [[Septiennealimmal clan]] (P8/9, P5) |
| These all have period 1/31 of an octave.
| | : 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. |
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| ===[[Turkish maqam music temperaments]]=== | | === Clans defined by a 2.3.11 comma === |
| 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.
| | Color name: ila. See also [[subgroup temperaments]]. |
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| ===[[Very low accuracy temperaments]]===
| | ; Lulubiti clan (P8/2, P5) |
| All hope abandon ye who enter here.
| | : 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. |
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| ===[[Very high accuracy temperaments]]===
| | ; [[Rastmic clan]] (P8, P5/2) |
| Microtemperaments which don't fit in elsewhere.
| | : 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. |
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| ===[[High badness temperaments]]===
| | ; [[Nexus clan]] (P8/3, P4/2) |
| High in badness, but worth cataloging for one reason or another.
| | : 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. |
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| ===[[11-limit comma temperaments]]===
| | ; Alphaxenic or Laquadloti clan (P8/2, M2/4) |
| | : This 2.3.11 clan tempers out the [[Alpharabian comma]] {{monzo| -17 2 0 0 4 }}. Its half-octave period is ~363/256, and its generator is ~33/32. Four generators equals ~9/8. 3/2 is equated to a period plus 2 generators, and 11/8 is equated to a period minus a generator. This clan includes a strong extension to the comic or Saquadyobiti temperament, which is in the jubilismic clan. Its color name is Laquadloti. |
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| =Rank 3 temperaments= | | === Clans defined by a 2.3.13 comma === |
| | Color name: tha. See also [[subgroup temperaments]]. |
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| Even less familiar than rank 2 temperaments are the [[Planar Temperament|rank 3 temperaments]], based on 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.
| | ; Thuthuti clan (P8, P5/2) |
| | : This 2.3.13 clan tempers out [[512/507]] ({{monzo| 9 -1 0 0 0 -2 }}). Its generator is ~16/13. Two generators equals ~3/2. 13/8 is equated to 1 octave minus 1 generator. This clan includes as a strong extension the [[dicot]] temperament, which is in the dicot family. |
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| ===[[Marvel family]]===
| | ; Satrithoti clan (P8, P11/3) |
| The head of the marvel family is marvel, which tempers out 225/224. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.
| | : 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. |
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| ===[[Starling family]]=== | | === Clans defined by a 2.5.7 comma === |
| Starling tempers out 126/125, and like marvel it has the same generators as the 5-limit. An excellent tuning for starling is [[77edo]], but 31, 46 or 58 also work nicely.
| | 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. |
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| ===[[Gamelismic family]]===
| | ; [[Jubilismic clan]] (P8/2, M3) |
| 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, 1029/1024.
| | : 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. |
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| ===[[Breed family]]===
| | ; [[Bapbo clan]] (P8, M3/2) |
| Breed is a 7-limit microtemperament which tempers out 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 your choice of 8/7 or 10/7.
| | : 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. |
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| ===[[Ragisma family]]===
| | ; [[Hemimean clan]] (P8, M3/2) |
| The 7-limit rank three microtemperament which tempers out the ragisma, 4375/4374, extends to various higher limit rank three temperaments such as thor.
| | : 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. |
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| ===[[Landscape family]]===
| | ; Mabilismic clan (P8, cM3/3) |
| The 7-limit rank three microtemperament which tempers out the lanscape comma, 250047/250000, extends to various higher limit rank three temperaments such as tyr and odin.
| | : 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. |
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| ===[[Hemifamity family]]===
| | ; Vorwell clan (P8, m6/3) |
| The hemifamity family of rank three temperaments tempers out the hemifamity comma, 5120/5103.
| | : 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. |
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| ===[[Porwell family]]===
| | ; Quinzo-atriyoti Nowa clan (P8, M3/5) |
| The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125.
| | : 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). |
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| ===[[Horwell family]]===
| | ; [[Llywelynsmic clan]] (P8, cM3/7) |
| The horwell family of rank three temperaments tempers out the horwell comma, 65625/65536.
| | : 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. |
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| ===[[Hemimage family]]===
| | ; [[Quince clan]] (P8, m6/7) |
| The hemimage family of rank three temperaments tempers out the hemimage comma, 10976/10935.
| | : 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. |
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| ===[[Sensamagic family]]===
| | ; Slither clan (P8, ccm6/9) |
| These temper out 245/243.
| | : 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. |
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| ===[[Keemic family]]=== | | === Clans defined by a 3.5.7 comma === |
| These temper out 875/864. | | 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. |
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| ===[[Sengic family]]===
| | ; Rutribiyoti Noca clan (P12, M6) |
| These temper out the senga, 686/675.
| | : 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. |
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| ===[[Orwellismic family]]===
| | ; [[Sensamagic clan]] (P12, M6/2) |
| These temper out 1728/1715.
| | : 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. |
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| ===[[Nuwell family]]===
| | ; [[Gariboh clan]] (P12, M6/3) |
| These temper out the nuwell comma, 2430/2401.
| | : This 3.5.7 clan tempers out the [[gariboh comma]] {{monzo| 0 -2 5 -3 }} (3125/3087). The generator is ~25/21, two generators equals ~7/5, and three generators equals the classical major sixth (~5/3). Its color name is Triru-aquinyoti Noca. |
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| ===[[Octagar family]]===
| | ; [[Mirkwai clan]] (P12, cm7/5) |
| The octagar family of rank three temperaments tempers out the octagar comma, 4000/3969.
| | : 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. |
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| ===[[Mirkwai family]]===
| | ; Sasepzo-atriguti Noca clan (P12, m7/7) |
| The mirkwai family of rank three temperaments tempers out the mirkwai comma, 16875/16807. | | : 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). |
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| ===[[Hemimean family]]===
| | ; Satritrizo-aguguti Noca clan (P12, c<sup>3</sup>M6/9) |
| The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125. | | : 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). |
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| ===[[Mirwomo family]]===
| | ; Saquadtrizo-asepguti Noca clan (P12, c<sup>5</sup>m7/12) |
| The mirwomo family of rank three temperaments tempers out the mirwomo comma, 33075/32768. | | : 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. |
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| ===[[Dimcomp family]]=== | | === Temperaments defined by a 2.3.5.7 comma === |
| The dimcomp family of rank three temperaments tempers out the dimcomp comma, 390625/388962.
| | These are defined by a full 7-limit (color name: yaza) comma. |
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| ===[[Tolermic family]]===
| | ; [[Septisemi temperaments]] |
| These temper out the tolerma, 179200/177147. | | : 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. |
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| ===[[Kleismic rank three family]]===
| | ; [[Greenwoodmic temperaments]] |
| These are the rank three temperaments tempering out the kleisma, 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. | | : These temper out the [[greenwoodma]], {{monzo| -3 4 1 -2 }} (405/392). Its color name is Ruruyoti. |
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| ===[[Diaschismic rank three family]]===
| | ; [[Keegic temperaments]] |
| These are the rank three temperaments tempering out the dischisma, 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.
| | : Keegic rank-2 temperaments temper out the [[keega]], {{monzo| -3 1 -3 3 }} (1029/1000). Its color name is Trizoguti. |
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| ===[[Didymus rank three family]]===
| | ; [[Mint temperaments]] |
| These are the rank three temperaments tempering out the didymus 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.
| | : 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. |
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| ===[[Porcupine rank three family]]===
| | ; [[Avicennmic temperaments]] |
| These are the rank three temperaments tempering out the porcupine comma or maximal diesis, 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. | | : These temper out the [[avicennma]], {{monzo| -9 1 2 1 }} (525/512), also known as Avicenna's enharmonic diesis. Its color name is Zoyoyoti. |
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| ===[[Archytas family]]===
| | ; Sengic temperaments |
| Archytas temperament tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord.
| | : Sengic rank-2 temperaments temper out the [[senga]], {{monzo| 1 -3 -2 3 }} (686/675). Its color name is Trizo-aguguti. |
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| ===[[Jubilismic family]]===
| | ; [[Keemic temperaments]] |
| Jubilismic temperament tempers out 50/49 and thereby identifies the two septimal tritones, 7/5 and 10/7.
| | : Keemic rank-2 temperaments temper out the [[keema]], {{monzo| -5 -3 3 1 }} (875/864). Its color name is Zotriyoti. |
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| ===[[Semiphore family]]===
| | ; Secanticorn temperaments |
| 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".
| | : Secanticorn rank-2 temperaments temper out the [[secanticornisma]], {{monzo| -3 11 -5 -1 }} (177147/175000). Its color name is Laruquinguti. |
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| ===[[Mint family]]===
| | ; Nuwell temperaments |
| The mint temperament tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7.
| | : Nuwell rank-2 temperaments temper out the [[nuwell comma]], {{monzo| 1 5 1 -4 }} (2430/2401). Its color name is Quadru-ayoti. |
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| ===[[Valinorismic temperaments]]===
| | ; Mermismic temperaments |
| These temper out the valinorsma, 176/175.
| | : Mermismic rank-2 temperaments temper out the [[mermisma]], {{monzo| 5 -1 7 -7 }} (2500000/2470629). Its color name is Sepruyoti. |
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| ===[[Werckismic temperaments]]===
| | ; Negricorn temperaments |
| These temper out the werckisma, 441/440.
| | : Negricorn rank-2 temperaments temper out the [[negricorn comma]], {{monzo| 6 -5 -4 4 }} (153664/151875). Its color name is Saquadzoguti. |
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| ===[[Swetismic temperaments]]===
| | ; Tolermic temperaments |
| These temper out the swetisma, 540/539. | | : These temper out the [[tolerma]], {{monzo| 10 -11 2 1 }} (179200/177147). Its color name is Sazoyoyoti. |
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| ===[[Lehmerismic temperaments]]===
| | ; Valenwuer temperaments |
| These temper out the lehmerisma, 3025/3024.
| | : Valenwuer rank-2 temperaments temper out the [[valenwuer comma]], {{monzo| 12 3 -6 -1 }} (110592/109375). Its color name is Sarutribiguti. |
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| ===[[Kalismic temperaments]]===
| | ; [[Mirwomo temperaments]] |
| These temper out the kalisma, 9801/9800.
| | : Mirwomo rank-2 temperaments temper out the [[mirwomo comma]], {{monzo| -15 3 2 2 }} (33075/32768). Its color name is Labizoyoti. |
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| =[[Rank four temperaments|Rank 4 temperaments]]=
| | ; Catasyc temperaments |
| | : Catasyc rank-2 temperaments temper out the [[catasyc comma]], {{monzo| -11 -3 8 -1 }} (390625/387072). Its color name is Laruquadbiyoti. |
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| 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.
| | ; Compass temperaments |
| | : Compass rank-2 temperaments temper out the [[compass comma]], {{monzo| -6 -2 10 -5 }} (9765625/9680832). Its color name is Quinruyoyoti. |
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| =[[Subgroup temperaments]]= | | ; Trimyna temperaments |
| | : Trimyna rank-2 temperaments temper out the [[trimyna comma]], {{monzo| -4 1 -5 5 }} (50421/50000). Its color name is Quinzoguti. |
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| | ; [[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. |
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| | ; [[Octagar temperaments]] |
| | : Octagar rank-2 temperaments temper out the [[octagar comma]], {{monzo| 5 -4 3 -2 }} (4000/3969). Its color name is Rurutriyoti. |
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| | ; [[Orwellismic temperaments]] |
| | : Orwellismic rank-2 temperaments temper out [[orwellisma]], {{monzo| 6 3 -1 -3 }} (1728/1715). Its color name is Triru-aguti. |
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| | ; Mynaslendric temperaments |
| | : Mynaslendric rank-2 temperaments temper out the [[mynaslender comma]], {{monzo| 11 4 1 -7 }} (829440/823543). Its color name is Sepru-ayoti. |
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| | ; [[Mistismic temperaments]] |
| | : Mistismic rank-2 temperaments temper out the [[mistisma]], {{monzo| 16 -6 -4 1 }} (458752/455625). Its color name is Sazoquadguti. |
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| | ; [[Varunismic temperaments]] |
| | : Varunismic rank-2 temperaments temper out the [[varunisma]], {{monzo| -9 8 -4 2 }} (321489/320000). Its color name is Labizoguguti. |
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| | ; [[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. |
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| | ; Dimcomp temperaments |
| | : Dimcomp rank-2 temperaments temper out the [[dimcomp comma]], {{monzo| -1 -4 8 -4 }} (390625/388962). Its color name is Quadruyoyoti. |
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| | ; [[Cataharry temperaments]] |
| | : Cataharry rank-2 temperaments temper out the [[cataharry comma]], {{monzo| -4 9 -2 -2 }} (19683/19600). Its color name is Labiruguti. |
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| | ; [[Canousmic temperaments]] |
| | : Canousmic rank-2 temperaments temper out the [[canousma]], {{monzo| 4 -14 3 4 }} (4802000/4782969). Its color name is Saquadzo-atriyoti. |
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| | ; [[Triwellismic temperaments]] |
| | : Triwellismic rank-2 temperaments temper out the [[triwellisma]], {{monzo| 1 -1 -7 6 }} (235298/234375). Its color name is Tribizo-asepguti. |
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| | ; [[Hemimage temperaments]] |
| | : Hemimage rank-2 temperaments temper out the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} (10976/10935). Its color name is Satrizo-aguti. |
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| | ; [[Hemifamity temperaments]] |
| | : Hemifamity rank-2 temperaments temper out the [[hemifamity comma]], {{monzo| 10 -6 1 -1 }} (5120/5103). Its color name is Saruyoti. |
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| | ; [[Parkleiness temperaments]] |
| | : Parkleiness rank-2 temperaments temper out the [[parkleiness comma]], {{monzo| 7 7 -9 1 }} (1959552/1953125). Its color name is Zotritriguti. |
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| | ; [[Porwell temperaments]] |
| | : Porwell rank-2 temperaments temper out the [[porwell comma]], {{monzo| 11 1 -3 -2 }} (6144/6125). Its color name is Sarurutriguti. |
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| | ; [[Cartoonismic temperaments]] |
| | : Cartoonismic rank-2 temperaments temper out the [[cartoonisma]], {{monzo| 12 -3 -14 9 }} (165288374272/164794921875). Its color name is Satritrizo-asepbiguti. |
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| | ; [[Hemfiness temperaments]] |
| | : Hemfiness rank-2 temperaments temper out the [[hemfiness comma]], {{monzo| 15 -5 3 -5 }} (4096000/4084101). Its color name is Saquinru-atriyoti. |
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| | ; [[Hewuermera temperaments]] |
| | : Hewuermera rank-2 temperaments temper out the [[hewuermera comma]], {{monzo| 16 2 -1 -6 }} (589824/588245). Its color name is Satribiru-aguti. |
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| | ; [[Lokismic temperaments]] |
| | : Lokismic rank-2 temperaments temper out the [[lokisma]], {{monzo| 21 -8 -6 2 }} (102760448/102515625). Its color name is Sasa-bizotriguti. |
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| | ; Decovulture temperaments |
| | : Decovulture rank-2 temperaments temper out the [[decovulture comma]], {{monzo| 26 -7 -4 -2 }} (67108864/66976875). Its color name is Sasabiruguguti. |
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| | ; Pontiqak temperaments |
| | : Pontiqak rank-2 temperaments temper out the [[pontiqak comma]], {{monzo| -17 -6 9 2 }} (95703125/95551488). Its color name is Lazozotritriyoti. |
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| | ; [[Mitonismic temperaments]] |
| | : Mitonismic rank-2 temperaments temper out the [[mitonisma]], {{monzo| -20 7 -1 4 }} (5250987/5242880). Its color name is Laquadzo-aguti. |
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| | ; [[Horwell temperaments]] |
| | : Horwell rank-2 temperaments temper out the [[horwell comma]], {{monzo| -16 1 5 1 }} (65625/65536). Its color name is Lazoquinyoti. |
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| | ; Neptunismic temperaments |
| | : Neptunismic rank-2 temperaments temper out the [[neptunisma]], {{monzo| -12 -5 11 -2 }} (48828125/48771072). Its color name is Laruruleyoti. |
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| | ; [[Metric microtemperaments]] |
| | : Metric rank-2 temperaments temper out the [[meter]], {{monzo| -11 2 7 -3 }} (703125/702464). Its color name is Latriru-asepyoti. |
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| | ; [[Wizmic microtemperaments]] |
| | : Wizmic rank-2 temperaments temper out the [[wizma]], {{monzo| -6 -8 2 5 }} (420175/419904). Its color name is Quinzo-ayoyoti. |
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| | ; [[Supermatertismic temperaments]] |
| | : Supermatertismic rank-2 temperaments temper out the [[supermatertisma]], {{monzo| -6 3 9 -7 }} (52734375/52706752). Its color name is Lasepru-atritriyoti. |
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| | ; [[Breedsmic temperaments]] |
| | : Breedsmic rank-2 temperaments temper out the [[breedsma]], {{monzo| -5 -1 -2 4 }} (2401/2400). Its color name is Bizozoguti. |
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| | ; Supermasesquartismic temperaments |
| | : Supermasesquartismic rank-2 temperaments temper out the [[supermasesquartisma]], {{monzo| -5 10 5 -8 }} (184528125/184473632). Its color name is Laquadbiru-aquinyoti. |
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| | ; [[Ragismic microtemperaments]] |
| | : Ragismic rank-2 temperaments temper out the [[ragisma]], {{monzo| -1 -7 4 1 }} (4375/4374). Its color name is Zoquadyoti. |
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| | ; 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. |
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| | ; [[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. |
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| | == 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. |
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| | === 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: |
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| | ; [[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. |
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| | ; [[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. |
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| | ; [[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. |
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| | ; [[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. |
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| | === 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: |
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| | ; [[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. |
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| | ; [[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. |
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| | ; 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. |
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| | ; [[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. |
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| | ; [[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. |
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| | ; 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. |
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| | === Families defined by a 2.3.5.7 comma === |
| | Color name: yaza. |
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| | ; [[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. |
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| | ; [[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. |
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| | ; [[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. |
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| | ; 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. |
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| | ; 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. |
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| | ; [[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. |
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| | ; [[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. |
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| | ; [[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. |
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| | ; [[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. |
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| | ; [[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. |
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| | ; [[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. |
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| | ; [[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. |
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| | ; [[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. |
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| | ; 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. |
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| | ; [[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. |
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| | ; [[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. |
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| | ; [[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. |
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| | ; [[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. |
| | |
| | ; [[The Quartercache]] |
| | : This is the commatic realm of the 11-limit comma 117440512/117406179 ({{monzo| 24 -6 0 1 -5 }}), the [[quartisma]]. Its color name is Saquinlu-azoti. |
| | |
| | == Miscellaneous other temperaments == |
| | ; [[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 [[Arabic, Turkish, Persian music|makam (maqam) music]] in a systematic way. This includes, in effect, certain linear temperaments. |
|
| |
|
| By a //commatic realm// is meant the whole collection of regular temperaments of various ranks and for both full groups and [[Just intonation subgroups|subgroups]] tempering out a given comma. For some commas, looking at the full commatic realm seems the best approach to discussing associated temperaments.
| | ; [[Very low accuracy temperaments]] |
| | : All hope abandon ye who enter here. |
|
| |
|
| ==[[Orgonia]]==
| | ; [[Very high accuracy temperaments]] |
| Orgonia is the commatic realm of the [[11-limit]] comma 65536/65219 = |16 0 0 -2 -3>, the orgonisma.
| | : Microtemperaments which do not fit in elsewhere. |
|
| |
|
| ==[[The Biosphere]]==
| | ; Middle Path tables |
| The Biosphere is the name given to the commatic realm of the 13-limit comma 91/90.
| | : 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]] |
|
| |
|
| ==[[The Archipelago]]== | | == Maps of temperaments == |
| The Archipelago is a name which has been given to the commatic realm of the [[13-limit]] comma 676/675.
| | * [[Map of rank-2 temperaments]], sorted by generator size |
| | * [[Catalog of rank two temperaments]] |
| | ** [[Catalog of seven-limit rank two temperaments]] |
| | ** [[Catalog of eleven-limit rank two temperaments]] |
| | ** [[Catalog of thirteen-limit rank two temperaments]] |
| | * [[List of rank two temperaments by generator and period]] |
| | * [[Rank-2 temperaments by mapping of 3]] |
| | * [[Temperaments for MOS shapes]] |
| | * [[Tree of rank two temperaments]] |
|
| |
|
| | == Temperament nomenclature == |
| | * [[Temperament naming]] |
|
| |
|
| | == External links == |
| | * [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:Lists of temperaments]] <!-- main article --> |
| =Links=
| |
| * [[http://en.wikipedia.org/wiki/Regular_temperament|Regular temperaments - Wikipedia]]</pre></div>
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| <h4>Original HTML content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Tour of Regular Temperaments</title></head><body><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/Regul%C3%A4re%20Temperaturen">Deutsch</a><br />
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| <!-- ws:start:WikiTextTocRule:236:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:236 --><!-- ws:start:WikiTextTocRule:237: --><a href="#Regular temperaments">Regular temperaments</a><!-- ws:end:WikiTextTocRule:237 --><!-- ws:start:WikiTextTocRule:238: --><!-- ws:end:WikiTextTocRule:238 --><!-- ws:start:WikiTextTocRule:239: --><!-- ws:end:WikiTextTocRule:239 --><!-- ws:start:WikiTextTocRule:240: --> | <a href="#Equal temperaments">Equal temperaments</a><!-- ws:end:WikiTextTocRule:240 --><!-- ws:start:WikiTextTocRule:241: --> | <a href="#Rank 2 (including &quot;linear&quot;) temperaments">Rank 2 (including &quot;linear&quot;) temperaments</a><!-- ws:end:WikiTextTocRule:241 --><!-- ws:start:WikiTextTocRule:242: --><!-- ws:end:WikiTextTocRule:242 --><!-- ws:start:WikiTextTocRule:243: --><!-- 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ws:start:WikiTextTocRule:284: --><!-- ws:end:WikiTextTocRule:284 --><!-- ws:start:WikiTextTocRule:285: --><!-- ws:end:WikiTextTocRule:285 --><!-- ws:start:WikiTextTocRule:286: --><!-- ws:end:WikiTextTocRule:286 --><!-- ws:start:WikiTextTocRule:287: --><!-- ws:end:WikiTextTocRule:287 --><!-- ws:start:WikiTextTocRule:288: --> | <a href="#Temperaments for a given comma">Temperaments for a given comma</a><!-- ws:end:WikiTextTocRule:288 --><!-- ws:start:WikiTextTocRule:289: --><!-- ws:end:WikiTextTocRule:289 --><!-- ws:start:WikiTextTocRule:290: --><!-- ws:end:WikiTextTocRule:290 --><!-- ws:start:WikiTextTocRule:291: --><!-- ws:end:WikiTextTocRule:291 --><!-- ws:start:WikiTextTocRule:292: --><!-- ws:end:WikiTextTocRule:292 --><!-- ws:start:WikiTextTocRule:293: --><!-- ws:end:WikiTextTocRule:293 --><!-- ws:start:WikiTextTocRule:294: --><!-- ws:end:WikiTextTocRule:294 --><!-- ws:start:WikiTextTocRule:295: --><!-- ws:end:WikiTextTocRule:295 --><!-- ws:start:WikiTextTocRule:296: --><!-- 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--> | <a href="#Subgroup temperaments">Subgroup temperaments</a><!-- ws:end:WikiTextTocRule:349 --><!-- ws:start:WikiTextTocRule:350: --> | <a href="#Commatic realms">Commatic realms</a><!-- ws:end:WikiTextTocRule:350 --><!-- ws:start:WikiTextTocRule:351: --><!-- ws:end:WikiTextTocRule:351 --><!-- ws:start:WikiTextTocRule:352: --><!-- ws:end:WikiTextTocRule:352 --><!-- ws:start:WikiTextTocRule:353: --><!-- ws:end:WikiTextTocRule:353 --><!-- ws:start:WikiTextTocRule:354: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:354 --><!-- ws:start:WikiTextTocRule:355: -->
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Regular temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Regular temperaments</h1>
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| <br />
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| Regular temperaments are non-Just tunings wherein the infinite number of intervals in p-limit Just intonation (or any subgroup thereof) are mapped to a smaller (though still infinite) set of tempered intervals, by &quot;tempering&quot; (deliberately mistuning) some of the ratios such that a comma (or set of commas) &quot;vanishes&quot; 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 &quot;puns&quot; as commas are tempered out. Temperaments effectively reduce the &quot;dimensionality&quot; 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 would be located by a four-dimensional set of coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which commas (and how many) are tempered out, and 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, or in other words the &quot;rank&quot; of the temperament).<br />
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| A rank r <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_temperament" rel="nofollow">regular temperament</a> 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 <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a> can be defined in various ways, for instance by giving a set of <a class="wiki_link" href="/comma">commas</a> tempered out by the temperament, or a set of r independent <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">vals</a> defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the <a class="wiki_link" href="/comma%20pump%20examples">comma pumps</a> 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.<br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Regular temperaments-Why would I want to use a regular temperament?"></a><!-- ws:end:WikiTextHeadingRule:2 -->Why would I want to use a regular temperament?</h2>
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| 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, which are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical &quot;puns&quot;, which are melodies or chord progressions that exploit the multiplicity of &quot;meanings&quot; of tempered intervals.<br />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Regular temperaments-What do I need to know to understand all the numbers on the pages for individual regular temperaments?"></a><!-- ws:end:WikiTextHeadingRule:4 -->What do I need to know to understand all the numbers on the pages for individual regular temperaments?</h2>
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| 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 as a first order of business. These concepts are rather straightforward and require very little math to understand.<br />
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| 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 &quot;optimal&quot; tuning for the generator. The two most frequently used forms of optimization are POTE (&quot;Pure-Octave Tenney-Euclidean&quot;) and <a class="wiki_link" href="/Top%20tuning">TOP</a> (&quot;Tenney OPtimal&quot;, or &quot;Tempered Octaves, Please&quot;). Optimization is rather<br />
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| intensive mathematically, but it is seldom (if ever) left as an exercise to the reader; most temperaments are presented here in their optimal forms.<br />
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| <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:6 --><a class="wiki_link" href="/edo">Equal temperaments</a></h1>
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| <a class="wiki_link" href="/Equal%20Temperaments">Equal temperaments</a> (abbreviated ET or tET) and 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 &quot;Rank 1&quot;) 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 &quot;equal division&quot; 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 &quot;fun&quot; results). The familiar 12-note equal temperament (12-ET) 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-tET.<br />
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| <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Rank 2 (including &quot;linear&quot;) temperaments"></a><!-- ws:end:WikiTextHeadingRule:8 -->Rank 2 (including &quot;linear&quot;) temperaments<!-- ws:start:WikiTextAnchorRule:356:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@lineartemperaments&quot; title=&quot;Anchor: lineartemperaments&quot;/&gt; --><a name="lineartemperaments"></a><!-- ws:end:WikiTextAnchorRule:356 --></h1>
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| Regular temperaments of ranks two and three are cataloged <a class="wiki_link" href="/Optimal%20patent%20val">here</a>. Other pages listing them are <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a>'s <a class="wiki_link" href="/Catalog%20of%20five-limit%20rank%20two%20temperaments">catalog of 5-limit rank two temperaments</a>, and a <a class="wiki_link" href="/Proposed%20names%20for%20rank%202%20temperaments">page</a> listing higher limit rank two temperaments. There is also <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>'s <a class="wiki_link_ext" href="http://x31eq.com/catalog2.html" rel="nofollow">giant list of regular temperaments</a>.<br />
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| P-limit Rank 2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators (though they may have any number of step-sizes). This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. Rank-2 temperaments can be reduced to a related rank-1 temperaments 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-TET by tempering out the Pythagorean comma.<br />
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| <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families"></a><!-- ws:end:WikiTextHeadingRule:10 -->Families</h2>
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| 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 <a class="wiki_link" href="/Normal%20lists">normal comma list</a> of the various temperaments, where a <strong>comma</strong> is a small interval, not a square or cube or other power, which is tempered out by the temperament.<br />
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| Meantone is a familiar historical temperament based on a chain of fifths (or fourths), but it is only one of many possibilities for temperaments based on a chain of generating intervals. These are referred to as &quot;rank 2&quot; temperaments, since they are based on a set of two linearly independent intervals. One of these intervals (typically an octave or fraction of an octave) can be selected as the &quot;period&quot;, and another interval, usually chosen to be smaller than the period, is referred to as the &quot;generator&quot;.<br />
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| <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Meantone family"></a><!-- ws:end:WikiTextHeadingRule:12 --><a class="wiki_link" href="/Meantone%20family">Meantone family</a></h3>
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| The meantone family tempers out 81/80, also called the syntonic comma. 81/80 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 some sort of average or &quot;mean&quot; of the two tones. It has a flattened fifth (or sharpened fourth) as generator. Some meantone tunings are <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/31edo">31edo</a>, <a class="wiki_link" href="/43edo">43edo</a>, <a class="wiki_link" href="/50edo">50edo</a>, <a class="wiki_link" href="/55edo">55edo</a> and <a class="wiki_link" href="/81edo">81edo</a>. 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 syntonic comma (the 81/80 interval.)<br />
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| <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Schismatic family"></a><!-- ws:end:WikiTextHeadingRule:14 --><a class="wiki_link" href="/Schismatic%20family">Schismatic family</a></h3>
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| The schismatic family tempers out the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the syntonic comma (81/80). The 5-limit version of the temperament is a <a class="wiki_link" href="/Microtempering">microtemperament</a> 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 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/12edo">12edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/29edo">29edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/41edo">41edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/53edo">53edo</a>, and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/118edo">118edo</a>.<br />
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| <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Kleismic family"></a><!-- ws:end:WikiTextHeadingRule:16 --><a class="wiki_link" href="/Kleismic%20family">Kleismic family</a></h3>
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| The kleismic family of temperaments tempers out the kleisma of 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 <a class="wiki_link" href="/15edo">15edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/34edo">34edo</a>, <a class="wiki_link" href="/49edo">49edo</a>, <a class="wiki_link" href="/53edo">53edo</a>, <a class="wiki_link" href="/72edo">72edo</a>, <a class="wiki_link" href="/87edo">87edo</a> and <a class="wiki_link" href="/140edo">140edo</a> among its possible tunings.<br />
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| <!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Magic family"></a><!-- ws:end:WikiTextHeadingRule:18 --><a class="wiki_link" href="/Magic%20family">Magic family</a></h3>
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| The magic family tempers out 3125/3072, known as the magic comma or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/2. The magic generator is itself an approximate 5/4. The magic family includes <a class="wiki_link" href="/16edo">16edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/22edo">22edo</a>, <a class="wiki_link" href="/25edo">25edo</a>, and <a class="wiki_link" href="/41edo">41edo</a> among its possible tunings, with the latter being near-optimal.<br />
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| <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Diaschismic family"></a><!-- ws:end:WikiTextHeadingRule:20 --><a class="wiki_link" href="/Diaschismic%20family">Diaschismic family</a></h3>
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| The diaschismic family tempers out 2048/2025, the <a class="wiki_link" href="/diaschisma">diaschisma</a>, which tempers things 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 <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/22edo">22edo</a>, <a class="wiki_link" href="/34edo">34edo</a>, <a class="wiki_link" href="/46edo">46edo</a>, <a class="wiki_link" href="/56edo">56edo</a>, <a class="wiki_link" href="/58edo">58edo</a> and <a class="wiki_link" href="/80edo">80edo</a>. A noted 7-limit extension to diaschismic is<a class="wiki_link" href="/pajara"> pajara</a> temperament, where the intervals 50/49 and 64/63 are tempered out, and of which <a class="wiki_link" href="/22edo">22edo</a> is an excellent tuning.<br />
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| <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Pelogic family"></a><!-- ws:end:WikiTextHeadingRule:22 --><a class="wiki_link" href="/Pelogic%20family">Pelogic family</a></h3>
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| This tempers out the pelogic comma, 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, the accumulated flatness leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates 2L5s &quot;anti-diatonic&quot; scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include <a class="wiki_link" href="/9edo">9edo</a>, <a class="wiki_link" href="/16edo">16edo</a>, <a class="wiki_link" href="/23edo">23edo</a>, and <a class="wiki_link" href="/25edo">25edo</a>.<br />
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| <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Porcupine family"></a><!-- ws:end:WikiTextHeadingRule:24 --><a class="wiki_link" href="/Porcupine%20family">Porcupine family</a></h3>
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| The porcupine family tempers out 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 <a class="wiki_link" href="/15edo">15edo</a>, <a class="wiki_link" href="/22edo">22edo</a>, <a class="wiki_link" href="/37edo">37edo</a>, and <a class="wiki_link" href="/59edo">59edo</a>.<br />
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| <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Würschmidt family"></a><!-- ws:end:WikiTextHeadingRule:26 --><a class="wiki_link" href="/W%C3%BCrschmidt%20family">Würschmidt family</a></h3>
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| The würschmidt (or wuerschmidt) family tempers out Würschmidt's 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 MOS's as <a class="wiki_link" href="/magic%20family">magic temperament</a>, but is tuned slightly more accurately. Both <a class="wiki_link" href="/31edo">31edo</a> and <a class="wiki_link" href="/34edo">34edo</a> can be used as würschmidt tunings, as can <a class="wiki_link" href="/65edo">65edo</a>, which is quite accurate.<br />
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| <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Augmented family"></a><!-- ws:end:WikiTextHeadingRule:28 --><a class="wiki_link" href="/Augmented%20family">Augmented family</a></h3>
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| The augmented family tempers out the diesis of 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 <a class="wiki_link" href="/12edo">12edo</a>, which is an excellent tuning for augmented. It is the temperament that results in what is commonly called the &quot;augmented scale&quot; (3L3s) in common 12-based music theory, as well as what is commonly called &quot;<a class="wiki_link_ext" href="http://www.tcherepnin.com/alex/basic_elem1.htm#9step" rel="nofollow" target="_blank">Tcherepnin's scale</a>&quot; (3L6s).<br />
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| <!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Dimipent family"></a><!-- ws:end:WikiTextHeadingRule:30 --><a class="wiki_link" href="/Dimipent%20family">Dimipent family</a></h3>
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| The dimipent family tempers out the major diesis aka diminished comma, 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 <a class="wiki_link" href="/12edo">12edo</a>.<br />
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| <!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Dicot family"></a><!-- ws:end:WikiTextHeadingRule:32 --><a class="wiki_link" href="/Dicot%20family">Dicot family</a></h3>
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| 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. <a class="wiki_link" href="/7edo">7edo</a> makes for a &quot;good&quot; dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the &quot;neutral&quot; dicot 3rds span a 3/2. Tunings include <a class="wiki_link" href="/7edo">7edo</a>, <a class="wiki_link" href="/10edo">10edo</a>, and <a class="wiki_link" href="/17edo">17edo</a>.<br />
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| <!-- ws:start:WikiTextHeadingRule:34:&lt;h3&gt; --><h3 id="toc17"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Tetracot family"></a><!-- ws:end:WikiTextHeadingRule:34 --><a class="wiki_link" href="/Tetracot%20family">Tetracot family</a></h3>
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| 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 20000/19683, the minimal diesis or tetracot comma. <a class="wiki_link" href="/7edo">7edo</a> can also be considered a tetracot tuning, as can <a class="wiki_link" href="/20edo">20edo</a>, <a class="wiki_link" href="/27edo">27edo</a>, <a class="wiki_link" href="/34edo">34edo</a>, and <a class="wiki_link" href="/41edo">41edo</a>.<br />
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| <!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Sensipent family"></a><!-- ws:end:WikiTextHeadingRule:36 --><a class="wiki_link" href="/Sensipent%20family">Sensipent family</a></h3>
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| This tempers out the sensipent comma, 78732/78125, also known as the medium semicomma. Tunings include <a class="wiki_link" href="http://xenharmonic.wikispaces.com/8edo">8edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/19edo">19edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/46edo">46edo</a>, and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo">65edo</a>.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:38:&lt;h3&gt; --><h3 id="toc19"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Orwell and the semicomma family"></a><!-- ws:end:WikiTextHeadingRule:38 --><a class="wiki_link" href="/Semicomma%20family">Orwell and the semicomma family</a></h3>
| |
| The semicomma (also known as <strong>Fokker's comma)</strong> 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 <a class="wiki_link" href="/orwell">orwell</a> temperament.<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:40:&lt;h3&gt; --><h3 id="toc20"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Pythagorean family"></a><!-- ws:end:WikiTextHeadingRule:40 --><a class="wiki_link" href="/Pythagorean%20family">Pythagorean family</a></h3>
| |
| The Pythagorean family tempers out the Pythagorean comma, |-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 temperament and catler temperament. 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.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:42:&lt;h3&gt; --><h3 id="toc21"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Apotome family"></a><!-- ws:end:WikiTextHeadingRule:42 --><a class="wiki_link" href="/Apotome%20family">Apotome family</a></h3>
| |
| This family tempers out the apotome, 2187/2048, which is a 3-limit comma.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:44:&lt;h3&gt; --><h3 id="toc22"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Gammic family"></a><!-- ws:end:WikiTextHeadingRule:44 --><a class="wiki_link" href="/Gammic%20family">Gammic family</a></h3>
| |
| 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 <a class="wiki_link" href="/Carlos%20Gamma">Carlos Gamma</a>. Another member is Neptune temperament.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:46:&lt;h3&gt; --><h3 id="toc23"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Minortonic family"></a><!-- ws:end:WikiTextHeadingRule:46 --><a class="wiki_link" href="/Minortonic%20family">Minortonic family</a></h3>
| |
| 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).<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:48:&lt;h3&gt; --><h3 id="toc24"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Bug family"></a><!-- ws:end:WikiTextHeadingRule:48 --><a class="wiki_link" href="/Bug%20family">Bug family</a></h3>
| |
| This tempers out 27/25, the large limma or bug comma.<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:50:&lt;h3&gt; --><h3 id="toc25"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Father family"></a><!-- ws:end:WikiTextHeadingRule:50 --><a class="wiki_link" href="/Father%20family">Father family</a></h3>
| |
| This tempers out 16/15, the just diatonic semitone.<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:52:&lt;h3&gt; --><h3 id="toc26"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Sycamore family"></a><!-- ws:end:WikiTextHeadingRule:52 --><a class="wiki_link" href="/Sycamore%20family">Sycamore family</a></h3>
| |
| 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.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:54:&lt;h3&gt; --><h3 id="toc27"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Escapade family"></a><!-- ws:end:WikiTextHeadingRule:54 --><a class="wiki_link" href="/Escapade%20family">Escapade family</a></h3>
| |
| This tempers out the escapade comma, |32 -7 -9&gt;, which is the difference between nine just major thirds and seven just fourths.<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:56:&lt;h3&gt; --><h3 id="toc28"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Amity family"></a><!-- ws:end:WikiTextHeadingRule:56 --><a class="wiki_link" href="/Amity%20family">Amity family</a></h3>
| |
| This tempers out the amity comma, 1600000/1594323 = |9 -13 5&gt;.<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:58:&lt;h3&gt; --><h3 id="toc29"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Vulture family"></a><!-- ws:end:WikiTextHeadingRule:58 --><a class="wiki_link" href="/Vulture%20family">Vulture family</a></h3>
| |
| This tempers out the vulture comma, |24 -21 4&gt;.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:60:&lt;h3&gt; --><h3 id="toc30"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Vishnuzmic family"></a><!-- ws:end:WikiTextHeadingRule:60 --><a class="wiki_link" href="/Vishnuzmic%20family">Vishnuzmic family</a></h3>
| |
| 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.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:62:&lt;h3&gt; --><h3 id="toc31"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Luna Family"></a><!-- ws:end:WikiTextHeadingRule:62 --><a class="wiki_link" href="/Luna%20Family">Luna Family</a></h3>
| |
| This tempers out the Luna comma; |38 -2 -15&gt; (274877906944/274658203125)<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:64:&lt;h3&gt; --><h3 id="toc32"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Laconic Family"></a><!-- ws:end:WikiTextHeadingRule:64 --><a class="wiki_link" href="/Laconic%20Family">Laconic Family</a></h3>
| |
| This tempers out the laconic comma, 2187/2000, which is the difference between three 10/9's and one 3/2. Laconic is supported by 16-EDO, 21-EDO, and 37b-EDO, among others.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:66:&lt;h3&gt; --><h3 id="toc33"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Immunity family"></a><!-- ws:end:WikiTextHeadingRule:66 --><a class="wiki_link" href="/Immunity%20family">Immunity family</a></h3>
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| This tempers out the immunity comma, 1638400/1594323.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:68:&lt;h3&gt; --><h3 id="toc34"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Ditonmic family"></a><!-- ws:end:WikiTextHeadingRule:68 --><a class="wiki_link" href="/Ditonmic%20family">Ditonmic family</a></h3>
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| This tempers out the ditonma, 1220703125/1207959552.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:70:&lt;h3&gt; --><h3 id="toc35"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Shibboleth family"></a><!-- ws:end:WikiTextHeadingRule:70 --><a class="wiki_link" href="/Shibboleth%20family">Shibboleth family</a></h3>
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| This tempers out the shibboleth comma, 1953125/1889568.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:72:&lt;h3&gt; --><h3 id="toc36"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Comic family"></a><!-- ws:end:WikiTextHeadingRule:72 --><a class="wiki_link" href="/Comic%20family">Comic family</a></h3>
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| This tempers out the comic comma, 5120000/4782969.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:74:&lt;h3&gt; --><h3 id="toc37"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Wesley family"></a><!-- ws:end:WikiTextHeadingRule:74 --><a class="wiki_link" href="/Wesley%20family">Wesley family</a></h3>
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| This tempers out the wesley comma, 78125/73728.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:76:&lt;h3&gt; --><h3 id="toc38"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Fifive family"></a><!-- ws:end:WikiTextHeadingRule:76 --><a class="wiki_link" href="/Fifive%20family">Fifive family</a></h3>
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| This tempers out the fifive comma, 9765625/9565938.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:78:&lt;h3&gt; --><h3 id="toc39"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Maja family"></a><!-- ws:end:WikiTextHeadingRule:78 --><a class="wiki_link" href="/Maja%20family">Maja family</a></h3>
| |
| This tempers out the maja comma, 762939453125/753145430616.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:80:&lt;h3&gt; --><h3 id="toc40"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Mutt family"></a><!-- ws:end:WikiTextHeadingRule:80 --><a class="wiki_link" href="/Mutt%20family">Mutt family</a></h3>
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| This tempers out the mutt comma, |-44 -3 21&gt;, leading to some strange properties.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:82:&lt;h2&gt; --><h2 id="toc41"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans"></a><!-- ws:end:WikiTextHeadingRule:82 -->Clans</h2>
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:84:&lt;h3&gt; --><h3 id="toc42"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Gamelismic clan"></a><!-- ws:end:WikiTextHeadingRule:84 --><a class="wiki_link" href="/Gamelismic%20clan">Gamelismic clan</a></h3>
| |
| If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> 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. Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. We can modify the definition of <a class="wiki_link" href="/Normal%20lists">normal comma list</a> for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.<br />
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| <br />
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| Particularly noteworthy as member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds.<br />
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| <br />
| |
| Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called &quot;blackjack&quot; and a 31-note scale called &quot;canasta&quot; have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:86:&lt;h3&gt; --><h3 id="toc43"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Trienstonic clan"></a><!-- ws:end:WikiTextHeadingRule:86 --><a class="wiki_link" href="/Trienstonic%20clan">Trienstonic clan</a></h3>
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| This clan tempers out the septimal third-tone, <a class="wiki_link" href="/28_27">28/27</a>, a triprime comma with factors of 2, 3 and 7.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:88:&lt;h3&gt; --><h3 id="toc44"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Slendro clan"></a><!-- ws:end:WikiTextHeadingRule:88 --><a class="wiki_link" href="/Slendro%20clan">Slendro clan</a></h3>
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| This clan tempers out the slendro diesis, <a class="wiki_link" href="/49_48">49/48</a>, a triprime comma with factors of 2, 3 and 7.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:90:&lt;h3&gt; --><h3 id="toc45"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Jubilismic clan"></a><!-- ws:end:WikiTextHeadingRule:90 --><a class="wiki_link" href="/Jubilismic%20clan">Jubilismic clan</a></h3>
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| This tempers out the jubilisma, <a class="wiki_link" href="/50_49">50/49</a>, E.A. the difference between 10/7 and 7/5.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:92:&lt;h3&gt; --><h3 id="toc46"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Archytas clan"></a><!-- ws:end:WikiTextHeadingRule:92 --><a class="wiki_link" href="/Archytas%20clan">Archytas clan</a></h3>
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| This clan tempers out the Archytas comma, <a class="wiki_link" href="/64_63">64/63</a>, which is a triprime comma with factors of 2, 3 and 7. The clan consists of rank two temperaments, and should not be confused with the <a class="wiki_link" href="/Archytas%20family">Archytas family</a> of rank three temperaments.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:94:&lt;h3&gt; --><h3 id="toc47"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Sensamagic clan"></a><!-- ws:end:WikiTextHeadingRule:94 --><a class="wiki_link" href="/Sensamagic%20clan">Sensamagic clan</a></h3>
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| This clan tempers out 245/243, the sensamagic comma.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:96:&lt;h3&gt; --><h3 id="toc48"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Hemimean clan"></a><!-- ws:end:WikiTextHeadingRule:96 --><a class="wiki_link" href="/Hemimean%20clan">Hemimean clan</a></h3>
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| This tempers out the hemimean comma, 3136/3125, a no-threes comma.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:98:&lt;h3&gt; --><h3 id="toc49"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Mirkwai clan"></a><!-- ws:end:WikiTextHeadingRule:98 --><a class="wiki_link" href="/Mirkwai%20clan">Mirkwai clan</a></h3>
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| This tempers out the mirkwai comma, |0 3 4 -5&gt; = 16875/16807, a no-twos comma (ratio of odd numbers.)<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:100:&lt;h3&gt; --><h3 id="toc50"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Clans-Quince clan"></a><!-- ws:end:WikiTextHeadingRule:100 --><a class="wiki_link" href="/Quince%20clan">Quince clan</a></h3>
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| This tempers out quince, the no-threes comma |-15 0 -2 7&gt; = 823543/819200.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:102:&lt;h1&gt; --><h1 id="toc51"><a name="Temperaments for a given comma"></a><!-- ws:end:WikiTextHeadingRule:102 -->Temperaments for a given comma</h1>
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:104:&lt;h3&gt; --><h3 id="toc52"><a name="Temperaments for a given comma--Septisemi temperaments"></a><!-- ws:end:WikiTextHeadingRule:104 --><a class="wiki_link" href="/Septisemi%20temperaments">Septisemi temperaments</a></h3>
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| These are very low complexity temperaments tempering out the minor septimal semitone, 21/20 and hence equating 5/3 with 7/4.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:106:&lt;h3&gt; --><h3 id="toc53"><a name="Temperaments for a given comma--Mint temperaments"></a><!-- ws:end:WikiTextHeadingRule:106 --><a class="wiki_link" href="/Mint%20temperaments">Mint temperaments</a></h3>
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| These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:108:&lt;h3&gt; --><h3 id="toc54"><a name="Temperaments for a given comma--Greenwoodmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:108 --><a class="wiki_link" href="/Greenwoodmic%20temperaments">Greenwoodmic temperaments</a></h3>
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| These temper out the greenwoodma, |-3 4 1 -2&gt; = 405/392.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:110:&lt;h3&gt; --><h3 id="toc55"><a name="Temperaments for a given comma--Avicennmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:110 --><a class="wiki_link" href="/Avicennmic%20temperaments">Avicennmic temperaments</a></h3>
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| These temper out the avicennma, |-9 1 2 1&gt; = 525/512, also known as Avicenna's enharmonic diesis.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:112:&lt;h3&gt; --><h3 id="toc56"><a name="Temperaments for a given comma--Keemic temperaments"></a><!-- ws:end:WikiTextHeadingRule:112 --><a class="wiki_link" href="/Keemic%20temperaments">Keemic temperaments</a></h3>
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| These temper out the keema, |-5 -3 3 1&gt; = 875/864.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:114:&lt;h3&gt; --><h3 id="toc57"><a name="Temperaments for a given comma--Starling temperaments"></a><!-- ws:end:WikiTextHeadingRule:114 --><a class="wiki_link" href="/Starling%20temperaments">Starling temperaments</a></h3>
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| Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma (<span class="commentBody">the difference between three 6/5s plus one 7/6, and an octave) </span>is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:116:&lt;h3&gt; --><h3 id="toc58"><a name="Temperaments for a given comma--Marvel temperaments"></a><!-- ws:end:WikiTextHeadingRule:116 --><a class="wiki_link" href="/Marvel%20temperaments">Marvel temperaments</a></h3>
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| These temper out |-5 2 2 -1&gt; = 225/224, the marvel comma, and 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.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:118:&lt;h3&gt; --><h3 id="toc59"><a name="Temperaments for a given comma--Orwellismic temperaments"></a><!-- ws:end:WikiTextHeadingRule:118 --><a class="wiki_link" href="/Orwellismic%20temperaments">Orwellismic temperaments</a></h3>
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| These temper out |6 3 -1 -3&gt; = 1728/1715, the orwellisma.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:120:&lt;h3&gt; --><h3 id="toc60"><a name="Temperaments for a given comma--Octagar temperaments"></a><!-- ws:end:WikiTextHeadingRule:120 --><a class="wiki_link" href="/Octagar%20temperaments">Octagar temperaments</a></h3>
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| Octagar temperaments temper out the octagar comma, |5 -4 3 -2&gt; = 4000/3969.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:122:&lt;h3&gt; --><h3 id="toc61"><a name="Temperaments for a given comma--Hemifamity temperaments"></a><!-- ws:end:WikiTextHeadingRule:122 --><a class="wiki_link" href="/Hemifamity%20temperaments">Hemifamity temperaments</a></h3>
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| Hemifamity temperaments temper out the hemifamity comma, |10 -6 1 -1&gt; = 5120/5103.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:124:&lt;h3&gt; --><h3 id="toc62"><a name="Temperaments for a given comma--Porwell temperaments"></a><!-- ws:end:WikiTextHeadingRule:124 --><a class="wiki_link" href="/Porwell%20temperaments">Porwell temperaments</a></h3>
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| Porwell temperaments temper out the porwell comma, |11 1 -3 -2&gt; = 6144/6125.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:126:&lt;h3&gt; --><h3 id="toc63"><a name="Temperaments for a given comma--Stearnsmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:126 --><a class="wiki_link" href="/Stearnsmic%20temperaments">Stearnsmic temperaments</a></h3>
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| Stearnsmic temperaments temper out the stearnsma, |1 10 0 -6&gt; = 118098/117649.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:128:&lt;h3&gt; --><h3 id="toc64"><a name="Temperaments for a given comma--Hemimage temperaments"></a><!-- ws:end:WikiTextHeadingRule:128 --><a class="wiki_link" href="/Hemimage%20temperaments">Hemimage temperaments</a></h3>
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| Hemimage temperaments temper out the hemimage comma, |5 -7 -1 3&gt; = 10976/10935.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:130:&lt;h3&gt; --><h3 id="toc65"><a name="Temperaments for a given comma--Horwell temperaments"></a><!-- ws:end:WikiTextHeadingRule:130 --><a class="wiki_link" href="/Horwell%20temperaments">Horwell temperaments</a></h3>
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| Horwell temperaments temper out the horwell comma, |-16 1 5 1&gt; = 65625/65536.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:132:&lt;h3&gt; --><h3 id="toc66"><a name="Temperaments for a given comma--Breedsmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:132 --><a class="wiki_link" href="/Breedsmic%20temperaments">Breedsmic temperaments</a></h3>
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| A breedsmic temperament is one which tempers out the breedsma, |-5 -1 -2 4&gt; = 2401/2400. Some which do not get discussed elsewhere are collected on a page <a class="wiki_link" href="/Breedsmic%20temperaments">here</a>.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:134:&lt;h3&gt; --><h3 id="toc67"><a name="Temperaments for a given comma--Ragismic microtemperaments"></a><!-- ws:end:WikiTextHeadingRule:134 --><a class="wiki_link" href="/Ragismic%20microtemperaments">Ragismic microtemperaments</a></h3>
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| A ragismic temperament is one which tempers out |-1 -7 4 1&gt; = 4375/4374. 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.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:136:&lt;h3&gt; --><h3 id="toc68"><a name="Temperaments for a given comma--Landscape microtemperaments"></a><!-- ws:end:WikiTextHeadingRule:136 --><a class="wiki_link" href="/Landscape%20microtemperaments">Landscape microtemperaments</a></h3>
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| A landscape temperament is one which tempers out |-4 6 -6 3&gt; = 250047/250000.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:138:&lt;h2&gt; --><h2 id="toc69"><a name="Temperaments for a given comma-Wizmic microtemperaments"></a><!-- ws:end:WikiTextHeadingRule:138 --><a class="wiki_link" href="/Wizmic%20microtemperaments">Wizmic microtemperaments</a></h2>
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| A wizmic temperament is one which tempers out the wizma, | -6 -8 2 5 &gt; = 420175/419904.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:140:&lt;h3&gt; --><h3 id="toc70"><a name="Temperaments for a given comma-Wizmic microtemperaments-31 comma temperaments"></a><!-- ws:end:WikiTextHeadingRule:140 --><a class="wiki_link" href="/31%20comma%20temperaments">31 comma temperaments</a></h3>
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| These all have period 1/31 of an octave.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:142:&lt;h3&gt; --><h3 id="toc71"><a name="Temperaments for a given comma-Wizmic microtemperaments-Turkish maqam music temperaments"></a><!-- ws:end:WikiTextHeadingRule:142 --><a class="wiki_link" href="/Turkish%20maqam%20music%20temperaments">Turkish maqam music temperaments</a></h3>
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| Various theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish <a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian">makam (maqam) music</a> in a systematic way. This includes, in effect, certain linear temperaments.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:144:&lt;h3&gt; --><h3 id="toc72"><a name="Temperaments for a given comma-Wizmic microtemperaments-Very low accuracy temperaments"></a><!-- ws:end:WikiTextHeadingRule:144 --><a class="wiki_link" href="/Very%20low%20accuracy%20temperaments">Very low accuracy temperaments</a></h3>
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| All hope abandon ye who enter here.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:146:&lt;h3&gt; --><h3 id="toc73"><a name="Temperaments for a given comma-Wizmic microtemperaments-Very high accuracy temperaments"></a><!-- ws:end:WikiTextHeadingRule:146 --><a class="wiki_link" href="/Very%20high%20accuracy%20temperaments">Very high accuracy temperaments</a></h3>
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| Microtemperaments which don't fit in elsewhere.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:148:&lt;h3&gt; --><h3 id="toc74"><a name="Temperaments for a given comma-Wizmic microtemperaments-High badness temperaments"></a><!-- ws:end:WikiTextHeadingRule:148 --><a class="wiki_link" href="/High%20badness%20temperaments">High badness temperaments</a></h3>
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| High in badness, but worth cataloging for one reason or another.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:150:&lt;h3&gt; --><h3 id="toc75"><a name="Temperaments for a given comma-Wizmic microtemperaments-11-limit comma temperaments"></a><!-- ws:end:WikiTextHeadingRule:150 --><a class="wiki_link" href="/11-limit%20comma%20temperaments">11-limit comma temperaments</a></h3>
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:152:&lt;h1&gt; --><h1 id="toc76"><a name="Rank 3 temperaments"></a><!-- ws:end:WikiTextHeadingRule:152 -->Rank 3 temperaments</h1>
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| <br />
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| Even less familiar than rank 2 temperaments are the <a class="wiki_link" href="/Planar%20Temperament">rank 3 temperaments</a>, based on 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.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:154:&lt;h3&gt; --><h3 id="toc77"><a name="Rank 3 temperaments--Marvel family"></a><!-- ws:end:WikiTextHeadingRule:154 --><a class="wiki_link" href="/Marvel%20family">Marvel family</a></h3>
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| The head of the marvel family is marvel, which tempers out 225/224. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:156:&lt;h3&gt; --><h3 id="toc78"><a name="Rank 3 temperaments--Starling family"></a><!-- ws:end:WikiTextHeadingRule:156 --><a class="wiki_link" href="/Starling%20family">Starling family</a></h3>
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| Starling tempers out 126/125, and like marvel it has the same generators as the 5-limit. An excellent tuning for starling is <a class="wiki_link" href="/77edo">77edo</a>, but 31, 46 or 58 also work nicely.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:158:&lt;h3&gt; --><h3 id="toc79"><a name="Rank 3 temperaments--Gamelismic family"></a><!-- ws:end:WikiTextHeadingRule:158 --><a class="wiki_link" href="/Gamelismic%20family">Gamelismic family</a></h3>
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| 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, 1029/1024.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:160:&lt;h3&gt; --><h3 id="toc80"><a name="Rank 3 temperaments--Breed family"></a><!-- ws:end:WikiTextHeadingRule:160 --><a class="wiki_link" href="/Breed%20family">Breed family</a></h3>
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| Breed is a 7-limit microtemperament which tempers out 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 your choice of 8/7 or 10/7.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:162:&lt;h3&gt; --><h3 id="toc81"><a name="Rank 3 temperaments--Ragisma family"></a><!-- ws:end:WikiTextHeadingRule:162 --><a class="wiki_link" href="/Ragisma%20family">Ragisma family</a></h3>
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| The 7-limit rank three microtemperament which tempers out the ragisma, 4375/4374, extends to various higher limit rank three temperaments such as thor.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:164:&lt;h3&gt; --><h3 id="toc82"><a name="Rank 3 temperaments--Landscape family"></a><!-- ws:end:WikiTextHeadingRule:164 --><a class="wiki_link" href="/Landscape%20family">Landscape family</a></h3>
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| The 7-limit rank three microtemperament which tempers out the lanscape comma, 250047/250000, extends to various higher limit rank three temperaments such as tyr and odin.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:166:&lt;h3&gt; --><h3 id="toc83"><a name="Rank 3 temperaments--Hemifamity family"></a><!-- ws:end:WikiTextHeadingRule:166 --><a class="wiki_link" href="/Hemifamity%20family">Hemifamity family</a></h3>
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| The hemifamity family of rank three temperaments tempers out the hemifamity comma, 5120/5103.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:168:&lt;h3&gt; --><h3 id="toc84"><a name="Rank 3 temperaments--Porwell family"></a><!-- ws:end:WikiTextHeadingRule:168 --><a class="wiki_link" href="/Porwell%20family">Porwell family</a></h3>
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| The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:170:&lt;h3&gt; --><h3 id="toc85"><a name="Rank 3 temperaments--Horwell family"></a><!-- ws:end:WikiTextHeadingRule:170 --><a class="wiki_link" href="/Horwell%20family">Horwell family</a></h3>
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| The horwell family of rank three temperaments tempers out the horwell comma, 65625/65536.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:172:&lt;h3&gt; --><h3 id="toc86"><a name="Rank 3 temperaments--Hemimage family"></a><!-- ws:end:WikiTextHeadingRule:172 --><a class="wiki_link" href="/Hemimage%20family">Hemimage family</a></h3>
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| The hemimage family of rank three temperaments tempers out the hemimage comma, 10976/10935.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:174:&lt;h3&gt; --><h3 id="toc87"><a name="Rank 3 temperaments--Sensamagic family"></a><!-- ws:end:WikiTextHeadingRule:174 --><a class="wiki_link" href="/Sensamagic%20family">Sensamagic family</a></h3>
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| These temper out 245/243.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:176:&lt;h3&gt; --><h3 id="toc88"><a name="Rank 3 temperaments--Keemic family"></a><!-- ws:end:WikiTextHeadingRule:176 --><a class="wiki_link" href="/Keemic%20family">Keemic family</a></h3>
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| These temper out 875/864.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:178:&lt;h3&gt; --><h3 id="toc89"><a name="Rank 3 temperaments--Sengic family"></a><!-- ws:end:WikiTextHeadingRule:178 --><a class="wiki_link" href="/Sengic%20family">Sengic family</a></h3>
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| These temper out the senga, 686/675.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:180:&lt;h3&gt; --><h3 id="toc90"><a name="Rank 3 temperaments--Orwellismic family"></a><!-- ws:end:WikiTextHeadingRule:180 --><a class="wiki_link" href="/Orwellismic%20family">Orwellismic family</a></h3>
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| These temper out 1728/1715.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:182:&lt;h3&gt; --><h3 id="toc91"><a name="Rank 3 temperaments--Nuwell family"></a><!-- ws:end:WikiTextHeadingRule:182 --><a class="wiki_link" href="/Nuwell%20family">Nuwell family</a></h3>
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| These temper out the nuwell comma, 2430/2401.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:184:&lt;h3&gt; --><h3 id="toc92"><a name="Rank 3 temperaments--Octagar family"></a><!-- ws:end:WikiTextHeadingRule:184 --><a class="wiki_link" href="/Octagar%20family">Octagar family</a></h3>
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| The octagar family of rank three temperaments tempers out the octagar comma, 4000/3969.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:186:&lt;h3&gt; --><h3 id="toc93"><a name="Rank 3 temperaments--Mirkwai family"></a><!-- ws:end:WikiTextHeadingRule:186 --><a class="wiki_link" href="/Mirkwai%20family">Mirkwai family</a></h3>
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| The mirkwai family of rank three temperaments tempers out the mirkwai comma, 16875/16807.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:188:&lt;h3&gt; --><h3 id="toc94"><a name="Rank 3 temperaments--Hemimean family"></a><!-- ws:end:WikiTextHeadingRule:188 --><a class="wiki_link" href="/Hemimean%20family">Hemimean family</a></h3>
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| The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:190:&lt;h3&gt; --><h3 id="toc95"><a name="Rank 3 temperaments--Mirwomo family"></a><!-- ws:end:WikiTextHeadingRule:190 --><a class="wiki_link" href="/Mirwomo%20family">Mirwomo family</a></h3>
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| The mirwomo family of rank three temperaments tempers out the mirwomo comma, 33075/32768.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:192:&lt;h3&gt; --><h3 id="toc96"><a name="Rank 3 temperaments--Dimcomp family"></a><!-- ws:end:WikiTextHeadingRule:192 --><a class="wiki_link" href="/Dimcomp%20family">Dimcomp family</a></h3>
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| The dimcomp family of rank three temperaments tempers out the dimcomp comma, 390625/388962.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:194:&lt;h3&gt; --><h3 id="toc97"><a name="Rank 3 temperaments--Tolermic family"></a><!-- ws:end:WikiTextHeadingRule:194 --><a class="wiki_link" href="/Tolermic%20family">Tolermic family</a></h3>
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| These temper out the tolerma, 179200/177147.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:196:&lt;h3&gt; --><h3 id="toc98"><a name="Rank 3 temperaments--Kleismic rank three family"></a><!-- ws:end:WikiTextHeadingRule:196 --><a class="wiki_link" href="/Kleismic%20rank%20three%20family">Kleismic rank three family</a></h3>
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| These are the rank three temperaments tempering out the kleisma, 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.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:198:&lt;h3&gt; --><h3 id="toc99"><a name="Rank 3 temperaments--Diaschismic rank three family"></a><!-- ws:end:WikiTextHeadingRule:198 --><a class="wiki_link" href="/Diaschismic%20rank%20three%20family">Diaschismic rank three family</a></h3>
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| These are the rank three temperaments tempering out the dischisma, 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.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:200:&lt;h3&gt; --><h3 id="toc100"><a name="Rank 3 temperaments--Didymus rank three family"></a><!-- ws:end:WikiTextHeadingRule:200 --><a class="wiki_link" href="/Didymus%20rank%20three%20family">Didymus rank three family</a></h3>
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| These are the rank three temperaments tempering out the didymus 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.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:202:&lt;h3&gt; --><h3 id="toc101"><a name="Rank 3 temperaments--Porcupine rank three family"></a><!-- ws:end:WikiTextHeadingRule:202 --><a class="wiki_link" href="/Porcupine%20rank%20three%20family">Porcupine rank three family</a></h3>
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| These are the rank three temperaments tempering out the porcupine comma or maximal diesis, 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.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:204:&lt;h3&gt; --><h3 id="toc102"><a name="Rank 3 temperaments--Archytas family"></a><!-- ws:end:WikiTextHeadingRule:204 --><a class="wiki_link" href="/Archytas%20family">Archytas family</a></h3>
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| Archytas temperament tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:206:&lt;h3&gt; --><h3 id="toc103"><a name="Rank 3 temperaments--Jubilismic family"></a><!-- ws:end:WikiTextHeadingRule:206 --><a class="wiki_link" href="/Jubilismic%20family">Jubilismic family</a></h3>
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| Jubilismic temperament tempers out 50/49 and thereby identifies the two septimal tritones, 7/5 and 10/7.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:208:&lt;h3&gt; --><h3 id="toc104"><a name="Rank 3 temperaments--Semiphore family"></a><!-- ws:end:WikiTextHeadingRule:208 --><a class="wiki_link" href="/Semiphore%20family">Semiphore family</a></h3>
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| 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 &quot;semi-fourth&quot;.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:210:&lt;h3&gt; --><h3 id="toc105"><a name="Rank 3 temperaments--Mint family"></a><!-- ws:end:WikiTextHeadingRule:210 --><a class="wiki_link" href="/Mint%20family">Mint family</a></h3>
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| The mint temperament tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:212:&lt;h3&gt; --><h3 id="toc106"><a name="Rank 3 temperaments--Valinorismic temperaments"></a><!-- ws:end:WikiTextHeadingRule:212 --><a class="wiki_link" href="/Valinorismic%20temperaments">Valinorismic temperaments</a></h3>
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| These temper out the valinorsma, 176/175.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:214:&lt;h3&gt; --><h3 id="toc107"><a name="Rank 3 temperaments--Werckismic temperaments"></a><!-- ws:end:WikiTextHeadingRule:214 --><a class="wiki_link" href="/Werckismic%20temperaments">Werckismic temperaments</a></h3>
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| These temper out the werckisma, 441/440.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:216:&lt;h3&gt; --><h3 id="toc108"><a name="Rank 3 temperaments--Swetismic temperaments"></a><!-- ws:end:WikiTextHeadingRule:216 --><a class="wiki_link" href="/Swetismic%20temperaments">Swetismic temperaments</a></h3>
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| These temper out the swetisma, 540/539.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:218:&lt;h3&gt; --><h3 id="toc109"><a name="Rank 3 temperaments--Lehmerismic temperaments"></a><!-- ws:end:WikiTextHeadingRule:218 --><a class="wiki_link" href="/Lehmerismic%20temperaments">Lehmerismic temperaments</a></h3>
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| These temper out the lehmerisma, 3025/3024.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:220:&lt;h3&gt; --><h3 id="toc110"><a name="Rank 3 temperaments--Kalismic temperaments"></a><!-- ws:end:WikiTextHeadingRule:220 --><a class="wiki_link" href="/Kalismic%20temperaments">Kalismic temperaments</a></h3>
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| These temper out the kalisma, 9801/9800.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:222:&lt;h1&gt; --><h1 id="toc111"><a name="Rank 4 temperaments"></a><!-- ws:end:WikiTextHeadingRule:222 --><a class="wiki_link" href="/Rank%20four%20temperaments">Rank 4 temperaments</a></h1>
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| <br />
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| 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 <a class="wiki_link" href="/Hobbits">hobbit scales</a> can be constructed for them.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:224:&lt;h1&gt; --><h1 id="toc112"><a name="Subgroup temperaments"></a><!-- ws:end:WikiTextHeadingRule:224 --><a class="wiki_link" href="/Subgroup%20temperaments">Subgroup temperaments</a></h1>
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| <br />
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| A wide-open field. These are regular temperaments of various ranks which temper <a class="wiki_link" href="/just%20intonation%20subgroups">just intonation subgroups</a>.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:226:&lt;h1&gt; --><h1 id="toc113"><a name="Commatic realms"></a><!-- ws:end:WikiTextHeadingRule:226 -->Commatic realms</h1>
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| <br />
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| By a <em>commatic realm</em> is meant the whole collection of regular temperaments of various ranks and for both full groups and <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroups</a> tempering out a given comma. For some commas, looking at the full commatic realm seems the best approach to discussing associated temperaments.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:228:&lt;h2&gt; --><h2 id="toc114"><a name="Commatic realms-Orgonia"></a><!-- ws:end:WikiTextHeadingRule:228 --><a class="wiki_link" href="/Orgonia">Orgonia</a></h2>
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| Orgonia is the commatic realm of the <a class="wiki_link" href="/11-limit">11-limit</a> comma 65536/65219 = |16 0 0 -2 -3&gt;, the orgonisma.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:230:&lt;h2&gt; --><h2 id="toc115"><a name="Commatic realms-The Biosphere"></a><!-- ws:end:WikiTextHeadingRule:230 --><a class="wiki_link" href="/The%20Biosphere">The Biosphere</a></h2>
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| The Biosphere is the name given to the commatic realm of the 13-limit comma 91/90.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:232:&lt;h2&gt; --><h2 id="toc116"><a name="Commatic realms-The Archipelago"></a><!-- ws:end:WikiTextHeadingRule:232 --><a class="wiki_link" href="/The%20Archipelago">The Archipelago</a></h2>
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| The Archipelago is a name which has been given to the commatic realm of the <a class="wiki_link" href="/13-limit">13-limit</a> comma 676/675.<br />
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| <br />
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| <br />
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| <br />
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| <hr />
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| <!-- ws:start:WikiTextHeadingRule:234:&lt;h1&gt; --><h1 id="toc117"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:234 -->Links</h1>
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| <ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_temperament" rel="nofollow">Regular temperaments - Wikipedia</a></li></ul></body></html></pre></div>
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