Tour of regular temperaments: Difference between revisions
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Added color names, pergens, intro paragraphs, monzos for all commas with > 2 digits. Changed | to [ in monzos (more readable, | looks like 1). Added prime subgroups to clans, sorted rank-3 section by prime subgroup, made new section "Misc temperaments" |
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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_tuning|POTE]] ("Pure-Octave Tenney-Euclidean") and [[TOP_tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please"). Optimization is rather intensive mathematically, but it is seldom left as an exercise to the reader; most temperaments are presented here in their optimal forms. | Another recent contribution to the field of temperament is the concept of optimization, which can take many forms. The point of optimization is to minimize the difference between a temperament and JI by finding an optimal tuning for the generator. The two most frequently used forms of optimization are [[POTE_tuning|POTE]] ("Pure-Octave Tenney-Euclidean") and [[TOP_tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please"). Optimization is rather intensive mathematically, but it is seldom left as an exercise to the reader; most temperaments are presented here in their optimal forms. | ||
=Equal temperaments= | Yet another recent development is the concept of a [[pergen]], appearing here as (P8, P5/2) or somesuch. Every rank-2, rank-3, rank-4, etc. temperament has a pergen, which specifies the period and the generator. Assuming the prime subgroup includes both 2 and 3, the period is either an octave or some fraction of it, and the generator is either some 3-limit interval or some fraction of one. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every strong extension of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but don't uniquely identify them. "W" in a pergen means "widened by one 8ve", e.g. WWP5 is a 5th plus two 8ves, or 6/1. | ||
Each temperament has two names: a traditional name and a [[Color notation|color name]]. The traditional names are [[Temperament Names|arbitrary and ideosyncratic]], but the color names are systematic and rigorous, and the comma can be deduced from the color name. Wa = 3-limit, yo = 5-over, gu = 5-under, zo = 7-over, and ru = 7-under. See also [[Color notation/Temperament Names|Color Notation/Temperament Names]]. | |||
==Equal temperaments (Rank-1 temperaments)== | |||
[[Equal-step tuning|Equal temperaments]] (abbreviated ET or tET) and [[EDO|equal divisions of the octave]] (abbreviated EDO or ED2) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator, making it a rank-1 temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval, let alone the octave, and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO, although one can also use unsupported vals or poorly-supported vals to achieve "fun" results. The familiar 12-note equal temperament, or 12edo, reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-ET. | [[Equal-step tuning|Equal temperaments]] (abbreviated ET or tET) and [[EDO|equal divisions of the octave]] (abbreviated EDO or ED2) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator, making it a rank-1 temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval, let alone the octave, and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO, although one can also use unsupported vals or poorly-supported vals to achieve "fun" results. The familiar 12-note equal temperament, or 12edo, reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-ET. | ||
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Regular temperaments of ranks two and three are cataloged on the [[Optimal patent val]] page. Rank-2 temperaments are also listed at [[Proposed names for rank 2 temperaments]] by their generator mappings, and at [[Map of rank-2 temperaments]] by their generator size. See also the [[pergen]]s page. There is also [[Graham Breed]]'s [http://x31eq.com/catalog2.html giant list of regular temperaments]. | Regular temperaments of ranks two and three are cataloged on the [[Optimal patent val]] page. Rank-2 temperaments are also listed at [[Proposed names for rank 2 temperaments]] by their generator mappings, and at [[Map of rank-2 temperaments]] by their generator size. See also the [[pergen]]s page. There is also [[Graham Breed]]'s [http://x31eq.com/catalog2.html giant list of regular temperaments]. | ||
==Families== | == Families == | ||
As we go up in rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) The same comment applies to 7-limit temperaments and rank three, etc. Members of families and their relationships can be classified by the [[Normal_lists|normal comma list]] of the various temperaments. Families include weak extensions as well as strong, in other words, the pergen shown here may change. | |||
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 | |||
===[[Meantone family]]=== | ===[[Meantone family|Meantone or Gu family]] (P8, P5) === | ||
The meantone family tempers out [[81/80]], also called the syntonic comma. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are [[12edo]], [[19edo]], [[31edo]], [[43edo]], [[50edo]], [[55edo]] and [[81edo]]. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma. | The meantone family tempers out [[81/80]], also called the syntonic comma. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are [[12edo]], [[19edo]], [[31edo]], [[43edo]], [[50edo]], [[55edo]] and [[81edo]]. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma. | ||
===[[Schismatic family]]=== | ===[[Schismatic family|Schismatic or Layo family]] (P8, P5)=== | ||
The schismatic family tempers out the schisma of [[32805/32768]], which is the amount by which the Pythagorean comma exceeds the syntonic comma. The 5-limit version of the temperament is a [[Microtempering|microtemperament]] which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity; whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4. Tunings include [[12edo]], [[29edo]], [[41edo]], [[53edo]], and [[118edo]]. | The schismatic family tempers out the schisma of [-15 8 1> = [[32805/32768]], which is the amount by which the Pythagorean comma exceeds the syntonic comma. The 5-limit version of the temperament is a [[Microtempering|microtemperament]] which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity; whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4. Tunings include [[12edo]], [[29edo]], [[41edo]], [[53edo]], and [[118edo]]. | ||
===[[Kleismic family]]=== | ===[[Kleismic family|Kleismic or Tribiyo family]] (P8, P12/6)=== | ||
The kleismic family of temperaments tempers out the [[kleisma]] | The kleismic family of temperaments tempers out the [[kleisma]] [-6 -5 6> = 15625/15552, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. The kleismic family includes [[15edo]], [[19edo]], [[34edo]], [[49edo]], [[53edo]], [[72edo]], [[87edo]] and [[140edo]] among its possible tunings. | ||
===[[Magic family]]=== | ===[[Magic family|Magic or Laquinyo family]] (P8, P12/5)=== | ||
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 3/1. The generator is itself an approximate 5/4. The magic family includes [[16edo]], [[19edo]], [[22edo]], [[25edo]], and [[41edo]] among its possible tunings, with the latter being near-optimal. | The magic family tempers out [-10 -1 5> (3125/3072), known as the [[magic comma]] or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/1. The generator is itself an approximate 5/4. The magic family includes [[16edo]], [[19edo]], [[22edo]], [[25edo]], and [[41edo]] among its possible tunings, with the latter being near-optimal. | ||
===[[Diaschismic family]]=== | ===[[Diaschismic family|Diaschismic or Sagugu family]] (P8/2, P5)=== | ||
The diaschismic family tempers out the [[diaschisma]], 2048/2025, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. A noted 7-limit extension to diaschismic is [[pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out, of which [[22edo]] is an excellent tuning. | The diaschismic family tempers out the [[diaschisma]], [11 -4 -2> or 2048/2025, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. A noted 7-limit extension to diaschismic is [[pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out, of which [[22edo]] is an excellent tuning. | ||
===[[Pelogic family]]=== | ===[[Pelogic family|Pelogic or Layobi family]] (P8, P5)=== | ||
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, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates [[2L 5s]] "anti-diatonic" scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo]], [[16edo]], [[23edo]], and [[25edo]]. | This tempers out the pelogic comma, [-7 3 1> = [[135/128]], also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates [[2L 5s]] "anti-diatonic" scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo]], [[16edo]], [[23edo]], and [[25edo]]. | ||
===[[Porcupine family]]=== | ===[[Porcupine family|Porcupine or Triyo family]] (P8, P4/3)=== | ||
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 porcupine family tempers out [1 -5 3> = [[250/243]], the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include [[15edo]], [[22edo]], [[37edo]], and [[59edo]]. | ||
===[[Würschmidt family]]=== | ===[[Würschmidt family|Würschmidt or Saquadbigu family]] (P8, WWP5/8)=== | ||
The würschmidt (or wuerschmidt) family tempers out the [[Würschmidt comma]], 393216/390625 = | The würschmidt (or wuerschmidt) family tempers out the [[Würschmidt comma]], 393216/390625 = [17 1 -8>. Würschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a perfect 5th two octaves up); that is, (5/4)^8 * (393216/390625) = 6. It tends to generate the same MOSs as [[Magic_family|magic temperament]], but is tuned slightly more accurately. Both [[31edo]] and [[34edo]] can be used as würschmidt tunings, as can [[65edo]], which is quite accurate. | ||
===[[Augmented_family|Augmented family]]=== | ===[[Augmented_family|Augmented or Trigu family]] (P8/3, 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" ([[3L 3s]]) in common 12-based music theory, as well as what is commonly called "[http://www.tcherepnin.com/alex/basic_elem1.htm#9step Tcherepnin's scale]" ([[3L 6s]]). | The augmented family tempers out the diesis of [7 0 -3> = [[128/125]], the difference between three 5/4 major thirds and a 2/1 octave, and so identifies the major third with the third-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]], which is an excellent tuning for augmented. It is the temperament that results in what is commonly called the "augmented scale" ([[3L 3s]]) in common 12-based music theory, as well as what is commonly called "[http://www.tcherepnin.com/alex/basic_elem1.htm#9step Tcherepnin's scale]" ([[3L 6s]]). | ||
===[[Dimipent family]]=== | ===[[Dimipent family|Dimipent or Quadgu family]] (P8/4, P5)=== | ||
The dimipent (or diminished) family tempers out the major diesis or 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]]. | The dimipent (or diminished) family tempers out the major diesis or diminished comma, [3 4 -4> or [[648/625]], the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as [[12edo]]. | ||
===[[Dicot family]]=== | ===[[Dicot family|Dicot or Yoyo family]] (P8, P5/2)=== | ||
The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, [[25/24]] (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into 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 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]]. | ||
===[[Tetracot family]]=== | ===[[Tetracot family|Tetracot or Saquadyo family]] (P8, P5/4)=== | ||
The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by 20000/19683, the minimal diesis or [[tetracot comma]]. [[7edo]] can also be considered a tetracot tuning, as can [[20edo]], [[27edo]], [[34edo]], and [[41edo]]. | The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by [5 -9 4> (20000/19683), the minimal diesis or [[tetracot comma]]. [[7edo]] can also be considered a tetracot tuning, as can [[20edo]], [[27edo]], [[34edo]], and [[41edo]]. | ||
===[[Sensipent family]]=== | ===[[Sensipent family|Sensipent or Sepgu family]] (P8, WWP5/7)=== | ||
The sensipent (sensi) family tempers out the [[sensipent comma]], 78732/78125, also known as the medium semicomma. Tunings include [[8edo]], [[19edo]], [[46edo]], and [[65edo]]. | The sensipent (sensi) family tempers out the [[sensipent comma]], [2 9 -7> (78732/78125), also known as the medium semicomma. Tunings include [[8edo]], [[19edo]], [[46edo]], and [[65edo]]. | ||
===[[Semicomma_family|Orwell and the semicomma family]]=== | ===[[Semicomma_family|Orwell or Sepru, and the semicomma or Lasepyo family]] (P8, P12/7)=== | ||
The [[semicomma]] (also known as Fokker's comma), 2109375/2097152 = | 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. | ||
===[[Pythagorean family]]=== | ===[[Pythagorean family|Pythagorean or Lalawa family]] (P8/12, ^1)=== | ||
The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = | The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = [-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 and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate chains of 12-equal, offset from one another justly tuned 5/4. | ||
===[[Apotome family]]=== | ===[[Apotome family|Apotome or Lawa family]] (P8/7, ^1)=== | ||
This family tempers out the apotome, 2187/2048, which is a 3-limit comma. | This family tempers out the apotome, [-11 7 0> = 2187/2048, which is a 3-limit comma. | ||
===[[Gammic family]]=== | ===[[Gammic family|Gammic or Laquinquadyo family]] (P8, P5/20)=== | ||
The gammic family tempers out the gammic comma, | 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. | ||
===[[Minortonic family]]=== | ===[[Minortonic family|Minortonic or Trila-segu family]] (P8, WWP5/17)=== | ||
This tempers out the minortone comma, | 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). | ||
===[[Bug family]]=== | ===[[Bug family|Bug or Gugu family]] (P8, P4/2)=== | ||
This tempers out [[27/25]], the large limma or bug comma. | This tempers out [[27/25]], the large limma or bug comma. | ||
===[[Father family]]=== | ===[[Father family|Father or Gubi family]] (P8, P5)=== | ||
This tempers out [[16/15]], the just diatonic semitone. | This tempers out [[16/15]], the just diatonic semitone. | ||
===[[Sycamore family]]=== | ===[[Sycamore family|Sycamore or Laleyo family]] (P8, P5/11)=== | ||
The sycamore family tempers out the sycamore comma, | 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. | ||
===[[Escapade family]]=== | ===[[Escapade family|Escapade or Sasa-tritrigu family]] (P8, P4/9)=== | ||
This tempers out the [[escapade comma]], | This tempers out the [[escapade comma]], [32 -7 -9>, which is the difference between nine just major thirds and seven just fourths. | ||
===[[Amity family]]=== | ===[[Amity family|Amity or Saquinyo family]] (P8, P11/5)=== | ||
This tempers out the [[amity comma]], 1600000/1594323 = | This tempers out the [[amity comma]], 1600000/1594323 = [9 -13 5>. | ||
===[[Vulture family]]=== | ===[[Vulture family|Vulture or Sasa-quadyo family]] (P8, P12/4)=== | ||
This tempers out the [[vulture comma]], | This tempers out the [[vulture comma]], [24 -21 4>. | ||
===[[Vishnuzmic family]]=== | ===[[Vishnuzmic family|Vishnuzmic or Sasepbigu family]] (P8/2, P4/7)=== | ||
This tempers out the vishnuzma, | 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. | ||
===[[Luna family]]=== | ===[[Luna family|Luna or Sasa-quintrigu family]] (P8, WWP4/15)=== | ||
This tempers out the luna comma, | This tempers out the luna comma, [38 -2 -15> (274877906944/274658203125) | ||
===[[Laconic family]]=== | ===[[Laconic family|Laconic or Latrigubi family]] (P8, P5/3)=== | ||
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 [[16edo]], [[21edo]], and [[37edo]] (using the 37b mapping), among others. | This tempers out the laconic comma, [-4 7 -3> (2187/2000), which is the difference between three 10/9's and one 3/2. Laconic is supported by [[16edo]], [[21edo]], and [[37edo]] (using the 37b mapping), among others. | ||
===[[Immunity family]]=== | ===[[Immunity family|Immunity or Sasa-yoyo family]] (P8, P4/2)=== | ||
This tempers out the immunity comma, 1638400/1594323. | This tempers out the immunity comma, [16 -13 2> (1638400/1594323). | ||
===[[Ditonmic family]]=== | ===[[Ditonmic family|Ditonmic or Lala-theyo family]] (P8, W<sup>4</sup>P4/13)=== | ||
This tempers out the ditonma, 1220703125/1207959552. | This tempers out the ditonma, [-27 -2 13> = 1220703125/1207959552. | ||
===[[Shibboleth family]]=== | ===[[Shibboleth family|Shibboleth or Tritriyo family]] (P8, WWP4/9)=== | ||
This tempers out the shibboleth comma, 1953125/1889568. | This tempers out the shibboleth comma, [-5 -10 9> = 1953125/1889568. | ||
===[[Comic family]]=== | ===[[Comic family|Comic or Saquadyobi family]] (P8/2, M2/4)=== | ||
This tempers out the comic comma, 5120000/4782969. | This tempers out the comic comma, [13 -14 4> = 5120000/4782969. | ||
===[[Wesley family]]=== | ===[[Wesley family|Wesley or Lasepyobi family]] (P8, WWP4/7)=== | ||
This tempers out the wesley comma, 78125/73728. | This tempers out the wesley comma, [-13 -2 7> = 78125/73728. | ||
===[[Fifive family]]=== | ===[[Fifive family|Fifive or Saquinbiyo family]] (P8/2, P5/5)=== | ||
This tempers out the fifive comma, 9765625/9565938. | This tempers out the fifive comma, [-1 -14 10> = 9765625/9565938. | ||
===[[Maja family]]=== | ===[[Maja family|Maja or Saseyo family]] (P8, W<sup>6</sup>P4/17)=== | ||
This tempers out the maja comma, 762939453125/753145430616. | This tempers out the maja comma, [-3 -23 17> = 762939453125/753145430616. | ||
===[[Pental family]]=== | ===[[Pental family|Pental or Trila-quingu family]] (P8/5, P5)=== | ||
This tempers out the pental comma, 847288609443/838860800000 = | This tempers out the pental comma, 847288609443/838860800000 = [-28 25 -5>. | ||
===[[Qintosec family]]=== | ===[[Qintosec family|Qintosec or Quadsa-quinbigu family]] (P8/5, P5/2)=== | ||
This tempers out the qintosec comma, 140737488355328/140126044921875 = | This tempers out the qintosec comma, 140737488355328/140126044921875 = [47 -15 -10>. | ||
===[[Trisedodge family]]=== | ===[[Trisedodge family|Trisedodge or Saquintrigu family]] (P8/5, P4/3)=== | ||
This tempers out the trisedodge comma, 30958682112/30517578125 = | This tempers out the trisedodge comma, 30958682112/30517578125 = [19 10 -15>. | ||
===[[Maquila family]]=== | ===[[Maquila family|Maquila or Trisa-segu family]] (P8, W<sup>7</sup>P5/17)=== | ||
This tempers out the maquila comma, 562949953421312/556182861328125 = | This tempers out the maquila comma, 562949953421312/556182861328125 = [49 -6 -17>. | ||
===[[Mutt family]]=== | ===[[Mutt family|Mutt or Trila-septriyo family]] (P8/3, WWP4/7)=== | ||
This tempers out the [[mutt_comma|mutt comma]], | This tempers out the [[mutt_comma|mutt comma]], [-44 -3 21>, leading to some strange properties. | ||
==Clans== | ==Clans== | ||
| Line 152: | Line 155: | ||
If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another [[Just_intonation_subgroups|subgroup]] of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. We can modify the definition of [[Normal_lists|normal comma list]] for clans by changing the ordering of prime numbers, and using this to sort out clan relationships. | If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another [[Just_intonation_subgroups|subgroup]] of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. We can modify the definition of [[Normal_lists|normal comma list]] for clans by changing the ordering of prime numbers, and using this to sort out clan relationships. | ||
===[[ | ===[[Trienstonic clan|Trienstonic or Zo clan]] (P8, P5) === | ||
This 2.3.7 clan tempers out the septimal third-tone, [[28/27]], a no-fives comma. | |||
===[[ | ===[[Slendro clan|Slendro or Zozo clan]] (P8, P4/2)=== | ||
This clan tempers out the | This 2.3.7 clan tempers out the slendro diesis, [[49/48]], a no-fives comma. | ||
===[[ | ===[[Archytas clan|Archytas or Ru clan]] (P8, P5)=== | ||
This clan tempers out the | This 2.3.7 clan tempers out the Archytas comma, [[64/63]], a no-fives comma. The clan consists of rank two temperaments, and should not be confused with the [[Archytas family]] of rank three temperaments. | ||
===[[ | ===[[Gamelismic clan|Gamelismic or Latrizo clan]] (P8, P5/3)=== | ||
This tempers out the | This 2.3.7 clan tempers out the gamelisma, [-10 1 0 3> = 1029/1024, a no-fives comma. Three 8/7 generators equals a 5th. A particularly noteworthy member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds. Miracle temperament divides the fifth into 6 equal steps, thus it's a weak extension. Its 21-note scale called "blackjack" and 31-note scale called "canasta" have some useful properties. It is the most efficient 11-limit temperament for many purposes, with a tuning close to 72-EDO. | ||
===[[ | ===[[Jubilismic clan|Jubilismic or Biruyo Nowa clan]] (P8/2, M3)=== | ||
This clan tempers out the | This 2.5.7 clan tempers out the jubilisma, [[50/49]], a no-threes comma which is the difference between 10/7 and 7/5. The pergen's M3 generator equals 5/4. The half-octave period equals 7/5. | ||
===[[ | ===[[Hemimean clan|Hemimean or Zozoquingu Nowa clan]] (P8, M2)=== | ||
This clan tempers out | This 2.5.7 clan tempers out the hemimean comma, [6 0 -5 2> = 3136/3125, a no-threes comma. Two M2 generators equals 5/4, and five of them equals 7/4. | ||
===[[ | ===[[Quince clan|Quince or Lasepzo-agugu Nowa clan]] (P8, M2/2)=== | ||
This tempers out the | This 2.5.7 clan tempers out the quince, [-15 0 -2 7> = 823543/819200, a no-threes comma. Two generators equals 8/7 (a M2), and seven generators equals 8/5. | ||
===[[ | ===[[Sensamagic clan|Sensamagic or Zozoyo Noca clan]] (P12, M3)=== | ||
This tempers out the | This 3.5.7 clan tempers out the sensamagic comma [0 -5 1 2> = 245/243, a no-twos comma (ratio of odd numbers). The M3 generator = 9/7, and two generators equals 5/3. | ||
===[[ | ===[[Mirkwai clan|Mirkwai or Quinru-aquadyo Noca clan]] (P12, WM7/4)=== | ||
This tempers out the | This 3.5.7 clan tempers out the mirkwai comma, [0 3 4 -5> = 16875/16807, a no-twos comma. Four generators = a major 14th = 27/7. | ||
=Temperaments for a given comma= | =Temperaments for a given comma= | ||
===[[Septisemi temperaments]]=== | ===[[Septisemi temperaments|Septisemi or Zogu temperaments]] (P8, P5, ^1)=== | ||
These are very low complexity temperaments tempering out the minor septimal semitone, [[21/20]] and hence equating 5/3 with 7/4. | These are very low complexity temperaments tempering out the minor septimal semitone, [[21/20]] and hence equating 5/3 with 7/4. ^1 = 81/80. | ||
===[[Wizmic microtemperaments|Wizmic or Quinzo-ayoyo microtemperaments]] (P8, P5, vm7/2)=== | |||
A wizmic temperament is one which tempers out the wizma, [ -6 -8 2 5 > = 420175/419904. vm7 = 7/4. | |||
===[[ | ===[[Greenwoodmic temperaments|Greenwoodmic or Ruruyo temperaments]] (P8, P5, ^1)=== | ||
These | These temper out the greenwoodma, [-3 4 1 -2> = 405/392. ^1 = 64/63. | ||
===[[ | ===[[Avicennmic temperaments|Avicennmic or Zoyoyo temperaments]] (P8, P5, ^1)=== | ||
These temper out the | These temper out the avicennma, [-9 1 2 1> = 525/512, also known as Avicenna's enharmonic diesis. ^1 = 81/80. | ||
===[[ | ===[[Garischismic temperaments|Garischismic or Sasaru temperaments]] (P8, P5, ^1)=== | ||
A garischismic temperament is one which tempers out the garischisma, [25 -14 0 -1> = 33554432/33480783. ^1 = 81/80. | |||
===[[ | ===[[Stearnsmic temperaments|Stearnsmic or Latribiru temperaments]] (P8/2, P4/3, ^1)=== | ||
Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6> = 118098/117649. ^1 = 81/80. | |||
===[[ | ===[[Cataharry temperaments|Cataharry or Labirugu temperaments]] (P8, P12/2, ^1)=== | ||
Cataharry temperaments temper out the cataharry comma, [-4 9 -2 -2> = 19683/19600. ^1 = 81/80. | |||
===[[ | ===''[[Mint temperaments|Mint or Rugu temperaments]] (P8, P5, ^1)''=== | ||
These | ''These are low complexity, high error temperaments tempering out the septimal quarter-tone, [[36/35]].'' | ||
===[[ | ===''[[Keemic temperaments|Keemic or Zotriyo temperaments]] (P8, P5, ^1)''=== | ||
These temper out | ''These temper out the keema, [-5 -3 3 1> = 875/864.'' | ||
===[[ | ===''[[Starling temperaments|Starling or Zotrigu temperaments]] (P8, P5, ^1)''=== | ||
''These temper out [1 2 -3 1> = [[126/125]], the septimal semicomma or starling comma the difference between three 6/5s plus one 7/6, and an octave), and include myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.'' | |||
===[[ | ===''[[Marvel temperaments|Marvel or Ruyoyo temperaments]] (P8, P5, ^1)''=== | ||
''These temper out [-5 2 2 -1> = [[225/224]], the marvel comma.'' | |||
===[[ | ===''[[Orwellismic temperaments|Orwellismic or Satriru-agu temperaments]] (P8, P5, ^1)''=== | ||
''These temper out [6 3 -1 -3> = 1728/1715, the orwellisma.'' | |||
===[[ | ===[[Octagar temperaments|''Octagar or Rurutriyo temperaments'']]=== | ||
''Octagar temperaments temper out the octagar comma, [5 -4 3 -2> = 4000/3969.'' | |||
===[[ | ===''[[Hemifamity temperaments|Hemifamity or Saruyo temperaments]] (P8, P5, ^1)''=== | ||
''Hemifamity temperaments temper out the hemifamity comma, [10 -6 1 -1> = 5120/5103.'' | |||
===[[ | ===''[[Porwell temperaments|Porwell or Sarurutrigu temperaments]]'' === | ||
''Porwell temperaments temper out the porwell comma, [11 1 -3 -2> = 6144/6125.'' | |||
===[[ | ===''[[Hemimage temperaments|Hemimage or Satrizo-agu temperaments]] (P8, P5, ^1)''=== | ||
''Hemimage temperaments temper out the hemimage comma, [5 -7 -1 3> = 10976/10935.'' | |||
===[[Horwell temperaments]]=== | ===''[[Horwell temperaments|Horwell or Lazoquinyo temperaments]] (P8, P5, ^1)''=== | ||
Horwell temperaments temper out the horwell comma, | ''Horwell temperaments temper out the horwell comma, [-16 1 5 1> = 65625/65536.'' | ||
===[[Breedsmic temperaments]]=== | ===''[[Breedsmic temperaments|Breedsmic or Bizozogu temperaments]] (P8, P5/2, /1)''=== | ||
A breedsmic temperament is one which tempers out the breedsma, | ''A breedsmic temperament is one which tempers out the breedsma, [-5 -1 -2 4> = 2401/2400.'' | ||
===[[Ragismic microtemperaments]]=== | ===''[[Ragismic microtemperaments|Ragismic or Zoquadyo microtemperaments]] (P8, P5, ^1)''=== | ||
A ragismic temperament is one which tempers out | ''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.'' | ||
===[[Landscape microtemperaments]]=== | ===''[[Landscape microtemperaments|Landscape or Trizogugu microtemperaments]] (P8/3, P5, ^1)''=== | ||
A landscape temperament is one which tempers out | ''A landscape temperament is one which tempers out [-4 6 -6 3> = 250047/250000.'' | ||
= | =Rank-3 temperaments= | ||
Even less familiar than rank-2 temperaments are the [[Planar_Temperament|rank-3 temperaments]], generated by a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank-3 temperaments by the commas they temper out. They can also be identified by pergen. The 3rd element of a rank-3 pergen is usually a comma, but sometimes it's some fraction of a 5-limit or 7-limit interval. | |||
===[[ | ===[[Didymus rank three family|Didymus or Gu rank three family]] (P8, P5, ^1)=== | ||
These are the rank three temperaments tempering out the didymus or meantone comma, 81/80. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth. The meantone comma equates every 5-limit interval to some 3-limit interval, therefore the generators are the same as for 2.3.7 JI: 2/1, 3/1 and 7/1. These may be reduced to 2/1, 3/2 and 7/4, and 7/4 may be reduced further to 64/63. Hence in the pergen,^1 = 64/63. | |||
===[[ | ===[[Porcupine rank three family|Porcupine or Triyo rank three family]] (P8, P4/3, /1)=== | ||
These are the rank three temperaments tempering out the porcupine comma or maximal diesis, [1 -5 3> = 250/243. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth. /1 = 64/63. | |||
===[[ | ===[[Diaschismic rank three family|Diaschismic or Sagugu rank three family]] (P8/2, P5, /1)=== | ||
These are the rank three temperaments tempering out the dischisma, [11 -4 -2> = 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth. /1 = 64/63. | |||
===[[ | ===[[Kleismic rank three family|Kleismic or Tribiyo rank three family]] (P8, P12/6, /1)=== | ||
These are the rank three temperaments tempering out the kleisma, [-6 -5 6> = 15625/15552. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth. /1 = 64/63. | |||
===[[ | ===[[Archytas family|Archytas or Ru family]] (P8, P5, ^1)=== | ||
These | Archytas temperament tempers out 64/63. This comma equates every 7-limit interval to some 3-limit interval, therefore the generators are the same as for 2.3.5 JI: 2/1, 3/1 and 5/1. These may be reduced to 2/1, 3/2 and 5/4, and 5/4 may be reduced further to 81/80. Hence in the pergen, ^1 = 81/80. If 81/80 were tempered out too, the otonal tetrad 4:5:6:7 would be identified with the 3-dominant seventh chord. | ||
===[[Semiphore family|Semiphore or Zozo family]] (P8, P4/2, ^1)=== | |||
Semiphore temperament tempers out 49/48 and thereby identifies the septimal minor third, 7/6 and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like "semi-fourth". ^1 = 81/80. | |||
= | ===[[Gamelismic family|Gamelismic or Latrizo family]] (P8, P5/3, ^1)=== | ||
Not to be confused with the gamelismic clan of rank two temperaments, the gamelismic family are those rank three temperaments which temper out the gamelisma, [-10 1 0 3> = 1029/1024. ^1 = 81/80. | |||
===[[Marvel family|Marvel or Ruyoyo family]] (P8, P5, ^1)=== | |||
The head of the marvel family is marvel, which tempers out [-5 2 2 -1> = [[225/224]]. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle. Other family members include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton. | |||
The marvel comma equates every 7-limit interval to some 5-limit interval, therefore the generators are the same as for 5-limit JI: 2/1, 3/1 and 5/1. These may be reduced to 2/1, 3/2 and 5/4, and 5/4 may be reduced further to 81/80. Hence in the pergen, ^1 = 81/80. | |||
The | |||
===[[Starling family]]=== | ===[[Starling family|Starling or Zotrigu family]] (P8, P5, ^1)=== | ||
Starling tempers out [[126/125]], and | Starling tempers out the septimal semicomma or starling comma [1 2 -3 1> = [[126/125]], the difference between three 6/5s plus one 7/6, and an octave. Like marvel, it has the same generators as 5-limit JI. An excellent tuning for starling is [[77edo]], but 31, 46 or 58 also work nicely. Its family includes myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum. ^1 = 81/80. | ||
===[[ | ===[[Breed family|Breed or Bizozogu family]] (P8, P5/2, /1)=== | ||
Breed is a 7-limit microtemperament which tempers out [-5 -1 -2 4> = 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749edo will certainly do the trick. Breed has generators of 2/1, a 49/40-cum-60/49 neutral third, and 64/63. | |||
===[[ | ===[[Ragisma family|Ragisma or Zoquadyo family]] (P8, P5, ^1)=== | ||
The 7-limit rank three microtemperament which tempers out the ragisma, [-1 -7 4 1> = 4375/4374, extends to various higher limit rank three temperaments such as thor. These are not by any means all microtemperaments, but those which are not highly accurate are probably best discussed under another heading. Accurate ones include ennealimmal, supermajor, enneadecal, amity, mitonic, parakleismic, gamera and vishnu. Pontiac belongs on the list but falls under the schismatic family rubric. ^1 = 81/80. | |||
===[[ | ===[[Landscape family|Landscape or Trizogugu family]] (P8/3, P5, ^1)=== | ||
The 7-limit rank three microtemperament which tempers out the | The 7-limit rank three microtemperament which tempers out the lanscape comma, [-4 6 -6 3> = 250047/250000, extends to various higher limit rank three temperaments such as tyr and odin. The period is 63/50 and ^1 = 81/80. | ||
===[[ | ===[[Hemifamity family|Hemifamity or Saruyo family]] (P8, P5, ^1)=== | ||
The | The hemifamity family of rank three temperaments tempers out the hemifamity comma, [10 -6 1 -1> = 5120/5103. ^1 = 81/80. | ||
===[[ | ===[[Porwell family|Porwell or Sarurutrigu family]] (P8, P5, vM6/2)=== | ||
The | The porwell family of rank three temperaments tempers out the porwell comma, [11 1 -3 -2> = 6144/6125. vM6 = a downmajor 6th = 5/3. | ||
===[[ | ===[[Horwell family|Horwell or Lazoquinyo family]] (P8, P5, ^1)=== | ||
The | The horwell family of rank three temperaments tempers out the horwell comma, [-16 1 5 1> = 65625/65536. ^1 = 81/80. | ||
===[[ | ===[[Hemimage family|Hemimage or Satrizo-agu family]] (P8, P5, ^1)=== | ||
The | The hemimage family of rank three temperaments tempers out the hemimage comma, [5 -7 -1 3> = 10976/10935. ^1 = 64/63. | ||
===[[ | ===[[Sensamagic family|Sensamagic or Zozoyo family]] (P8, P5, ^1)=== | ||
These temper out [0 -5 1 2> = 245/243. ^1 = 64/63. | |||
===[[ | ===[[Keemic family|Keemic or Zotriyo family]] (P8, P5, ^1)=== | ||
These temper out | These temper out the keema [-5 -3 3 1> = 875/864. ^1 = 81/80. | ||
===[[ | ===[[Sengic family|Sengic or Trizo-agugu family]] (P8, P5, vm3/2)=== | ||
These temper out | These temper out the senga, [1 -3 -2 3> = 686/675. One generator =15/14, two = 7/6 (a downminor 3rd), and three = 6/5. | ||
===[[ | ===[[Orwellismic family|Orwellismic or Triru-agu family]] (P8, P5, ^1)=== | ||
These temper out | These temper out [6, 3, -1, -3> = 1728/1715. ^1 = 64/63. | ||
===[[ | ===[[Nuwell family|Nuwell or Quadru-ayo family]] (P8, P5, ^1)=== | ||
These temper out | These temper out the nuwell comma, [1, 5, 1, -4> = 2430/2401. ^1 = 64/63. | ||
===[[ | ===[[Octagar family|Octagar or Rurutriyo family]] (P8, P5, ^m6/2)=== | ||
The octagar family of rank three temperaments tempers out the octagar comma, [5 -4 3 -2> = 4000/3969. ^m6 = an upminor 6th = 8/5. | |||
===[[ | ===[[Mirkwai family|Mirkwai or Quinru-aquadyo family]] (P8, P5, W^M7/4)=== | ||
The | The mirkwai family of rank three temperaments tempers out the mirkwai comma, [0 3 4 -5> = 16875/16807. W^M7 = a wide upmajor 7th = 27/7. | ||
===[[ | ===[[Hemimean family|Hemimean or Zozoquingu family]] (P8, P5, vM3/2)=== | ||
The | The hemimean family of rank three temperaments tempers out the hemimean comma, [6 0 -5 2> = 3136/3125. vM3 = a downmajor 3rd = 5/4. | ||
===[[ | ===[[Mirwomo family|Mirwomo or Labizoyo family]] (P8, P5/2, ^1)=== | ||
The | The mirwomo family of rank three temperaments tempers out the mirwomo comma, [-15 3 2 2> = 33075/32768. P5/2 = 128/105 and ^1 = 81/80. | ||
===[[ | ===[[Dimcomp family|Dimcomp or Quadruyoyo family]] (P8/4, P5, ^1)=== | ||
The | The dimcomp family of rank three temperaments tempers out the dimcomp comma, [-1 -4 8 -4> = 390625/388962. The period = 25/21. | ||
===[[ | ===[[Tolermic family|Tolermic or Sazoyoyo family]] (P8, P5, ^1)=== | ||
These temper out the tolerma, [10 -11 2 1> = 179200/177147. ^1 = 81/80. | |||
===[[ | ===[[Jubilismic family|Jubilismic or Biruyo family]] (P8/2, P5, ^1)=== | ||
Jubilismic temperament tempers out 50/49 and thereby identifies the two septimal tritones, 7/5 and 10/7. The period is 7/5, and ^1 = 81/80. | |||
===[[ | ===[[Mint family|Mint or Rugu family]] (P8, P5, ^1)=== | ||
The mint temperament is a low complexity, high error temperament, tempering out the septimal quarter-tone 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7. ^1 = 81/80 or 64/63. | |||
=[[Rank_four_temperaments|Rank-4 temperaments]]= | |||
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. | |||
===[[ | ===[[Valinorismic temperaments|Valinorismic or Lorugugu temperaments]] === | ||
These | These temper out the valinorsma, [4 0 -2 -1 1> = 176/175. | ||
===[[ | ===[[Rastmic temperaments|Rastmic or Lulu temperaments]]=== | ||
These temper out the rastma, [-1 5 0 0 -2> = 243/242. As a no-fives no-sevens rank-2 temperament, it's (P8, P5/2). | |||
===[[ | ===[[Werckismic temperaments|Werckismic or Luzozogu temperaments]]=== | ||
These temper out the werckisma, [-3 2 -1 2 -1> = 441/440. | |||
===[[ | ===[[Swetismic temperaments|Swetismic or Lururuyo temperaments]]=== | ||
These temper out the swetisma, [2 3 1 -2 -1> = 540/539. | |||
===[[ | ===[[Lehmerismic temperaments|Lehmerismic or Loloruyoyo temperaments]]=== | ||
These temper out the lehmerisma, [-4 -3 2 -1 2> = 3025/3024. | |||
===[[ | ===[[Kalismic temperaments|Kalismic or Bilorugu temperaments]]=== | ||
These temper out the | These temper out the kalisma, [-3 4 -2 -2 2> = 9801/9800. | ||
=[[Subgroup temperaments]]= | |||
A wide-open field. These are regular temperaments of various ranks which temper [[just intonation subgroups]]. | |||
== | =Commatic realms= | ||
By a ''commatic realm'' is meant the whole collection of regular temperaments of various ranks and for 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. | |||
==[[Orgonia|Orgonia or Satrilu-aruru]]== | |||
Orgonia is the commatic realm of the [[11-limit]] comma 65536/65219 = [16 0 0 -2 -3>, the orgonisma. | |||
=[[ | ==[[The Biosphere|The Biosphere or Thozogu]] == | ||
The Biosphere is the name given to the commatic realm of the 13-limit comma 91/90. | |||
==[[The Archipelago|The Archipelago or Bithogu]]== | |||
The Archipelago is a name which has been given to the commatic realm of the [[13-limit]] comma [2 -3 -2 0 0 2> = 676/675. | |||
= | = Miscellaneous other temperaments = | ||
===[[31 comma temperaments]]=== | |||
These all have period 1/31 of an octave. | |||
= | ===[[Turkish maqam music temperaments]]=== | ||
Various theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish [[Arabic,_Turkish,_Persian|makam (maqam) music]] in a systematic way. This includes, in effect, certain linear temperaments. | |||
===[[Very low accuracy temperaments]]=== | |||
All hope abandon ye who enter here. | |||
==[[ | ===[[Very high accuracy temperaments]]=== | ||
Microtemperaments which don't fit in elsewhere. | |||
==[[ | ===[[High badness temperaments]]=== | ||
High in badness, but worth cataloging for one reason or another. | |||
==[[ | ===[[11-limit comma temperaments]]=== | ||
These temperaments go to 11... | |||
=Links= | =Links= | ||