Tour of regular temperaments: Difference between revisions
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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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">[[toc|flat]] | <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">[[toc|flat]] | ||
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=Regular temperaments= | =Regular temperaments= | ||
Regular temperaments are irrational tunings, wherein the infinite number of intervals in n-limit Just intonation (or any subgroup thereof) are mapped to a smaller (though possibly 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 mainly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI). Temperaments effectively reduce the "dimensionality" of JI, thus 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). | |||
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. 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 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. 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. | ||
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=[[edo|Equal temperaments]]= | =[[edo|Equal temperaments]]= | ||
[[Equal Temperaments|Equal temperaments]] (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. An | [[Equal Temperaments|Equal temperaments]] (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. An n-limit ET is simply an n-limit temperament that uses a single generator (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. | ||
=Rank 2 (including "linear") temperaments[[#lineartemperaments]]= | =Rank 2 (including "linear") temperaments[[#lineartemperaments]]= | ||
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. | 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. | ||
N-limit Rank 2 temperaments map all intervals of n-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. | |||
As we go up from 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.) 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. | As we go up from 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.) 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. | ||
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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. | 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. | ||
===[[Sensipent family]]=== | ===[[Sensipent family]]=== | ||
This tempers out the sensipent comma, 78732/78125. | This tempers out the sensipent comma, 78732/78125. | ||
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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. | 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. | ||
===[[Apotome family]]=== | ===[[Apotome family]]=== | ||
This family tempers out the apotome, 2187/2048, which is a 3-limit comma. | This family tempers out the apotome, 2187/2048, which is a 3-limit comma. | ||
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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. | 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]]=== | ||
This tempers out the minortone comma, |-16 35 -17>. The head of the family is minortonic temperament, with generator a minor tone. | This tempers out the minortone comma, |-16 35 -17>. The head of the family is minortonic temperament, with generator a minor tone. | ||
===[[Sycamore family]]=== | ===[[Sycamore family]]=== | ||
The sycamore family tempers out the sycamore comma, |-16 -6 11> = 48838125/47775744, which is the amount by which five stacked chromatic semitones, 25/24, exceed 6/5, and hence also the amount six exceeds 5/4. | The sycamore family tempers out the sycamore comma, |-16 -6 11> = 48838125/47775744, which is the amount by which five stacked chromatic semitones, 25/24, exceed 6/5, and hence also the amount six exceeds 5/4. | ||
===[[Mutt family]]=== | ===[[Mutt family]]=== | ||
This tempers out the mutt comma, |-44 -3 21>, leading to some strange properties. | This tempers out the mutt comma, |-44 -3 21>, leading to some strange properties. | ||
===[[Escapade family]]=== | ===[[Escapade family]]=== | ||
This tempers out escapade, |32 -7 -9>. | This tempers out escapade, |32 -7 -9>. | ||
===[[Vulture family]]=== | ===[[Vulture family]]=== | ||
This tempers out the vulture comma, |24 -21 4>. | This tempers out the vulture comma, |24 -21 4>. | ||
===[[Vishnuzmic family]]=== | ===[[Vishnuzmic family]]=== | ||
This tempers out the vishnuzma, |23 6 -14>. | This tempers out the vishnuzma, |23 6 -14>. | ||
===[[Pelogic family]]=== | ===[[Pelogic family]]=== | ||
This tempers out the pelogic comma, 135/128, also known as the major chroma or major limma. | This tempers out the pelogic comma, 135/128, also known as the major chroma or major limma. | ||
===[[Bug family]]=== | ===[[Bug family]]=== | ||
This tempers out 27/25, the large limma or bug comma. | This tempers out 27/25, the large limma or bug comma. | ||
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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. | 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. | ||
===[[Archytas clan]]=== | ===[[Archytas clan]]=== | ||
This clan tempers out the Archytas comma, 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. | This clan tempers out the Archytas comma, 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. | ||
===[[Trienstonic clan]]=== | ===[[Trienstonic clan]]=== | ||
This clan tempers out the septimal third-tone, 28/27, a triprime comma with factors of 2, 3 and 7. | This clan tempers out the septimal third-tone, 28/27, a triprime comma with factors of 2, 3 and 7. | ||
===[[Sensamagic clan]]=== | ===[[Sensamagic clan]]=== | ||
This clan tempers out 245/243, the sensamagic comma. | This clan tempers out 245/243, the sensamagic comma. | ||
===[[Jubilismic clan]]=== | ===[[Jubilismic clan]]=== | ||
This tempers out the jubilisma, 50/49. | This tempers out the jubilisma, 50/49. | ||
===[[Hemimean clan]]=== | ===[[Hemimean clan]]=== | ||
This tempers out the hemimean comma, 3136/3125, a no-threes comma. | This tempers out the hemimean comma, 3136/3125, a no-threes comma. | ||
===[[Mirkwai clan]]=== | ===[[Mirkwai clan]]=== | ||
This tempers out the mirkwai comma, 16875/16807, a no-twos comma (ratio of odd numbers.) | This tempers out the mirkwai comma, 16875/16807, a no-twos comma (ratio of odd numbers.) | ||
===[[Quartonic temperaments]]=== | ===[[Quartonic temperaments]]=== | ||
These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35. | These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35. | ||
===[[Avicennmic temperaments]]=== | ===[[Avicennmic temperaments]]=== | ||
These temper out the avicennma, 525/512, also known as Avicenna's enharmonic diesis. | These temper out the avicennma, 525/512, also known as Avicenna's enharmonic diesis. | ||
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These temper out 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. | These temper out 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. | ||
===[[Hemifamity temperaments]]=== | ===[[Hemifamity temperaments]]=== | ||
The hemifamity temperaments temper out the hemifamity comma, 5120/5103. | The hemifamity temperaments temper out the hemifamity comma, 5120/5103. | ||
===[[Porwell temperaments]]=== | ===[[Porwell temperaments]]=== | ||
The porwell temperaments temper out the porwell comma, 6144/6125. | The porwell temperaments temper out the porwell comma, 6144/6125. | ||
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===[[Gamelismic family]]=== | ===[[Gamelismic family]]=== | ||
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. | 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. | ||
===[[Breed family]]=== | ===[[Breed family]]=== | ||
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, 2749et will certainly do the trick. Breed has generators of 2, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7. | 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, 2749et will certainly do the trick. Breed has generators of 2, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7. | ||
===[[Ragisma family]]=== | ===[[Ragisma family]]=== | ||
The 7-limit rank three microtemperament which tempers out the ragisma, 4375/4374, extends to various higher limit rank three temperaments such as thor. | The 7-limit rank three microtemperament which tempers out the ragisma, 4375/4374, extends to various higher limit rank three temperaments such as thor. | ||
===[[Hemifamity family]]=== | ===[[Hemifamity family]]=== | ||
The hemifamity family of rank three temperaments tempers out the hemifamity comma, 5120/5103. | The hemifamity family of rank three temperaments tempers out the hemifamity comma, 5120/5103. | ||
===[[Porwell family]]=== | ===[[Porwell family]]=== | ||
The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125. | The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125. | ||
===[[Horwell family]]=== | ===[[Horwell family]]=== | ||
The horwell family of rank three temperaments tempers out the horwell comma, 65625/65536. | The horwell family of rank three temperaments tempers out the horwell comma, 65625/65536. | ||
===[[Hemimage family]]=== | ===[[Hemimage family]]=== | ||
The hemimage family of rank three temperaments tempers out the hemimage comma, 10976/10935. | The hemimage family of rank three temperaments tempers out the hemimage comma, 10976/10935. | ||
===[[Sensamagic family]]=== | ===[[Sensamagic family]]=== | ||
These temper out 245/243. | These temper out 245/243. | ||
===[[Keemic family]]=== | ===[[Keemic family]]=== | ||
These temper out 875/864. | These temper out 875/864. | ||
===[[Sengic family]]=== | ===[[Sengic family]]=== | ||
These temper out the senga, 686/675. | These temper out the senga, 686/675. | ||
===[[Orwellismic family]]=== | ===[[Orwellismic family]]=== | ||
These temper out 1728/1715. | These temper out 1728/1715. | ||
===[[Nuwell family]]=== | ===[[Nuwell family]]=== | ||
These temper out the nuwell comma, 2430/2401. | These temper out the nuwell comma, 2430/2401. | ||
===[[Octagar family]]=== | ===[[Octagar family]]=== | ||
The octagar family of rank three temperaments tempers out the octagar comma, 4000/3969. | The octagar family of rank three temperaments tempers out the octagar comma, 4000/3969. | ||
===[[Mirkwai family]]=== | ===[[Mirkwai family]]=== | ||
The mirkwai family of rank three temperaments tempers out the mirkwai comma, 16875/16807. | The mirkwai family of rank three temperaments tempers out the mirkwai comma, 16875/16807. | ||
===[[Hemimean family]]=== | ===[[Hemimean family]]=== | ||
The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125. | The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125. | ||
===[[Kleismic rank three family]]=== | ===[[Kleismic rank three family]]=== | ||
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 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. | ||
===[[Diaschismic rank three family]]=== | ===[[Diaschismic rank three family]]=== | ||
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. | 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. | ||
===[[Didymus rank three family]]=== | ===[[Didymus rank three family]]=== | ||
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. | 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. | ||
===[[Porcupine rank three family]]=== | ===[[Porcupine rank three family]]=== | ||
These are the rank three temperaments tempering out the porcupine comma or aximal 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 are the rank three temperaments tempering out the porcupine comma or aximal 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. | ||
===[[Archytas family]]=== | ===[[Archytas family]]=== | ||
Archytas temperament tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord. | Archytas temperament tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord. | ||
===[[Jubilismic family]]=== | ===[[Jubilismic family]]=== | ||
Jubilismic temperament tempers out 50/49 and thereby identifies the two septimal tritones, 7/5 and 10/7. | Jubilismic temperament tempers out 50/49 and thereby identifies the two septimal tritones, 7/5 and 10/7. | ||
===[[Semaphore family]]=== | ===[[Semaphore family]]=== | ||
Semaphore 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". | Semaphore 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". | ||
===[[Quartonic family]]=== | ===[[Quartonic family]]=== | ||
The quartonic temperament tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7. | The quartonic temperament tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7. | ||
===[[Werckismic temperaments]]=== | ===[[Werckismic temperaments]]=== | ||
These temper out the werckisma, 441/440. | These temper out the werckisma, 441/440. | ||
===[[Swetismic temperaments]]=== | ===[[Swetismic temperaments]]=== | ||
These temper out the swetisma, 540/539. | These temper out the swetisma, 540/539. | ||
===[[Lehmerismic temperaments]]=== | ===[[Lehmerismic temperaments]]=== | ||
These temper out the lehmerisma, 3025/3024. | These temper out the lehmerisma, 3025/3024. | ||
===[[Kalismic temperaments]]=== | ===[[Kalismic temperaments]]=== | ||
These temper out the kalisma, 9801/9800. | These temper out the kalisma, 9801/9800. | ||
=[[Rank four temperaments|Rank 4 temperaments]]= | =[[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. | 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. | ||
=[[Subgroup temperaments]]= | =[[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 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. | 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]]== | ||
By //Orgonia// is meant the commatic realm of the [[11-limit]] comma 65536/65219 = |16 0 0 -2 -3>, the orgonisma. | By //Orgonia// is meant the commatic realm of the [[11-limit]] comma 65536/65219 = |16 0 0 -2 -3>, the orgonisma. | ||
==[[The Biosphere]]== | ==[[The Biosphere]]== | ||
The Biosphere is the name given to the commatic realm of the 13-limit comma 91/90. | The Biosphere is the name given to the commatic realm of the 13-limit comma 91/90. | ||
==[[The Archipelago]]== | ==[[The Archipelago]]== | ||
The Archipelago is a name which has been given to the commatic realm of the [[13-limit]] comma 676/675. | The Archipelago is a name which has been given to the commatic realm of the [[13-limit]] comma 676/675. | ||
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<br /> | |||
Regular temperaments are irrational tunings, wherein the infinite number of intervals in n-limit Just intonation (or any subgroup thereof) are mapped to a smaller (though possibly 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 mainly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI). Temperaments effectively reduce the &quot;dimensionality&quot; of JI, thus 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. 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 /> | 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. 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;h1&gt; --><h1 id="toc1"><a name="Equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 --><a class="wiki_link" href="/edo">Equal temperaments</a></h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 --><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) are similar concepts, although there are distinctions in the way these terms are used. An | <a class="wiki_link" href="/Equal%20Temperaments">Equal temperaments</a> (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. An n-limit ET is simply an n-limit temperament that uses a single generator (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:4:&lt;h1&gt; --><h1 id="toc2"><a name="Rank 2 (including &quot;linear&quot;) temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Rank 2 (including &quot;linear&quot;) temperaments<!-- ws:start:WikiTextAnchorRule:239:&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:239 --></h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Rank 2 (including &quot;linear&quot;) temperaments"></a><!-- ws:end:WikiTextHeadingRule:4 -->Rank 2 (including &quot;linear&quot;) temperaments<!-- ws:start:WikiTextAnchorRule:239:&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:239 --></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.<br /> | 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.<br /> | ||
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N-limit Rank 2 temperaments map all intervals of n-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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As we go up from 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.) 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 /> | As we go up from 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.) 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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<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Sensipent family"></a><!-- ws:end:WikiTextHeadingRule:26 --><a class="wiki_link" href="/Sensipent%20family">Sensipent family</a></h3> | <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Sensipent family"></a><!-- ws:end:WikiTextHeadingRule:26 --><a class="wiki_link" href="/Sensipent%20family">Sensipent family</a></h3> | ||
This tempers out the sensipent comma, 78732/78125.<br /> | This tempers out the sensipent comma, 78732/78125.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Orwell and the semicomma family"></a><!-- ws:end:WikiTextHeadingRule:28 --><a class="wiki_link" href="/Semicomma%20family">Orwell and the semicomma family</a></h3> | <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Orwell and the semicomma family"></a><!-- ws:end:WikiTextHeadingRule:28 --><a class="wiki_link" href="/Semicomma%20family">Orwell and the semicomma family</a></h3> | ||
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<!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Apotome family"></a><!-- ws:end:WikiTextHeadingRule:32 --><a class="wiki_link" href="/Apotome%20family">Apotome family</a></h3> | <!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Apotome family"></a><!-- ws:end:WikiTextHeadingRule:32 --><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 /> | This family tempers out the apotome, 2187/2048, which is a 3-limit comma.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:34:&lt;h3&gt; --><h3 id="toc17"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Gammic family"></a><!-- ws:end:WikiTextHeadingRule:34 --><a class="wiki_link" href="/Gammic%20family">Gammic family</a></h3> | <!-- ws:start:WikiTextHeadingRule:34:&lt;h3&gt; --><h3 id="toc17"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Gammic family"></a><!-- ws:end:WikiTextHeadingRule:34 --><a class="wiki_link" href="/Gammic%20family">Gammic family</a></h3> | ||
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<!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Minortonic family"></a><!-- ws:end:WikiTextHeadingRule:36 --><a class="wiki_link" href="/Minortonic%20family">Minortonic family</a></h3> | <!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Minortonic family"></a><!-- ws:end:WikiTextHeadingRule:36 --><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 generator a minor tone.<br /> | This tempers out the minortone comma, |-16 35 -17&gt;. The head of the family is minortonic temperament, with generator a minor tone.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:38:&lt;h3&gt; --><h3 id="toc19"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Sycamore family"></a><!-- ws:end:WikiTextHeadingRule:38 --><a class="wiki_link" href="/Sycamore%20family">Sycamore family</a></h3> | <!-- ws:start:WikiTextHeadingRule:38:&lt;h3&gt; --><h3 id="toc19"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Sycamore family"></a><!-- ws:end:WikiTextHeadingRule:38 --><a class="wiki_link" href="/Sycamore%20family">Sycamore family</a></h3> | ||
The sycamore family tempers out the sycamore comma, |-16 -6 11&gt; = 48838125/47775744, 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 /> | The sycamore family tempers out the sycamore comma, |-16 -6 11&gt; = 48838125/47775744, 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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<!-- ws:start:WikiTextHeadingRule:40:&lt;h3&gt; --><h3 id="toc20"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Mutt family"></a><!-- ws:end:WikiTextHeadingRule:40 --><a class="wiki_link" href="/Mutt%20family">Mutt family</a></h3> | <!-- ws:start:WikiTextHeadingRule:40:&lt;h3&gt; --><h3 id="toc20"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Mutt family"></a><!-- ws:end:WikiTextHeadingRule:40 --><a class="wiki_link" href="/Mutt%20family">Mutt family</a></h3> | ||
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<!-- ws:start:WikiTextHeadingRule:42:&lt;h3&gt; --><h3 id="toc21"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Escapade family"></a><!-- ws:end:WikiTextHeadingRule:42 --><a class="wiki_link" href="/Escapade%20family">Escapade family</a></h3> | <!-- ws:start:WikiTextHeadingRule:42:&lt;h3&gt; --><h3 id="toc21"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Escapade family"></a><!-- ws:end:WikiTextHeadingRule:42 --><a class="wiki_link" href="/Escapade%20family">Escapade family</a></h3> | ||
This tempers out escapade, |32 -7 -9&gt;.<br /> | This tempers out escapade, |32 -7 -9&gt;.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:44:&lt;h3&gt; --><h3 id="toc22"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Vulture family"></a><!-- ws:end:WikiTextHeadingRule:44 --><a class="wiki_link" href="/Vulture%20family">Vulture family</a></h3> | <!-- ws:start:WikiTextHeadingRule:44:&lt;h3&gt; --><h3 id="toc22"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Vulture family"></a><!-- ws:end:WikiTextHeadingRule:44 --><a class="wiki_link" href="/Vulture%20family">Vulture family</a></h3> | ||
This tempers out the vulture comma, |24 -21 4&gt;.<br /> | This tempers out the vulture comma, |24 -21 4&gt;.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:46:&lt;h3&gt; --><h3 id="toc23"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Vishnuzmic family"></a><!-- ws:end:WikiTextHeadingRule:46 --><a class="wiki_link" href="/Vishnuzmic%20family">Vishnuzmic family</a></h3> | <!-- ws:start:WikiTextHeadingRule:46:&lt;h3&gt; --><h3 id="toc23"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Vishnuzmic family"></a><!-- ws:end:WikiTextHeadingRule:46 --><a class="wiki_link" href="/Vishnuzmic%20family">Vishnuzmic family</a></h3> | ||
This tempers out the vishnuzma, |23 6 -14&gt;.<br /> | This tempers out the vishnuzma, |23 6 -14&gt;.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:48:&lt;h3&gt; --><h3 id="toc24"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Pelogic family"></a><!-- ws:end:WikiTextHeadingRule:48 --><a class="wiki_link" href="/Pelogic%20family">Pelogic family</a></h3> | <!-- ws:start:WikiTextHeadingRule:48:&lt;h3&gt; --><h3 id="toc24"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Pelogic family"></a><!-- ws:end:WikiTextHeadingRule:48 --><a class="wiki_link" href="/Pelogic%20family">Pelogic family</a></h3> | ||
This tempers out the pelogic comma, 135/128, also known as the major chroma or major limma.<br /> | This tempers out the pelogic comma, 135/128, also known as the major chroma or major limma.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:50:&lt;h3&gt; --><h3 id="toc25"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Bug family"></a><!-- ws:end:WikiTextHeadingRule:50 --><a class="wiki_link" href="/Bug%20family">Bug family</a></h3> | <!-- ws:start:WikiTextHeadingRule:50:&lt;h3&gt; --><h3 id="toc25"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Bug family"></a><!-- ws:end:WikiTextHeadingRule:50 --><a class="wiki_link" href="/Bug%20family">Bug family</a></h3> | ||
This tempers out 27/25, the large limma or bug comma.<br /> | This tempers out 27/25, the large limma or bug comma.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:52:&lt;h3&gt; --><h3 id="toc26"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Gamelismic clan"></a><!-- ws:end:WikiTextHeadingRule:52 --><a class="wiki_link" href="/Gamelismic%20clan">Gamelismic clan</a></h3> | <!-- ws:start:WikiTextHeadingRule:52:&lt;h3&gt; --><h3 id="toc26"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Gamelismic clan"></a><!-- ws:end:WikiTextHeadingRule:52 --><a class="wiki_link" href="/Gamelismic%20clan">Gamelismic clan</a></h3> | ||
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<!-- ws:start:WikiTextHeadingRule:54:&lt;h3&gt; --><h3 id="toc27"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Archytas clan"></a><!-- ws:end:WikiTextHeadingRule:54 --><a class="wiki_link" href="/Archytas%20clan">Archytas clan</a></h3> | <!-- ws:start:WikiTextHeadingRule:54:&lt;h3&gt; --><h3 id="toc27"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Archytas clan"></a><!-- ws:end:WikiTextHeadingRule:54 --><a class="wiki_link" href="/Archytas%20clan">Archytas clan</a></h3> | ||
This clan tempers out the Archytas comma, 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 <a class="wiki_link" href="/Archytas%20family">Archytas family</a> of rank three temperaments.<br /> | This clan tempers out the Archytas comma, 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 <a class="wiki_link" href="/Archytas%20family">Archytas family</a> of rank three temperaments.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:56:&lt;h3&gt; --><h3 id="toc28"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Trienstonic clan"></a><!-- ws:end:WikiTextHeadingRule:56 --><a class="wiki_link" href="/Trienstonic%20clan">Trienstonic clan</a></h3> | <!-- ws:start:WikiTextHeadingRule:56:&lt;h3&gt; --><h3 id="toc28"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Trienstonic clan"></a><!-- ws:end:WikiTextHeadingRule:56 --><a class="wiki_link" href="/Trienstonic%20clan">Trienstonic clan</a></h3> | ||
This clan tempers out the septimal third-tone, 28/27, a triprime comma with factors of 2, 3 and 7.<br /> | This clan tempers out the septimal third-tone, 28/27, a triprime comma with factors of 2, 3 and 7.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:58:&lt;h3&gt; --><h3 id="toc29"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Sensamagic clan"></a><!-- ws:end:WikiTextHeadingRule:58 --><a class="wiki_link" href="/Sensamagic%20clan">Sensamagic clan</a></h3> | <!-- ws:start:WikiTextHeadingRule:58:&lt;h3&gt; --><h3 id="toc29"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Sensamagic clan"></a><!-- ws:end:WikiTextHeadingRule:58 --><a class="wiki_link" href="/Sensamagic%20clan">Sensamagic clan</a></h3> | ||
This clan tempers out 245/243, the sensamagic comma.<br /> | This clan tempers out 245/243, the sensamagic comma.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:60:&lt;h3&gt; --><h3 id="toc30"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Jubilismic clan"></a><!-- ws:end:WikiTextHeadingRule:60 --><a class="wiki_link" href="/Jubilismic%20clan">Jubilismic clan</a></h3> | <!-- ws:start:WikiTextHeadingRule:60:&lt;h3&gt; --><h3 id="toc30"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Jubilismic clan"></a><!-- ws:end:WikiTextHeadingRule:60 --><a class="wiki_link" href="/Jubilismic%20clan">Jubilismic clan</a></h3> | ||
This tempers out the jubilisma, 50/49.<br /> | This tempers out the jubilisma, 50/49.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:62:&lt;h3&gt; --><h3 id="toc31"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Hemimean clan"></a><!-- ws:end:WikiTextHeadingRule:62 --><a class="wiki_link" href="/Hemimean%20clan">Hemimean clan</a></h3> | <!-- ws:start:WikiTextHeadingRule:62:&lt;h3&gt; --><h3 id="toc31"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Hemimean clan"></a><!-- ws:end:WikiTextHeadingRule:62 --><a class="wiki_link" href="/Hemimean%20clan">Hemimean clan</a></h3> | ||
This tempers out the hemimean comma, 3136/3125, a no-threes comma.<br /> | This tempers out the hemimean comma, 3136/3125, a no-threes comma.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:64:&lt;h3&gt; --><h3 id="toc32"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Mirkwai clan"></a><!-- ws:end:WikiTextHeadingRule:64 --><a class="wiki_link" href="/Mirkwai%20clan">Mirkwai clan</a></h3> | <!-- ws:start:WikiTextHeadingRule:64:&lt;h3&gt; --><h3 id="toc32"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Mirkwai clan"></a><!-- ws:end:WikiTextHeadingRule:64 --><a class="wiki_link" href="/Mirkwai%20clan">Mirkwai clan</a></h3> | ||
This tempers out the mirkwai comma, 16875/16807, a no-twos comma (ratio of odd numbers.)<br /> | This tempers out the mirkwai comma, 16875/16807, a no-twos comma (ratio of odd numbers.)<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:66:&lt;h3&gt; --><h3 id="toc33"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Quartonic temperaments"></a><!-- ws:end:WikiTextHeadingRule:66 --><a class="wiki_link" href="/Quartonic%20temperaments">Quartonic temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:66:&lt;h3&gt; --><h3 id="toc33"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Quartonic temperaments"></a><!-- ws:end:WikiTextHeadingRule:66 --><a class="wiki_link" href="/Quartonic%20temperaments">Quartonic temperaments</a></h3> | ||
These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35.<br /> | These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:68:&lt;h3&gt; --><h3 id="toc34"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Avicennmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:68 --><a class="wiki_link" href="/Avicennmic%20temperaments">Avicennmic temperaments</a></h3> | <!-- ws:start:WikiTextHeadingRule:68:&lt;h3&gt; --><h3 id="toc34"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Avicennmic temperaments"></a><!-- ws:end:WikiTextHeadingRule:68 --><a class="wiki_link" href="/Avicennmic%20temperaments">Avicennmic temperaments</a></h3> | ||
These temper out the avicennma, 525/512, also known as Avicenna's enharmonic diesis.<br /> | These temper out the avicennma, 525/512, also known as Avicenna's enharmonic diesis.<br /> | ||
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The hemifamity temperaments temper out the hemifamity comma, 5120/5103.<br /> | The hemifamity temperaments temper out the hemifamity comma, 5120/5103.<br /> | ||
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The porwell temperaments temper out the porwell comma, 6144/6125.<br /> | The porwell temperaments temper out the porwell comma, 6144/6125.<br /> | ||
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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 /> | 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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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 /> | 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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The hemifamity family of rank three temperaments tempers out the hemifamity comma, 5120/5103.<br /> | The hemifamity family of rank three temperaments tempers out the hemifamity comma, 5120/5103.<br /> | ||
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The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125.<br /> | The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125.<br /> | ||
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The horwell family of rank three temperaments tempers out the horwell comma, 65625/65536.<br /> | The horwell family of rank three temperaments tempers out the horwell comma, 65625/65536.<br /> | ||
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The hemimage family of rank three temperaments tempers out the hemimage comma, 10976/10935.<br /> | The hemimage family of rank three temperaments tempers out the hemimage comma, 10976/10935.<br /> | ||
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These temper out 245/243.<br /> | These temper out 245/243.<br /> | ||
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These temper out 875/864.<br /> | These temper out 875/864.<br /> | ||
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These temper out the senga, 686/675.<br /> | These temper out the senga, 686/675.<br /> | ||
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These temper out 1728/1715.<br /> | These temper out 1728/1715.<br /> | ||
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These temper out the nuwell comma, 2430/2401.<br /> | These temper out the nuwell comma, 2430/2401.<br /> | ||
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The octagar family of rank three temperaments tempers out the octagar comma, 4000/3969.<br /> | The octagar family of rank three temperaments tempers out the octagar comma, 4000/3969.<br /> | ||
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The mirkwai family of rank three temperaments tempers out the mirkwai comma, 16875/16807.<br /> | The mirkwai family of rank three temperaments tempers out the mirkwai comma, 16875/16807.<br /> | ||
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The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125.<br /> | The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125.<br /> | ||
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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 /> | 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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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 /> | 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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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 /> | 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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These are the rank three temperaments tempering out the porcupine comma or aximal 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 /> | These are the rank three temperaments tempering out the porcupine comma or aximal 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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Archytas temperament tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord.<br /> | Archytas temperament tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord.<br /> | ||
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Jubilismic temperament tempers out 50/49 and thereby identifies the two septimal tritones, 7/5 and 10/7.<br /> | Jubilismic temperament tempers out 50/49 and thereby identifies the two septimal tritones, 7/5 and 10/7.<br /> | ||
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Semaphore 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 /> | Semaphore 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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The quartonic temperament tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7.<br /> | The quartonic temperament tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7.<br /> | ||
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These temper out the werckisma, 441/440.<br /> | These temper out the werckisma, 441/440.<br /> | ||
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These temper out the swetisma, 540/539.<br /> | These temper out the swetisma, 540/539.<br /> | ||
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These temper out the lehmerisma, 3025/3024.<br /> | These temper out the lehmerisma, 3025/3024.<br /> | ||
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These temper out the kalisma, 9801/9800.<br /> | These temper out the kalisma, 9801/9800.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:144:&lt;h1&gt; --><h1 id="toc72"><a name="Rank 4 temperaments"></a><!-- ws:end:WikiTextHeadingRule:144 --><a class="wiki_link" href="/Rank%20four%20temperaments">Rank 4 temperaments</a></h1> | <!-- ws:start:WikiTextHeadingRule:144:&lt;h1&gt; --><h1 id="toc72"><a name="Rank 4 temperaments"></a><!-- ws:end:WikiTextHeadingRule:144 --><a class="wiki_link" href="/Rank%20four%20temperaments">Rank 4 temperaments</a></h1> | ||
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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 /> | 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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<!-- ws:start:WikiTextHeadingRule:146:&lt;h1&gt; --><h1 id="toc73"><a name="Subgroup temperaments"></a><!-- ws:end:WikiTextHeadingRule:146 --><a class="wiki_link" href="/Subgroup%20temperaments">Subgroup temperaments</a></h1> | <!-- ws:start:WikiTextHeadingRule:146:&lt;h1&gt; --><h1 id="toc73"><a name="Subgroup temperaments"></a><!-- ws:end:WikiTextHeadingRule:146 --><a class="wiki_link" href="/Subgroup%20temperaments">Subgroup temperaments</a></h1> | ||
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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 /> | 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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<!-- ws:start:WikiTextHeadingRule:148:&lt;h1&gt; --><h1 id="toc74"><a name="Commatic realms"></a><!-- ws:end:WikiTextHeadingRule:148 -->Commatic realms</h1> | <!-- ws:start:WikiTextHeadingRule:148:&lt;h1&gt; --><h1 id="toc74"><a name="Commatic realms"></a><!-- ws:end:WikiTextHeadingRule:148 -->Commatic realms</h1> | ||
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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 /> | 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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<!-- ws:start:WikiTextHeadingRule:150:&lt;h2&gt; --><h2 id="toc75"><a name="Commatic realms-Orgonia"></a><!-- ws:end:WikiTextHeadingRule:150 --><a class="wiki_link" href="/Orgonia">Orgonia</a></h2> | <!-- ws:start:WikiTextHeadingRule:150:&lt;h2&gt; --><h2 id="toc75"><a name="Commatic realms-Orgonia"></a><!-- ws:end:WikiTextHeadingRule:150 --><a class="wiki_link" href="/Orgonia">Orgonia</a></h2> | ||
By <em>Orgonia</em> is meant 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 /> | By <em>Orgonia</em> is meant 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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<!-- ws:start:WikiTextHeadingRule:152:&lt;h2&gt; --><h2 id="toc76"><a name="Commatic realms-The Biosphere"></a><!-- ws:end:WikiTextHeadingRule:152 --><a class="wiki_link" href="/The%20Biosphere">The Biosphere</a></h2> | <!-- ws:start:WikiTextHeadingRule:152:&lt;h2&gt; --><h2 id="toc76"><a name="Commatic realms-The Biosphere"></a><!-- ws:end:WikiTextHeadingRule:152 --><a class="wiki_link" href="/The%20Biosphere">The Biosphere</a></h2> | ||
The Biosphere is the name given to the commatic realm of the 13-limit comma 91/90.<br /> | The Biosphere is the name given to the commatic realm of the 13-limit comma 91/90.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:154:&lt;h2&gt; --><h2 id="toc77"><a name="Commatic realms-The Archipelago"></a><!-- ws:end:WikiTextHeadingRule:154 --><a class="wiki_link" href="/The%20Archipelago">The Archipelago</a></h2> | <!-- ws:start:WikiTextHeadingRule:154:&lt;h2&gt; --><h2 id="toc77"><a name="Commatic realms-The Archipelago"></a><!-- ws:end:WikiTextHeadingRule:154 --><a class="wiki_link" href="/The%20Archipelago">The Archipelago</a></h2> | ||
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 /> | 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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