Tour of regular temperaments: Difference between revisions
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[[de:Reguläre_Temperaturen]] [[ja:レギュラーテンペラメントとランクrテンペラメント]] | [[de:Reguläre_Temperaturen]] | ||
[[ja:レギュラーテンペラメントとランクrテンペラメント]] | |||
=Regular temperaments= | =Regular temperaments= | ||
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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]. | ||
== 2.3 | == Families defined by a 2.3 (wa) comma == | ||
These are families defined by a comma that uses | These are families defined by a comma that uses a wa or 3-limit comma. If prime 5 is assumed to be part of the subgroup, and no other comma is tempered out, the comma creates a rank-2 temperament. The comma can also stand as parent to a 7-limit or higher family when other commas are tempered out as well, or when prime 7 is assumed to be part of the subgroup. Any temperament in these families can be thought of as consisting of mutiple "copies" of an EDO separated by a small comma. This small comma is represented in the pergen by ^1. | ||
===[[Blackwood|Blackwood or Sawa family]] (P8/5, ^1)=== | ===[[Blackwood|Blackwood or Sawa family]] (P8/5, ^1)=== | ||
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The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = [-19 12 0>, which implies [[12-edo]]. This family includes the compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave The 5-limit compton temperament can be thought of as multiple rings of 12-edo, offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12-edo instruments slightly detuned from each other provide an easy way to make music with these temperaments. | The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = [-19 12 0>, which implies [[12-edo]]. This family includes the compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave The 5-limit compton temperament can be thought of as multiple rings of 12-edo, offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12-edo instruments slightly detuned from each other provide an easy way to make music with these temperaments. | ||
== 2.3.5 | == Families defined by a 2.3.5 (ya) comma == | ||
These are families defined by a comma | These are families defined by a ya or 5-limit comma. As we go up in rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) The same comment applies to 7-limit temperaments and rank three, etc. Members of families and their relationships can be classified by the [[Normal_lists|normal comma list]] of the various temperaments. Families include weak extensions as well as strong, in other words, the pergen shown here may change. | ||
===[[Meantone family|Meantone or Gu family]] (P8, P5) === | ===[[Meantone family|Meantone or Gu family]] (P8, P5) === | ||
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The gammic family tempers out the gammic comma, [-29 -11 20>. Nine generators of about 35¢ equals ~6/5, eleven equal ~5/4 and twenty equal ~3/2. 34-edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is Neptune temperament. | The gammic family tempers out the gammic comma, [-29 -11 20>. Nine generators of about 35¢ equals ~6/5, eleven equal ~5/4 and twenty equal ~3/2. 34-edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is Neptune temperament. | ||
==2.3.7 | ==Clans defined by a 2.3.7 (za) comma== | ||
These are defined by a | These are defined by a za or 7-limit-no-fives comma. See also [[subgroup temperaments]]. | ||
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. | ||
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This clan tempers out the Sepru comma [7 8 0 -7> = 33.8¢. Its generator is ~7/6. Seven generators equals ~3/1. 7/4 is equated to 8 generators minus 1 octave. This clan includes as a strong extension the [[Orwell]] temperament, which is in the Semicomma family. | This clan tempers out the Sepru comma [7 8 0 -7> = 33.8¢. Its generator is ~7/6. Seven generators equals ~3/1. 7/4 is equated to 8 generators minus 1 octave. This clan includes as a strong extension the [[Orwell]] temperament, which is in the Semicomma family. | ||
== 2.3.11 | == Clans defined by a 2.3.11 (ila) comma == | ||
See also [[subgroup temperaments]]. | |||
=== [[Rastmic temperaments|Rastmic or Neutral or Lulu clan]] (P8, P5/2) === | === [[Rastmic temperaments|Rastmic or Neutral or Lulu clan]] (P8, P5/2) === | ||
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=== Laquadlo clan (P8/2, M2/4) === | === Laquadlo clan (P8/2, M2/4) === | ||
This 2.3.11 clan tempers out the Laquadlo comma [-17 2 0 0 4>. Its half-ocave period is ~363/256, and its generator is ~33/32. Four generators equals ~9/8. 3/2 is equated to a period plus 2 generators, and 11/8 is equated to a period minus a generator. This clan includes as a strong extension the Comic aka Saquadyobi temperament, which is in the Comic family. | This 2.3.11 clan tempers out the Laquadlo comma [-17 2 0 0 4>. Its half-ocave period is ~363/256, and its generator is ~33/32. Four generators equals ~9/8. 3/2 is equated to a period plus 2 generators, and 11/8 is equated to a period minus a generator. This clan includes as a strong extension the Comic aka Saquadyobi temperament, which is in the Comic family. | ||
== Clans defined by a 2.3.13 (tha) comma == | |||
See also [[subgroup temperaments]]. | |||
=== [[Hemif|Hemif or Thuthu clan]] (P8, P5/2) === | === [[Hemif|Hemif or Thuthu clan]] (P8, P5/2) === | ||
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This 2.3.13 clan tempers out the Satritho comma 512/507 = [0 -7 0 0 0 3>. Its generator is ~18/13. Three generators equals ~8/3. 13/8 is equated to 7 generators minus three octaves. This clan is related to the Latriru clan. | This 2.3.13 clan tempers out the Satritho comma 512/507 = [0 -7 0 0 0 3>. Its generator is ~18/13. Three generators equals ~8/3. 13/8 is equated to 7 generators minus three octaves. This clan is related to the Latriru clan. | ||
== 2.5.7 | == Clans defined by a 2.5.7 (yaza nowa) comma == | ||
These are defined by a | These are defined by a yaza nowa or 7-limit-no-threes comma. See also [[subgroup temperaments]]. | ||
=== [[Jubilismic clan|Jubilismic or Biruyo Nowa clan]] (P8/2, M3) === | === [[Jubilismic clan|Jubilismic or Biruyo Nowa clan]] (P8/2, M3) === | ||
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This clan tempers out the quince, [-15 0 -2 7> = 823543/819200. The generator is ~343/320 = ~116¢. Two generators equals ~8/7 (a M2), and seven generators equals ~8/5. An obvious 5-limit interpretation of the generator is 16/15, leading to the [[Magic]] temperament, which is in the Magic family. | This clan tempers out the quince, [-15 0 -2 7> = 823543/819200. The generator is ~343/320 = ~116¢. Two generators equals ~8/7 (a M2), and seven generators equals ~8/5. An obvious 5-limit interpretation of the generator is 16/15, leading to the [[Magic]] temperament, which is in the Magic family. | ||
== 3.5.7 | == Clans defined by a 3.5.7 (yaza noca) comma == | ||
These are defined by a comma | These are defined by a yaza noca or 7-limit-no-twos comma. Any no-twos comma is a ratio of odd numbers. If no other commas are tempered out, It is a non-octave tuning, with a period of a perfect 12th or tritave, 3/1 (or some fraction of that). See also [[subgroup temperaments]]. | ||
===[[Sensamagic clan|Sensamagic or Zozoyo Noca clan]] (P12, M3)=== | ===[[Sensamagic clan|Sensamagic or Zozoyo Noca clan]] (P12, M3)=== | ||
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Even less familiar than rank-2 temperaments are the [[Planar_Temperament|rank-3 temperaments]], generated by a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank-3 temperaments by the commas they temper out. They can also be identified by pergen. The 3rd generator in a rank-3 pergen is usually a comma, but sometimes it's some fraction of a 5-limit or 7-limit interval. | Even less familiar than rank-2 temperaments are the [[Planar_Temperament|rank-3 temperaments]], generated by a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank-3 temperaments by the commas they temper out. They can also be identified by pergen. The 3rd generator in a rank-3 pergen is usually a comma, but sometimes it's some fraction of a 5-limit or 7-limit interval. | ||
== 2.3.5 | == Families defined by a 2.3.5 (ya) comma == | ||
Every ya or 5-limit comma defines a rank-3 family, thus every comma in the list of rank-two 2.3.5 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a yaza or 7-limit temperament in which one of the generators is ~7/1. This generator can be reduced to ~7/4, which can be reduced further to ~64/63. Hence in all the pergens below, the ^1 or /1 generator is ~64/63. An additional 7-limit or 11-limit comma creates a yazala or 11-limit temperament, and so forth. All these examples are yaza: | |||
===[[Didymus rank three family|Didymus or Gu rank three family]] (P8, P5, ^1)=== | ===[[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 | These are the rank three temperaments tempering out the didymus or meantone comma, 81/80. | ||
===[[Diaschismic rank three family|Diaschismic or Sagugu rank three family]] (P8/2, P5, /1)=== | ===[[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. | These are the rank three temperaments tempering out the dischisma, [11 -4 -2> = 2048/2025. The half-octave period is ~45/32. | ||
===[[Porcupine rank three family|Porcupine or Triyo rank three family]] (P8, P4/3, /1)=== | ===[[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. In the pergen, | These are the rank three temperaments tempering out the porcupine comma or maximal diesis, [1 -5 3> = 250/243. In the pergen, P4/3 is ~10/9. | ||
===[[Kleismic rank three family|Kleismic or Tribiyo rank three family]] (P8, P12/6, /1)=== | ===[[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. In the pergen, / | These are the rank three temperaments tempering out the kleisma, [-6 -5 6> = 15625/15552. In the pergen, P12/6 is ~6/5. | ||
== 2.3.7 | == Families defined by a 2.3.7 (za) comma == | ||
Every za or 7-limit-no-fives comma defines a rank-3 family, thus every comma in the list of rank-two 2.3.7 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a yaza or 7-limit temperament in which one of the generators is ~5/1. This generator can be reduced to ~5/4, which may be reduced further to ~81/80. Hence in all the pergens below, ^1 = ~81/80. An additional 5-limit or 11-limit comma creates a yazala or 11-limit temperament, and so forth. All these examples are yaza: | |||
===[[Archytas family|Archytas or Ru family]] (P8, P5, ^1)=== | ===[[Archytas family|Archytas or Ru family]] (P8, P5, ^1)=== | ||
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===[[Gamelismic family|Gamelismic or Latrizo family]] (P8, P5/3, ^1)=== | ===[[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. | 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. In the pergen, P5/3 is ~8/7. | ||
===[[Stearnsmic temperaments|Stearnsmic or Latribiru family]] (P8/2, P4/3, ^1)=== | ===[[Stearnsmic temperaments|Stearnsmic or Latribiru family]] (P8/2, P4/3, ^1)=== | ||
Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6> = 118098/117649. | Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6> = 118098/117649. In the pergen, P8/2 is 343/243 and P4/3 is ~54/49. | ||
== Families defined by a 2.3.5.7 (yaza) comma == | |||
===[[Marvel family|Marvel or Ruyoyo family]] (P8, P5, ^1)=== | ===[[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 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. | ||
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===[[Rastmic temperaments|Rastmic or Lulu temperaments]]=== | ===[[Rastmic temperaments|Rastmic or Lulu temperaments]]=== | ||
These temper out the rastma, [-1 5 0 0 -2> = 243/242. As | These temper out the rastma, [-1 5 0 0 -2> = 243/242. As an ila (11-limit no-fives no-sevens) rank-2 temperament, it's (P8, P5/2). | ||
===[[Werckismic temperaments|Werckismic or Luzozogu temperaments]]=== | ===[[Werckismic temperaments|Werckismic or Luzozogu temperaments]]=== | ||
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= Miscellaneous other temperaments = | = Miscellaneous other temperaments = | ||
===[[31 comma temperaments]]=== | ===[[26th-octave temperaments]]=== | ||
These all have period 1/31 of an octave. | These temperaments all have a period of 1/26 of an octave. | ||
===[[31 comma temperaments|31-comma temperaments]]=== | |||
These all have a period of 1/31 of an octave. | |||
===[[Turkish maqam music temperaments]]=== | ===[[Turkish maqam music temperaments]]=== | ||
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===[[High badness temperaments]]=== | ===[[High badness temperaments]]=== | ||
High in badness, but worth cataloging for one reason or another. | High in badness, but worth cataloging for one reason or another. | ||
=Links= | =Links= | ||