Regular temperament: Difference between revisions

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== FAQ ==

=== Why would I want to use a regular temperament? ===

Regular temperaments are of most use to musicians who want their music to sound as much as possible like stacking-based [[just intonation]], but without the difficulties normally associated with it, such as wolf ~~intervals~~, ~~commas~~, and [[comma pump]]s. Specifically, if your chord progression [[pump]]s a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral ~~3rds~~, without caring much what ratio they are tuned to. Thus one might use ~~Rastmic~~ even though no commas are ~~[[pump]]ed~~.

Regular temperaments are of most use to musicians who want their music to sound as much as possible like stacking-based [[just intonation]], but without the difficulties normally associated with it, such as [[wolf interval]]s, [[comma]]s, and [[comma pump]]s. Specifically, if your chord progression [[pump]]s a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral thirds, without caring much what ratio they are tuned to. Thus one might use rastmic even though no commas are pumped.

=== How does regular temperament theory helps me compose music? ===

The skill of music composition is acquired by studying the disciplines such as {{w|harmony}}, {{w|musical form|form}}, {{w|orchestration}}, in addition to extensive listening. One common misconception is that learning regular temperament theory can be a substitute for any of those. Regular temperament theory does indeed present you with numerous tuning systems, and provide the tools to help you compare and choose between them based on some common goals. It also tells you how harmonic resources are available in each tuning system, though the question of putting them together to a piece of work is really up to you to experiment with. In other words, one may think of the relationship between regular temperament theory and composition as this: regular temperament theory tells you how to ''choose'' a tuning, while composition regards how to ''use'' a chosen tuning.

=== What do I need to know to understand all the numbers on the pages for individual regular temperaments? ===

Although the concept of regular temperament is centuries old and predates much of modern mathematics, members of the Yahoo! Alternative Tuning List have developed a particular form of numerical shorthand for describing the properties of temperaments. The most important of these are [[val]]s ([[mapping]]s), [[monzo]]s and [[tempering out|tempering out comma]]s, which any student of the modern regular temperament paradigm should become familiar with. These concepts are rather straightforward and require little math to understand.

The [[rank]] of a temperament is its dimension. It equals the number of [[formal prime]]s ~~in the subgroup~~ minus the number of independent commas that are tempered out.

The [[rank]] of a temperament is its dimension. It equals the number of [[formal prime]]s being used minus the number of independent commas that are tempered out.

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"), [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please") and more recently [[CTE]] ("Constained Tenney-Euclidean"), which has become the new standard instead of POTE since POTE is meant to be an approximation. 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 in terms of POTE and CTE generators. In addition, for each temperament there is a [[~~Optimal~~ ET sequence|sequence of equal temperaments]] showing possible [[equal-step tuning]]s in the order of better absolute accuracy to JI. The most common browser tools used for finding optimal tunings (useful for investigating new temperaments) are [[Graham Breed]]'s [http://x31eq.com/temper/ Temperament Finder] and [[User:Sintel|sintel]]'s [https://sintel.pythonanywhere.com/ Temperament Calculator]; the former gives temperament names (usually consistent with the wiki) and implements a wide variety of features like finding related temperaments while the latter implements [[CTE]] and more complex types of subgroups (like allowing ratios as generators) and supports an alternative notation to [[warts]] that is more convenient for arbitrary subgroups.

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"), [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please") and more recently [[CTE]] ("Constained Tenney-Euclidean"), which has become the new standard instead of POTE since POTE is meant to be an approximation. 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 in terms of POTE and CTE generators. In addition, for each temperament there is a [[optimal ET sequence|sequence of equal temperaments]] showing possible [[equal-step tuning]]s in the order of better absolute accuracy to JI. The most common browser tools used for finding optimal tunings (useful for investigating new temperaments) are [[Graham Breed]]'s [http://x31eq.com/temper/ Temperament Finder] and [[User:Sintel|sintel]]'s [https://sintel.pythonanywhere.com/ Temperament Calculator]; the former gives temperament names (usually consistent with the wiki) and implements a wide variety of features like finding related temperaments while the latter implements CTE and more complex types of subgroups (like allowing ratios as generators) and supports an alternative notation to [[warts]] that is more convenient for arbitrary subgroups.

Each temperament has two names: a traditional name and a [[~~Color~~ notation|color name]]. The traditional names are diverse in ~~terms of~~ [[~~Temperament~~ names|sources]], whereas the color names are systematic and rigorous, and the comma(s) can be deduced from the color name. {{nowrap|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~~]].

Each temperament has two names: a traditional name and a [[color notation|color name]]. The traditional names are diverse in [[temperament names|sources]], whereas the color names are systematic and rigorous, and the comma(s) can be deduced from the color name. {{nowrap|Wa {{=}} 3-limit|yo {{=}} 5-over|gu {{=}} 5-under|zo {{=}} 7-over|and ru {{=}} 7-under}}. See also [[Color notation/Temperament names]].

Yet another recent development is the concept of a [[pergen]], appearing [[~~Tour~~ of ~~Regular Temperaments|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(s). Assuming the prime subgroup includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a ~~5th~~ or some fraction of some 3-limit interval. 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. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.

Yet another recent development is the concept of a [[pergen]], appearing in our [[tour of regular temperaments]] 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(s). Assuming the prime subgroup includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. 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. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.

Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyo]] has the 5th sharpened by one-fifth of [[250/243]] = {{monzo| 1 -5 3 }}. Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Trigu tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name.

Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyo]] has the 5th sharpened by one-fifth of [[250/243]] ({{monzo| 1 -5 3 }}). Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Trigu tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name.

== Further reading ==

@@ Line 36: / Line 36: @@
 == FAQ ==
 === Why would I want to use a regular temperament? ===
-Regular temperaments are of most use to musicians who want their music to sound as much as possible like stacking-based [[just intonation]], but without the difficulties normally associated with it, such as wolf intervals, commas, and [[comma pump]]s. Specifically, if your chord progression [[pump]]s a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral 3rds, without caring much what ratio they are tuned to. Thus one might use Rastmic even though no commas are [[pump]]ed.
+Regular temperaments are of most use to musicians who want their music to sound as much as possible like stacking-based [[just intonation]], but without the difficulties normally associated with it, such as [[wolf interval]]s, [[comma]]s, and [[comma pump]]s. Specifically, if your chord progression [[pump]]s a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral thirds, without caring much what ratio they are tuned to. Thus one might use rastmic even though no commas are pumped.
+=== How does regular temperament theory helps me compose music? ===
+The skill of music composition is acquired by studying the disciplines such as {{w|harmony}}, {{w|musical form|form}}, {{w|orchestration}}, in addition to extensive listening. One common misconception is that learning regular temperament theory can be a substitute for any of those. Regular temperament theory does indeed present you with numerous tuning systems, and provide the tools to help you compare and choose between them based on some common goals. It also tells you how harmonic resources are available in each tuning system, though the question of putting them together to a piece of work is really up to you to experiment with. In other words, one may think of the relationship between regular temperament theory and composition as this: regular temperament theory tells you how to ''choose'' a tuning, while composition regards how to ''use'' a chosen tuning.
 === What do I need to know to understand all the numbers on the pages for individual regular temperaments? ===
 Although the concept of regular temperament is centuries old and predates much of modern mathematics, members of the Yahoo! Alternative Tuning List have developed a particular form of numerical shorthand for describing the properties of temperaments. The most important of these are [[val]]s ([[mapping]]s), [[monzo]]s and [[tempering out|tempering out comma]]s, which any student of the modern regular temperament paradigm should become familiar with. These concepts are rather straightforward and require little math to understand.
-The [[rank]] of a temperament is its dimension. It equals the number of [[formal prime]]s in the subgroup minus the number of independent commas that are tempered out.
+The [[rank]] of a temperament is its dimension. It equals the number of [[formal prime]]s being used minus the number of independent commas that are tempered out.
-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"), [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please") and more recently [[CTE]] ("Constained Tenney-Euclidean"), which has become the new standard instead of POTE since POTE is meant to be an approximation. 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 in terms of POTE and CTE generators. In addition, for each temperament there is a [[Optimal ET sequence|sequence of equal temperaments]] showing possible [[equal-step tuning]]s in the order of better absolute accuracy to JI. The most common browser tools used for finding optimal tunings (useful for investigating new temperaments) are [[Graham Breed]]'s [http://x31eq.com/temper/ Temperament Finder] and [[User:Sintel|sintel]]'s [https://sintel.pythonanywhere.com/ Temperament Calculator]; the former gives temperament names (usually consistent with the wiki) and implements a wide variety of features like finding related temperaments while the latter implements [[CTE]] and more complex types of subgroups (like allowing ratios as generators) and supports an alternative notation to [[warts]] that is more convenient for arbitrary subgroups.
+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"), [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please") and more recently [[CTE]] ("Constained Tenney-Euclidean"), which has become the new standard instead of POTE since POTE is meant to be an approximation. 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 in terms of POTE and CTE generators. In addition, for each temperament there is a [[optimal ET sequence|sequence of equal temperaments]] showing possible [[equal-step tuning]]s in the order of better absolute accuracy to JI. The most common browser tools used for finding optimal tunings (useful for investigating new temperaments) are [[Graham Breed]]'s [http://x31eq.com/temper/ Temperament Finder] and [[User:Sintel|sintel]]'s [https://sintel.pythonanywhere.com/ Temperament Calculator]; the former gives temperament names (usually consistent with the wiki) and implements a wide variety of features like finding related temperaments while the latter implements CTE and more complex types of subgroups (like allowing ratios as generators) and supports an alternative notation to [[warts]] that is more convenient for arbitrary subgroups.
-Each temperament has two names: a traditional name and a [[Color notation|color name]]. The traditional names are diverse in terms of [[Temperament names|sources]], whereas the color names are systematic and rigorous, and the comma(s) can be deduced from the color name. {{nowrap|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]].
+Each temperament has two names: a traditional name and a [[color notation|color name]]. The traditional names are diverse in [[temperament names|sources]], whereas the color names are systematic and rigorous, and the comma(s) can be deduced from the color name. {{nowrap|Wa {{=}} 3-limit|yo {{=}} 5-over|gu {{=}} 5-under|zo {{=}} 7-over|and ru {{=}} 7-under}}. See also [[Color notation/Temperament names]].
-Yet another recent development is the concept of a [[pergen]], appearing [[Tour of Regular Temperaments|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(s). Assuming the prime subgroup includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a 5th or some fraction of some 3-limit interval. 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. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.
+Yet another recent development is the concept of a [[pergen]], appearing in our [[tour of regular temperaments]] 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(s). Assuming the prime subgroup includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. 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. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.
-Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyo]] has the 5th sharpened by one-fifth of [[250/243]] = {{monzo| 1 -5 3 }}. Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Trigu tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name.
+Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyo]] has the 5th sharpened by one-fifth of [[250/243]] ({{monzo| 1 -5 3 }}). Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Trigu tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name.
 == Further reading ==