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]]~~

~~= Regular temperaments =~~

'''Regular temperaments''' are non-Just tunings in which the infinite number of intervals in [[Harmonic Limit|''p''-limit]] [[just intonation]], or any [[Just intonation subgroups|subgroup]] thereof, are mapped to a smaller, though still infinite, set of [[tempering out|tempered]] intervals. This is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the "dimensionality" of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, 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. The dimensionality is the rank of the temperament.

A rank ''r'' regular temperament in a particular tuning may be defined by giving ''r'' multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank ''r'' temperament will be defined by ''r'' generators, and thus ''r'' [[vals]]. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of ''r'' independent 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.

~~== 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 low-overtone just intonation, but without the difficulties normally associated with [[low-overtone JI]], such as wolf intervals, commas, and comma pumps. Specifically, if your chord progression pumps 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 pumped.

~~== 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 (aka mappings) 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 equals the number of primes in the subgroup minus the number of linearly independent (i.e. non-redundant) 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") and [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please"). Optimization is rather intensive mathematically, but it is seldom left as an exercise to the reader; most temperaments are presented here in their optimal forms in terms of POTE generators. In addition, for each temperament there is a list of EDOs showing possible EDO tunings in the order of better accuracy.

Yet another recent development is the concept of a [[pergen]], appearing here as (P8, P5/2) or somesuch. Every rank-2, rank-3, rank-4, etc. temperament has a pergen, which specifies the period and the generator(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.

Each temperament has two names: a traditional name and a [[Color notation|color name]]. The traditional names are [[Temperament Names|arbitrary]], but the color names are systematic and rigorous, and the comma can be deduced from the color name. Wa = 3-limit, yo = 5-over, gu = 5-under, zo = 7-over, and ru = 7-under. See also [[Color notation/Temperament Names|Color Notation/Temperament Names]].

~~=Equal temperaments (Rank-1 temperaments)=~~

[[Equal-step tuning|Equal temperaments]] (abbreviated ET or tET) and [[EDO|equal divisions of the octave]] (abbreviated EDO or ED2) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator, making it a rank-1 temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval, let alone the octave, and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO, although one can also use unsupported vals or poorly-supported vals to achieve "fun" results. The familiar 12-note equal temperament, or 12edo, reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-ET.

~~=Rank-2 (including linear) temperaments=~~

A p-limit rank-2 temperament maps all intervals of p-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a '''linear temperament'''. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-ET by tempering out the Pythagorean comma.

Regular temperaments of ranks two and three are cataloged on the [[Optimal patent val]] page. Rank-2 temperaments are also listed at [[Proposed names for rank 2 temperaments]] by their generator mappings, and at [[Map of rank-2 temperaments]] by their generator size. See also the [[pergen]]s page. There is also [[Graham Breed]]'s [http://x31eq.com/catalog2.html giant list of regular temperaments].

== Families defined by a 2.3 (wa) comma ==

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= Links =

* [[Wikipedia: Regular temperament]]

* [http://www.huygens-fokker.org/docs/lintemps.html List of temperaments] in [http://www.huygens-fokker.org/scala Scala] with ready to use values

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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]]
-= Regular temperaments =
-'''Regular temperaments''' are non-Just tunings in which the infinite number of intervals in [[Harmonic Limit|''p''-limit]] [[just intonation]], or any [[Just intonation subgroups|subgroup]] thereof, are mapped to a smaller, though still infinite, set of [[tempering out|tempered]] intervals. This is done by deliberately mistuning some of the ratios such that a [[comma]] or set of commas vanishes by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the "dimensionality" of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, 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. The dimensionality is the rank of the temperament.
-A rank ''r'' regular temperament in a particular tuning may be defined by giving ''r'' multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank ''r'' temperament will be defined by ''r'' generators, and thus ''r'' [[vals]]. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of ''r'' independent 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.
-== 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 low-overtone just intonation, but without the difficulties normally associated with [[low-overtone JI]], such as wolf intervals, commas, and comma pumps. Specifically, if your chord progression pumps 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 pumped.
-== 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 (aka mappings) 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 equals the number of primes in the subgroup minus the number of linearly independent (i.e. non-redundant) 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") and [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please"). Optimization is rather intensive mathematically, but it is seldom left as an exercise to the reader; most temperaments are presented here in their optimal forms in terms of POTE generators. In addition, for each temperament there is a list of EDOs showing possible EDO tunings in the order of better accuracy.
-Yet another recent development is the concept of a [[pergen]], appearing here as (P8, P5/2) or somesuch. Every rank-2, rank-3, rank-4, etc. temperament has a pergen, which specifies the period and the generator(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.
-Each temperament has two names: a traditional name and a [[Color notation|color name]]. The traditional names are [[Temperament Names|arbitrary]], but the color names are systematic and rigorous, and the comma can be deduced from the color name. Wa = 3-limit, yo = 5-over, gu = 5-under, zo = 7-over, and ru = 7-under. See also [[Color notation/Temperament Names|Color Notation/Temperament Names]].
-=Equal temperaments (Rank-1 temperaments)=
-[[Equal-step tuning|Equal temperaments]] (abbreviated ET or tET) and [[EDO|equal divisions of the octave]] (abbreviated EDO or ED2) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator, making it a rank-1 temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval, let alone the octave, and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO, although one can also use unsupported vals or poorly-supported vals to achieve "fun" results. The familiar 12-note equal temperament, or 12edo, reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-ET.
-=Rank-2 (including linear) temperaments=
-A p-limit rank-2 temperament maps all intervals of p-limit JI using a set of 2-dimensional coordinates, thus a rank-2 temperament is said to have two generators, though it may have any number of step-sizes. This means that a rank-2 temperament is defined by a period-generator mapping, a set of 2 vals, one val for each generator. The larger generator is called the period, as the temperament will repeat at that interval, and is often a fraction of an octave; if it is exactly an octave, the temperament is said to be a '''linear temperament'''. Rank-2 temperaments can be reduced to a related rank-1 temperament by tempering out an additional comma. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-ET by tempering out the Pythagorean comma.
-Regular temperaments of ranks two and three are cataloged on the [[Optimal patent val]] page. Rank-2 temperaments are also listed at [[Proposed names for rank 2 temperaments]] by their generator mappings, and at [[Map of rank-2 temperaments]] by their generator size. See also the [[pergen]]s page. There is also [[Graham Breed]]'s [http://x31eq.com/catalog2.html giant list of regular temperaments].
 == Families defined by a 2.3 (wa) comma ==
@@ Line 537: / Line 506: @@
 = Links =
-* [[Wikipedia: Regular temperament]]
 * [http://www.huygens-fokker.org/docs/lintemps.html List of temperaments] in [http://www.huygens-fokker.org/scala Scala] with ready to use values