Saturation, torsion, and contorsion: Difference between revisions

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| en = Saturation, torsion, and contorsion

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| ja = 飽和、ねじれ、contorsion

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: ''This is a general introduction to this concept; for a more mathematical take on this, see [[Mathematical theory of saturation]].''

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[[Category:Terms]]

[[Category:Math]]

In [[regular temperament theory]], a [[temperament]] is saturated if its set of available intervals matches what is suggested by its mapping or comma basis. A temperament's mapping can fail with respect to saturation by being contorted, and its comma basis can fail through torsion.

In [[regular temperament theory]], a [[temperament]] is '''saturated''' or ''defactored'' if its set of available intervals matches what is suggested by its mapping or comma basis. A temperament's mapping can fail with respect to saturation by being contorted, and its comma basis can fail through torsion.

This article briefly explains these issues; for lattice-based visualizations and intuitive explanations, see [[Pathology of enfactoring|Pathology of saturation]].

== Contorsion ==

A temperament (more specifically, its [[mapping]]) displays '''contorsion''' if there is some generatable interval which no [[just intonation]] interval maps to. This generatable interval is a '''contorted generator''', which has the property that every just interval's mapping has a multiple of ''c'' of that generator, where ''c'', the '''contorsion order''', is greater than one. In a contorted temperament, all generator bases will contain at least one contorted generator. The ~~largest~~ contorsion order is ~~called~~ the ~~'''greatest factor'''~~<ref>~~This term is inspired by~~ H. J. S. Smith's [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], ~~where Smith describes~~ the GCD of a matrix's minor determinants ~~as its~~ "greatest divisor~~". "Divisor" and "factor" are synonyms and they prefer "factor" for its connection with the term "defactor~~".</ref>.

A temperament (more specifically, its [[mapping]]) displays '''contorsion''' or '''enfactoring''' if there is some generatable interval which no [[just intonation]] interval maps to. This generatable interval is a '''contorted generator''', which has the property that under any appropriate generator basis, every just interval's mapping has a multiple of ''c'' of that generator, where ''c'', the '''contorsion order''', is greater than one. In a contorted temperament, all generator bases will contain at least one contorted generator. The overall contorsion order of the temperament is the product of all the seperate orders.<ref>H. J. S. Smith [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], the overall order is equivalent to the GCD of a matrix's minor determinants, which Smith calls the "greatest divisor".</ref>.

For example, [[5-limit]] [[36edo|36et]] (with mapping {{mapping|36 57 84}}) uses 12 of its pitches per octave (the ones within [[12edo]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of [[Generator|generators]] that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contorsion order 3. For a higher-rank example, the 13-limit {{nowrap|87 & 111}} temperament Hemimist, with mapping [{{mapping|3 0 26 56 8}}, {{mapping|0 2 -8 -20 1}}], when restricted to the 2.5.7.11 subgroup, has no just intonation interval corresponding to the period or the square of the period, although there is a just intonation interval (namely, 2/1) corresponding to the cube of the period. ~~Therefore~~, this a contorted generator with contorsion order 3.

For example, [[5-limit]] [[36edo|36et]] (with mapping {{mapping|36 57 84}}) uses 12 of its pitches per octave (the ones within [[12edo]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of [[Generator|generators]] that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contorsion order 3. For a higher-rank example, the 13-limit {{nowrap|87 & 111}} temperament Hemimist, with mapping [{{mapping|3 0 26 56 8}}, {{mapping|0 2 -8 -20 1}}], when restricted to the 2.5.7.11 subgroup, has no just intonation interval corresponding to the period or the square of the period, although there is a just intonation interval (namely, 2/1) corresponding to the cube of the period. Thus, this a contorted generator with contorsion order 3.

If a temperament has a subgroup which is contorted, especially a subgroup with small primes (for example, the 11-limit subgroup of 23-limit [[44edo|44et]]), that temperament will likely be easier to traverse than the number of generators required according to the mapping would suggest.

== Torsion in temperaments ==

A temperament (more specifically, its [[comma basis]]) displays '''torsion''' if there is some interval mapped to zero which is not formable by multiplying commas in the basis. This interval is a '''comma with torsion''', which has the property that commas in the basis can be multiplied to form the ''c''th power of this ratio, but not that ratio itself or any smaller power, where ''c'' is the '''torsion order'''.

A temperament (more specifically, its [[comma basis]]) displays '''torsion''' or ''enfactoring'' if there is some interval mapped to zero which is not formable by multiplying commas in the basis. This interval is a '''comma with torsion''', which has the property that commas in the basis can be multiplied to form the ''c''th power of this ratio, but not that ratio itself or any smaller power, where ''c'' is the '''torsion order'''.

For instance, in a temperament with comma basis {[[6561/6250]], [[128/125]]}, there is an interval 81/80 which is not formable by multiplying commas in the basis, but is nevertheless forced to be mapped to zero because {{nowrap|(81/80)^2 {{=}} (6561/6250)/(128/125)}} is part of the basis. Thus, 81/80 displays torsion with torsion order 2.

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+{{interwiki
+| de =
+| en = Saturation, torsion, and contorsion
+| es =
+| ja = 飽和、ねじれ、contorsion
+}}
 : ''This is a general introduction to this concept; for a more mathematical take on this, see [[Mathematical theory of saturation]].''
@@ Line 4: / Line 10: @@
 [[Category:Terms]]
 [[Category:Math]]
-In [[regular temperament theory]], a [[temperament]] is saturated if its set of available intervals matches what is suggested by its mapping or comma basis. A temperament's mapping can fail with respect to saturation by being contorted, and its comma basis can fail through torsion.
+In [[regular temperament theory]], a [[temperament]] is '''saturated''' or ''defactored'' if its set of available intervals matches what is suggested by its mapping or comma basis. A temperament's mapping can fail with respect to saturation by being contorted, and its comma basis can fail through torsion.
 This article briefly explains these issues; for lattice-based visualizations and intuitive explanations, see [[Pathology of enfactoring|Pathology of saturation]].
 == Contorsion ==
-A temperament (more specifically, its [[mapping]]) displays '''contorsion''' if there is some generatable interval which no [[just intonation]] interval maps to. This generatable interval is a '''contorted generator''', which has the property that every just interval's mapping has a multiple of ''c'' of that generator, where ''c'', the '''contorsion order''', is greater than one. In a contorted temperament, all generator bases will contain at least one contorted generator. The largest contorsion order is called the '''greatest factor'''<ref>This term is inspired by H. J. S. Smith's [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], where Smith describes the GCD of a matrix's minor determinants as its "greatest divisor". "Divisor" and "factor" are synonyms and they prefer "factor" for its connection with the term "defactor".</ref>.
+A temperament (more specifically, its [[mapping]]) displays '''contorsion''' or '''enfactoring''' if there is some generatable interval which no [[just intonation]] interval maps to. This generatable interval is a '''contorted generator''', which has the property that under any appropriate generator basis, every just interval's mapping has a multiple of ''c'' of that generator, where ''c'', the '''contorsion order''', is greater than one. In a contorted temperament, all generator bases will contain at least one contorted generator. The overall contorsion order of the temperament is the product of all the seperate orders.<ref>H. J. S. Smith [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], the overall order is equivalent to the GCD of a matrix's minor determinants, which Smith calls the "greatest divisor".</ref>.
-For example, [[5-limit]] [[36edo|36et]] (with mapping {{mapping|36 57 84}}) uses 12 of its pitches per octave (the ones within [[12edo]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of [[Generator|generators]] that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contorsion order 3. For a higher-rank example, the 13-limit {{nowrap|87 &amp; 111}} temperament Hemimist, with mapping [{{mapping|3 0 26 56 8}}, {{mapping|0 2 -8 -20 1}}], when restricted to the 2.5.7.11 subgroup, has no just intonation interval corresponding to the period or the square of the period, although there is a just intonation interval (namely, 2/1) corresponding to the cube of the period. Therefore, this a contorted generator with contorsion order 3.
+For example, [[5-limit]] [[36edo|36et]] (with mapping {{mapping|36 57 84}}) uses 12 of its pitches per octave (the ones within [[12edo]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of [[Generator|generators]] that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contorsion order 3. For a higher-rank example, the 13-limit {{nowrap|87 &amp; 111}} temperament Hemimist, with mapping [{{mapping|3 0 26 56 8}}, {{mapping|0 2 -8 -20 1}}], when restricted to the 2.5.7.11 subgroup, has no just intonation interval corresponding to the period or the square of the period, although there is a just intonation interval (namely, 2/1) corresponding to the cube of the period. Thus, this a contorted generator with contorsion order 3.
 If a temperament has a subgroup which is contorted, especially a subgroup with small primes (for example, the 11-limit subgroup of 23-limit [[44edo|44et]]), that temperament will likely be easier to traverse than the number of generators required according to the mapping would suggest.
 == Torsion in temperaments ==
-A temperament (more specifically, its [[comma basis]]) displays '''torsion''' if there is some interval mapped to zero which is not formable by multiplying commas in the basis. This interval is a '''comma with torsion''', which has the property that commas in the basis can be multiplied to form the ''c''th power of this ratio, but not that ratio itself or any smaller power, where ''c'' is the '''torsion order'''.
+A temperament (more specifically, its [[comma basis]]) displays '''torsion''' or ''enfactoring'' if there is some interval mapped to zero which is not formable by multiplying commas in the basis. This interval is a '''comma with torsion''', which has the property that commas in the basis can be multiplied to form the ''c''th power of this ratio, but not that ratio itself or any smaller power, where ''c'' is the '''torsion order'''.
 For instance, in a temperament with comma basis {[[6561/6250]], [[128/125]]}, there is an interval 81/80 which is not formable by multiplying commas in the basis, but is nevertheless forced to be mapped to zero because {{nowrap|(81/80)^2 {{=}} (6561/6250)/(128/125)}} is part of the basis. Thus, 81/80 displays torsion with torsion order 2.