Saturation, torsion, and contorsion: Difference between revisions

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== 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 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 ~~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''' 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's [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], the overall order is quivalent 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. Thus, this a contorted generator with contorsion order 3.

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 == 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 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 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''' 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's [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], the overall order is quivalent 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. Thus, this a contorted generator with contorsion order 3.