Saturation, torsion, and contorsion: Difference between revisions

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In [[regular temperament theory]], a [[~~mapping~~]] or [[~~comma basis~~]] is '''~~saturated~~''' if ~~it represents~~ a ~~temperament without~~ any ~~redundancies due to~~ a ~~common factor~~, ~~such that~~ the ~~temperament~~ is ~~represented with~~ the ~~simplest matrix~~. '''~~Torsion~~''~~' and~~ '''contorsion''' ~~are~~ the ~~opposite~~ of ~~saturation~~, ~~but torsion only regards comma bases, whereas~~ contorsion ~~only regards mappings~~. ~~In other words~~, a comma basis ~~is either saturated, or it~~ ''~~has~~ torsion''~~, and~~ a ~~mapping is either saturated, or it~~ ''~~has contorsion~~'' (~~or ''~~is ~~contorted'')~~. ~~The greatest factor~~ of a ~~mapping has been called its~~ '''~~contorted order~~'''.

In [[regular temperament theory]], a [[temperament]] (more specifically, its [[mapping]]) - displays '''contorsion''' if there are some pitches which no [[just intonation]] interval (within the temperament's [[subgroup]]) maps to. For example, the rank-1 [[5-limit]] temperament described by [[24edo|24et]] is fairly accurate but only uses 12 of its pitches per octave (the ones within [[12edo|12et]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 12 pitches, making 24et contorted in the 5-limit. For a higher-rank example, [[septimal meantone]] in the [[7-limit]] maps [[3/1|harmonic 3]] to 1 meantone [[3/2|fifth]], [[5/1|harmonic 5]] to 4 fifths, and [[7/1|harmonic 7]] to 10 fifths up. But if it is restricted to the subgroup [[2.5.7 subgroup|2.5.7]], all just intonation intervals within that subgroup occur at ''even'' numbers of fifths up or down, because both 4 and 10 are even numbers, and so pitches located at odd numbers of fifths up or down do not have a representation in the 2.5.7 subgroup. The temperament containing the half of notes that occur at even fifths is in fact [[didacus]], generated by the 2-fifth interval (in other words, a meantone whole tone, identified here as [[28/25]]), and so we can say that septimal meantone is contorted in the 2.5.7 subgroup, inheriting this subgroup's representation from didacus.

The above examples depicted a situation where the intervals of a given subgroup occur every 2 pitches in the underlying temperament; but they can occur every 3, or 4, or any other number of [[generator]]s apart. However many generators are needed to step from one interval in your chosen subgroup to the next is the '''contorsion order''' of that subgroup within the temperament. (In the case of higher-rank temperaments, it is possible for different generators to have different contorsion orders.)

A temperament (more specifically, its [[comma basis]]) displays '''torsion''' if it [[tempers out]] a ''power'' of some ratio, but does not temper out that ratio. For instance, in a temperament with comma basis {[[6561/6250]], [[128/125]]}, (81/80)^2 = (6561/6250)/(128/125) is tempered out but [[81/80]] is not explicitly tempered out. In this temperament, there is no clear way to assign a pitch to 81/80; for this reason, torted temperaments are not particularly useful. Similarly to the concept of contorsion order, torsion order can be defined as the lowest power of a generic just intonation interval that is necessarily part of the temperament's lattice.

A temperament is '''saturated''' if it is neither torted nor contorted.

In general, being unsaturated is a bad thing<ref>Technically speaking, saturation is a property of lattices, not the matrices that generate them, and is only "bad" when referring to a comma basis or a lattice of supporting maps.</ref>, as the redundancy means that the same temperament information can be represented in a simpler way. There are other manners in which unsaturation is bad, and these depend on whether the matrix is a comma basis or a mapping, discussed below. For all these reasons, unsaturated matrices are typically considered to not truly represent temperaments.

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 {{Beginner|Mathematical theory of saturation}}
-In [[regular temperament theory]], a [[mapping]] or [[comma basis]] is '''saturated''' if it represents a temperament without any redundancies due to a common factor, such that the temperament is represented with the simplest matrix. '''Torsion''' and '''contorsion''' are the opposite of saturation, but torsion only regards comma bases, whereas contorsion only regards mappings. In other words, a comma basis is either saturated, or it ''has torsion'', and a mapping is either saturated, or it ''has contorsion'' (or ''is contorted''). The greatest factor of a mapping has been called its '''contorted order'''.
+In [[regular temperament theory]], a [[temperament]] (more specifically, its [[mapping]]) - displays '''contorsion''' if there are some pitches which no [[just intonation]] interval (within the temperament's [[subgroup]]) maps to. For example, the rank-1 [[5-limit]] temperament described by [[24edo|24et]] is fairly accurate but only uses 12 of its pitches per octave (the ones within [[12edo|12et]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 12 pitches, making 24et contorted in the 5-limit. For a higher-rank example, [[septimal meantone]] in the [[7-limit]] maps [[3/1|harmonic 3]] to 1 meantone [[3/2|fifth]], [[5/1|harmonic 5]] to 4 fifths, and [[7/1|harmonic 7]] to 10 fifths up. But if it is restricted to the subgroup [[2.5.7 subgroup|2.5.7]], all just intonation intervals within that subgroup occur at ''even'' numbers of fifths up or down, because both 4 and 10 are even numbers, and so pitches located at odd numbers of fifths up or down do not have a representation in the 2.5.7 subgroup. The temperament containing the half of notes that occur at even fifths is in fact [[didacus]], generated by the 2-fifth interval (in other words, a meantone whole tone, identified here as [[28/25]]), and so we can say that septimal meantone is contorted in the 2.5.7 subgroup, inheriting this subgroup's representation from didacus.
+The above examples depicted a situation where the intervals of a given subgroup occur every 2 pitches in the underlying temperament; but they can occur every 3, or 4, or any other number of [[generator]]s apart. However many generators are needed to step from one interval in your chosen subgroup to the next is the '''contorsion order''' of that subgroup within the temperament. (In the case of higher-rank temperaments, it is possible for different generators to have different contorsion orders.)
+A temperament (more specifically, its [[comma basis]]) displays '''torsion''' if it [[tempers out]] a ''power'' of some ratio, but does not temper out that ratio. For instance, in a temperament with comma basis {[[6561/6250]], [[128/125]]}, (81/80)^2 = (6561/6250)/(128/125) is tempered out but [[81/80]] is not explicitly tempered out. In this temperament, there is no clear way to assign a pitch to 81/80; for this reason, torted temperaments are not particularly useful. Similarly to the concept of contorsion order, torsion order can be defined as the lowest power of a generic just intonation interval that is necessarily part of the temperament's lattice.
+A temperament is '''saturated''' if it is neither torted nor contorted.
 In general, being unsaturated is a bad thing<ref>Technically speaking, saturation is a property of lattices, not the matrices that generate them, and is only "bad" when referring to a comma basis or a lattice of supporting maps.</ref>, as the redundancy means that the same temperament information can be represented in a simpler way. There are other manners in which unsaturation is bad, and these depend on whether the matrix is a comma basis or a mapping, discussed below. For all these reasons, unsaturated matrices are typically considered to not truly represent temperaments.