Saturation, torsion, and contorsion: Difference between revisions

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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 (more specifically, its [[comma basis]]) displays '''torsion''' if it [[temper out|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.

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 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 (more specifically, its [[comma basis]]) displays '''torsion''' if it [[temper out|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.