Saturation, torsion, and contorsion: Difference between revisions

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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.

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, ''inheriting'' its 5-limit representation from 12et. 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.)

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 {{Beginner|Mathematical theory of saturation}}
-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.
+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, ''inheriting'' its 5-limit representation from 12et. 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.)