Saturation, torsion, and contorsion: Difference between revisions

Line 1:

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

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

Revision as of 00:16, 13 March 2025 view source Godtone (talk \| contribs) autopatrolled, emailconfirmed 2,449 edits m remove "beginner" while the page isnt ready for a beginner audience ← Older edit		Revision as of 19:59, 13 March 2025 view source Sintel (talk \| contribs) autopatrolled, emailconfirmed 1,636 edits m italics for ref to math article Newer edit →
Line 1:		Line 1:
	This is a general introduction to this concept; for a more mathematical take on this, see [[Mathematical theory of saturation]].		: ''This is a general introduction to this concept; for a more mathematical take on this, see [[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, ''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.		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.