Saturation, torsion, and contorsion: Difference between revisions

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Contorted mappings represent a different sort of pathology from comma bases with torsion: they involve tuning systems for which some pitches are unmapped, i.e. no just intonation interval maps to them. Contorted mappings can be useful in a way that unsaturated comma bases can not: these matrices do at least represent temperament-like systems with sensible notions of pitch. When compared to the temperament that is represented by the saturated version of the same mapping, they simply have these extra unmapped pitches that no just ratio tempers to.

In practical terms, a contorted edo does a relatively good job of approximating a given JI lattice, but it doesn't actually use all notes from the tuning, instead looping back to 1/1 before using up all the notes. For example, 24edo isn't too bad at approximating 2.3.5, but it only uses its 12edo subset to do that, since the odd degrees of 24edo fall outside of the 2.3.5 approximation.

The term was invented for RTT in 2002 by [[Paul Erlich]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456 Yahoo! Tuning Group | ''My top 5--for Paul'']</ref>, as a play on the word "co-torsion", being dual to the situation with "torsion" above.

'''''Enfactoring''''' is a proposed replacement for both ''torsion'' and ''contorsion''. So, a mapping or comma basis is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion).

== See also ==

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 Contorted mappings represent a different sort of pathology from comma bases with torsion: they involve tuning systems for which some pitches are unmapped, i.e. no just intonation interval maps to them. Contorted mappings can be useful in a way that unsaturated comma bases can not: these matrices do at least represent temperament-like systems with sensible notions of pitch. When compared to the temperament that is represented by the saturated version of the same mapping, they simply have these extra unmapped pitches that no just ratio tempers to.
+In practical terms, a contorted edo does a relatively good job of approximating a given JI lattice, but it doesn't actually use all notes from the tuning, instead looping back to 1/1 before using up all the notes. For example, 24edo isn't too bad at approximating 2.3.5, but it only uses its 12edo subset to do that, since the odd degrees of 24edo fall outside of the 2.3.5 approximation.
 The term was invented for RTT in 2002 by [[Paul Erlich]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456 Yahoo! Tuning Group | ''My top 5--for Paul'']</ref>, as a play on the word "co-torsion", being dual to the situation with "torsion" above.
 '''''Enfactoring''''' is a proposed replacement for both ''torsion'' and ''contorsion''. So, a mapping or comma basis is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion).
 == See also ==