Saturation, torsion, and contorsion: Difference between revisions

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In [[regular temperament theory]], a [[temperament]] is saturated if its set of available intervals matches what is suggested by its mapping or comma basis. A temperament's mapping can fail with respect to saturation by being contorted, and its comma basis can fail through torsion.

~~For a detailed discussion on~~ these issues, see [[Pathology of enfactoring|Pathology of saturation]].

This article briefly explains these issues; for lattice-based visualizations and intuitive explanations, see [[Pathology of enfactoring|Pathology of saturation]].

==Contorsion==

A temperament (more specifically, its [[mapping]]) displays '''contorsion''' if there is some generatable interval which no [[just intonation]] interval maps to. This generatable interval is a '''contorted generator''', which has the property that every just interval's mapping has a multiple of c of that generator, where c, the '''contorsion order''', is greater than one. In a contorted temperament, all generator bases will contain at least one contorted generator. The largest contorsion order is called the '''greatest factor'''<ref>This term is inspired by H. J. S. Smith's [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], where Smith describes the GCD of a matrix's minor determinants as its "greatest divisor". "Divisor" and "factor" are synonyms and they prefer "factor" for its connection with the term "defactor".</ref>.

@@ Line 6: / Line 6: @@
 In [[regular temperament theory]], a [[temperament]] is saturated if its set of available intervals matches what is suggested by its mapping or comma basis. A temperament's mapping can fail with respect to saturation by being contorted, and its comma basis can fail through torsion.
-For a detailed discussion on these issues, see [[Pathology of enfactoring|Pathology of saturation]].
+This article briefly explains these issues; for lattice-based visualizations and intuitive explanations, see [[Pathology of enfactoring|Pathology of saturation]].
 ==Contorsion==
 A temperament (more specifically, its [[mapping]]) displays '''contorsion''' if there is some generatable interval which no [[just intonation]] interval maps to. This generatable interval is a '''contorted generator''', which has the property that every just interval's mapping has a multiple of c of that generator, where c, the '''contorsion order''', is greater than one. In a contorted temperament, all generator bases will contain at least one contorted generator. The largest contorsion order is called the '''greatest factor'''<ref>This term is inspired by H. J. S. Smith's [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], where Smith describes the GCD of a matrix's minor determinants as its "greatest divisor". "Divisor" and "factor" are synonyms and they prefer "factor" for its connection with the term "defactor".</ref>.