Saturation, torsion, and contorsion: Difference between revisions

{{Beginner|Mathematical theory of saturation}}

A temperament matrix is '''saturated''' when it represents a temperament without any redundancies due to a common factor. A [[mapping]] is saturated when no common factor is found in its rows (i.e. generator maps). A [[comma basis]], the dual of a mapping, is saturated when no common factor is found in its columns (i.e. comma vectors).

 

To be more specific, a mapping is saturated if no [[Wikipedia: Coprime integers|coprime]]<ref>If the multiples used on the linear combinations themselves have a GCD > 1, the resulting row will always have a GCD > 1, and such a linear combination therefore can not be used to demonstrate unsaturation. For example, consider the matrix {{ket| {{map| 1 0 -4 }}, {{map| 0 1 4 }} }}; we can find the linear combination of rows 2 × {{map| 1 0 -4 }} + 4 × {{map| 0 1 4 }} = {{map| 2 4 8 }}, which has a GCD of 2, but that is clearly a result of the fact that we used 2× and 4× of the original rows, and 2 and 4 have a GCD of 2. So this linear combination does not prove that the matrix is unsaturated. It is still possible that another linear combination might prove it, but this one does not.</ref> integer linear combination of its rows can produce another row whose entries have a common factor other than 1. For example, {{ket| {{map| 3 0 -1 }} {{map| 0 3 5 }} }} is ''not'' saturated, because {{map| 3 0 -1 }} - {{map| 0 3 5 }} = {{map| 3 -3 6 }}, which has a common factor of 3. A mapping which consists of a single row with a common factor, such as {{ket| {{map| 24 38 56 }} }} with a visible common factor of 2, is also not saturated.

 

The greatest factor possible to create as a GCD of a coprime linear combination of rows in this way is called the mapping's '''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>, and so an alternative definition of saturation would be that the mapping must have a greatest factor of 1.

Being unsaturated is, in most cases, a bad thing<ref>Technically speaking, saturation is a property of lattices, not the matrices that generate them, and is only "bad" when referring to a comma basis or a lattice of supporting maps.</ref>. The redundancy means that the same temperament information can be represented in a simpler way. There are other manners in which unsaturation is bad, and these depend on whether the matrix is a comma basis, in which case the unsaturation is called [[#Torsion and contorsion|torsion]], or a mapping, in which case the unsaturation is called [[#Torsion and contorsion|contorsion]]; both of these cases are defined below. For all these reasons, unsaturated matrices are typically considered to not truly represent temperaments. For a more detailed discussion on these issues, see [[Pathology of enfactoring|Pathology of saturation]].

Saturation algorithms correct for this problem, ensuring our ability to most simply — and thereby uniquely — identify temperaments using only matrices. This need can otherwise be satisfied using [[wedgie]]s. The simplest and fastest algorithm for saturating matrices is called [[column Hermite defactoring]]. For more information on such algorithms, see [[Defactoring algorithms|Saturation algorithms]].

The term saturation was coined by Nicolas Bourbaki in 1972<ref>[https://pdfcoffee.com/commutative-algebra-bourbaki-pdf-free.html Nicolas Bourbaki. ''Commutative Algebra'']</ref>, working in the field of commutative algebra. It came to RTT via [[Gene Ward Smith]] and [[Graham Breed]]'s observations of the work of the mathematician William Stein and his Sage software<ref>It may also have come through PARI/GT.</ref>. The earliest identified terminology for this concept was in 1861 by H. J. S. Smith<ref>H. J. S. Smith is the creator of the [[Wikipedia: Smith normal form|Smith normal form]] used in [[Defactoring algorithms #Precedent: Smith defactoring|Gene Ward Smith's saturation algorithm]].</ref> who called saturated matrices "prime matrices"<ref>Also from ''On Systems of Linear Indeterminate Equations and Congruences'', linked above. Neither ''prime matrix'' nor ''greatest divisor'' seems to have caught on in the mathematical community.</ref>.

'''''Defactoring''''' is the term used in the writings of [[Dave Keenan]] and [[Douglas Blumeyer]]<ref>A list of which can be found in [[Douglas Blumeyer #Some of his work here on the wiki]]</ref> as a proposed replacement for ''saturation''.  

== Torsion and contorsion ==

'''Torsion''' is the opposite of saturation, but only regarding comma bases; in other words, a comma basis is either saturated, or it ''has torsion''.

A comma basis with torsion is rarely useful at all. It states that a power of a ratio is made to [[vanish]] but does not explicitly state that the ratio itself is made to vanish (for instance, (81/80)² is made to vanish but 81/80 is not). From a mathematical standpoint, there are multiple ways to interpret this situation. Historically, a group theory formalism was used, leading to the interpretation that the ratio itself was indeed ''not'' made to vanish, despite this being musically absurd. Using a linear algebra formalism as is preferred now, however, no such absurdity is suggested. This historical usage of the group theory formalism is another reason why it is bad practice to use comma bases with torsion.

Torsion also occurs in a similar situation where the list of commas defines a [[periodicity block]]; in fact, this was its original use case.

The term has been used since at least as early as 1932<ref>[https://scholar.google.com/scholar?q=%22torsion+group%22&hl=en&as_sdt=0%2C5&as_ylo=1900&as_yhi=1940 Google Scholar: Torsion group]</ref><ref>[https://math.stackexchange.com/questions/300586/where-does-the-word-torsion-in-algebra-come-from Stack Exchange | ''Where does the word "torsion" in algebra come from?'']</ref> and came to RTT from the mathematical field of group theory.

'''Contorsion''' is the opposite of saturation, but only regarding mappings; in other words, a mapping is either saturated, or it ''has contorsion'' (or we can say that it ''is contorted''). The greatest factor of a mapping has been called its '''contorted order'''.

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.

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).  

* [[Defactoring terminology proposal]]

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