Saturation, torsion, and contorsion: Difference between revisions

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This is a general introduction to this concept; for a more mathematical take on this, see [[Mathematical theory of saturation]].
{{interwiki
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| en = Saturation, torsion, and contorsion
| es =
| ja = 飽和、ねじれ、contorsion
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: ''This is a general introduction to this concept; for a more mathematical take on this, see [[Mathematical theory of saturation]].''


Saturation, torsion, contorsion, and defactoring are all terms for the same effect in [[RTT]], but used in slightly different ways.
[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]
In [[regular temperament theory]], a [[temperament]] is '''saturated''' or ''defactored'' 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.


== Saturation ==
This article briefly explains these issues; for lattice-based visualizations and intuitive explanations, see [[Pathology of enfactoring|Pathology of saturation]].
Suppose that we have a lattice of vectors of some kind; this could be a lattice of vals, monzos, or just vectors in general. Then a sublattice of this lattice is said to be '''saturated''' if it has the property that, if it contains a *multiple* of some vector from the original lattice, it also contains the vector itself. This definition can be a mouthful, so we'll unpack it with some examples.


Suppose our main lattice is the lattice of monzos; these all have integer coefficients. Then we can look at matrices in which the columns are monzos, which can represent a set of vanishing commas. Then we can obtain a sublattice of monzos by looking at the set of all integer-weighted linear combinations of the columns, which represent all of the vanishing commas in some temperament; this is called the '''column lattice''' or '''integer column span''' of our matrix. This sublattice is said to be '''saturated''' if it doesn't contain any comma whose monzo coefficients have a GCD greater than one, unless it also contains the comma obtained by dividing the monzo through by this common divisor. In plain English, this means that the lattice of vanishing commas cannot have any vanishing comma which is the square, or cube, etc, of some other comma, unless that other comma is in the lattice of vanishing commas as well. In this situation, we say that the matrix is saturated as well and that it properly represents a temperament. If not, the matrix is conventionally thought to either not represent a temperament at all, or at least be a pathological representation of a temperament, as explained in the section on '''torsion''' below.
== Contorsion ==
A temperament (more specifically, its [[mapping]]) displays '''contorsion''' or '''enfactoring''' 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 under any appropriate generator basis, 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 overall contorsion order of the temperament is the product of all the seperate orders.<ref>H. J. S. Smith [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], the overall order is equivalent to the GCD of a matrix's minor determinants, which Smith calls the "greatest divisor".</ref>.


Similarly, suppose our main lattice is the lattice of vals; these again have integer coefficients. Then given any [[mapping|mapping matrix]], which will have each row equal to some val, we can obtain a sublattice of vals by looking at the set of all integer-weighted linear combinations of the rows of our mapping matrix, which happens to be the sublattice of all vals that support the temperament in question. This is called the '''row lattice''' or '''integer row span''' of our matrix. This sublattice is said to be '''saturated''' if it doesn't contain any val whose coefficients have a greatest common divisor greater than one, unless it also contains the corresponding val obtained by dividing the coefficients through by this greatest common divisor. If the mapping is *not* saturated, it is conventionally thought to not represent a temperament at all, but rather a pathological object which is called "contorted," as explained in the section on '''contorsion''' below.
For example, [[5-limit]] [[36edo|36et]] (with mapping {{mapping|36 57 84}}) uses 12 of its pitches per octave (the ones within [[12edo]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of [[Generator|generators]] that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contorsion order 3. For a higher-rank example, the 13-limit {{nowrap|87 &amp; 111}} temperament Hemimist, with mapping [{{mapping|3 0 26 56 8}}, {{mapping|0 2 -8 -20 1}}], when restricted to the 2.5.7.11 subgroup, has no just intonation interval corresponding to the period or the square of the period, although there is a just intonation interval (namely, 2/1) corresponding to the cube of the period. Thus, this a contorted generator with contorsion order 3.


For example, {{ket|{{map|3 0 -1}} {{map|0 3 5}}}} is ''not'' a saturated matrix, because {{map|3 0 -1}} - {{map|0 3 5}} = {{map|3 -3 6}}, which has a common factor of 3, and there is no integer linear combination which can produce the corresponding val with the factor of 3 removed, which would be {{map|1 -1 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, as there is of course no way to produce the GCD-reduced version of {{map|12 19 28}} by simply multiplying by an integer.
If a temperament has a subgroup which is contorted, especially a subgroup with small primes (for example, the 11-limit subgroup of 23-limit [[44edo|44et]]), that temperament will likely be easier to traverse than the number of generators required according to the mapping would suggest.


For the purposes of temperament representation, being unsaturated is typically a bad thing. If the matrix is a comma basis, the unsaturation is called [[Saturation, torsion, contorsion, and defactoring#Torsion|torsion]], and if it a mapping, the unsaturation is called [[Saturation, torsion, contorsion, and defactoring#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 [[The pathology of saturation]].
== Torsion in temperaments ==
A temperament (more specifically, its [[comma basis]]) displays '''torsion''' or ''enfactoring'' if there is some interval mapped to zero which is not formable by multiplying commas in the basis. This interval is a '''comma with torsion''', which has the property that commas in the basis can be multiplied to form the ''c''th power of this ratio, but not that ratio itself or any smaller power, where ''c'' is the '''torsion order'''.


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 [[Saturation algorithms]].
For instance, in a temperament with comma basis {[[6561/6250]], [[128/125]]}, there is an interval 81/80 which is not formable by multiplying commas in the basis, but is nevertheless forced to be mapped to zero because {{nowrap|(81/80)^2 {{=}} (6561/6250)/(128/125)}} is part of the basis. Thus, 81/80 displays torsion with torsion order 2.


The term saturation was coined by Nicolas Bourbaki in 1972<ref>https://pdfcoffee.com/commutative-algebra-bourbaki-pdf-free.html</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 [[Smith Normal Form]] used in [[Defactoring_algorithms#Precedent:_Smith_defactoring|Gene Ward Smith's saturation algorithm]].</ref> who called saturated matrices "prime matrices"<ref>See "On Systems of Linear Indeterminate Equations and Congruences", which can be accessed here: https://www.jstor.org/stable/pdf/108738.pdf, where Smith also describes the GCD of a matrix's minor determinants as its "greatest divisor" (though neither of this term nor "prime matrix" seems to have caught on).</ref>.
== Torsion in periodicity blocks ==
A comma basis in the context of periodicity blocks displays torsion if it displays torsion as a temperament—precisely when there is some comma with torsion where commas in the basis can be multiplied to form the ''c''th power of this ratio, but not that ratio itself or any smaller power, where ''c'' is the torsion order.


== Torsion ==
Within periodicity blocks, no mapping needs to be defined from a comma basis, so comma bases with torsion are able to form periodicity blocks where the smallest comma with torsion is not tempered out.
'''Torsion''' is used to refer to 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 tempered out but does not explicitly state that the ratio itself is tempered out (for instance, (81/80)^2 is tempered out 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 is indeed ''not'' tempered out, but somehow the power of the ratio is; thus within this formalism unsaturated comma bases represent musically absurd pathological objects. Within a linear algebra formalism, there isn't quite as absurd an interpretation, but the historical usage of the group theory formalism is one reason why it is still viewed as bad practice to use comma bases with torsion.
== Saturation algorithms ==
 
An unsaturated mapping or comma basis can be made saturated, ensuring our ability to most simply—and thereby uniquely—identify temperaments using only matrices. The simplest and fastest algorithm for saturating matrices is called [[column Hermite defactoring]]. For more information on such algorithms, see [[Defactoring algorithms]].
Torsion also refers to a similar situation that occurs for the list of commas defining a [[periodicity block]]; in fact, this was its original use case.
 
The term torsion 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</ref><ref>https://math.stackexchange.com/questions/300586/where-does-the-word-torsion-in-algebra-come-from</ref> and came to RTT from the mathematical field of group theory.
 
== Contorsion ==
'''Contorsion''' is used to refer to 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'').


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.
== History and terminology ==
The term ''saturation'' was coined by {{w|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 {{w|William A. Stein|William Stein}} and his {{w|SageMath}} software<ref>It may also have come through PARI/GT<!-- typo of PARI/GP? -->.</ref>. The earliest identified terminology for this concept was in 1861 by {{w|Henry John Stephen Smith|H. J. S. Smith}}<ref>H. J. S. Smith is the creator of the {{w|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>.


The term contorsion was invented for RTT in 2002 by [[Paul Erlich]]<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456</ref>, as a play on the word "co-torsion", being dual to the situation with "torsion" above.
The term ''torsion'' 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. Historically, a group-theory formalism was used to analyze comma bases with torsion, where the smallest comma displaying torsion was not made to vanish although a power of that comma was, which is musically impossible; using a linear algebra formalism as is preferred now, no such impossibility is suggested. The term ''contorsion'' 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.


== Defactoring ==
In the case of temperaments, [[Dave Keenan]] and [[Douglas Blumeyer]] have proposed<ref>See [[Defactoring terminology proposal]] for details.</ref> and used '''defactoring''' as a replacement for ''saturation'' and '''enfactoring''' as a replacement for both ''torsion'' and ''contorsion''. So, a mapping or comma basis of a temperament is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion).
'''Defactoring''' is a synonym for saturation, so it applies to either mappings or comma bases. Its antonym is enfactoring. So, a mapping or comma basis is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion).  


Defactored and enfactored were coined by [[Dave Keenan]] in collaboration with [[Douglas Blumeyer]] in 2021 as replacement terms for saturation, torsion, and contorsion; for more information, see [[Defactoring terminology proposal]].
== See also ==
* [http://www.tonalsoft.com/enc/t/torsion.aspx Tonalsoft's page on torsion]
* [http://www.tonalsoft.com/enc/c/contortion.aspx Tonalsoft's page on contorsion]


== References ==
== References and footnotes ==
<references />

Latest revision as of 00:10, 28 June 2025

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 is saturated or defactored 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.

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

Contorsion

A temperament (more specifically, its mapping) displays contorsion or enfactoring 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 under any appropriate generator basis, 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 overall contorsion order of the temperament is the product of all the seperate orders.[1].

For example, 5-limit 36et (with mapping [⟨36 57 84]]) uses 12 of its pitches per octave (the ones within 12edo) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of generators that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contorsion order 3. For a higher-rank example, the 13-limit 87 & 111 temperament Hemimist, with mapping [[⟨3 0 26 56 8]], [⟨0 2 -8 -20 1]]], when restricted to the 2.5.7.11 subgroup, has no just intonation interval corresponding to the period or the square of the period, although there is a just intonation interval (namely, 2/1) corresponding to the cube of the period. Thus, this a contorted generator with contorsion order 3.

If a temperament has a subgroup which is contorted, especially a subgroup with small primes (for example, the 11-limit subgroup of 23-limit 44et), that temperament will likely be easier to traverse than the number of generators required according to the mapping would suggest.

Torsion in temperaments

A temperament (more specifically, its comma basis) displays torsion or enfactoring if there is some interval mapped to zero which is not formable by multiplying commas in the basis. This interval is a comma with torsion, which has the property that commas in the basis can be multiplied to form the cth power of this ratio, but not that ratio itself or any smaller power, where c is the torsion order.

For instance, in a temperament with comma basis {6561/6250, 128/125}, there is an interval 81/80 which is not formable by multiplying commas in the basis, but is nevertheless forced to be mapped to zero because (81/80)^2 = (6561/6250)/(128/125) is part of the basis. Thus, 81/80 displays torsion with torsion order 2.

Torsion in periodicity blocks

A comma basis in the context of periodicity blocks displays torsion if it displays torsion as a temperament—precisely when there is some comma with torsion where commas in the basis can be multiplied to form the cth power of this ratio, but not that ratio itself or any smaller power, where c is the torsion order.

Within periodicity blocks, no mapping needs to be defined from a comma basis, so comma bases with torsion are able to form periodicity blocks where the smallest comma with torsion is not tempered out.

Saturation algorithms

An unsaturated mapping or comma basis can be made saturated, ensuring our ability to most simply—and thereby uniquely—identify temperaments using only matrices. The simplest and fastest algorithm for saturating matrices is called column Hermite defactoring. For more information on such algorithms, see Defactoring algorithms.

History and terminology

The term saturation was coined by Nicolas Bourbaki in 1972[2], 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 SageMath software[3]. The earliest identified terminology for this concept was in 1861 by H. J. S. Smith[4] who called saturated matrices "prime matrices"[5].

The term torsion has been used since at least as early as 1932[6][7] and came to RTT from the mathematical field of group theory. Historically, a group-theory formalism was used to analyze comma bases with torsion, where the smallest comma displaying torsion was not made to vanish although a power of that comma was, which is musically impossible; using a linear algebra formalism as is preferred now, no such impossibility is suggested. The term contorsion was invented for RTT in 2002 by Paul Erlich[8], as a play on the word "co-torsion", being dual to the situation with "torsion" above.

In the case of temperaments, Dave Keenan and Douglas Blumeyer have proposed[9] and used defactoring as a replacement for saturation and enfactoring as a replacement for both torsion and contorsion. So, a mapping or comma basis of a temperament is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion).

See also

References and footnotes

  1. ↑ H. J. S. Smith On Systems of Linear Indeterminate Equations and Congruences, the overall order is equivalent to the GCD of a matrix's minor determinants, which Smith calls the "greatest divisor".
  2. ↑ Nicolas Bourbaki. Commutative Algebra
  3. ↑ It may also have come through PARI/GT.
  4. ↑ H. J. S. Smith is the creator of the Smith normal form used in Gene Ward Smith's saturation algorithm.
  5. ↑ 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.
  6. ↑ Google Scholar: Torsion group
  7. ↑ Stack Exchange | Where does the word "torsion" in algebra come from?
  8. ↑ Yahoo! Tuning Group | My top 5--for Paul
  9. ↑ See Defactoring terminology proposal for details.
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