Xenharmonic Wiki

Saturation, torsion, and contorsion: Difference between revisions

← Older editNewer edit →
VisualWikitext
Sintel (talk | contribs)
autopatrolled, emailconfirmed
1,632 edits
m italics for ref to math article
Hkm (talk | contribs)
autopatrolled
982 edits
edits for restructuring, clarity, and brevity
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.
[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]
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.


The above examples depicted a situation where the intervals of a given subgroup occur every 2 pitches in the underlying temperament; but they can occur every 3, or 4, or any other number of [[generator]]s apart. However many generators are needed to step from one interval in your chosen subgroup to the next is the '''contorsion order''' of that subgroup within the temperament. (In the case of higher-rank temperaments, it is possible for different generators to have different contorsion orders.)
For a detailed discussion on these issues, 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>.


A temperament (more specifically, its [[comma basis]]) displays '''torsion''' if it [[temper out|tempers out]] a ''power'' of some ratio, but does not temper out that ratio. For instance, in a temperament with comma basis {[[6561/6250]], [[128/125]]}, (81/80)^2 = (6561/6250)/(128/125) is tempered out but [[81/80]] is not explicitly tempered out. In this temperament, there is no clear way to assign a pitch to 81/80; for this reason, torted temperaments are not particularly useful. Similarly to the concept of contorsion order, torsion order can be defined as the lowest power of a generic just intonation interval that is necessarily part of the temperament's lattice.
For example, [[5-limit]] [[36edo|36et]] (with mapping [36 57 84]) 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 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 87&111 temperament Hemimist, with mapping [⟨3 0 26 56 8], ⟨0 2 -8 -20 1]], when restricted to the 2.5.7.11.13 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. Therefore, this a contorted generator with contorsion order 3.


A temperament is '''saturated''' if it is neither torted nor contorted.  
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.
==Torsion in temperaments==
A temperament (more specifically, its [[comma basis]]) displays '''torsion''' 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'''.


In general, being unsaturated is 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>, as 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 or a mapping, discussed below. For all these reasons, unsaturated matrices are typically considered to not truly represent temperaments.  
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 ''c''th power of this ratio, but not that ratio itself or any smaller power, where c is the torsion order.


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)<sup>2</sup> 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.
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 ==
Torsion also occurs in a similar situation where the list of commas defines a [[periodicity block]]; in fact, this was its original use case.
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|Saturation algorithms]].
 
==History and terminology==
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 cannot: 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 may do a relatively good job of approximating a given JI lattice, but it does not actually use all notes from the tuning, instead looping back to 1/1 before using up all the notes. For example, 24edo is not too bad at approximating the 5-limit, but it only uses its 12edo subset to do that, since the odd degrees of 24edo fall outside of the 5-limit approximation.
 
For a more detailed discussion on these issues, see [[Pathology of enfactoring|Pathology of saturation]].
 
== Mathematical definition ==
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 {{w|coprime integers|coprime}}<ref>If the multiples used on the linear combinations themselves have a GCD >&nbsp;1, the resulting row will always have a GCD >&nbsp;1, and such a linear combination therefore can not be used to demonstrate unsaturation. For example, consider the matrix {{mapping| 1 0 -4 | 0 1 4 }}; we can find the linear combination of rows {{nowrap| 2 × {{val| 1 0 -4 }} + 4 × {{val| 0 1 4 }} {{=}} {{val| 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 {{w|linear combination}} of its rows can produce another row whose entries have a common factor other than 1. For example, {{mapping| 3 0 -1 | 0 3 5 }} is ''not'' saturated, because {{nowrap|{{val| 3 0 -1 }} − {{val| 0 3 5 }} {{=}} {{val| 3 -3 6 }}}}, which has a common factor of 3. A mapping which consists of a single row with a common factor, such as {{mapping| 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.
 
== 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. 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]].  
 
== 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|Sage}} 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 ''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|Sage}} 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 ''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. 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.  
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.
 
[[Dave Keenan]] and [[Douglas Blumeyer]] have proposed '''''defactoring''''' as a replacement for ''saturation'', and '''''enfactoring''''' as a replacement for both ''torsion'' and ''contorsion''. So, a mapping or comma basis is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion). These terms are used in their writings. See [[Defactoring terminology proposal]] for details.
 
== References and footnotes ==
<references/>


[[Category:Regular temperament theory]]
In the case of temperaments, [[Dave Keenan]] and [[Douglas Blumeyer]] have proposed 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).<ref>See [[Defactoring terminology proposal]] for details.</ref> Keenan and Blumeyer reserve the word "torsion" for the case of periodicity blocks.
[[Category:Terms]]
==References and footnotes==
[[Category:Math]]
<references />
Retrieved from "https://en.xen.wiki/w/Saturation,_torsion,_and_contorsion"