Saturation, torsion, and contorsion: Difference between revisions
Cmloegcmluin (talk | contribs) →Defactoring: correction about replacement, and move tangential note to footnote |
Cmloegcmluin (talk | contribs) a simpler, less mathematically involved way to explain the exact condition of saturation more accurately |
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== Saturation == | == 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 mappings). 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 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<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's 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 doesn't prove that the matrix is unsaturated. It's still possible that another linear combination might prove it, but this one does not.</ref>. 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. | |||
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 [[Saturation, torsion, contorsion, and defactoring#Torsion|torsion]], or a mapping, in which case 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]]. | |||
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]]. | 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]]. | ||
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'''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''. | '''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 | 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 was indeed ''not'' tempered out, 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 refers to a similar situation that occurs for the list of commas defining a [[periodicity block]]; in fact, this was its original use case. | 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. | ||
Revision as of 18:00, 17 November 2021
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.
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 mappings). 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 linear combination of its rows can produce another row whose entries have a common factor (other than 1). For example, [⟨3 0 -1] ⟨0 3 5]⟩ is not saturated, because ⟨3 0 -1] - ⟨0 3 5] = ⟨3 -3 6], which has a common factor of 3[1]. A mapping which consists of a single row with a common factor, such as [⟨24 38 56]⟩ with a visible common factor of 2, is also not saturated.
Being unsaturated is, in most cases, a bad thing[2]. 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, or a mapping, in which case the unsaturation is called 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.
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 wedgies. The simplest and fastest algorithm for saturating matrices is called column Hermite defactoring. For more information on such algorithms, see Saturation algorithms.
The term saturation was coined by Nicolas Bourbaki in 1972[3], 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[4]. The earliest identified terminology for this concept was in 1861 by H. J. S. Smith[5] who called saturated matrices "prime matrices"[6].
Torsion
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 was indeed not tempered out, 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 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[7][8] 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.
The term contorsion was invented for RTT in 2002 by Paul Erlich[9], as a play on the word "co-torsion", being dual to the situation with "torsion" above.
Defactoring
Defactoring is the term used in the writings of Dave Keenan and Douglas Blumeyer[10] as a proposed replacement for saturation. Like saturation, it applies to either mappings or comma bases. Its antonym is enfactoring and is likewise 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 Defactoring terminology proposal.
References
- ↑ 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 [⟨1 0 -4] ⟨0 1 4]⟩; we can find the linear combination of rows 2×⟨1 0 -4] + 4×⟨0 1 4] = ⟨2 4 8], which has a GCD of 2, but that's 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 doesn't prove that the matrix is unsaturated. It's still possible that another linear combination might prove it, but this one does not.
- ↑ 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.
- ↑ https://pdfcoffee.com/commutative-algebra-bourbaki-pdf-free.html
- ↑ It may also have come through PARI/GT.
- ↑ H. J. S. Smith is the creator of the Smith Normal Form used in Gene Ward Smith's saturation algorithm.
- ↑ 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).
- ↑ https://scholar.google.com/scholar?q=%22torsion+group%22&hl=en&as_sdt=0%2C5&as_ylo=1900&as_yhi=1940
- ↑ https://math.stackexchange.com/questions/300586/where-does-the-word-torsion-in-algebra-come-from
- ↑ https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456
- ↑ a list of which can be found here