Saturation, torsion, and contorsion: Difference between revisions

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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 "On Systems of Linear Indeterminate Equations and Congruences", which can be accessed here: https://www.jstor.org/stable/pdf/108738.pdf, where Smith describes the GCD of a matrix's minor determinants as its "greatest divisor". "Divisor" and "factor" are synonyms and we 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 [[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]].

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 enfactoring|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 [[Defactoring algorithms|Saturation algorithms]].

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>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>.

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 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 "On Systems of Linear Indeterminate Equations and Congruences", which can be accessed here: https://www.jstor.org/stable/pdf/108738.pdf, where Smith describes the GCD of a matrix's minor determinants as its "greatest divisor". "Divisor" and "factor" are synonyms and we 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 [[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]].
+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 enfactoring|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 [[Defactoring algorithms|Saturation algorithms]].
 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>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>.