Saturation, torsion, and contorsion: Difference between revisions

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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'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> 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.

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 GCF>1, the resulting row will always have a GCF>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 GCF 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 GCF 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> 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 GCF 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 GCF 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]].

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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]].

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>.

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>.

== Torsion ==

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 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'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> 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.
+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 GCF>1, the resulting row will always have a GCF>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 GCF 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 GCF 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> 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 GCF 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 GCF 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]].
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 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]].
-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>.
+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>.
 == Torsion ==