Saturation, torsion, and contorsion: Difference between revisions

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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. 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, generally speaking, a bad thing. The redundancy means that the same temperament information can be represented in a simpler way. For this reason, unsaturated matrices are sometimes considered to not truly represent temperaments. An unsaturated mapping can be useful: it represents a temperament-like pitch system which is audibly different than the temperament that is represented by the saturated version of the same mapping; it is essentially the same as that temperament, except with extra pitches that no input pitch tempers to. On the other hand, an unsaturated comma basis is not useful; it simply represents the same temperament as the saturated version, but in an inefficient and confusing way. For more information on this, see [[The pathology of saturation]].

Being unsaturated is, generally speaking, 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. For this reason, unsaturated matrices are sometimes considered to not truly represent temperaments. An unsaturated mapping can be useful: it represents a temperament-like pitch system which is audibly different than the temperament that is represented by the saturated version of the same mapping; it is essentially the same as that temperament, except with extra pitches that no input pitch tempers to. On the other hand, an unsaturated comma basis is not useful; it simply represents the same temperament as the saturated version, but in an inefficient and confusing way. For more information on this, 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]].

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>.

~~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.~~

== Torsion ==

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 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. 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, generally speaking, a bad thing. The redundancy means that the same temperament information can be represented in a simpler way. For this reason, unsaturated matrices are sometimes considered to not truly represent temperaments. An unsaturated mapping can be useful: it represents a temperament-like pitch system which is audibly different than the temperament that is represented by the saturated version of the same mapping; it is essentially the same as that temperament, except with extra pitches that no input pitch tempers to. On the other hand, an unsaturated comma basis is not useful; it simply represents the same temperament as the saturated version, but in an inefficient and confusing way. For more information on this, see [[The pathology of saturation]].
+Being unsaturated is, generally speaking, 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. For this reason, unsaturated matrices are sometimes considered to not truly represent temperaments. An unsaturated mapping can be useful: it represents a temperament-like pitch system which is audibly different than the temperament that is represented by the saturated version of the same mapping; it is essentially the same as that temperament, except with extra pitches that no input pitch tempers to. On the other hand, an unsaturated comma basis is not useful; it simply represents the same temperament as the saturated version, but in an inefficient and confusing way. For more information on this, 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]].
 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>.
-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.
 == Torsion ==