Saturation, torsion, and contorsion: Difference between revisions

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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>.

== 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''.

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</ref><ref>https://math.stackexchange.com/questions/300586/where-does-the-word-torsion-in-algebra-come-from</ref> 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'').

The term contorsion was invented for RTT in 2002 by [[Paul Erlich]]<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456</ref>.

== Defactoring ==

'''Defactoring''' is a synonym for saturation, so it applies to either mappings or comma bases. Its antonym is enfactoring. So, a mapping or comma basis is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion). Defactored and enfactored ~~have been proposed~~ as replacement terms for saturation, torsion, and contorsion; for more information, see [[Defactoring terminology proposal]].

'''Defactoring''' is a synonym for saturation, so it applies to either mappings or comma bases. Its antonym is enfactoring. So, a mapping or comma basis is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion).

Defactored and enfactored were coined by [[Dave Keenan]] in 2021 as replacement terms for saturation, torsion, and contorsion; for more information, see [[Defactoring terminology proposal]].

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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>.
 == 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''.
+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</ref><ref>https://math.stackexchange.com/questions/300586/where-does-the-word-torsion-in-algebra-come-from</ref> 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'').
+The term contorsion was invented for RTT in 2002 by [[Paul Erlich]]<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456</ref>.
 == Defactoring ==
-'''Defactoring''' is a synonym for saturation, so it applies to either mappings or comma bases. Its antonym is enfactoring. So, a mapping or comma basis is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion). Defactored and enfactored have been proposed as replacement terms for saturation, torsion, and contorsion; for more information, see [[Defactoring terminology proposal]].
+'''Defactoring''' is a synonym for saturation, so it applies to either mappings or comma bases. Its antonym is enfactoring. So, a mapping or comma basis is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion).
+Defactored and enfactored were coined by [[Dave Keenan]] in 2021 as replacement terms for saturation, torsion, and contorsion; for more information, see [[Defactoring terminology proposal]].