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, 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>.

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]].

~~An unsaturated comma basis is rarely useful at all. It~~ means that ~~you have, for instance, some power of~~ a ~~ratio being tempered out without explicitly stating that you~~ are ~~tempering out the ratio itself (for instance~~, ~~tempering out (81/80)^2 without tempering out 81/80). The interpretation of this situation depends somewhat~~ on the ~~mathematical formalism of regular temperament theory: historically~~, ~~these are viewed as leading to very strange mathematical objects~~ which ~~have intervals which are *not* tempered out~~, ~~but somehow become tempered out only when you add them to themselves some number of times (a situation called '''~~torsion~~''') - a musical absurdity. In the linear algebra-centric formalism we present here~~, ~~these strange objects don't arise~~, ~~but we still view it as bad practice to give an unsaturated basis of commas for the kernel~~ or ~~null space of some temperament for this historical reason.~~

~~Unsaturated~~ mapping ~~matrices~~, on the ~~other hand~~, ~~represent a different sort of pathology: they involve tuning systems for which some pitches are "unmapped"~~ and ~~have no rational interpretation or relationship to just intonation at all, a situation called '''~~contorsion~~''' (as a sort~~ of ~~play on the word "co-torsion", being dual to the situation with "torsion" above)~~. For ~~this reason~~, unsaturated matrices are ~~also~~ typically considered to not truly represent temperaments. ~~On the other hand, unsaturated mappings can also be useful in~~ a way that unsaturated comma bases are not: unlike the situation with torsion above, these matrices do at least represent temperament-like tuning 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 extra "unmapped" pitches that no just ratio tempers to.

~~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]].

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== 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 was 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.

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'''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>.

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]]<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456</ref>, as a play on the word "co-torsion", being dual to the situation with "torsion" above.

== Defactoring ==

@@ Line 8: / Line 8: @@
 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, 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>.
+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]].
-An unsaturated comma basis is rarely useful at all. It means that you have, for instance, some power of a ratio being tempered out without explicitly stating that you are tempering out the ratio itself (for instance, tempering out (81/80)^2 without tempering out 81/80). The interpretation of this situation depends somewhat on the mathematical formalism of regular temperament theory: historically, these are viewed as leading to very strange mathematical objects which have intervals which are *not* tempered out, but somehow become tempered out only when you add them to themselves some number of times (a situation called '''torsion''') - a musical absurdity. In the linear algebra-centric formalism we present here, these strange objects don't arise, but we still view it as bad practice to give an unsaturated basis of commas for the kernel or null space of some temperament for this historical reason.
-Unsaturated mapping matrices, on the other hand, represent a different sort of pathology: they involve tuning systems for which some pitches are "unmapped" and have no rational interpretation or relationship to just intonation at all, a situation called '''contorsion''' (as a sort of play on the word "co-torsion", being dual to the situation with "torsion" above). For this reason, unsaturated matrices are also typically considered to not truly represent temperaments. On the other hand, unsaturated mappings can also be useful in a way that unsaturated comma bases are not: unlike the situation with torsion above, these matrices do at least represent temperament-like tuning 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 extra "unmapped" pitches that no just ratio tempers to.
-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]].
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 == 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 was 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.
@@ Line 30: / Line 26: @@
 '''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>.
+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]]<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456</ref>, as a play on the word "co-torsion", being dual to the situation with "torsion" above.
 == Defactoring ==