User:Hkm: Difference between revisions

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DRAFT BELOW

In [[regular temperament theory]], a [[temperament]] (more specifically, its [[mapping]]) - displays '''contorsion''' if there are some interval which no [[just intonation]] interval maps to. If a temperament is contorted, there is some '''contorted generator''' where every just interval's mapping has a multiple of c of that generator, where c, the '''contortion order''', is greater than one. The largest contortion order is called the '''greatest factor'''. For example, the [[5-limit]] [[36edo|36et]] (with mapping [36 57 84]) uses 12 of its pitches per octave (the ones within [[12edo|12et]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of [[Generator|generators]] that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contortion order 3. For a higher-rank example, the 3.5.7 restriction of [[Sensipent family#Septimal%20sensi|septimal sensi]], with mapping [⟨-1 -1 -2], ⟨7 9 13]] and mapping generators "~2", ~9/7, where "~2" is in quotes because it is no longer in the subgroup. This temperament has the interval corresponding to the period which no just intonation maps to. We can thus find a contorted generator; namely, [FIGURE THIS OUT]

In [[regular temperament theory]], a [[temperament]] (more specifically, its [[mapping]]) displays '''contorsion''' if there are some interval which no [[just intonation]] interval maps to. If a temperament is contorted, there is some '''contorted generator''' where every just interval's mapping has a multiple of c of that generator, where c, the '''contortion order''', is greater than one. The largest contortion order is called the '''greatest factor'''. For example, the [[5-limit]] [[36edo|36et]] (with mapping [36 57 84]) uses 12 of its pitches per octave (the ones within [[12edo|12et]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of [[Generator|generators]] that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contortion order 3. For a higher-rank example, the 3.5.7 restriction of [[Sensipent family#Septimal%20sensi|septimal sensi]], with mapping [⟨-1 -1 -2], ⟨7 9 13]] and mapping generators "~2", ~9/7, where "~2" is in quotes because it is no longer in the subgroup. This temperament has the interval corresponding to the period which no just intonation maps to. We can thus find a contorted generator; namely, [FIGURE THIS OUT]

A temperament (more specifically, its [[comma basis]]) displays '''torsion''' if there is no mapping which describes it. If a temperament has torsion, it [[temper out|tempers out]] a ''power'' c of some ratio, the '''contorted comma''', but does not temper out that ratio, where c is the '''torsion order'''. For instance, in a temperament with comma basis {[[6561/6250]], [[128/125]]}, (81/80)^2 = (6561/6250)/(128/125) is tempered out but [[81/80]] is not explicitly tempered out. In this temperament, there is no clear way to assign a pitch to 81/80; for this reason, temperaments with torsion are not particularly useful. Similarly to the concept of contorsion order, torsion order can be defined as the lowest power of a generic just intonation interval that is necessarily part of the temperament's lattice.

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For a more detailed discussion on these issues, see [[Pathology of enfactoring|Pathology of saturation]].

==Saturation algorithms==

An unsaturated mapping or comma basis can be made saturated, 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]].

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 DRAFT BELOW
-In [[regular temperament theory]], a [[temperament]] (more specifically, its [[mapping]]) - displays '''contorsion''' if there are some interval which no [[just intonation]] interval maps to. If a temperament is contorted, there is some '''contorted generator''' where every just interval's mapping has a multiple of c of that generator, where c, the '''contortion order''', is greater than one. The largest contortion order is called the '''greatest factor'''. For example, the [[5-limit]] [[36edo|36et]] (with mapping [36 57 84]) uses 12 of its pitches per octave (the ones within [[12edo|12et]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of [[Generator|generators]] that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contortion order 3. For a higher-rank example, the 3.5.7 restriction of [[Sensipent family#Septimal%20sensi|septimal sensi]], with mapping [⟨-1 -1 -2], ⟨7 9 13]] and mapping generators "~2", ~9/7, where "~2" is in quotes because it is no longer in the subgroup. This temperament has the interval corresponding to the period which no just intonation maps to. We can thus find a contorted generator; namely, [FIGURE THIS OUT]
+In [[regular temperament theory]], a [[temperament]] (more specifically, its [[mapping]]) displays '''contorsion''' if there are some interval which no [[just intonation]] interval maps to. If a temperament is contorted, there is some '''contorted generator''' where every just interval's mapping has a multiple of c of that generator, where c, the '''contortion order''', is greater than one. The largest contortion order is called the '''greatest factor'''. For example, the [[5-limit]] [[36edo|36et]] (with mapping [36 57 84]) uses 12 of its pitches per octave (the ones within [[12edo|12et]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of [[Generator|generators]] that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contortion order 3. For a higher-rank example, the 3.5.7 restriction of [[Sensipent family#Septimal%20sensi|septimal sensi]], with mapping [⟨-1 -1 -2], ⟨7 9 13]] and mapping generators "~2", ~9/7, where "~2" is in quotes because it is no longer in the subgroup. This temperament has the interval corresponding to the period which no just intonation maps to. We can thus find a contorted generator; namely, [FIGURE THIS OUT]
 A temperament (more specifically, its [[comma basis]]) displays '''torsion''' if there is no mapping which describes it. If a temperament has torsion, it [[temper out|tempers out]] a ''power'' c of some ratio, the '''contorted comma''', but does not temper out that ratio, where c is the '''torsion order'''. For instance, in a temperament with comma basis {[[6561/6250]], [[128/125]]}, (81/80)^2 = (6561/6250)/(128/125) is tempered out but [[81/80]] is not explicitly tempered out. In this temperament, there is no clear way to assign a pitch to 81/80; for this reason, temperaments with torsion are not particularly useful. Similarly to the concept of contorsion order, torsion order can be defined as the lowest power of a generic just intonation interval that is necessarily part of the temperament's lattice.
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 For a more detailed discussion on these issues, see [[Pathology of enfactoring|Pathology of saturation]].
 ==Saturation algorithms==
 An unsaturated mapping or comma basis can be made saturated, 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]].