Saturation, torsion, and contorsion: Difference between revisions

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, ~~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]].

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>.

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]].

@@ 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, 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]].
+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>.
+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]].