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Saturation, torsion, and contorsion: Difference between revisions

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→Defactoring: correction about replacement, and move tangential note to footnote
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a simpler, less mathematically involved way to explain the exact condition of saturation more accurately
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== Saturation ==
== Saturation ==
Suppose that we have a lattice of vectors of some kind; this could be a lattice of vals, monzos, or just vectors in general. Then a sublattice of this lattice is said to be '''saturated''' if it has the property that, if it contains a *multiple* of some vector from the original lattice, it also contains the vector itself. This definition can be a mouthful, so we'll unpack it with some examples.
A temperament matrix is '''saturated''' when it represents a temperament without any redundancies due to a common factor. A [[mapping]] is saturated when no common factor is found in its rows (i.e. generator mappings). A [[comma basis]], the dual of a mapping, is saturated when no common factor is found in its columns (i.e. comma vectors).


Suppose our main lattice is the lattice of monzos; these all have integer coefficients. Then we can look at matrices in which the columns are monzos, which can represent a set of vanishing commas. Then we can obtain a sublattice of monzos by looking at the set of all integer-weighted linear combinations of the columns, which represent all of the vanishing commas in some temperament; this is called the '''column lattice''' or '''integer column span''' of our matrix. This sublattice is said to be '''saturated''' if it doesn't contain any comma whose monzo coefficients have a GCD greater than one, unless it also contains the comma obtained by dividing the monzo through by this common divisor. In plain English, this means that the lattice of vanishing commas cannot have any vanishing comma which is the square, or cube, etc, of some other comma, unless that other comma is in the lattice of vanishing commas as well. In this situation, we say that the matrix is saturated as well and that it properly represents a temperament. If not, the matrix is conventionally thought to either not represent a temperament at all, or at least be a pathological representation of a temperament, as explained in the section on '''torsion''' below.
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<ref>If the multiples used on the linear combinations themselves have a GCD>1, the resulting row will always have a GCD>1, and such a linear combination therefore can not be used to demonstrate unsaturation. For example, consider the matrix {{ket|{{map|1 0 -4}} {{map|0 1 4}}}}; we can find the linear combination of rows 2×{{map|1 0 -4}} + 4×{{map|0 1 4}} = {{map|2 4 8}}, which has a GCD of 2, but that's clearly a result of the fact that we used 2× and 4× of the original rows, and 2 and 4 have a GCD of 2. So this linear combination doesn't prove that the matrix is unsaturated. It's still possible that another linear combination might prove it, but this one does not.</ref>. 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.


Similarly, suppose our main lattice is the lattice of vals; these again have integer coefficients. Then given any [[mapping|mapping matrix]], which will have each row equal to some val, we can obtain a sublattice of vals by looking at the set of all integer-weighted linear combinations of the rows of our mapping matrix, which happens to be the sublattice of all vals that support the temperament in question. This is called the '''row lattice''' or '''integer row span''' of our matrix. This sublattice is said to be '''saturated''' if it doesn't contain any val whose coefficients have a greatest common divisor greater than one, unless it also contains the corresponding val obtained by dividing the coefficients through by this greatest common divisor. If the mapping is *not* saturated, it is conventionally thought to not represent a temperament at all, but rather a pathological object which is called "contorted," as explained in the section on '''contorsion''' below.
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]].
 
For example, {{ket|{{map|3 0 -1}} {{map|0 3 5}}}} is ''not'' a saturated matrix, because {{map|3 0 -1}} - {{map|0 3 5}} = {{map|3 -3 6}}, which has a common factor of 3, and there is no integer linear combination which can produce the corresponding val with the factor of 3 removed, which would be {{map|1 -1 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, as there is of course no way to produce the GCD-reduced version of {{map|12 19 28}} by simply multiplying by an integer.
 
For the purposes of temperament representation, being unsaturated is typically a bad thing. If the matrix is a comma basis, the unsaturation is called [[Saturation, torsion, contorsion, and defactoring#Torsion|torsion]], and if it a mapping, 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]].


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]].
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''' 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''.
'''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 that the ratio itself is indeed ''not'' tempered out, but somehow the power of the ratio is; thus within this formalism unsaturated comma bases represent musically absurd pathological objects. Within a linear algebra formalism, there isn't quite as absurd an interpretation, but the historical usage of the group theory formalism is one reason why it is still viewed as bad practice to use comma bases with 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 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.
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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