Saturation, torsion, and contorsion

This is a general introduction to this concept; for a more mathematical take on this, see Mathematical theory of saturation.

Saturation, torsion, contorsion, and defactoring are all terms for the same effect in RTT, but used in slightly different ways.

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.

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.

Similarly, suppose our main lattice is the lattice of vals; these again have integer coefficients. Then given any 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.

For example, [⟨3 0 -1] ⟨0 3 5]⟩ is not a saturated matrix, because ⟨3 0 -1] - ⟨0 3 5] = ⟨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 ⟨1 -1 3]. A mapping which consists of a single row with a common factor, such as [⟨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 ⟨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 torsion, and if it a mapping, the unsaturation is called 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 wedgies. 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, 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. The earliest identified terminology for this concept was in 1861 by H. J. S. Smith who called saturated matrices "prime matrices".

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.

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.

The term torsion has been used since at least as early as 1932 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).

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, as a play on the word "co-torsion", being dual to the situation with "torsion" above.

Defactoring

Defactoring is the term used in the writings of Dave Keenan and Douglas Blumeyer, a list of which can be found here, as a proposed replacement for saturation, torsion, and contorsion. Like saturation, 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). See also Defactoring terminology proposal.