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

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).

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, [⟨3 0 -1] ⟨0 3 5]⟩ is not saturated, because ⟨3 0 -1] - ⟨0 3 5] = ⟨3 -3 6], which has a common factor of 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.

Being unsaturated is, generally speaking, a bad thing. 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.

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.

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.

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).

Defactoring

Defactoring is a synonym for saturation, so 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). Defactored and enfactored have been proposed as replacement terms for saturation, torsion, and contorsion; for more information, see Defactoring terminology proposal.