Saturation, torsion, and contorsion
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 maps). 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 coprime integer 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.
The greatest factor possible to create as a GCD of a coprime linear combination of rows in this way is called the mapping's greatest factor, and so an alternative definition of saturation would be that the mapping must have a greatest factor of 1.
Being unsaturated is, in most cases, a bad thing. 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 torsion, or a mapping, in which case 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 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 and contorsion
Torsion is 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 made to vanish but does not explicitly state that the ratio itself is made to vanish (for instance, (81/80)2 is made to vanish 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 made to vanish, 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 occurs in a similar situation where the list of commas defines a periodicity block; in fact, this was its original use case.
Contorsion is 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). The greatest factor of a mapping has been called its contorted order.
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.
Enfactoring is a proposed replacement for both torsion and contorsion. So, a mapping or comma basis is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion).
References and footnotes
- 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 [⟨1 0 -4], ⟨0 1 4]⟩; we can find the linear combination of rows 2 × ⟨1 0 -4] + 4 × ⟨0 1 4] = ⟨2 4 8], which has a GCD of 2, but that is 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 does not prove that the matrix is unsaturated. It is still possible that another linear combination might prove it, but this one does not.
- This term is inspired by H. J. S. Smith's On Systems of Linear Indeterminate Equations and Congruences, where Smith describes the GCD of a matrix's minor determinants as its "greatest divisor". "Divisor" and "factor" are synonyms and they prefer "factor" for its connection with the term "defactor".
- 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.
- Nicolas Bourbaki. Commutative Algebra
- It may also have come through PARI/GT.
- H. J. S. Smith is the creator of the Smith normal form used in Gene Ward Smith's saturation algorithm.
- Also from On Systems of Linear Indeterminate Equations and Congruences, linked above. Neither prime matrix nor greatest divisor seems to have caught on in the mathematical community.
- A list of which can be found in Douglas Blumeyer #Some of his work here on the wiki
- Google Scholar: Torsion group
- Stack Exchange | Where does the word "torsion" in algebra come from?
- Yahoo! Tuning Group | My top 5--for Paul