Saturation, torsion, and contorsion

This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily. The corresponding expert page for this topic is Mathematical theory of saturation.

In regular temperament theory, a temperament (more specifically, its mapping) - displays contorsion if there are some pitches which no just intonation interval (within the temperament's subgroup) maps to. For example, the rank-1 5-limit temperament described by 24et is fairly accurate but only uses 12 of its pitches per octave (the ones within 12et) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 12 pitches, making 24et contorted in the 5-limit, inheriting its 5-limit representation from 12et. For a higher-rank example, septimal meantone in the 7-limit maps harmonic 3 to 1 meantone fifth, harmonic 5 to 4 fifths, and harmonic 7 to 10 fifths up. But if it is restricted to the subgroup 2.5.7, all just intonation intervals within that subgroup occur at even numbers of fifths up or down, because both 4 and 10 are even numbers, and so pitches located at odd numbers of fifths up or down do not have a representation in the 2.5.7 subgroup. The temperament containing the half of notes that occur at even fifths is in fact didacus, generated by the 2-fifth interval (in other words, a meantone whole tone, identified here as 28/25), and so we can say that septimal meantone is contorted in the 2.5.7 subgroup, inheriting this subgroup's representation from didacus.

The above examples depicted a situation where the intervals of a given subgroup occur every 2 pitches in the underlying temperament; but they can occur every 3, or 4, or any other number of generators apart. However many generators are needed to step from one interval in your chosen subgroup to the next is the contorsion order of that subgroup within the temperament. (In the case of higher-rank temperaments, it is possible for different generators to have different contorsion orders.)

A temperament (more specifically, its comma basis) displays torsion if it tempers out a power of some ratio, but does not temper out that ratio. For instance, in a temperament with comma basis {6561/6250, 128/125}, (81/80)^2 = (6561/6250)/(128/125) is tempered out but 81/80 is not explicitly tempered out. In this temperament, there is no clear way to assign a pitch to 81/80; for this reason, torted temperaments are not particularly useful. Similarly to the concept of contorsion order, torsion order can be defined as the lowest power of a generic just intonation interval that is necessarily part of the temperament's lattice.

A temperament is saturated if it is neither torted nor contorted.

In general, being unsaturated is a bad thing, as 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 or a mapping, discussed below. For all these reasons, unsaturated matrices are typically considered to not truly represent temperaments.

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.

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 cannot: 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.

In practical terms, a contorted edo may do a relatively good job of approximating a given JI lattice, but it does not actually use all notes from the tuning, instead looping back to 1/1 before using up all the notes. For example, 24edo is not too bad at approximating the 5-limit, but it only uses its 12edo subset to do that, since the odd degrees of 24edo fall outside of the 5-limit approximation.

For a more detailed discussion on these issues, see Pathology of saturation.

Mathematical definition

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.

Saturation algorithms

An unsaturated mapping or comma basis can be made saturated, 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.

Terminology

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".

The term torsion has been used since at least as early as 1932 and came to RTT from the mathematical field of group theory. 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.

Dave Keenan and Douglas Blumeyer have proposed defactoring as a replacement for saturation, and enfactoring as a replacement for both torsion and contorsion. So, a mapping or comma basis is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion). These terms are used in their writings. See Defactoring terminology proposal for details.