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[1]. 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.
The term saturation was coined by Nicolas Bourbaki in 1972[2], 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[3]. The earliest identified terminology for this concept was in 1861 by H. J. S. Smith[4] who called saturated matrices "prime matrices"[5].
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.
The term torsion has been used since at least as early as 1932[6][7] 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).
The term contorsion was invented for RTT in 2002 by Paul Erlich[8].
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 were coined by Dave Keenan in collaboration with Douglas Blumeyer in 2021 as replacement terms for saturation, torsion, and contorsion; for more information, see Defactoring terminology proposal.
References
- ↑ 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.
- ↑ https://pdfcoffee.com/commutative-algebra-bourbaki-pdf-free.html
- ↑ 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.
- ↑ See "On Systems of Linear Indeterminate Equations and Congruences", which can be accessed here: https://www.jstor.org/stable/pdf/108738.pdf, where Smith also describes the GCD of a matrix's minor determinants as its "greatest divisor" (though neither of this term nor "prime matrix" seems to have caught on).
- ↑ https://scholar.google.com/scholar?q=%22torsion+group%22&hl=en&as_sdt=0%2C5&as_ylo=1900&as_yhi=1940
- ↑ https://math.stackexchange.com/questions/300586/where-does-the-word-torsion-in-algebra-come-from
- ↑ https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456