Saturation, torsion, and contorsion
- This is a general introduction to this concept; for a more mathematical take on this, see Mathematical theory of saturation.
In regular temperament theory, a temperament is saturated or defactored if its set of available intervals matches what is suggested by its mapping or comma basis. A temperament's mapping can fail with respect to saturation by being contorted, and its comma basis can fail through torsion.
This article briefly explains these issues; for lattice-based visualizations and intuitive explanations, see Pathology of saturation.
Contorsion
A temperament (more specifically, its mapping) displays contorsion or enfactoring if there is some generatable interval which no just intonation interval maps to. This generatable interval is a contorted generator, which has the property that under any appropriate generator basis, every just interval's mapping has a multiple of c of that generator, where c, the contorsion order, is greater than one. In a contorted temperament, all generator bases will contain at least one contorted generator. The overall contorsion order of the temperament is the product of all the seperate orders.[1].
For example, 5-limit 36et (with mapping [⟨36 57 84]]) uses 12 of its pitches per octave (the ones within 12edo) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of generators that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contorsion order 3. For a higher-rank example, the 13-limit 87 & 111 temperament Hemimist, with mapping [[⟨3 0 26 56 8]], [⟨0 2 -8 -20 1]]], when restricted to the 2.5.7.11 subgroup, has no just intonation interval corresponding to the period or the square of the period, although there is a just intonation interval (namely, 2/1) corresponding to the cube of the period. Thus, this a contorted generator with contorsion order 3.
If a temperament has a subgroup which is contorted, especially a subgroup with small primes (for example, the 11-limit subgroup of 23-limit 44et), that temperament will likely be easier to traverse than the number of generators required according to the mapping would suggest.
Torsion in temperaments
A temperament (more specifically, its comma basis) displays torsion or enfactoring if there is some interval mapped to zero which is not formable by multiplying commas in the basis. This interval is a comma with torsion, which has the property that commas in the basis can be multiplied to form the cth power of this ratio, but not that ratio itself or any smaller power, where c is the torsion order.
For instance, in a temperament with comma basis {6561/6250, 128/125}, there is an interval 81/80 which is not formable by multiplying commas in the basis, but is nevertheless forced to be mapped to zero because (81/80)^2 = (6561/6250)/(128/125) is part of the basis. Thus, 81/80 displays torsion with torsion order 2.
Torsion in periodicity blocks
A comma basis in the context of periodicity blocks displays torsion if it displays torsion as a temperament—precisely when there is some comma with torsion where commas in the basis can be multiplied to form the cth power of this ratio, but not that ratio itself or any smaller power, where c is the torsion order.
Within periodicity blocks, no mapping needs to be defined from a comma basis, so comma bases with torsion are able to form periodicity blocks where the smallest comma with torsion is not tempered out.
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. The simplest and fastest algorithm for saturating matrices is called column Hermite defactoring. For more information on such algorithms, see Defactoring algorithms.
History and terminology
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 SageMath 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].
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. Historically, a group-theory formalism was used to analyze comma bases with torsion, where the smallest comma displaying torsion was not made to vanish although a power of that comma was, which is musically impossible; using a linear algebra formalism as is preferred now, no such impossibility is suggested. The term contorsion was invented for RTT in 2002 by Paul Erlich[8], as a play on the word "co-torsion", being dual to the situation with "torsion" above.
In the case of temperaments, Dave Keenan and Douglas Blumeyer have proposed[9] and used defactoring as a replacement for saturation and enfactoring as a replacement for both torsion and contorsion. So, a mapping or comma basis of a temperament is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion).
See also
References and footnotes
- ↑ H. J. S. Smith On Systems of Linear Indeterminate Equations and Congruences, the overall order is equivalent to the GCD of a matrix's minor determinants, which Smith calls the "greatest divisor".
- ↑ 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.
- ↑ Google Scholar: Torsion group
- ↑ Stack Exchange | Where does the word "torsion" in algebra come from?
- ↑ Yahoo! Tuning Group | My top 5--for Paul
- ↑ See Defactoring terminology proposal for details.