User:Hkm

fan of 72edo and contorted edos

I plan to make pages more accessible to beginners, and also create an introduction page for the wiki and xen in general

I've created a map of regular temperaments, which I might put on the wiki soon

DRAFT BELOW

In regular temperament theory, a temperament (more specifically, its mapping) - displays contorsion if there are some interval which no just intonation interval maps to. If a temperament is contorted, there is some contorted generator where every just interval's mapping has a multiple of c of that generator, where c, the contortion order, is greater than one. The largest contortion order is called the greatest factor.

For example, the 5-limit 36et (with mapping [36 57 84]) 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 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 contortion order 3. For a higher-rank example, the 3.5.7 restriction of septimal sensi, with mapping [⟨-1 -1 -2], ⟨7 9 13]] and mapping generators "~2", ~9/7, where "~2" is in quotes because it is no longer in the subgroup. This temperament has the interval corresponding to the period which no just intonation maps to. We can thus find a contorted generator; namely, [FIGURE THIS OUT]

A temperament (more specifically, its comma basis) displays torsion if there is no mapping which describes it. If a temperament has torsion, it tempers out a power c of some ratio, the contorted comma, but does not temper out that ratio, where c is the torsion order. 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, temperaments with torsion 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 shows neither torsion or contortion.

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

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.

History and 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. 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, 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.