Defactoring: Difference between revisions

Elsewhere, [[Normal_lists|Integer Reduced Row Echelon Form]], or IRREF, has been proposed as a normal form for mappings. It has a similar problem as HNF does, however, in that it does not always defactor matrices. Worse, even, sometimes IRREF introduces enfactoring where before there was none! For example, consider this mapping for 5-limit porcupine, {{vector|{{map|7 11 16}} {{map|22 35 51}}}}. This mapping is not enfactored, but its IRREF is {{vector|{{map|3 0 -1}} {{map|0 3 5}}}}, which is 3-enfactored. More on this later.

If you've studied RTT extensively, you've probably encountered the terms [[Saturation|"saturated" and "contorted"]] that are sometimes used to describe mappings. These two terms each have several flaws, and so this article presents alternative terms that are clearer and more descriptive: "defactored" and "enfactored", respectively. These new terms were coined by [[Dave Keenan]] in collaboration with [[Douglas Blumeyer]] in June of 2021<ref>Many, many other terms were considered before arriving at defactored and enfactored, including but not limited to: repeated, (up/down)sampled, decimated, divided/multiplied, divisive/multiplicative, completed/depleted/repleted/pleated, efficient, brown, dry, spongy/holey, fluffy, corrugated, copied, shredded, tupled, tupletted, enphactored (where the ph stood for possibly hidden)...</ref>.  

Several concerns with the term "saturation" may be identified:

Indeed 18 is the common factor in {{vector|{{map|11 16 21}} {{map|19 26 33}}}}.</ref>. We think we should avoid propagating Sage's decision to overload matrix saturation with a second meaning.  

# Furthermore, there is another common but conflicting sense of saturation for matrices which clamps entry values to between -1 and 1<ref>See https://math.stackexchange.com/questions/1964814/linear-transformation-of-a-saturated-vector and https://faculty.uml.edu//thu/tcs01-june.pdf</ref>.  

As for the term "contorsion", the concerns with it are:

# It is prone to spelling confusion. People commonly refer to temperaments with contorsion as "contorted". But contorted is the adjective form of a different word, contortion, with a t, not an s. The proper adjective form of contorsion would be contorsioned. Would you use "torted" instead of torsioned? Or would people prefer "torsional" and "contorsional", even though that suggests only of or pertaining to in general rather than having the effect applied.<ref>If it was meant to most strongly evoke duality with torsion, it should have been spelled "cotorsion". Naming it "contorsion" is an annoying step toward "contortion" but stopping halfway there. But this isn't a strong point, because duality with torsion was the false assumption mentioned above.</ref>  

# Due to its similarity with the word "contortion", the word contorsion evokes bending, twisting, and knotting. But there is nothing bendy, twisty, or knotted about the effect it has on JI lattices or tuning space.

In accordance with this research and reasoning, this article henceforth will eschew the terms saturation and contorsion in favor of defactored and enfactored.