User:Ganaram inukshuk/Notes: Difference between revisions

== On the Origin of MOS Recursion ==

[[Recursive structure of MOS scales|MOS recursion]] describes a set of properties that all moment-of-symmetry scales share that, among other things, allows us to create a few algorithms for determining whether an arbitrary scale of large and small steps has those properties.

* s->L

Applying ruleset 1 to itself n times produces ruleset 3, where L produces an L followed by n s's:

* s->L

Reversing the order of L's and s's of ruleset 2 produces this intermediate ruleset:

#* s->sLL...L (n L's)

The chunking operation is contingent on rulesets 5 and 6 since there must be two unique chunks whose string sizes differ by exactly one L or one s. Repeatedly applying these rules as reduction rules on a valid moment-of-symmetry scale will reduce the scale to a progenitor scale of either Ls or sL. The reduction rules can be denoted as such:  

== On Modal Brightness and Numeric Encoding ==

Modal brightness typically refers to how "bright" or "dark" the usual diatonic modes are (lydian, ionian, mixolydian, dorian, aeolian, phrygian, locrian). Since diatonic (5L 2s) is one of many moment-of-symmetry scales, the idea of modal brightness can be generalized using [[UDP notation]].

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Therefore, to produce the modes of a MOS in descending modal brightness, start with the scale code, produce all of its possible shifts, interpret them as binary numbers, and sort them in descending order. It should be noted that the characters "L" and "s", when sorted in lexicographic order (IE, alphabetical order), equivalently represent the binary representations in descending order, so the conversion to binary numbers is technically not necessary.

As an example, consider [[3L 4s]] represented as sLsLsLs. Its six other shifts are LsLsLss, sLsLssL, LsLssLs, sLssLsL, LssLsLs, and ssLsLsL. Sorting them produces LsLsLss, LsLssLs, LssLsLs, sLsLsLs, sLsLssL, sLssLsL, and ssLsLsL, and are enumerated using UDP notation from 6|0 to 0|6 accordingly. Again, the binary representation (and decimal forms) gives an intuitive sense of what it means for a scale to be bright. As of writing, the article on 3L 4s is written using sLsLsLs (UDP 3|3) as the "default" mode, or the mode represented using middle C as the root (or TAMNAMS middle J); in comparison, the default mode for diatonic is ionian (UDP 5|1, or LLsLLLs). UDP notation gives a sense of how many modes are brighter or darker starting from the default mode, though these sortings (and thereby binary encodings) provide that sense without any notion of a "default" mode.

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=== Including the Modes of More than One MOS ===

As a curiosity, there are 128 possible 7-bit numbers (0000000 to 1111111) representing the unsigned integer values of 0 to 127. Among the 6 possible heptatonic MOSses (1L 6s, 2L 5s, 4L 3s, 3L 4s, 5L 2s, and 6L 1s), there are therefore 42 modes total. For our purposes, we include equiheptatonic (7 equal divisions of the octave) as being represented by both 0000000 and 1111111 (or simultaneously being both 0L 7s and 7L 0s) for a total of 43 (or 44) scales.

=== Including Assigned Values for L and s ===