Music Theory for Conworlders: Other MOS structures

Okay, so I've established a handful of MOS structures this far:

ssLsL (pentatonic major, s = major second, L = minor third)
LLsLLLs (diatonic major, L = major second, s = minor second)
ssLsssL ('anti-diatonic' mavila, s = slightly wide minor second, L = slightly wide major second)

I am not sure if I gave any 'generally' applicable MOS patterns, but there's one pretty simple one. Examples of it will appear in any set:

AA*B 

where * is the kleene star. This reads out 'one or more A before one B'. Let A, B ∈ {L,s} and A≠B. Since we permit 'rotating' the set, M = A*BA*, where |M| > 2 also is a reasonable description - viz. any string that is at least two symbols long and has at most one B, and an arbitrary number of As before and/or after the B.

This gives us things such as LLLLs, ssssL, LLLLLs, sssssssssL, ... the reason why these are all MOSes should be fairly obvious - all seconds except one will be of the form A (and one of the form B), all thirds will be AA, except two - one will be AB, one will be BA, but the length of AB and BA is the same so we can consider them both to have the length |A|+|B|; for fourths, we get ABA, BAA and AAB as well as AAA, etc.

But these are only a subset of possible MOSes. We can look at an example that is in some use in Bohlen-Pierce:

LssLsLsLs

Or at the 11 edo version of a temperament by the name 'orgone':

LsLsLsL

and another one which I have not found a name for:

ssLssLsL

The plot thickens. Apparently, (AB)*A or (AB)*A(AB)* or somesuch qualifies - Any number of ABs, where A, B ∈ {L,s} and A≠B, and one A at the end - again, permitting rotation of these as well.
The last example also indicates that maybe things like (AB^N)*AB^N-1? I am not actually entirely sure on whether all scales generated by that form will qualify, although all the previous forms given here do qualify.

And of course, I have previously mentioned the trick of pretending that a MOS is really an equal temperament, and then picking out notes within that MOS by numbering the notes as a sequence and then picking such notes by use of MOSes. I.e. Pentatonic-in-seven as MOSes within the diatonic scale, here given with the 'root' for each 'basic' rotation in large print. Keep in mind that one can further of course 'rotate' within the subset itself:

C	D	E	F	G	A	B
*	*	*		*	*
	*	*	*		*	*
*		*	*	*		*
*	*		*	*	*
	*	*		*	*	*
*		*	*		*	*
*	*		*	*		*

If we were to look at each of these scales in terms of 12-tone equal temperament intervals from the scale root, we'd get:

min2	maj2	min3	maj3	p4	aug4	p5	min6	maj6	min7	maj7
	*		*			*		*
	*	*				*		*
*		*				*	*
	*		*			*		*
	*		*			*		*
	*	*				*	*
*		*			*		*

And all of these have their own rotations as well, except of course the three ones I've marked with red, which have identical structures and thus we can ignore the rotations two of them contribute, and the same goes for the green pair. A short not very rigorous investigation suggests to me that the rotations. I probably should code a thing to generate and rotate these for me and then output them as html tables, but I am a bit too tired today.