MOS substitution

MOS substitution is a procedure for obtaining a ternary (3 step sizes) scale from two MOS patterns. It consists of taking one MOS pattern (called the template MOS), choosing a step size, and overwriting all instances of that step size using the step pattern of another MOS pattern (called the filling MOS), step size for step size. A MOS substitution scale may have any combination of step sizes and is meant as a way to somewhat evenly distribute three step sizes.

Convention

MOS substitution scales are denoted using the notation "subst ax(bycz)" or just "ax(bycz)", called the scale's (MOS substitution) signature. Here the template MOS is ax(b+c)X where X is the slot letter, and the filling MOS is (a mode of) bycz. Any particular scale of a given MOS substitution type is said to be "a subst ax(bycz)" or "a scale of type ax(bycz)". A specific MOS substitution scale may be denoted template_MOS_with_slot_letter_X(filling_MOS).

The three subst 2L(1m2s) scales

Simplified UDP for filling MOS	Filling MOS	Step pattern		Denoted as
		Template MOS:	LXLXX
2|0	mss	LmLss		LXLXX(mss)
1|1	sms	LsLms		LXLXX(sms)
0|2	ssm	LsLsm		LXLXX(ssm)

Examples

Simple examples

Many familiar ternary scales can be constructed as MOS substitutions.

LMsLMLs = XXsXXXs(LMLML), LXXLXLX(MsMs), XMXXMXX(LsLLs)

LLmLLLs = LLXLLLX(ms)

LMsLM = XXsXX(LMLM)

Aberrismic scales

Diasem: LmLsLmLsL = LXLXLXLXL(msms)

Blackdye: sLmLsLmLsL = XLXLXLXLXL(smsms)

The three standard diaslen scale patterns, denoted 4sL (LsLsLmLsLsm), 4sC (LmLsLsLmLss), and 4sR (LsLmLsLsLms), may be constructed via MOS substitution, giving generator sequences for the three scales:

diaslen scales as MOS substitution scales

chirality	filling MOS	step pattern		generator sequence
		template MOS:	LXLXLXLXLXX	intvl. class of gen.:	2-steps
4sC	mssmss	LmLsLsLmLss		GS(L+m, L+s, L+s)
4sR	smssms	LsLmLsLsLms		GS(L+s, L+m, L+s)
4sL	ssmssm	LsLsLmLsLsm		GS(L+s, L+s, L+m)

More general classes of MOS substitutions

• Ternary interleaved-MOS scales such as blackdye (5L2m3s, sLmLsLmLsL) and whitedye (5L2m7s, sLsLsLsmsLsLsm) are MOS substitution scales. In those particular examples, the template MOS is of the form ky kz.
• MOS substitution scales where the filling MOS is of the form ky kz are of the following two types:
  • Odd-regular MV3 scales are subst ax(ky kz)⟨something⟩ scales where a is odd. Such scales come in a chiral pair.
    • Example: Diasem (5L(2m2s)) has chiral variants LmLsLmLsL and LsLmLsLmL.
  • Even-regular MV3 scales (such as 5sC diachrome) are subst ax(ky kz)⟨something⟩ scales where a is even and k is odd. In this case, "ax(ky kz)" specifies the scale uniquely.
    • Example: 5sC diachrome is a subst 2m(5L5s): LsLsLmsLsLsm.
• A ternary scale is always a MOS substitution scale if the L = m and s = 0 monotone-MOS properties are satisfied.

Properties

If the template MOS is single-period, the MOS substitution scale has a generator sequence using only two distinct intervals. The length of this GS is the length of the period of the filling MOS. Hence if you want a short generator sequence, you should use a multiperiod filling MOS, like for diaslen.

Caveats

MOS substitution scales in general still have somewhat messy lattices. If you want an especially nice lattice (with two distinct generators in the GS, and only one generator different in that GS) you should additionally want the number of slot letters in the template MOS's generator to be the generator interval class of the filling MOS, again like for diaslen.