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)". For example: blackdye is a (in fact, the) subst 5L(2m3s). This means that ignoring the distinction between m and s, the five L steps and the two remaining step sizes are distributed in a 5L5X MOS pattern, but instead of using X, the MOS pattern for 2m3s is used.

For example, if the template MOS is LXLXX, and the filling MOS is mss, then the resulting MOS substitution scales are LmLss, LsLms, and LsLsm. The first scale is denoted "subst 2L(1m2s 2|0)", using UDP notation for the filling MOS, and is said to be a "subst 2L(1m2s)". For the purpose of naming a specific subst 2L(1m2s) scale, we always substitute into the brightest mode of the template MOS, where X is treated as the smaller step.

The three subst 2L(1m2s) scales

Simplified UDP for filling MOS	Filling MOS	Step pattern		Denoted as
		Template MOS:	LXLXX
2|0	mss	LmLss		subst 2L(1m2s 2|0)
1|1	sms	LsLms		subst 2L(1m2s 1|1)
0|2	ssm	LsLsm		subst 2L(1m2s 0|2)

Examples

Simple examples

Many familiar ternary scales can be constructed as MOS substitutions.

LMsLMLs <- LLsLLLs x LMLML, LssLsLs x MsMs, sMssMss x LsLLs

LLmLLLs <- LLsLLLs x ms

LMsLM <- LLsLL x LMLM

Aberrismic scales

Diasem: LmLsLmLsL <- LsLsLsLsL x msms

Blackdye: sLmLsLmLsL <- sLsLsLsLsL x 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.
    • Example: Diasem is a subst 5L(2m2s): LmLsLmLsL.
  • 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, "subst 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.