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, even when "perfectly" even distribution (in the sense of every step size occupying MOS positions) may be impossible.

Convention

MOS substitution scales are described using the notation "subst ax(bycz)" or just "ax(bycz)", called the scale's (MOS substitution) type. Here the template MOS is ax(b+c)X where X is the slot letter, and the filling MOS is (a mode of) 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.
  • • 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.

Parallelogram substring scales

MOS substitution scales in general still have somewhat messy pitch-class lattices (or they may not have one at all, due to the equave being equally divided). However, some have nicer lattices in the sense that they are substrings of a row-by-row (lexicographical) parallelogram traversal. That is, the pitch classes form a parallelogram (a rectangle under some basis)

1. with all rows traversed in the same direction
2. with some number (possibly zero) of full middle rows
3. the first row is not required to be full but must be a suffix of a full row
4. the last row is not required to be full but must be a prefix of a full row.

We call such scales "parallelogram substring scales". Not all ternary parallelogram substring scales are MOS substitution scales.

If the template MOS is single-period and the number of slot letters in a template MOS's generator is a generator interval class of the filling MOS (e.g. diaslen), the MOS substitution scale is a parallelogram substring scale. What other MOS substitution scales satisfy the parallelogram substring property is still being investigated.