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 may be impossible.
Convention
MOS substitution scales are denoted using the notation "subst ax(bycz)" or just "ax(bycz)", called the scale's (MOS substitution) signature or (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).
| 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:
| 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.
- Odd-regular MV3 scales are subst ax(ky kz)⟨something⟩ scales where a is odd. Such scales come in a chiral pair.
- 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. However, some have nicer lattices in the sense that they are substrings of a parallelogram traversal. (We just call such scales "parallelogram substring scales".) If the template MOS is primitive, and the slot letters in the template MOS's generator to be the 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.
