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)". 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); to make this notation unique for a particular given MOS-substitution scale, the brightest mode for the template MOS is conventionally used, treating the slot letter X as the smaller step.
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.
| 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. 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.
