Ternary scale

A ternary scale is a scale with exactly 3 step sizes. This article is an overview of the theory of ternary scale patterns.

Relationship with rank

Ternary scales with independent step sizes are rank-3. However, rank-3 scales need not be ternary. Duodene (16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2/1) is an example of a rank-3 (2.3.5) quaternary scale.

MV3 ternary scales

A maximum variety 3 (MV3) scale is a scale whose ordinals come in at most 3 sizes.

Single-period MV3 ternary scale patterns come in the following types:

1. pairwise-MOS: scales such that identifying any two step sizes always results in a MOS
   • abacaba scales
   • odd-regular scales: MOS substitution scales of type ax(bybz) where a is odd and gcd(a, b) = 1 (example: diasem)
   • even-regular scales: MOS substitution scales of type ax(bybz) where a is even and gcd(a, b) = 1 (example: penslen)
2. abcba scales: the unique sporadic non-pairwise-MOS MV3 ternary pattern
3. twisted MV3 scales (see below): an infinite family of non-pairwise-MOS MV3 ternary scales

Odd-regular scales

Odd-regular scales have a structure where the generator chain of a 1-period MOS is detempered into a generator sequence of two alternating generators, g1 and g2, with a "bad" generator different from both on only one note. Odd-regular scales are chiral; switching g1 and g2 corresponds to using the other chirality.

Example: zarlino (RH LmsLmLs and LH LsmLsLm, 3L(2m2s)) is an odd-regular scale pattern; GS(5/4, 6/5) generates RH and GS(6/5, 5/4) generates LH.

Even-regular scales

Even-regular scales have two parallel generator chains offset by a len(scale)/2-step interval, with "bad" generators only occurring on 1 note per chain. This generator comes from detempering a generator of ax2bX (a 2-period MOS) for an even-regular scale of MOS substitution type ax(bybz). For example, achiral diachrome (LsLsLmsLsLsm, 2m(5L5s)) has a 5-step interval 2L + m + 2s (interpreted as 4/3) as a generator, and m + 4X (period-complement of X) is a generator of the MOS 2m10X. Even-regular scales are achiral.

Even-regular scales have an interleaving structure: if n > 4, an n-note even-regular scale is an interleaving of two copies of an n/2-note even-regular scale if n is a multiple of 4, and is an interleaving of the two opposite chiralities of an n/2-note odd-regular scale otherwise. Examples:

• Mosh3s (4m(3L3s), LmsLmsmLsm) is an interleaving of (Lm)(sL)(ms)(mL)(sm) = LmsLs and (ms)(Lm)(sm)(Ls)(mL) = sLsmL
• Penslen (6s(5L5m), msLsmLsmLsmsLmsL) is an interleaving of (ms)(Ls)(mL)(sm)(Ls)(ms)(Lm)(sL) = smLsmsLm and (sL)(sm)(Ls)(mL)(sm)(sL)(ms)(Lm) = msmLsmsL

Twisted MV3 scales

Twisted MV3 scales are much like MOS substitution scales, but the "template MOS" is not actually a MOS, but a "twisted" multiMOS, which is constructed as follows:

1. Take the brightest (or darkest) mode of a multiMOS, where the period (one MOS unit) has exactly one step count X even (e.g. 2X 1s, XXs -> 6X 3s, XXsXXsXXs)
2. Mark borders between units: XXs|XXs|XXs
3. Change s|X -> X|s (or vice versa) at some of the borders: XXX|sXs|XXs

Now do the letterwise substitution procedure, as in MOS substitution, with X as the slot letter, with filling MOS alternating: LmLsmsLms = XXXsXsXXs(LmLmLm).

Twisted MV3 scales always have step signature kaxkaykbz with k > 1.

Interleaving

If w(x, y) (a sequence of step sizes x and y) is a binary scale, then the ternary scale w(z(x-z), z(y-z)) (the same sequence but with substitutions x -> z(x-z), y -> z(y-z)) is an interleaving, namely cross-set(w(x, y), z).

Blackdye (sLmLsLmLsL, 5L(2m3s)) is an example: it is w(L+s, m+s) where w = 2L3s.

MOS substitution

Main article: MOS substitution

MOS substitution generalizes odd-regular and even-regular scales (as well as the above interleaving construction).

See also

• Aberrismic theory: A compositional theory using ternary scales