Ternary scale

A ternary scale is a scale with exactly 3 step sizes.

Ternary scales with independent step sizes are rank-3. However, rank-3 scales may not be ternary.

MV3 ternary scales

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

Classification

The MV3 ternary scale classification theorem states that 1-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: an infinite family of MOS substitution scales of type ax(bybz) where a is odd and gcd(a, b) = 1
   • even-regular scales: an infinite family MOS substitution scales of type ax(bybz) where a is even and gcd(a, b) = 1
2. abcba scales: the unique sporadic non-pairwise-MOS MV3 ternary pattern
3. twisted MV3 scales (see below): the one infinite family of non-pairwise-MOS MV3 ternary scales

(For a proof, see Bulgakova, Buzhinsky, and Goncharov's 2023 paper, "On balanced and abelian properties of circular words over a ternary alphabet".)

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. 2L 1s, LLs -> 6L 3s, LLsLLsLLs)
2. Put borders between units: LLs|LLs|LLs
3. Change s|L -> L|s at some of the borders: LLL|sLs|LLs

Now do the letterwise substitution procedure, as in MOS substitution, with X as the slot letter, with filling MOS alternating: LmLsmsLms. So we have a substitution word XXXsXsXXs[LmLmLm].

Twisted MV3 scales always have step signature kax kay kbz.