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.

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: an infinite family of MOS substitution scales of type ax(bybz) where a is odd and gcd(a, b) = 1 (example: diasem)
   • even-regular scales: an infinite family of 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 MV3 scales

Odd-regular MV3 scales have a structure where the generator chain of a 1-period MOS is detempered into a stack 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, 3L2m2s) is an odd-regular scale pattern; GS(5/4, 6/5) generates RH and GS(6/5, 5/4) generates LH.

Even-regular MV3 scales

Even-regular MV3 scales have two generator chains offset by a len(scale)/2-step interval, with "bad" generators only occurring on 2 notes. This generator comes from detempering a generator of ax2bX 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 + 2X is a generator of the MOS 2m10X.

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. Put 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.

MOS substitution

Main article: MOS substitution

MOS substitution generalizes odd-regular and even-regular scales.