Even-regular MV3 scale

An even-regular MV3 scale is a type of ternary scale with an even number of notes per period. An even-regular MV3 scale consists of two identical generator chains, where all generators are identical and subtend the same step class. The two chains are offset by an interval that subtends k steps in a 2k-note even-regular MV3 scale.

Notable even-regular MV3 scales

• Achiral diachrome (dia5s)
• Penslen (slen5m)
• Mosh3s
• Cthon5m

Properties

• Even-regular MV3 scales always satisfy all 3 of the monotone-MOS conditions.
• Another characterization of even-regular MV3 scales is that it is a ternary one-to-one detempering of a 2-period MOS word M(X, z) which has the form w(x, y, z)w(y, x, z) for some ternary word w and some permutation x, y, z of L, m, s where x and y always alternate in the scale. One even-regular MV3 scale is the achiral variant of diachrome.
• Even-regular MV3 scales are maximum variety 3 (MV3) but not strict variety 3 (SV3), and by the MV3 classification theorem a balanced single-period MV3 scale that has an even number of notes is always even-regular MV3 and has step signature aXaYbZ where a is odd and b is even.
• Even-regular MV3 scales are achiral. There is only one even-regular MV3 scale pattern for a given scale signature if it exists.
• If an even-regular MV3 is oddly even, then it is an interleaving of two odd-regular MV3's of opposite chiralities. If it is evenly even, then it is an interleaving of two copies of the same even-regular MV3, except in the trivial case xyxz where it is an interleaving of two 2-note MOSes (x+y)(x+z).

Terminology

The term even-regular MV3 has been coined by Inthar.

See also

• Odd-regular MV3 scale
• Ternary scale theorems