Generator-offset property

A scale satisfies the alternating generator property (also alt-gen or AG) if it satisfies the following equivalent properties:

• the scale can be built by stacking alternating generators
• the scale is generated by two chains of generators separated by a fixed interval, and the lengths of the chains differ by at most one.

The Zarlino (3L 2M 2S) JI scale is an example of an alt-gen scale, because it is built by stacking alternating 5/4 and 6/5 generators. Diasem (5L 2M 2S) is another example, with generators 7/6 and 8/7.

More formally, a cyclic word S (representing a periodic scale) of size n is alt-gen if it satisfies the following equivalent properties:

1. S can be built by stacking a single chain of alternating generators g₁ and g₂, resulting in a circle of the form either g₁ g₂ ... g₁ g₂ g₁ g₃ or g₁ g₂ ... g₁ g₂ g₃.
2. S is generated by two chains of generators separated by a fixed interval; either both chains are of size n/2, or one chain has size (n + 1)/2 and the second has size (n − 1)/2.

These are equivalent, since the separating interval can be taken to be g₁ and the generator of each chain = g₁ + g₂. This doesn't imply that g₁ and g₂ are the same number of scale steps. For example, 5-limit blackdye has g₁ = 9/5 (a 9-step) and g₂ = 5/3 (a 7-step).

Conjectures

Conjecture 2

If a non-multiperiod 3-step size scale word is

1. unconditionally MV3,
2. has odd cardinality,
3. is not of the form mx my mz,
4. and is not of the form xyxzxyx,

then it is alt-gen. (a converse to Theorem 1)