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.

Theorems

Theorem 1

Let S be a 3-step-size scale word in L, M, and s, and suppose S is alt-gen. Then:

1. S is unconditionally MV3 (i.e. MV3 regardless of tuning).
2. S is of the form ax by bz for some permutation (x, y, z) of (L, M, s).
3. The cardinality (size) of S is either odd, or 4 (and S is of the form xyxz).

Proof

Assuming alt-gen, we have two chains of generator g₀ (going right). The two cases are:

CASE 1: EVEN CARDINALITY
O-O-...-O (n/2 notes)
O-O-...-O (n/2 notes)

and

CASE 2: ODD CARDINALITY
O-O-O-...-O ((n+1)/2 notes)
O-O-...-O ((n-1)/2 notes).

Label the notes (1, j) and (2, j), 1 ≤ j ≤ (chain length), for notes in the upper and lower chain respectively.

In case 1, let g₁ = (2, 1) − (1, 1) and g₂ = (1, 2) − (2, 1). We have the chain g₁ g₂ g₁ g₂ ... g₁ g₃.

Let r be odd and r ≥ 3. Consider the following abstract sizes for the interval class (k-steps) reached by stacking r generators:

1. from g₁ ... g₁, we get a₁ = (r − 1)/2*g₀ + g₁ = (r + 1)/2 g₁ + (r − 1)/2 g₂
2. from g₂ ... g₂, we get a₂ = (r − 1)/2*g₀ + g₂ = (r − 1)/2 g₁ + (r + 1)/2 g₂
3. from g₂ (...even # of gens...) g₁ g₃ g₁ (...even # of gens...) g₂, we get a₃ = (r − 1)/2 g₁ + (r − 1)/2 g₂ + g₃
4. from g₁ (...odd # of gens...) g₁ g₃ g₁ (...odd # of gens...) g₁, we get a₄ = (r + 1)/2 g₁ + (r − 3)/2 g₂ + g₃.

Choose a tuning where g₁ and g₂ are both very close to but not exactly 1/2*g₀, resulting in a scale very close to the mos generated by 1/2 g₀. (i.e. g₁ and g₂ differ from 1/2*g₀ by ε, a quantity much smaller than the chroma of the n/2-note mos generated by g₀, which is |g₃ − g₂|). Thus we have 4 distinct sizes for k-steps:

1. a₁, a₂ and a₃ are clearly distinct.
2. a₄ − a₃ = g₁ − g₂ != 0, since the scale is a non-degenerate alt-gen scale.
3. a₄ − a₁ = g₃ − g₂ = (g₃ + g₁) − (g₂ + g₁) != 0. This is exactly the chroma of the mos generated by g₀.
4. a₄ − a₂ = g₁ − 2 g₂ + g₃ = (g₃ − g₂) + (g₁ − g₂) = (chroma ± ε) != 0 by choice of tuning.

By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus g₁ and g₂ must themselves be step sizes. Thus we see that an even-cardinality, unconditionally MV3, alt-gen scale must be of the form xy...xyxz. But this pattern is not unconditionally MV3 if n ≥ 6, since 3-steps come in 4 sizes: xyx, yxy, yxz and xzx. Thus n = 4 and the scale is xyxz.

In case 2, let (2, 1) − (1, 1) = g₁, (1, 2) − (2, 1) = g₂ be the two alternating generators. Let g₃ be the leftover generator after stacking alternating g₁ and g₂. Then the generator circle looks like g₁ g₂ g₁ g₂ ... g₁ g₂ g₃. Then the generators corresponding to a step are:

1. kg₁ + (k − 1)g₂
2. (k − 1)g₁ + kg₂
3. (k − 1)g₁ + (k − 1) g₂ + g₃,

if a step is an odd number of generators (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times.

(The above holds for any odd n ≥ 3.)

Now we only need to see that alt-gen + odd cardinality => unconditionally MV3. But the argument in case 2 above works for any interval class (unconditional MV3 wasn't used), hence any interval class comes in at most 3 sizes regardless of tuning.

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.