Generator-offset property

A scale satisfies the generator-offset property if it satisfies the following properties:

1. The scale is generated by two chains of stacked copies of an interval called the generator.
2. The two chains are separated by a different interval called the offset (the difference between the first note of the second chain and the first note of the first chain).
3. The lengths of the chains differ by at most one. (1-3 can be restated as: The scale can be built by stacking two alternating generators (called alternants) a₁ and a₂. Note that a₁ does not need to subtend, i.e. occur as, the same number of steps as a₂.)
4. The generator always occurs as the same number of steps. For example, the generator is never both a 2-step and a 3-step.

Plot of at-most-decatonic MV3 generator-offset scale regions in the plane.

The Zarlino (3L 2M 2S) JI scale is an example of a generator-offset scale, because it is built by stacking alternating 5/4 and 6/5 generators. The 7-limit version of diasem (5L 2M 2S) is another example, with generators 7/6 and 8/7.

Generator-offset scales generalize the notion of dipentatonic and diheptatonic scales where the pentatonic and heptatonic are MOS scales. A related but distinct notion is alternating generator sequence. While scales produced using the generator-offset procedure can be seen as a result of an alternating generator sequence of 2 alternants, the generator-offset perspective views the sum of the two alternants as the "canonical" generator, and the alternants as rather being possible choices of the offset which are effectively equivalent up to chirality. While a well-formed AGS scale requires each alternant in the AGS to subtend the same number of steps, the generator-offset property only requires each (aggregate) generator to subtend the same number of steps.

Mathematical definition

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

1. S is generated by two chains of stacked generators g separated by a fixed offset δ; either both chains are of size n/2 (implying n is even), or one chain has size (n + 1)/2 and the second has size (n − 1)/2 (implying n is odd).
2. The scale is well-formed with respect to g, i.e. all occurrences of the generator g are k-steps for a fixed k.

Open conjectures

Conjecture ("MV3 Sequences")

Given any two generators, we can iterate them to any number of notes and see what the maximum-variety of the resulting scale is. In particular, we can look at those scale sizes which are MV3, and thus compute the MV3 sequence for the pair of generators (similar to the "MOS sequence" one can compute for one generator). Thus, for any pair of generators, we can form the associated sequence of increasingly large MV3 scales.

Surprisingly, for almost all pairs of generators, this sequence seems to terminate after some (usually relatively small) scale. That is, if we simply take all possible pairs of generators between 0 and 1200 cents, and for each pair we compute the MV3 sequence for all generator pairs up to some maximum N, such as 1000, we can easily see that most points will have only a few entries in it, after which no MV3 scales are apparently generated. It would seem to be true that as the two generators get closer and closer in size, the MV3 sequence gets longer and longer, until when the two generators are equal you have an infinite-length sequence (corresponding to MOS).

It is pretty easy to see this behavior is true if we simply compute the MV3 sequences up to any very large N, far beyond the scale sizes we typically use in music theory, but it would be good to have a proof.

Open questions

This heading has those open questions for which no conjecture has yet been formed either way. (These can be updated as necessary)

1. Given any arbitrary MOS (or DE, etc) scale with at least three notes per period, is there *always* a MV3 generator-offset scale which can be derived as a "detempering" of that scale? Or is this only true for some MOS's? For instance, the MOS LLsLLLs has the MV3 generator-offset scale LmsLmLs as a detempering. Does a similar MV3 detempering exist for every possible DE scale with at least three notes per period, or at least for strict MOS's with one period per octave (e.g. well-formed scales)?
2. The scale tree is a great way to analyze MOS scales. For any generator, we can compute the various MOS's it forms if we simply look at the scale tree, and indeed MOS "words" like LLsLLLs can be identified with regions on the scale tree (in this situation the interval between 4/7 and 3/5). A similar "scale plane" should exist for generator-offset-MV3 scales, where given some word representing a generator-offset-MV3 scale, we can look at the set of points on the generator plane which generates it; these seem to often be triangles, with the lines corresponding to MOS's and the vertices corresponding to EDOs (though is this always true?). What is the big picture of this scale plane? Can we use Viggo Brun's algorithm for this, generalizing the theory of continued fractions? Is there some simple formula we can use to predict, given some generator-offset-MV3 scale, which region on the scale plane it corresponds to? Can we plot simple generator-size-proportions as points in this space? And so on.
3. In the theory of MOS, there is a second scale tree that is less frequently talked about, which Erv Wilson calls the "Rabbit Sequence" (Erv Wilson's original version, interactive version 1, interactive version 2). This is a tree for which each MOS word has two children, depending on if the MOS is "soft" (with L/s < 2) or "hard" (with L/s > 2). For instance, LsLss has the two children LLsLLLs and ssLsssL. Does a similar scale plane exist for these generator-offset-MV3 scales?