Generator-offset property: Difference between revisions

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

* A strengthening of the generator-offset property, tentatively named the ''swung-generator-alternant property'' (SGA), states that the alternants g₁ and g₂ can be taken to always subtend the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). All odd GO scales are SGA, and aside from odd GO scales, the only ternary scales to satisfy SGA are (xy)^''r''xz, ''r'' ≥ 1. The Zarlino and diasem scales above are both SGA. [[Blackdye]] is GO but not SGA.

In particular, odd GO scales always satisfy these properties (see Proposition 2 below).

[Note: This is not true with SGA replaced with GO; [[blackdye]] is a counterexample that is MV4.]

In case 1, let g₁ = (2, 1) − (1, 1), g₂ = (1, 2) − (2, 1), and g₃ = (1, 1) − (''n''/2, 2) = (−''n''/2*g₁ − g₁ − ''n''/2*g₂) mod e. We assume that g₁, g₂ and e are '''Z'''-linearly independent. We have the chain g₁ g₂ g₁ g₂ ... g₁ g₃ which visits every note in ''S''.  

Since ''S'' is GO it is well-formed with respect to g = (g₂ + g₁). Since g₁ and g₂ subtend the same number of steps, all multiples of the generator g must be even-steps, and those intervals that are "offset" by g₁ must be odd-steps. Letting ''M'' be the subset of all even-numbered notes (which are generated by g) and considering ''M'' as a scale by dividing degree indices in ''M'' by two, ''M'' is well-formed with respect to g, thus ''M'' (and its offset) must be a mos subset. Hence (g₃ + g₁), the imperfect generator of the mos generated by g, subtends the same number of steps as g. Thus g₂ and g₃ subtend the same number of steps, a fact we need in order to be able to substitute one instance of g₂ with g₃ in the next part.

# from g₁ ... g₁, we get a₁ = (''r'' − 1)/2*g₀ + g₁ = (''r'' + 1/2) g₁ + (''r'' − 1/2) g₂  

# from g₂ ... g₂, we get a₂ = (''r'' − 1)/2*g₀ + g₂ = (''r'' − 1/2) g₁ + (''r'' + 1/2) g₂

These are all distinct by '''Z'''-linear independence. 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-length, abstractly SV3, GO scale must be of the form (xy)^''r''xz. (Note that (xy)^''r''xz is not SV3, since it has only two kinds of 2-steps, xy and xz.) This proves (1).

In case 2, let (2, 1) − (1, 1) = g₁, (1, 2) − (2, 1) = g₂ be the two alternants. 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. Assuming that a step is an odd number of generators, the combinations of alternants corresponding to a step come in exactly 3 sizes:

=== Proposition 2 (Odd GO scales are SGA) ===

=== Proposition 3 (Properties of even GO scales) ===

A primitive GO scale of even size where the generator g is an even-step (i.e. g subtends an even number of steps) has the following properties:

Let ''S''(a, b, c) be a scale word in three '''Z'''-linearly independent step sizes a, b, c. Suppose ''S'' is pairwise well-formed (equivalently, all its projections are single-period mosses). Then ''S'' is SV3 and has an odd number of notes. Moreover, ''S'' is either GO or equivalent to the scale word abacaba.

# Given any arbitrary MOS (or DE, etc) scale with at least three notes per period, is there *always* a MV3 GO 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 GO 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)?

# 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 GO-MV3 scales, where given some word representing a GO-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 GO-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.

# In the theory of MOS, there is a second [[MOS Scale Family Tree|scale tree]] that is less frequently talked about, which Erv Wilson calls the "Rabbit Sequence" ([http://www.anaphoria.com/RabbitSequence.pdf Erv Wilson's original version], [https://mikebattagliamusic.com/MOSTree/MOSTreeab.html interactive version 1], [https://mikebattagliamusic.com/MOSTree/MOSTreeLs.html 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 GO-MV3 scales?

[[Category:GO scales| ]]<!--Main article-->