User:Arseniiv/Infinite MOS

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(Also this page contains mentions of me which isn’t very encyclopedic but I wonder how do I restate them in a neutral way without working on clarifying those places instead, which I have no time for, right now.)

An infinite (or aperiodic or quasiperiodic) MOS scale ^{[idiosyncratic term]} is a generalization of a MOS scale to aperiodic scales. Mathematically, an infinite MOS pattern is a Sturmian word (but extended in both directions), like a periodic MOS pattern is a Christoffel word.

Properties

Like a regular, periodic MOS scale, an infinite MOS scale has the following properties:

1. Maximum variety 2
2. Binary and balanced
3. Binary billiard scale

It is also, in a sense, distributionally even: if you take a limit, then it by all means should work but I haven’t tried. And it has “almost periods” with “almost generators”; an infinite family of MOS patterns increasing in length and including each other can be found in an aperiodic MOS pattern, in infinitely many places.

There are two degenerate cases of an infinite MOS pattern: ...sssssLsssss... and its dual ...LLLLLsLLLLL...; at first glance they can’t be realized as a cutting sequence, but they can either as a limiting case of slope approaching zero or infinity, or by a clever definition of cutting sequence which would include cases like this, or for example using dual numbers (or other infinitesimals).

Definition

To get a scale pattern, one just uses a Sturmian word by using any of its definitions which allows extension in both directions, for example as a cutting sequence of a straight line. Then step sizes are specified. As the scale is aperiodic, step sizes (and hardness) are divorced from the structure of the pattern, and “step abundance” (the slope of a cutting line) is its own variable.

When step abundance gets exactly rational, a Sturmian word becomes an infinitely repeated Christoffel word (and for some reason there doesn’t seem to exist a general term for this kind of cutting word; “Sturmian” is deemed to apply only to irrational slopes) and the scale becomes the usual periodic MOS scale. So, an aperiodic MOS scale is close to infinitely many periodic MOS scales, with periods larger and larger.

The Wikipedia page has a neat constuction for the pattern (though infinite only in one direction) which relates it with the continued fraction expansion of the cutting line’s slope (which equals step abundance, so that’s useful): if the continued fraction is [a₁ + 1, a₂, a₃, a₄ …], the pattern is constructed recursively in a Fibonacci-like way as follows:

p₀ = L

p₁ = s

p_{n + 2} = p_{n + 1}^a_n p_n

p₀ and p₁ may be switched places depending on convention. (Also I do wonder why there is a plus one in the first term of the continued fraction.)

When the fraction is finite, equivalently it means that an a_N = ∞ for some N, which means the resulting pattern is a previous one repeated infinitely, and each partial pattern p_k is a MOS pattern, so we get a regular MOS in case of rational slope, as stated above.

When the fraction is infinite, we just take a limit of this sequence of finite words, which does exist because each next word (except p₁, but we’re concerned only with eventuality) has the previous one as a prefix.

Quasiperiodicity

It is not just the case that you encounter MOS patterns p_k (or their rotations) infinitely often in an infinite pattern; if you translate the entire scale by one of them, you get “too many to be accidental” coincidences of notes between the old and the new scales. I’m not sure how to formulate it efficiently and clearly but it should be the same case of quasiperiodicity as in, like, a Penrose tiling. For example, many quasiperiodic tilings can result from cutting a higher-dimensional periodic tiling by a plane; and that applies to higher dimensions too, so this is the likely case for what happens in a lower dimension (probably the sole case, though one can try cutting the other two regular tilings of a plane by a line).

If the slope is a quadratic irrational, that is its continued fraction terms a_k eventually repeat, you can find a number S such that the scale being scaled by S happens to be a subscale of the original unscaled one. (No proof yet.) The best case of this behavior would be a metallic ratio. (Golden ratio gives a Fibonacci word for the scale pattern.)

Examples

An infinite Fibonacci word can be constructed from finite Fibonacci words by just stacking two previously generated ones together:

s

L

Ls = L ⋅ s

LsL = Ls ⋅ L

LsLLs = LsL ⋅ Ls

LsLLsLsL = LsLLs ⋅ LsL

LsLLsLsLLsLLs

LsLLsLsLLsLLsLsLLsLsL

LsLLsLsLLsLLsLsLLsLsLLsLLsLsLLsLLs

LsLLsLsLLsLLsLsLLsLsLLsLLsLsLLsLLsLsLLsLsLLsLLsLsLLsLsL

this way we can get a one-side-infinite FIbonacci word

LsLLsLsLLsLLsLsLLsLsLLsLLsLsLLsLLsLsLLsLsLLsLLsLsLLsLsL…

looking at it as a right side of a cutting sequence of a line going straight through (0, 0), we can add its inversion …sLLsLsLLsL and either Ls or sL for the (0, 0) which crosses the line both with a horizontal and a vertical grid lines (and this happens only once because the golden ratio is irrational, so we can safely pick the order here and not be afraid of having it not conform with some other place). In the end we get the entire scale pattern:

…LsLLsLLsLsLLsL Ls LsLLsLsLLsLLsL…

Note that there are also many different scale patterns for the same slope (for different positions of the cutting line), but they look essentially the same despite possibly not being equal to any transpose of this pattern.

As another example, there are infinitely many diatonic-ish aperiodic patterns, which you get by taking a slope between 2 and 3 (diatonic is 2.5 exactly, because it’s 5L2s).

Practice

In practice, as human hearing range is finite, one’s well off with a regular MOS scale which just has a giant period size. This way all the tools that work with MOS scales suffice, if only get a bit harder to fit this usage.

Any billiard scale generator can generate patterns for quasiperiodic scales by just allowing irrational (or just very complicated rational) velocities/slopes.