MOS Diagrams

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The moment-of-symmetry process of unfolding a scale takes, for most people, a conceptual leap or two. Below are visualizations of the process:

L and s

The mechanics of scale generation are such that — when iterating from one scale to the next densest one — all large steps in the preceding scale become one large step and one small step in the new scale.

Another way to think about this is that a small-step-sized chunk has been split off of each of the former large steps. The remainder can be either larger or smaller than the small step

  • If it is larger, then it stays the large step.
  • If it is smaller, then it becomes the new small step, and everything that used to be a small step is now a large step.

MOS iteration rules for L and s.png

We are reasoning about MOS concepts in the abstract here. These truths about large and small steps are true whether they are 100¢ or 4516.8¢, and all we really care about are their ratios. So if we treat our small steps’ size as [math]1[/math] then we can treat our large steps’ size as equal to the [math]L{:}s[/math] ratio.

So the [math]L{:}s[/math] ratio decreases by [math]1[/math] because if an [math]s[/math]-sized chunk has been sliced off [math]L[/math], and [math]s[/math]’s size is [math]1[/math], then [math]1[/math] should be subtracted from [math]L[/math].

When [math]L - s \gt s[/math]:

[math] \begin{align} L’{:}s’ &= (L - s){:}s \\ &= (L - 1){:}1 \\ &= L - 1 \end{align} [/math]

When [math]L - s \lt s[/math], the result is simply reciprocated:

[math] \begin{align} L’{:}s’ &= s{:}(L - s) \\ &= 1{:}(L - 1) \\ &= \frac{1}{L - 1} \end{align} [/math]