# MOS Diagrams

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.

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 $1$ then we can treat our large steps’ size as equal to the $L{:}s$ ratio.

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

When $L - s \gt s$:

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

When $L - s \lt s$, the result is simply reciprocated:

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