Talk:Periodic scale

The step form of a scale

I'd like the page to mention the two ways of writing a periodic scale as a function:

1. The cumulative form (the formalization in the article), a monotone increasing function [math]\displaystyle{ S: \mathbb{Z}\to \mathbb{R} }[/math] such that S(0) = 0 (unison) and S(k + len(S)) = S(k) + E where E is the equave.
2. The step form, the sequence of steps in the scale, related to the above via ΔS(k) = S(k + 1) − S(k), which is nonnegative and periodic in the usual math sense.

The latter form is implicit when regarding scales as scale words.

We could also mention scales considered more abstractly, [math]\displaystyle{ S: \mathbb{Z}\to A }[/math] where A is any torsion-free abelian group such as [math]\displaystyle{ \mathbb{R} }[/math] (log-frequency interval space), a free abelian group on steps, or a JI subgroup. Inthar (talk) 16:33, 8 February 2024 (UTC)