Talk:Periodic scale: Difference between revisions
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# The ''cumulative form'' (the formalization in the article), a monotone increasing function <math>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. | # The ''cumulative form'' (the formalization in the article), a monotone increasing function <math>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. | ||
# The ''step form'', the sequence of steps in the scale, related to the above via Δ''S''(''k'') = ''S''(''k'' + 1) − ''S''(''k''), which is nonnegative and periodic in the usual math sense. | # The ''step form'', the sequence of steps in the scale, related to the above via Δ''S''(''k'') = ''S''(''k'' + 1) − ''S''(''k''), which is nonnegative and periodic in the usual math sense. | ||
The latter form is | The latter form is implicit when regarding scales as scale words. | ||
We can also mention scales considered more abstractly, <math>S: \mathbb{Z}\to A</math> where ''A'' is any torsion-free abelian group such as <math>\mathbb{R}</math> (log-frequency interval space), a free abelian group on steps, or a JI subgroup. | We can also mention scales considered more abstractly, <math>S: \mathbb{Z}\to A</math> where ''A'' is any torsion-free abelian group such as <math>\mathbb{R}</math> (log-frequency interval space), a free abelian group on steps, or a JI subgroup. | ||
[[User:Inthar|Inthar]] ([[User talk:Inthar|talk]]) 16:33, 8 February 2024 (UTC) | [[User:Inthar|Inthar]] ([[User talk:Inthar|talk]]) 16:33, 8 February 2024 (UTC) | ||
Revision as of 21:49, 8 February 2024
The step form of a scale
I'd like the page to mention the two ways of writing a periodic scale as a function:
- 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.
- 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 can 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)