Talk:Periodic scale: Difference between revisions

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== The step form of a scale ==

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

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

# 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'')) = ''kE'' 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 latter form is ~~implicitly used~~ when regarding scales as scale words.

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 could 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)

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 == The step form of a scale ==
-I'd like the page to mention the two ways of writing a periodic scale as a function:
+I'd like the page to mention the two most standard ways of writing a periodic scale as a function:
-# 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'')) = ''kE'' 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) &minus; ''S''(''k''), which is nonnegative and periodic in the usual math sense.
-The latter form is implicitly used when regarding scales as scale words.
+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 could 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)