Talk:Periodic scale - Revision history

Inthar: /* The step form of a scale */

2024-02-08T23:17:16Z

The step form of a scale

← Older revision		Revision as of 23:17, 8 February 2024
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	== The step form of a scale ==		== The step form of a scale ==
	I'd like the page to mention the two most ~~canonical~~ 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'')) = ''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.

Inthar: /* The step form of a scale */

2024-02-08T23:17:04Z

The step form of a scale

← Older revision		Revision as of 23:17, 8 February 2024
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	== The step form of a 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:		I'd like the page to mention the two most canonical 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'')) = ''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.

Inthar: /* The step form of a scale */

2024-02-08T21:50:07Z

The step form of a scale

← Older revision		Revision as of 21:50, 8 February 2024
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	The latter form is implicit 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)		[[User:Inthar\|Inthar]] ([[User talk:Inthar\|talk]]) 16:33, 8 February 2024 (UTC)

Inthar: /* The step form of a scale */

2024-02-08T21:49:50Z

The step form of a scale

← Older revision		Revision as of 21:49, 8 February 2024
Line 3:		Line 3:
	# 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 ~~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 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)

Inthar at 21:32, 8 February 2024

2024-02-08T21:32:38Z

← Older revision		Revision as of 21:32, 8 February 2024
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	== The step form of a 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:		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>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 ''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 implicitly used when regarding scales as scale words.

Inthar: /* The step form of a scale */

2024-02-08T19:01:24Z

The step form of a scale

← Older revision		Revision as of 19:01, 8 February 2024
Line 1:		Line 1:
	== The step form of a 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:		I'd like the page to mention the two ways of writing a periodic scale as a function:
	# The ''cumulative form'' (the formalization on 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'')) = ''kE'' 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 implicitly used when regarding scales as scale words.		The latter form is implicitly used when regarding scales as scale words.

Inthar at 18:45, 8 February 2024

2024-02-08T18:45:38Z

← Older revision		Revision as of 18:45, 8 February 2024
Line 1:		Line 1:
			== 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 ways of writing a periodic scale as a function:
	# The ''cumulative form'' (the formalization on 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 on 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.

Inthar at 18:41, 8 February 2024

2024-02-08T18:41:44Z

← Older revision		Revision as of 18:41, 8 February 2024
Line 2:		Line 2:
	# The ''cumulative form'' (the formalization on 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 on 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 ''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 implicitly used 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)

Inthar at 16:51, 8 February 2024

2024-02-08T16:51:21Z

← Older revision		Revision as of 16:51, 8 February 2024
Line 2:		Line 2:
	# The ''cumulative form'' (the formalization on 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 on 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 ''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.
	We can also mention scales considered more abstractly, <math>S: \mathbb{Z}\to A</math> where ''A'' is any 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)

Inthar at 16:41, 8 February 2024

2024-02-08T16:41:32Z

← Older revision		Revision as of 16:41, 8 February 2024
Line 1:		Line 1:
	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 ways of writing a periodic scale as a function:
	# The ''cumulative form'' (the formalization on 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'')) = ''~~k'''''E'~~'' where '''E''' is the equave.		# The ''cumulative form'' (the formalization on 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 ''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.
	We can also mention scales considered more abstractly, <math>S: \mathbb{Z}\to A</math> where ''A'' is any 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 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)