ALS - Revision history

Cmloegcmluin: /* Examples */ update row headers per agreement at https://en.xen.wiki/w/Talk:APS

2023-10-19T20:37:13Z

Examples: update row headers per agreement at https://en.xen.wiki/w/Talk:APS

← Older revision		Revision as of 20:37, 19 October 2023
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	! 8		! 8
	\|-		\|-
	! frequency (f)		! frequency (''f'', ratio)
	\|(1)		\|(1)
	\|1.12		\|1.12
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	\|7.73		\|7.73
	\|-		\|-
	! pitch (~~log₂f~~)		! pitch (log₂''f'', octaves)
	\|(0)		\|(0)
	\|0.17		\|0.17
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	\|2.95		\|2.95
	\|-		\|-
	! length (1/f)		! length (1/''f'', ratio)
	\|(1)		\|(1)
	\|0.89		\|0.89
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	! 8		! 8
	\|-		\|-
	! frequency (f)		! frequency (''f'', ratio)
	\|(1)		\|(1)
	\|0.54		\|0.54
Line 112:		Line 112:
	\|0.13		\|0.13
	\|-		\|-
	! pitch (~~log₂f~~)		! pitch (log₂''f'', octaves)
	\|(0)		\|(0)
	\| -0.88		\| -0.88
Line 123:		Line 123:
	\| -2.95		\| -2.95
	\|-		\|-
	! length (1/f)		! length (1/''f'', ratio)
	\|(1 + 0/⁴√2)		\|(1 + 0/⁴√2)
	\|1 + 1/⁴√2		\|1 + 1/⁴√2

BudjarnLambeth: Added to category "xenharmonic series"

2023-04-26T05:16:32Z

Added to category "xenharmonic series"

← Older revision		Revision as of 05:16, 26 April 2023
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	[[Category:Subharmonic series‏‎]]		[[Category:Subharmonic series‏‎]]
	[[Category:Utonality]]		[[Category:Utonality]]
			[[Category:Xenharmonic series]]

Cmloegcmluin: /* Relationship to other tunings */ address Flora's recent concerns about rationality vs. irrationality in this system

2023-03-28T21:41:18Z

Relationship to other tunings: address Flora's recent concerns about rationality vs. irrationality in this system

← Older revision		Revision as of 21:41, 28 March 2023
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	=== As shifted undertone series ===		=== As shifted undertone series ===

	By varying the undertone series step size to some ~~rational~~ number (other than 1) you can produce ~~a US~~, and ~~by varying it to an irrational number~~ you can produce an ALS. In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes.		By varying the undertone series step size to some number other than 1, if that number is irrational you can produce an ALS, and if rational you can produce a US (which you could also call an ALS if you really wanted to). In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes.

	=== Vs. AFS ===		=== Vs. AFS ===
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	=== Vs. US ===		=== Vs. US ===

	A [[US\|US, or utonal sequence]], is the (rational) version of an ALS.		A [[US\|US, or utonal sequence]], is the rational version of an ALS.

	=== Vs. ELD ===		=== Vs. ELD ===

Cmloegcmluin: /* Vs. ELD */

2023-03-28T18:29:13Z

Vs. ELD

← Older revision		Revision as of 18:29, 28 March 2023
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	=== Vs. ELD ===		=== Vs. ELD ===

	By specifying n, your sequence will be equivalent to some [[ELD\|ELD (equal length division)]]; specifically n-ALSp = n-ELD((p-1)/n).		By specifying n, your sequence will be equivalent to one period of some [[ELD\|ELD (equal length division)]]; specifically n-ALSp = n-ELD((p-1)/n).

	== Examples ==		== Examples ==

Cmloegcmluin: fix section capitalization

2021-11-11T19:39:47Z

fix section capitalization

← Older revision		Revision as of 19:39, 11 November 2021
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	== Relationship to other tunings ==		== Relationship to other tunings ==

	=== as shifted undertone series ===		=== As shifted undertone series ===

	By varying the undertone series step size to some rational number (other than 1) you can produce a US, and by varying it to an irrational number you can produce an ALS. In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes.		By varying the undertone series step size to some rational number (other than 1) you can produce a US, and by varying it to an irrational number you can produce an ALS. In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes.

	=== vs. AFS ===		=== Vs. AFS ===

	The analogous otonal equivalent of an ALS is an [[AFS\|AFS (arithmetic frequency sequence)]].		The analogous otonal equivalent of an ALS is an [[AFS\|AFS (arithmetic frequency sequence)]].

	=== vs. US ===		=== Vs. US ===

	A [[US\|US, or utonal sequence]], is the (rational) version of an ALS.		A [[US\|US, or utonal sequence]], is the (rational) version of an ALS.

	=== vs. ELD ===		=== Vs. ELD ===

	By specifying n, your sequence will be equivalent to some [[ELD\|ELD (equal length division)]]; specifically n-ALSp = n-ELD((p-1)/n).		By specifying n, your sequence will be equivalent to some [[ELD\|ELD (equal length division)]]; specifically n-ALSp = n-ELD((p-1)/n).

Fredg999: Categories

2021-08-30T03:51:28Z

Cmloegcmluin: break up wall of information into helpful sections that are consistent across all arithmetic tuning pages

2021-03-24T20:38:31Z

break up wall of information into helpful sections that are consistent across all arithmetic tuning pages

@@ Line 1: / Line 1: @@
 An '''ALS''', or '''arithmetic length sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-Is full specification is (n-)ALSp: (n pitches of an) arithmetic length sequence adding by p. It is equivalent to an undertone series shifted ± frequency.
+== Specification ==
-The n is optional. If not provided, the sequence is open-ended. By specifying n, your sequence will be equivalent to some [[ELD|ELD (equal length division)]]; specifically n-ALSp = n-ELD((p-1)/n).
+Is full specification is (n-)ALSp: (n pitches of an) arithmetic length sequence adding by p. It is equivalent to an undertone series shifted ± frequency. The n is optional. If not provided, the sequence is open-ended.
-The analogous otonal equivalent of an ALS is an [[AFS|AFS (arithmetic frequency sequence)]].
+== Formula ==
 The formula for length <span><math>k</math></span> of an ALSp is:
@@ Line 16: / Line 12: @@
 L(k) = 1 + k⋅p
 </math>
 {| class="wikitable"

Cmloegcmluin: add formula for mathematician benefit

2021-03-23T22:51:24Z

add formula for mathematician benefit

← Older revision		Revision as of 22:51, 23 March 2021
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	Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.		Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.

			The formula for length <span><math>k</math></span> of an ALSp is:

			<math>
			L(k) = 1 + k⋅p
			</math>

	{\| class="wikitable"		{\| class="wikitable"

Cmloegcmluin: note about >1 lengths

2021-03-23T22:31:14Z

note about >1 lengths

← Older revision		Revision as of 22:31, 23 March 2021
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	A [[US\|US, or utonal sequence]], is a specific (rational) type of ALS. By varying the undertone series step size to some rational number (other than 1) you can produce a US, and by varying it to an irrational number you can produce an ALS. In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes.		A [[US\|US, or utonal sequence]], is a specific (rational) type of ALS. By varying the undertone series step size to some rational number (other than 1) you can produce a US, and by varying it to an irrational number you can produce an ALS. In other words, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but their relationship in pitch changes.

			Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.

	{\| class="wikitable"		{\| class="wikitable"

Cmloegcmluin: provide descending pitch version for utonal

2021-03-23T22:29:44Z

provide descending pitch version for utonal

@@ Line 11: / Line 11: @@
 {| class="wikitable"
 |+example: (1/⁴√2)-shifted undertone series segment = 8-ALS(1/⁴√2)
 |-
 ! quantity
@@ Line 55: / Line 56: @@
 |0.24
 |0.13
 |}

← Older revision		Revision as of 03:51, 30 August 2021
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	\|}		\|}

	~~[[Category:Undertone]]~~
	~~[[Category:Undertone series]]~~
	~~[[Category:Utonality]]~~
	[[Category:Subharmonic]]		[[Category:Subharmonic]]
	[[Category:Subharmonic series‏‎]]		[[Category:Subharmonic series‏‎]]
			[[Category:Utonality]]