US - Revision history

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

2023-10-19T20:37:09Z

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)
	\|(28/28)		\|(28/28)
	\|28/25		\|28/25
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	\|7/1		\|7/1
	\|-		\|-
	! pitch (~~log₂f~~)		! pitch (log₂''f'', octaves)
	\|(0)		\|(0)
	\|0.16		\|0.16
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	\|2.81		\|2.81
	\|-		\|-
	! length (1/f)		! length (1/''f'', ratio)
	\|(28/28)		\|(28/28)
	\|25/28		\|25/28
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	! 8		! 8
	\|-		\|-
	! frequency (f)		! frequency (''f'', ratio)
	\|(1/1)		\|(1/1)
	\|4/7		\|4/7
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	\|1/7		\|1/7
	\|-		\|-
	! pitch (~~log₂f~~)		! pitch (log₂''f'', octaves)
	\|(0)		\|(0)
	\| -0.81		\| -0.81
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	\| -2.81		\| -2.81
	\|-		\|-
	! length (1/f)		! length (1/''f'', ratio)
	\|(4/4)		\|(4/4)
	\|7/4		\|7/4

BudjarnLambeth: Added to category "xenharmonic series"

2023-04-26T05:19:18Z

Added to category "xenharmonic series"

← Older revision		Revision as of 05:19, 26 April 2023
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	[[Category:Subharmonic]]		[[Category:Subharmonic]]
	[[Category:Subharmonic series‏‎]]		[[Category:Subharmonic series‏‎]]
			[[Category:Xenharmonic series]]

Cmloegcmluin: add link

2023-03-30T15:30:50Z

add link

← Older revision		Revision as of 15:30, 30 March 2023
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	== Specification ==		== Specification ==

	The full specification of a US is (n-)USp: (n pitches of a) utonal sequence adding by p. The n is optional. If not provided, the sequence is open-ended.		The full specification of a US is (n-)USp: (n pitches of a) [[utonal]] sequence adding by p. The n is optional. If not provided, the sequence is open-ended.

	== Formula ==		== Formula ==

Cmloegcmluin: /* As shifted undertone series */ address Flora's recent concerns about rationality vs. irrationality in this system

2023-03-28T21:41:15Z

As shifted undertone series: 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 ~~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. UD ===		=== Vs. UD ===

Cmloegcmluin: fix section capitalization

2021-11-11T19:40:07Z

fix section capitalization

← Older revision		Revision as of 19:40, 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 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 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. UD ===		=== Vs. UD ===

	By specifying n, your sequence will be equivalent to some [[UD\|UD (utonal division)]]. E.g. 8-US3/4 = 8-UD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.		By specifying n, your sequence will be equivalent to some [[UD\|UD (utonal division)]]. E.g. 8-US3/4 = 8-UD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.

	=== vs. ALS ===		=== Vs. ALS ===

	A US is the rational version of [[ALS\|ALS, or arithmetic length sequence]].		A US is the rational version of [[ALS\|ALS, or arithmetic length sequence]].

Fredg999: Categories

2021-08-30T04:02:30Z

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

2021-03-24T20:38:42Z

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

← Older revision		Revision as of 20:38, 24 March 2021
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	A '''US''', or '''utonal sequence''', is a kind of [[Arithmetic tunings\|arithmetic]] and [[Harmonotonic tunings\|harmonotonic]] tuning.		A '''US''', or '''utonal sequence''', is a kind of [[Arithmetic tunings\|arithmetic]] and [[Harmonotonic tunings\|harmonotonic]] tuning.

	~~The full specification of a US is (n-)USp: (n pitches of a) utonal sequence adding by p.~~		== Specification ==

	The n is optional. If not provided, the sequence is open-ended~~. By specifying n, your sequence will be equivalent to some [[UD\|UD (utonal division)]]. E.g. 8-US3/4 = 8-UD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7~~.		The full specification of a US is (n-)USp: (n pitches of a) utonal sequence adding by p. The n is optional. If not provided, the sequence is open-ended.

	A US is a specific (rational) type of [[ALS\|ALS, or arithmetic length sequence]]. By varying the undertone series step size to some rational number (other than 1) you can produce a US, and 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.		== Formula ==

	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 a USp is:		The formula for length <span><math>k</math></span> of a USp is:
Line 14:		Line 12:
	L(k) = 1 + k⋅p		L(k) = 1 + k⋅p
	</math>		</math>

			== Tip about tunings based on length ==

			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.

			== Relationship to other tunings ==

			=== as shifted undertone series ===

			By varying the undertone series step size to some rational number (other than 1) you can produce a US, and 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. UD ===

			By specifying n, your sequence will be equivalent to some [[UD\|UD (utonal division)]]. E.g. 8-US3/4 = 8-UD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.

			=== vs. ALS ===

			A US is the rational version of [[ALS\|ALS, or arithmetic length sequence]].

			== Examples ==

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

Cmloegcmluin: add formula for mathematician benefit

2021-03-23T22:55:39Z

add formula for mathematician benefit

← Older revision		Revision as of 22:55, 23 March 2021
Line 8:		Line 8:

	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 a USp is:

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

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

Cmloegcmluin: note about >1 lengths

2021-03-23T22:31:05Z

note about >1 lengths

← Older revision		Revision as of 22:31, 23 March 2021
Line 6:		Line 6:

	A US is a specific (rational) type of [[ALS\|ALS, or arithmetic length sequence]]. By varying the undertone series step size to some rational number (other than 1) you can produce a US, and 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 is a specific (rational) type of [[ALS\|ALS, or arithmetic length sequence]]. By varying the undertone series step size to some rational number (other than 1) you can produce a US, and 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

2021-03-23T22:28:00Z

provide descending pitch version for utonal

@@ Line 9: / Line 9: @@
 {| class="wikitable"
 |+example: 8-US(3/4)
 |-
 ! quantity
@@ Line 24: / Line 25: @@
 |(28/28)
 |28/25
-|28/22
+|14/11
 |28/19
-|28/16
+|7/4
 |28/13
-|28/10
+|14/5
-|28/7
+|4/1
-|28/4
+|7/1
 |-
 ! pitch (log₂f)
@@ Line 53: / Line 54: @@
 |7/28
 |4/28
 |}

← Older revision		Revision as of 04:02, 30 August 2021
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	\|}		\|}

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