UD - Revision history

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

2023-10-19T20:36:59Z

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

← Older revision		Revision as of 20:36, 19 October 2023
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	! 4		! 4
	\|-		\|-
	! frequency (f)		! frequency (''f'', ratio)
	\|(1/1)		\|(1/1)
	\|8/7		\|8/7
Line 60:		Line 60:
	\|2/1		\|2/1
	\|-		\|-
	! pitch (~~log₂f~~)		! pitch (log₂''f'', octaves)
	\|(0)		\|(0)
	\|0.19		\|0.19
Line 67:		Line 67:
	\|1.00		\|1.00
	\|-		\|-
	! length (1/f)		! length (1/''f'', ratio)
	\|(8/8)		\|(8/8)
	\|7/8		\|7/8
Line 85:		Line 85:
	! 4		! 4
	\|-		\|-
	! frequency (f)		! frequency (''f'', ratio)
	\|(1/1)		\|(1/1)
	\|4/5		\|4/5
Line 92:		Line 92:
	\|1/2		\|1/2
	\|-		\|-
	! pitch (~~log₂f~~)		! pitch (log₂''f'', octaves)
	\|(0)		\|(0)
	\| -0.32		\| -0.32
Line 99:		Line 99:
	\| -1.00		\| -1.00
	\|-		\|-
	! length (1/f)		! length (1/''f'', ratio)
	\|(4/4)		\|(4/4)
	\|5/4		\|5/4

Cmloegcmluin: /* Vs ELD */ address Flora's recent concerns about rationality vs. irrationality in this system

2023-03-28T21:41:10Z

Vs ELD: 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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	=== Vs ELD ===		=== Vs ELD ===

	The only difference between n-UDp and [[ELD\|n-ELDp (equal length division)]] is that the p for UD is rational, while the p for ELD is irrational.		The only difference between n-UDp and [[ELD\|n-ELDp (equal length division)]] is that the p for UD must be rational, while the p for an ELD is probably irrational.

	=== Vs US ===		=== Vs US ===

Cmloegcmluin: fix section capitalization

2021-11-11T19:40:54Z

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 ==

	=== vs ED ===		=== Vs ED ===

	It is possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by length. You will have 12-ELDO.		It is possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by length. You will have 12-ELDO.

	=== vs under-n series ===		=== Vs under-n series ===

	An n-UDO is equivalent to the nth [[Overtone_scale#Next_Steps\|undertone mode, or under-n scale]].		An n-UDO is equivalent to the nth [[Overtone_scale#Next_Steps\|undertone mode, or under-n scale]].

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

	The only difference between n-UDp and [[ELD\|n-ELDp (equal length division)]] is that the p for UD is rational, while the p for ELD is irrational.		The only difference between n-UDp and [[ELD\|n-ELDp (equal length division)]] is that the p for UD is rational, while the p for ELD is irrational.

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

	A UD will be equivalent to some [[US\|US (utonal sequence)]]. E.g. 8-UD7 = 8-US3/4, because to get from 1 to 7 you cover 6 undertones, and 6 divided by 8 is 3/4.		A UD will be equivalent to some [[US\|US (utonal sequence)]]. E.g. 8-UD7 = 8-US3/4, because to get from 1 to 7 you cover 6 undertones, and 6 divided by 8 is 3/4.

	=== vs EDL ===		=== Vs EDL ===

	An [[EDL\|n-EDL]] is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).		An [[EDL\|n-EDL]] is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).

Fredg999: Categories

2021-08-30T04:02:08Z

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

2021-03-24T20:38:23Z

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

@@ Line 1: / Line 1: @@
 A '''UD''', or '''utonal division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-Its full specification is n-UDp: n utonal divisions of rational interval p. An n-UDO is equivalent to the nth [[Overtone_scale#Next_Steps|undertone mode, or under-n scale]].
+== Specification ==
-The only difference between n-UDp and [[ELD|n-ELDp (equal length division)]] is that the p for UD is rational, while the p for ELD is irrational.
+Its full specification is n-UDp: n utonal divisions of rational interval p.
-It is possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by length. You will have 12-ELDO. However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so ELD are typically reserved for irrational tunings, such as 12-ELDφ. So it would be more appropriate to name this tuning 12-UDO, for utonal divisions of the octave.
+== Formula ==
 To find the steps for an n-UDp, begin by recognizing that while the ratio between your root pitch's string length and the length you would pluck to get the lowest pitch is <span><math>p</math></span> (or <span><math>\frac p1</math></span>), if you are going to move arithmetically (by repeated addition) from <span><math>1</math></span> to <span><math>p</math></span>, then the difference in string length that you need to cover is not actually <span><math>p</math></span>, but only <span><math>p - 1</math></span>. And because you are dividing it into <span><math>n</math></span> parts, each step will have a size of <span><math>\frac{p-1}{n}</math></span>. So, the formula for the length of step <span><math>k</math></span> of an n-UDp is:
@@ Line 20: / Line 14: @@
 This way, when <span><math>k</math></span> is <span><math>0</math></span>, <span><math>L(k)</math></span> is simply <span><math>1</math></span>. And when <span><math>k</math></span> is <span><math>n</math></span>, <span><math>L(k)</math></span> is simply <span><math>1 + (p-1) = p</math></span>.
 {| class="wikitable"

Cmloegcmluin: add formula for mathematician benefit

2021-03-23T22:54:51Z

add formula for mathematician benefit

← Older revision		Revision as of 22:54, 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.

			To find the steps for an n-UDp, begin by recognizing that while the ratio between your root pitch's string length and the length you would pluck to get the lowest pitch is <span><math>p</math></span> (or <span><math>\frac p1</math></span>), if you are going to move arithmetically (by repeated addition) from <span><math>1</math></span> to <span><math>p</math></span>, then the difference in string length that you need to cover is not actually <span><math>p</math></span>, but only <span><math>p - 1</math></span>. And because you are dividing it into <span><math>n</math></span> parts, each step will have a size of <span><math>\frac{p-1}{n}</math></span>. So, the formula for the length of step <span><math>k</math></span> of an n-UDp is:

			<math>
			L(k) = 1 + (\frac kn)(p-1)
			</math>

			This way, when <span><math>k</math></span> is <span><math>0</math></span>, <span><math>L(k)</math></span> is simply <span><math>1</math></span>. And when <span><math>k</math></span> is <span><math>n</math></span>, <span><math>L(k)</math></span> is simply <span><math>1 + (p-1) = p</math></span>.

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

Cmloegcmluin: note about >1 lengths

2021-03-23T22:32:09Z

note about >1 lengths

← Older revision		Revision as of 22:32, 23 March 2021
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	An [[EDL\|n-EDL]] is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).		An [[EDL\|n-EDL]] is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).

			It is possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by length. You will have 12-ELDO. However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so ELD are typically reserved for irrational tunings, such as 12-ELDφ. So it would be more appropriate to name this tuning 12-UDO, for utonal divisions of the octave.

			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

2021-03-23T22:23:43Z

provide descending pitch version for utonal

← Older revision		Revision as of 22:23, 23 March 2021
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	{\| class="wikitable"		{\| class="wikitable"
	\|+example: 4-UDO = 4th undertone mode		\|+example: 4-UDO = 4th undertone mode ''(arranged so that the pitches are in ascending order and still begin on 1/1)''
	\|-		\|-
	! quantity		! quantity
Line 20:		Line 20:
	\|-		\|-
	! frequency (f)		! frequency (f)
	\|(8/8)		\|(1/1)
	\|8/7		\|8/7
	\|8/6		\|4/3
	\|8/5		\|8/5
	\|8/4		\|2/1
	\|-		\|-
	! pitch (log₂f)		! pitch (log₂f)
Line 34:		Line 34:
	\|-		\|-
	! length (1/f)		! length (1/f)
	\|(1/1)		\|(8/8)
	\|7/8		\|7/8
	\|3/4		\|6/8
	\|5/8		\|5/8
			\|4/8
			\|}

			{\| class="wikitable"
			\|+example: 4-UDO = 4th undertone mode ''(descending pitches)''
			\|-
			! quantity
			! (0)
			! 1
			! 2
			! 3
			! 4
			\|-
			! frequency (f)
			\|(1/1)
			\|4/5
			\|2/3
			\|4/7
	\|1/2		\|1/2
			\|-
			! pitch (log₂f)
			\|(0)
			\| -0.32
			\| -0.58
			\| -0.81
			\| -1.00
			\|-
			! length (1/f)
			\|(4/4)
			\|5/4
			\|6/4
			\|7/4
			\|8/4
	\|}		\|}


	[[Category:Undertone]]		[[Category:Undertone]]

Cmloegcmluin at 17:55, 23 March 2021

2021-03-23T17:55:41Z

← Older revision		Revision as of 17:55, 23 March 2021
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	The only difference between n-UDp and [[ELD\|n-ELDp (equal length division)]] is that the p for UD is rational, while the p for ELD is irrational.		The only difference between n-UDp and [[ELD\|n-ELDp (equal length division)]] is that the p for UD is rational, while the p for ELD is irrational.

			Your sequence will be equivalent to some [[US\|US (utonal sequence)]]. E.g. 8-UD7 = 8-US3/4, because to get from 1 to 7 you cover 6 undertones, and 6 divided by 8 is 3/4.

	An [[EDL\|n-EDL]] is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).		An [[EDL\|n-EDL]] is equivalent to a 2n-UDO (therefore EDL cannot be used to represent a UDO with an odd value for n).

Cmloegcmluin at 23:42, 22 March 2021

2021-03-22T23:42:02Z

← Older revision		Revision as of 23:42, 22 March 2021
Line 1:		Line 1:
	A '''UD''', or '''utonal division''', is a kind of [[Arithmetic tunings\|arithmetic]] and [[~~Monotonic~~ tunings\|~~monotonic~~]] tuning.		A '''UD''', or '''utonal division''', is a kind of [[Arithmetic tunings\|arithmetic]] and [[Harmonotonic tunings\|harmonotonic]] tuning.

	Its full specification is n-UDp: n utonal divisions of rational interval p. An n-UDO is equivalent to the nth [[Overtone_scale#Next_Steps\|undertone mode, or under-n scale]].		Its full specification is n-UDp: n utonal divisions of rational interval p. An n-UDO is equivalent to the nth [[Overtone_scale#Next_Steps\|undertone mode, or under-n scale]].

← 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]]