ELD - Revision history

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

2023-10-19T20:37:05Z

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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	! 4		! 4
	\|-		\|-
	! frequency (f)		! frequency (''f'', ratio)
	\|(1)		\|(1)
	\|1.11		\|1.11
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	\|φ		\|φ
	\|-		\|-
	! pitch (~~log₂f~~)		! pitch (log₂''f'', octaves)
	\|(0)		\|(0)
	\|0.14		\|0.14
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	\|0.69		\|0.69
	\|-		\|-
	! length (1/f)		! length (1/''f'', ratio)
	\|(1)		\|(1)
	\|0.90		\|0.90
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	! 4		! 4
	\|-		\|-
	! frequency (f)		! frequency (''f'', ratio)
	\|(1)		\|(1)
	\|0.87		\|0.87
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	\|1/φ		\|1/φ
	\|-		\|-
	! pitch (~~log₂f~~)		! pitch (log₂''f'', octaves)
	\|(0)		\|(0)
	\| -0.21		\| -0.21
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	\| -0.69		\| -0.69
	\|-		\|-
	! length (1/f)		! length (1/''f'', ratio)
	\|(1+(0/4)(φ-1)) = (0φ + 4)/4 = 1		\|(1+(0/4)(φ-1)) = (0φ + 4)/4 = 1
	\|1+(1/4)(φ-1) = (1φ + 3)/4		\|1+(1/4)(φ-1) = (1φ + 3)/4

FloraC: Rework to address alternative terms

2023-04-26T08:52:48Z

Rework to address alternative terms

← Older revision		Revision as of 08:52, 26 April 2023
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	An '''ELD''', or '''~~equal length~~ division''', is ~~a kind of~~ [[Arithmetic tunings\|arithmetic]] and [[~~Harmonotonic tunings\|harmonotonic~~]] tuning.		An '''ELD''' ('''equal length division'''), '''ALD''' ('''arithmetic length division'''), or '''IFD''' ('''inverse-arithmetic frequency division'''), is an [[Arithmetic tunings\|arithmetic]] and [[period]]ic [[tuning]] in which each period is divided to a number of steps of equal length difference.

	== Specification ==		== Specification ==

	Its full specification is n-~~ELDp:~~ n equal length divisions of ~~the interval~~ p.		Its full specification is ''n''-ELD-''p'' (''n'' equal length divisions of ''p''), or ''n''-ALD-''p'' (''n'' arithmetic length divisions of ''p''), or ''n''-IFD-''p'' (''n'' inverse-arithmetic frequency division of ''p'').

	== Formula ==		== Formula ==

	To find the steps for an n-~~ELDp~~, 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-ELDp is:		To find the steps for an ''n''-ELD-''p'', 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-ELDp is:

	<math>		<math>
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	=== Vs. UD ===		=== Vs. UD ===

	~~The only difference between an n-ELDp and an~~ [[UD\|n-~~UDp~~ (or utonal division)]] is that the p ~~for a utonal division~~ must be rational.		An [[UD\|''n''-UD-''p'' (or utonal division)]] is equivalent to an ''n''-ELD-''p'' except that the period ''p'' of the UD must be rational.

	=== Vs. EFD ===		=== Vs. EFD ===
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	=== Vs. ALS ===		=== Vs. ALS ===

	One period of an ELD will be equivalent to some [[ALS\|ALS (arithmetic length sequence)]]; specifically n-ELD((p-1)/n) = n-~~ALSp~~.		One period of an ELD will be equivalent to some [[ALS\|ALS (arithmetic length sequence)]]; specifically ''n''-ELD((''p'' - 1)/''n'') = ''n''-ALS-''p''.

	=== Vs. EDL ===		=== Vs. EDL ===

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

2023-03-28T21:41:12Z

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. EPD ===		=== Vs. EPD ===

	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 ~~can be~~ 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.		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.

	=== Vs. UD ===		=== Vs. UD ===

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

2023-03-28T21:41:04Z

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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	== Specification ==		== Specification ==

	Its full specification is n-ELDp: n equal length divisions of the ~~irrational~~ interval p.		Its full specification is n-ELDp: n equal length divisions of the interval p.

	== Formula ==		== Formula ==
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	=== Vs. EPD ===		=== Vs. EPD ===

	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.		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 can be 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.

	=== Vs. UD ===		=== Vs. UD ===

	The only difference between an n-ELDp and an [[UD\|n-UDp (or utonal division)]] is that the p for a utonal division is rational.		The only difference between an n-ELDp and an [[UD\|n-UDp (or utonal division)]] is that the p for a utonal division must be rational.

	=== Vs. EFD ===		=== Vs. EFD ===

Cmloegcmluin: /* Vs. ALS */ correction re: periodicity (thanks Flora!)

2023-03-28T21:23:01Z

Vs. ALS: correction re: periodicity (thanks Flora!)

← Older revision		Revision as of 21:23, 28 March 2023
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	=== Vs. ALS ===		=== Vs. ALS ===

	An ELD will be equivalent to some [[ALS\|ALS (arithmetic length sequence)]]; specifically n-ELD((p-1)/n) = n-ALSp.		One period of an ELD will be equivalent to some [[ALS\|ALS (arithmetic length sequence)]]; specifically n-ELD((p-1)/n) = n-ALSp.

	=== Vs. EDL ===		=== Vs. EDL ===

Cmloegcmluin: fix section capitalization

2021-11-11T19:40:35Z

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. EPD ===		=== Vs. EPD ===

	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.		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.

	=== vs. UD ===		=== Vs. UD ===

	The only difference between an n-ELDp and an [[UD\|n-UDp (or utonal division)]] is that the p for a utonal division is rational.		The only difference between an n-ELDp and an [[UD\|n-UDp (or utonal division)]] is that the p for a utonal division is rational.

	=== vs. EFD ===		=== Vs. EFD ===

	The analogous otonal equivalent of an ELD is an [[EFD\|EFD (equal frequency division)]].		The analogous otonal equivalent of an ELD is an [[EFD\|EFD (equal frequency division)]].

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

	An ELD will be equivalent to some [[ALS\|ALS (arithmetic length sequence)]]; specifically n-ELD((p-1)/n) = n-ALSp.		An ELD will be equivalent to some [[ALS\|ALS (arithmetic length sequence)]]; specifically n-ELD((p-1)/n) = n-ALSp.

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

	An ELD is not to be confused with [[EDL\|EDL, equal division of length]]. The latter term does not take an interval parameter because it is assumed to be the length of an entire string, and then only an octave subset of that is taken.		An ELD is not to be confused with [[EDL\|EDL, equal division of length]]. The latter term does not take an interval parameter because it is assumed to be the length of an entire string, and then only an octave subset of that is taken.

Fredg999: Categories

2021-08-30T03:59:36Z

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

2021-03-24T20:38:27Z

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

@@ Line 1: / Line 1: @@
 An '''ELD''', or '''equal length division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-Its full specification is n-ELDp: n equal length divisions of the irrational interval p. The only difference between an n-ELDp and an [[UD|n-UDp (or utonal division)]] is that the p for a utonal division is rational.
+== Specification ==
-The analogous otonal equivalent of an ELD is an [[EFD|EFD (equal frequency division)]].
+Its full specification is n-ELDp: n equal length divisions of the irrational 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-ELDp, 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-ELDp is:
@@ Line 18: / 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"
@@ Line 83: / Line 107: @@
 |1+(4/4)(φ-1) = (4φ + 0)/4 = φ
 |}
 [[Category:Undertone]]

Cmloegcmluin: add formula for mathematician benefit

2021-03-23T22:38:09Z

add formula for mathematician benefit

← Older revision		Revision as of 22:38, 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-ELDp, 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-ELDp 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: correct goof

2021-03-23T22:32:31Z

correct goof

← Older revision		Revision as of 22:32, 23 March 2021
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	An ELD will be equivalent to some [[ALS\|ALS (arithmetic length sequence)]]; specifically n-ELD((p-1)/n) = n-ALSp.		An ELD will be equivalent to some [[ALS\|ALS (arithmetic length sequence)]]; specifically n-ELD((p-1)/n) = n-ALSp.

	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 ~~otonal divisions of the octave and~~ utonal divisions of the octave~~, respectively~~.		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.		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.

← Older revision		Revision as of 03:59, 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]]