EFD - Revision history

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

2023-10-19T20:35:30Z

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

← Older revision		Revision as of 20:35, 19 October 2023
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	! 4		! 4
	\|-		\|-
	! frequency (f)		! frequency (''f'', ratio)
	\| (1 + (0/4)(φ - 1))<br>= (0φ + 4)/4<br>= 1		\| (1 + (0/4)(φ - 1))<br>= (0φ + 4)/4<br>= 1
	\| 1 + (1/4)(φ - 1)<br>= (1φ + 3)/4		\| 1 + (1/4)(φ - 1)<br>= (1φ + 3)/4
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	\| 1 + (4/4)(φ - 1)<br>= (4φ + 0)/4<br>= φ		\| 1 + (4/4)(φ - 1)<br>= (4φ + 0)/4<br>= 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)		\| (1)
	\| 4/(φ + 3) = 0.87		\| 4/(φ + 3) = 0.87

FloraC: Contract the intro. Arithmetic-ity implies step-monotonicity. Periodicity must be mentioned before the period

2023-04-26T08:35:35Z

Contract the intro. Arithmetic-ity implies step-monotonicity. Periodicity must be mentioned before the period

← Older revision		Revision as of 08:35, 26 April 2023
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	An '''EFD''' ('''equal frequency division''') or '''AFD''' ('''arithmetic frequency division''') is a kind of [[Arithmetic tunings\|arithmetic]] and [[~~Harmonotonic tunings\|harmonotonic~~]] tuning in which each period is divided to a number of steps of equal frequency difference~~. It is a [[period]]ic [[tuning system]]~~.		An '''EFD''' ('''equal frequency division''') or '''AFD''' ('''arithmetic frequency division''') is a kind of [[Arithmetic tunings\|arithmetic]] and [[period]]ic [[tuning]] in which each period is divided to a number of steps of equal frequency difference.

	== Specification ==		== Specification ==

FloraC: AD -> AFD

2023-04-26T08:30:03Z

AD -> AFD

← Older revision		Revision as of 08:30, 26 April 2023
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	An '''EFD''' ('''equal frequency division''') or '''AD''' ('''arithmetic division''') is a kind of [[Arithmetic tunings\|arithmetic]] and [[Harmonotonic tunings\|harmonotonic]] tuning in which each period is divided to a number of steps of equal frequency difference. It is a [[period]]ic [[tuning system]].		An '''EFD''' ('''equal frequency division''') or '''AFD''' ('''arithmetic frequency division''') is a kind of [[Arithmetic tunings\|arithmetic]] and [[Harmonotonic tunings\|harmonotonic]] tuning in which each period is divided to a number of steps of equal frequency difference. It is a [[period]]ic [[tuning system]].

	== Specification ==		== Specification ==

	Its full specification is ''n''-EFD-''p'' or ''n''-AD-''p'': ''n'' equal frequency divisions of ''p'', or ''n'' arithmetic divisions of ''p'' .		Its full specification is ''n''-EFD-''p'' or ''n''-AFD-''p'': ''n'' equal frequency divisions of ''p'', or ''n'' arithmetic frequency divisions of ''p'' .

	== Formula ==		== Formula ==
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	=== Vs. OD ===		=== Vs. OD ===

	An [[OD\|n-~~ODp~~]] is equivalent to an n-~~EFDp~~ except that the period <math>p</math> of the OD must be rational.		An [[OD\|''n''-OD-''p'']] is equivalent to an ''n''-EFD-''p'' except that the period <math>p</math> of the OD must be rational.

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

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

2023-03-28T21:23:03Z

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

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

	An EFD will be equivalent to some [[AFS\|AFS, or arithmetic frequency sequence]], which has had its count of pitches specified by prefixing "''n''-"; specifically, ''n''-efd-''p'' = ''n''-AFS((''p'' - 1)/''n'').		One period of an EFD will be equivalent to some [[AFS\|AFS, or arithmetic frequency sequence]], which has had its count of pitches specified by prefixing "''n''-"; specifically, ''n''-efd-''p'' = ''n''-AFS((''p'' - 1)/''n'').

	== Examples ==		== Examples ==

Cmloegcmluin: /* Examples */ include both phi-based and decimal values, so the descending nature can be understood

2023-03-22T18:03:23Z

Examples: include both phi-based and decimal values, so the descending nature can be understood

← Older revision		Revision as of 18:03, 22 March 2023
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	! length (1/f)		! length (1/f)
	\| (1)		\| (1)
	\| 4/(φ + 3)		\| 4/(φ + 3) = 0.87
	\| 2/(φ + 1)		\| 2/(φ + 1) = 0.76
	\| 4/(3φ + 1)		\| 4/(3φ + 1) = 0.68
	\| 1/φ		\| 1/φ = 0.62
	\|}		\|}

Cmloegcmluin: /* Examples */ parenthesize quantity for the base position, as is done for all other such tables in harmonotonic tuning pages

2023-03-22T18:01:44Z

Examples: parenthesize quantity for the base position, as is done for all other such tables in harmonotonic tuning pages

← Older revision		Revision as of 18:01, 22 March 2023
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	\|-		\|-
	! quantity		! quantity
	! 0		! (0)
	! 1		! 1
	! 2		! 2
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	\|-		\|-
	! frequency (f)		! frequency (f)
	\| 1 + (0/4)(φ - 1)<br>= (0φ + 4)/4<br>= 1		\| (1 + (0/4)(φ - 1))<br>= (0φ + 4)/4<br>= 1
	\| 1 + (1/4)(φ - 1)<br>= (1φ + 3)/4		\| 1 + (1/4)(φ - 1)<br>= (1φ + 3)/4
	\| 1 + (2/4)(φ - 1)<br>= (2φ + 2)/4		\| 1 + (2/4)(φ - 1)<br>= (2φ + 2)/4
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	\|-		\|-
	! pitch (log₂f)		! pitch (log₂f)
	\| 0		\| (0)
	\| 0.21		\| 0.21
	\| 0.39		\| 0.39
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	\|-		\|-
	! length (1/f)		! length (1/f)
	\| 1		\| (1)
	\| 4/(φ + 3)		\| 4/(φ + 3)
	\| 2/(φ + 1)		\| 2/(φ + 1)

Cmloegcmluin: /* Examples */ return the headings to how they are on other harmonotonic tuning page tables

2023-03-22T18:00:14Z

Examples: return the headings to how they are on other harmonotonic tuning page tables

← Older revision		Revision as of 18:00, 22 March 2023
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	\|+Example: 4-EFDφ		\|+Example: 4-EFDφ
	\|-		\|-
	! ~~Steps~~		! quantity
	! 0		! 0
	! 1		! 1
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	! 4		! 4
	\|-		\|-
	! ~~Frequency Ratios~~ (~~''r''~~)		! frequency (f)
	\| 1 + (0/4)(φ - 1)<br>= (0φ + 4)/4<br>= 1		\| 1 + (0/4)(φ - 1)<br>= (0φ + 4)/4<br>= 1
	\| 1 + (1/4)(φ - 1)<br>= (1φ + 3)/4		\| 1 + (1/4)(φ - 1)<br>= (1φ + 3)/4
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	\| 1 + (4/4)(φ - 1)<br>= (4φ + 0)/4<br>= φ		\| 1 + (4/4)(φ - 1)<br>= (4φ + 0)/4<br>= φ
	\|-		\|-
	! ~~Octaves~~ (~~log<sub>2</sub>''r''~~)		! pitch (log₂f)
	\| 0		\| 0
	\| 0.21		\| 0.21
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	\| 0.69		\| 0.69
	\|-		\|-
	! ~~Length Ratios~~ (1/~~''r''~~)		! length (1/f)
	\| 1		\| 1
	\| 4/(φ + 3)		\| 4/(φ + 3)

Cmloegcmluin: /* Vs. OD */ return some clarifying aspects

2023-03-22T17:57:38Z

Vs. OD: return some clarifying aspects

← Older revision		Revision as of 17:57, 22 March 2023
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	=== Vs. OD ===		=== Vs. OD ===

	OD is equivalent to ~~EFD,~~ except that the period of OD is rational.		An [[OD\|n-ODp]] is equivalent to an n-EFDp except that the period <math>p</math> of the OD must be rational.

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

Cmloegcmluin: /* Examples */ missed a spot

2023-03-22T17:54:00Z

Examples: missed a spot

← Older revision		Revision as of 17:54, 22 March 2023
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	{\| class="wikitable"		{\| class="wikitable"
	\|+Example: ~~4efdφ~~		\|+Example: 4-EFDφ
	\|-		\|-
	! Steps		! Steps

Cmloegcmluin: use formatting for initialisms that is consistent across all other harmonotonic tuning pages (and preferred by the original author of this article)

2023-03-22T17:51:15Z

use formatting for initialisms that is consistent across all other harmonotonic tuning pages (and preferred by the original author of this article)

← Older revision		Revision as of 17:51, 22 March 2023
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	== Specification ==		== Specification ==

	Its full specification is ''n''-~~efd~~-''p'' or ''n''-ad-''p'': ''n'' equal frequency divisions of ''p'', or ''n'' arithmetic divisions of ''p'' .		Its full specification is ''n''-EFD-''p'' or ''n''-AD-''p'': ''n'' equal frequency divisions of ''p'', or ''n'' arithmetic divisions of ''p'' .

	== Formula ==		== Formula ==

	To find the steps for an ''n''-~~efd~~-''p'', begin by recognizing that while the multiplicative interval relating your root position to the end position is <math>p</math> (or <math>\frac p1</math>), if you are going to move arithmetically (by repeated addition) from <math>1</math> to <math>p</math>, then the difference in frequency space that you are dividing up is not actually <math>p</math>, but <math>p - 1</math>. And because you are dividing it into <math>n</math> parts, each step will have a size of <math>\frac{p-1}{n}</math>. So within each period, the ratio ''c'' of the ''k''-th step of an ''n''-~~efd~~-''p'' is:		To find the steps for an ''n''-EFD-''p'', begin by recognizing that while the multiplicative interval relating your root position to the end position is <math>p</math> (or <math>\frac p1</math>), if you are going to move arithmetically (by repeated addition) from <math>1</math> to <math>p</math>, then the difference in frequency space that you are dividing up is not actually <math>p</math>, but <math>p - 1</math>. And because you are dividing it into <math>n</math> parts, each step will have a size of <math>\frac{p-1}{n}</math>. So within each period, the ratio ''c'' of the ''k''-th step of an ''n''-EFD-''p'' is:

	<math>		<math>
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	=== Vs. EPD ===		=== Vs. EPD ===

	Instead of equally dividing the octave into 12 equal parts by pitch, as is done for ~~12epdo~~, or ~~12edo~~ (because pitch can be assumed), standard tuning, you could divide it into 12 equal parts by ''frequency''. This would give you ~~12efdo~~.		Instead of equally dividing the octave into 12 equal parts by pitch, as is done for 12-EPDO, or 12-EDO (because pitch can be assumed), standard tuning, you could divide it into 12 equal parts by ''frequency''. This would give you 12-EFDO.

	=== Vs. OD ===		=== Vs. OD ===