5edf - Revision history

CompactStar at 00:38, 10 August 2025

2025-08-10T00:38:24Z

← Older revision		Revision as of 00:38, 10 August 2025
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	5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]] (also known as [[17ed4]]) 5edf has an extremely accurate approximation of the seventh harmonic for its size.		5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]] (also known as [[17ed4]]) 5edf has an extremely accurate approximation of the seventh harmonic for its size.

	5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. One must be wary of the 3/2-equivalence paradigm-there is no dominant, and major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as unlike the ~~convenyional~~ triads, they are not equivalent to dyads, and are somewhat more consonant than in [[12edo]] due to the laxer subtritone.		5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. One must be wary of the 3/2-equivalence paradigm-there is no dominant, and major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as unlike the conventional triads, they are not equivalent to dyads, and are somewhat more consonant than in [[12edo]] due to the laxer subtritone.
	=== Harmonics ===		=== Harmonics ===
	{{Harmonics in equal\|5\|3\|2\|columns=15}}		{{Harmonics in equal\|5\|3\|2\|columns=15}}

CompactStar at 00:38, 10 August 2025

2025-08-10T00:38:10Z

← Older revision		Revision as of 00:38, 10 August 2025
Line 5:		Line 5:
	5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]] (also known as [[17ed4]]) 5edf has an extremely accurate approximation of the seventh harmonic for its size.		5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]] (also known as [[17ed4]]) 5edf has an extremely accurate approximation of the seventh harmonic for its size.

	5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. One must be wary of the 3/2-equivalence paradigm-there is no dominant, and major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as they are not equivalent to dyads and are somewhat more consonant than in [[12edo]] due to the laxer subtritone.		5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. One must be wary of the 3/2-equivalence paradigm-there is no dominant, and major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as unlike the convenyional triads, they are not equivalent to dyads, and are somewhat more consonant than in [[12edo]] due to the laxer subtritone.
	=== Harmonics ===		=== Harmonics ===
	{{Harmonics in equal\|5\|3\|2\|columns=15}}		{{Harmonics in equal\|5\|3\|2\|columns=15}}

CompactStar at 00:37, 10 August 2025

2025-08-10T00:37:28Z

← Older revision		Revision as of 00:37, 10 August 2025
Line 5:		Line 5:
	5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]] (also known as [[17ed4]]) 5edf has an extremely accurate approximation of the seventh harmonic for its size.		5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]] (also known as [[17ed4]]) 5edf has an extremely accurate approximation of the seventh harmonic for its size.

	5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. One must be wary of the 3/2-equivalence paradigm-major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as they are not equivalent to dyads and are somewhat more consonant than in [[12edo]] due to the laxer subtritone.		5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. One must be wary of the 3/2-equivalence paradigm-there is no dominant, and major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as they are not equivalent to dyads and are somewhat more consonant than in [[12edo]] due to the laxer subtritone.
	=== Harmonics ===		=== Harmonics ===
	{{Harmonics in equal\|5\|3\|2\|columns=15}}		{{Harmonics in equal\|5\|3\|2\|columns=15}}

CompactStar: /* Theory */

2025-08-10T00:36:57Z

Theory

← Older revision		Revision as of 00:36, 10 August 2025
Line 5:		Line 5:
	5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]] (also known as [[17ed4]]) 5edf has an extremely accurate approximation of the seventh harmonic for its size.		5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]] (also known as [[17ed4]]) 5edf has an extremely accurate approximation of the seventh harmonic for its size.

	5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. ~~one~~ must be wary of the 3/2-equivalence paradigm-major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as they are not equivalent to dyads and are somewhat more consonant than in [[12edo]] due to the laxer subtritone.		5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. One must be wary of the 3/2-equivalence paradigm-major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as they are not equivalent to dyads and are somewhat more consonant than in [[12edo]] due to the laxer subtritone.
	=== Harmonics ===		=== Harmonics ===
	{{Harmonics in equal\|5\|3\|2\|columns=15}}		{{Harmonics in equal\|5\|3\|2\|columns=15}}

CompactStar: /* Theory */Add some helpful information I got from writing a sketch in this tuning.

2025-08-10T00:36:42Z

Theory: Add some helpful information I got from writing a sketch in this tuning.

← Older revision		Revision as of 00:36, 10 August 2025
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	== Theory ==		== Theory ==
	5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]]~~. 4 steps of 5edf is a fraction of a cent away to the seventh harmonic~~ (~~which is~~ [[~~112/81~~]] ~~instead of [[7/4]] since the equave is 3/2~~)~~, which is~~ an extremely accurate approximation for ~~the~~ size ~~of this scale~~.		5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]] (also known as [[17ed4]]) 5edf has an extremely accurate approximation of the seventh harmonic for its size.

			5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. one must be wary of the 3/2-equivalence paradigm-major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as they are not equivalent to dyads and are somewhat more consonant than in [[12edo]] due to the laxer subtritone.
	=== Harmonics ===		=== Harmonics ===
	{{Harmonics in equal\|5\|3\|2\|columns=15}}		{{Harmonics in equal\|5\|3\|2\|columns=15}}

MisterShafXen at 18:52, 25 June 2025

2025-06-25T18:52:03Z

← Older revision		Revision as of 18:52, 25 June 2025
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	\| 8		\| 8
	\| 1123		\| 1123
	\|		\| 44/23
	\| major seventh, minor octave		\| major seventh, minor octave
	\| E, Fb		\| E, Fb
Line 75:		Line 75:
	\| 9		\| 9
	\| 1264		\| 1264
	\|		\| 83/40
	\| major octave		\| major octave
	\| F		\| F
Line 82:		Line 82:
	\| 1404		\| 1404
	\| [[9/4]]		\| [[9/4]]
	\|		\| major ninth
	\| C		\| C
	\|}		\|}

	{{Todo\|expand}}		{{Todo\|expand}}

FloraC: +subsets and supersets; cleanup

2025-02-23T13:45:02Z

+subsets and supersets; cleanup

← Older revision		Revision as of 13:45, 23 February 2025
Line 6:		Line 6:

	=== Harmonics ===		=== Harmonics ===
	{{Harmonics in equal\|5\|3\|2}}		{{Harmonics in equal\|5\|3\|2\|columns=15}}

			=== Subsets and supersets ===
			5edf is the 3rd [[prime equal division\|prime edf]], after [[3edf]] and before [[7edf]].

	== Intervals ==		== Intervals ==
Line 13:		Line 16:
	! #		! #
	! Cents		! Cents
	! Approximate ~~Ratios~~		! Approximate ratios
	! colspan="2"\| [[1L 3s (fifth-equivalent)\|Neptunian]] ~~Notation~~		! colspan="2"\| [[1L 3s (fifth-equivalent)\|Neptunian]] notation
	\|-		\|-
	\| 0		\| 0
Line 23:		Line 26:
	\|-		\|-
	\| 1		\| 1
	\| 140.4		\| 140
	\| [[13/12]], [[49/45]]		\| [[13/12]], [[49/45]]
	\| augmented unison, minor second		\| augmented unison, minor second
Line 29:		Line 32:
	\|-		\|-
	\| 2		\| 2
	\| ~~280.8~~		\| 281
	\| [[75/64]], [[20/17]], [[13/11]]		\| [[13/11]], [[20/17]], [[75/64]]
	\| major second, minor third		\| major second, minor third
	\| D, Eb		\| D, Eb
	\|-		\|-
	\| 3		\| 3
	\| 421.2		\| 421
	\| [[14/11]], [[23/18]]		\| [[14/11]], [[23/18]]
	\| major third, diminished fourth		\| major third, diminished fourth
Line 41:		Line 44:
	\|-		\|-
	\| 4		\| 4
	\| ~~561.6~~		\| 562
	\| [[11/8]], [[18/13]], [[25/18]]		\| [[11/8]], [[18/13]], [[25/18]]
	\| perfect fourth		\| perfect fourth
Line 47:		Line 50:
	\|-		\|-
	\| 5		\| 5
	\| 702.0		\| 702
	\| [[3/2]]		\| [[3/2]]
	\| perfect fifth		\| perfect fifth
Line 53:		Line 56:
	\|-		\|-
	\| 6		\| 6
	\| 842.3		\| 842
	\| [[21/13]], [[13/8]], [[18/11]]		\| [[13/8]], [[18/11]], [[21/13]]
	\| augmented fifth, minor sixth		\| augmented fifth, minor sixth
	\| C#, Db		\| C#, Db
	\|-		\|-
	\| 7		\| 7
	\| ~~982.7~~		\| 983
	\| [[7/4]], [[30/17]]		\| [[7/4]], [[30/17]]
	\| major sixth, minor seventh		\| major sixth, minor seventh
Line 65:		Line 68:
	\|-		\|-
	\| 8		\| 8
	\| 1123.1		\| 1123
	\|		\|
	\| major seventh, minor octave		\| major seventh, minor octave
Line 71:		Line 74:
	\|-		\|-
	\| 9		\| 9
	\| ~~1263.5~~		\| 1264
	\|		\|
	\| major octave		\| major octave
	\| F		\| F
	\|-		\|-
	\| 10		\| 10
	\| ~~1403.9~~		\| 1404
	\|		\| [[9/4]]
	\|
	~~\| C~~
	\|-
	~~\| 11~~
	~~\| 1544.3~~
	\|
	\|
	~~\| C#, Db~~
	\|-
	~~\| 12~~
	~~\| 1684.7~~
	\|
	\|
	~~\| D, Eb~~
	\|-
	~~\| 13~~
	~~\| 1825.1~~
	\|
	\|
	~~\| E~~
	\|-
	~~\| 14~~
	~~\| 1965.5~~
	\|
	\|
	~~\| F~~
	\|-
	~~\| 15~~
	~~\| 2105.~~9
	\|
	\|		\|
	\| C		\| C
	\|-
	~~\| 16~~
	~~\| 2246.3~~
	\|
	\|
	~~\| C#, Db~~
	\|-
	~~\| 17~~
	~~\| 2386.6~~
	\|
	\|
	~~\| D~~
	\|}		\|}

	{{~~todo~~\|expand}}		{{Todo\|expand}}

BudjarnLambeth: {{todo|expand}}

2024-12-19T08:55:39Z

← Older revision		Revision as of 08:55, 19 December 2024
Line 124:		Line 124:
	\| D		\| D
	\|}		\|}

			{{todo\|expand}}

FloraC: Rework interval table, including removing irrelevant stuff

2024-05-10T06:14:35Z

Rework interval table, including removing irrelevant stuff

Show changes

FloraC: Adopt template: ED intro; +harmonics table; misc. cleanup

2024-05-10T06:09:12Z

Adopt template: ED intro; +harmonics table; misc. cleanup

← Older revision		Revision as of 06:09, 10 May 2024
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	{{Infobox ET}}		{{Infobox ET}}
	~~'''5EDF''' is the [[EDF\|equal division of the just perfect fifth]] into five parts of 140.391 [[cent\|cents]] each, corresponding~~ to 8.5476 [[edo]]. It is close to the [[~~Bleu~~]] generator chain and every second step of [[17edo]]. 4 steps of 5edf is a fraction of a cent away to the seventh harmonic (which is [[112/81]] instead of [[7/4]] since the equave is 3/2), which is an extremely accurate approximation for the size of this scale.		{{ED intro}} It corresponds to 8.5476 [[edo]].
	==Intervals==
			== Theory ==
			5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]]. 4 steps of 5edf is a fraction of a cent away to the seventh harmonic (which is [[112/81]] instead of [[7/4]] since the equave is 3/2), which is an extremely accurate approximation for the size of this scale.

			=== Harmonics ===
			{{Harmonics in equal\|5\|3\|2}}

			== Intervals ==
	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-