8edo - Revision history

Contribution: /* Intervals */

2026-06-17T20:54:53Z

Intervals

← Older revision		Revision as of 20:54, 17 June 2026
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	\| 11/6		\| 11/6
	\| 20/11, 11/6		\| 20/11, 11/6
	\|11/6, 14/13		\|11/6, 13/7
	\|9/5, 48/50		\|9/5, 48/50
	\|-		\|-

Contribution: /* Intervals */

2026-06-17T20:48:19Z

Intervals

← Older revision		Revision as of 20:48, 17 June 2026
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	\|}		\|}
	<nowiki />* Allows [[inversion]] by 2/1; other interpretations also possible		<nowiki />* Allows [[inversion]] by 2/1; other interpretations also possible

	<nowiki />** 16 integer limit, relative error < 15 %		<nowiki />** 16 integer limit, relative error < 15 %

Contribution: /* Intervals */

2026-06-17T20:47:57Z

Intervals

← Older revision		Revision as of 20:47, 17 June 2026
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	! 2.5/3.11/3.13/5*		! 2.5/3.11/3.13/5*
	! 10:11:12:13:14*		! 10:11:12:13:14*
	!val 8d~~, 16 integer limit, relative error < 15 %~~		!val 8d**
	!Patent val ⟨8 13 19]		!Patent val ⟨8 13 19]
	\|-		\|-
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	\|}		\|}
	<nowiki />* Allows [[inversion]] by 2/1; other interpretations also possible		<nowiki />* Allows [[inversion]] by 2/1; other interpretations also possible
			<nowiki />** 16 integer limit, relative error < 15 %

	== Notation ==		== Notation ==

Contribution: /* Intervals */

2026-06-17T20:46:53Z

Intervals

← Older revision		Revision as of 20:46, 17 June 2026
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	! rowspan="2" \| Steps		! rowspan="2" \| Steps
	! rowspan="2" \| Cents		! rowspan="2" \| Cents
	! colspan="3" \| JI approximation		! colspan="4" \| JI approximation
	!Other		!Other
	\|-		\|-
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	! 2.5/3.11/3.13/5*		! 2.5/3.11/3.13/5*
	! 10:11:12:13:14*		! 10:11:12:13:14*
			!val 8d, 16 integer limit, relative error < 15 %
	!Patent val ⟨8 13 19]		!Patent val ⟨8 13 19]
	\|-		\|-
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	\| 1/1		\| 1/1
	\| 1/1		\| 1/1
			\|1/1
	\|1/1, 16/15		\|1/1, 16/15
	\|-		\|-
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	\| 12/11		\| 12/11
	\| 12/11, 11/10		\| 12/11, 11/10
			\|14/13, 13/12, 12/11, 11/10
	\|10/9, 25/24		\|10/9, 25/24
	\|-		\|-
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	\| 6/5		\| 6/5
	\| 6/5		\| 6/5
			\|13/11, 6/5
	\|6/5, 9/8		\|6/5, 9/8
	\|-		\|-
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	\| 13/10		\| 13/10
	\| 13/10		\| 13/10
			\|9/7, 13/10
	\|4/3, 5/4		\|4/3, 5/4
	\|-		\|-
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	\|		\|
	\| 7/5, 10/7		\| 7/5, 10/7
			\|7/5, 10/7
	\|27/20, 25/18, 36/25		\|27/20, 25/18, 36/25
	\|-		\|-
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	\| 20/13		\| 20/13
	\| 20/13		\| 20/13
			\|14/9
	\|3/2, 8/5		\|3/2, 8/5
	\|-		\|-
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	\| 5/3		\| 5/3
	\| 5/3		\| 5/3
			\|5/3
	\|5/3, 16/9		\|5/3, 16/9
	\|-		\|-
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	\| 11/6		\| 11/6
	\| 20/11, 11/6		\| 20/11, 11/6
			\|11/6, 14/13
	\|9/5, 48/50		\|9/5, 48/50
	\|-		\|-
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	\| 2/1		\| 2/1
	\| 2/1		\| 2/1
			\|2/1
	\|2/1, 15/8		\|2/1, 15/8
	\|}		\|}

Overthink: /* Temperaments */ this isn't an edo join

2026-02-12T04:21:22Z

Temperaments: this isn't an edo join

← Older revision		Revision as of 04:21, 12 February 2026
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	=== Temperaments ===		=== Temperaments ===
	8edo is fairly composite, so the only step that generates a [[mos]] scale that covers every interval other than the 1 is the 3, producing scales of 332 and [[3L 2s\|21212]]. In terms of temperaments, in the 5-limit this is best interpreted as [[father]], as 8edo is the highest edo that tempers out the diatonic semitone in it's [[patent val]], merging 5/4 and 4/3 into a single interval, which is also the generator. This means major and minor chords are rotations of each other, making them inaccurate but very simple, with even the 5 note mos having 3 of both and providing a functional skeleton of 5-limit harmony, albeit with some very strange enharmonic equivalences. In terms of 7-limit extensions things get even more inaccurate, as the patent val supports [[mother]], but the ideal tuning for that is much closer to [[5edo]]. The 8d val supports septimal father and [[pater]], and is much closer to the ideal tuning for both, as the extremely sharp 7 works better with the ~~{{nowrap\|~~ 3 & 5 }}. In terms of multi-period temperaments, it makes for a near perfect [[walid]] or a much less accurate [[diminished (temperament)\|diminished]] scale.		8edo is fairly composite, so the only step that generates a [[mos]] scale that covers every interval other than the 1 is the 3, producing scales of 332 and [[3L 2s\|21212]]. In terms of temperaments, in the 5-limit this is best interpreted as [[father]], as 8edo is the highest edo that tempers out the diatonic semitone in it's [[patent val]], merging 5/4 and 4/3 into a single interval, which is also the generator. This means major and minor chords are rotations of each other, making them inaccurate but very simple, with even the 5 note mos having 3 of both and providing a functional skeleton of 5-limit harmony, albeit with some very strange enharmonic equivalences. In terms of 7-limit extensions things get even more inaccurate, as the patent val supports [[mother]], but the ideal tuning for that is much closer to [[5edo]]. The 8d val supports septimal father and [[pater]], and is much closer to the ideal tuning for both, as the extremely sharp 7 works better with the 3 and 5. In terms of multi-period temperaments, it makes for a near perfect [[walid]] or a much less accurate [[diminished (temperament)\|diminished]] scale.

	== Instruments ==		== Instruments ==

Lucius Chiaraviglio: /* Music */ Add Bryan Deister's ''microtonal improvisation in 8edo'' (2024)

2026-01-17T10:52:12Z

Music: Add Bryan Deister's ''microtonal improvisation in 8edo'' (2024)

← Older revision		Revision as of 10:52, 17 January 2026
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	; [[City of the Asleep]]		; [[City of the Asleep]]
	* [http://ia600607.us.archive.org/3/items/Transcendissonance/10Malebolge-CityOfTheAsleep.mp3 "Malebolge"], from [https://cityoftheasleep.bandcamp.com/album/transcendissonance ''Transcendissonance''] (2011)		* [http://ia600607.us.archive.org/3/items/Transcendissonance/10Malebolge-CityOfTheAsleep.mp3 "Malebolge"], from [https://cityoftheasleep.bandcamp.com/album/transcendissonance ''Transcendissonance''] (2011)

			; [[Bryan Deister]]
			* [https://www.youtube.com/shorts/qk_kMCCXpss ''microtonal improvisation in 8edo''] (2024)

	; [[Milan Guštar]]		; [[Milan Guštar]]

Overthink: /* Approximation to JI */ + more info

2025-12-19T03:34:53Z

Approximation to JI: + more info

← Older revision		Revision as of 03:34, 19 December 2025
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	=== Approximation to JI ===		=== Approximation to JI ===
	8edo forms an odd and even pitch set of two diminished seventh chords, which when used in combination yield dissonance. The system has been described as a "barbaric" harmonic system, containing no good approximation of harmonics 3, 5, 7, 11, 13, and 17; even so, it does a good job representing the [[just intonation subgroup]]s 2.11/3.13/5, with good intervals of [[13/10]] and an excellent version of [[11/6]]. Stacking the 450-cent interval can result in some semi-consonant chords such as 0-3-6 degrees, although these still are quite dissonant compared to standard root-3rd-P5 triads, which are unavailable in 8edo.		8edo forms an odd and even pitch set of two diminished seventh chords, which when used in combination yield dissonance. The system has been described as a "barbaric" harmonic system, containing no good approximation of harmonics 3, 5, 7, 11, 13, and 17; even so, it does a good job representing the [[just intonation subgroup]] 2.11/3.13/5, with good intervals of [[13/10]] and an excellent version of [[11/6]]. Stacking the 450-cent interval can result in some semi-consonant chords such as 0-3-6 degrees, although these still are quite dissonant compared to standard root-3rd-P5 triads, which are unavailable in 8edo.

	Another way of looking at 8edo is to treat a chord of 0-1-2-3-4 degrees (0-150-300-450-600 cents) as approximating harmonics 10:11:12:13:14 (~0-165-316-454-583 cents), which is not too implausible if you can buy that 12edo is a 5-limit temperament. This interpretation would imply that 121/120, 144/143, 169/168, and hence also 36/35 and 66/65, are tempered out.		Another way of looking at 8edo is to treat a chord of 0-1-2-3-4 degrees (0-150-300-450-600 cents) as approximating harmonics 10:11:12:13:14 (~0-165-316-454-583 cents), which is not too implausible if you can buy that 12edo is a 5-limit temperament. This interpretation would imply that 121/120, 144/143, 169/168, and hence also 36/35 and 66/65, are tempered out. The corresponding subgroup is 2.5/3.7/3.11/3.13/3. However, some intervals in this chord, such as [[14/11]] and [[7/6]], are tuned quite inaccurately (over 30 cents off). Nonetheless, the 8-form serves as an underlying structure in many [[non-over-1 temperament]]s.

	=== Relationship with the father comma ===		=== Relationship with the father comma ===

Squib: /* Subsets and supersets */added 72edo as superset

2025-11-16T14:25:31Z

Subsets and supersets: added 72edo as superset

← Older revision		Revision as of 14:25, 16 November 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	8edo contains [[2edo]] and [[4edo]] as subsets. Among its supersets are [[16edo]], [[24edo]], [[32edo]], ….		8edo contains [[2edo]] and [[4edo]] as subsets. Among its supersets are [[16edo]], [[24edo]], [[32edo]], … notably including [[72edo]], which expands its 2.11/3.13/5.17/3.19 subgroup into a full 19-limit temperament.

	== Intervals ==		== Intervals ==

BudjarnLambeth: Undo revision 208298 by BudjarnLambeth (talk)

2025-08-27T03:21:37Z

Undo revision 208298 by BudjarnLambeth (talk)

← Older revision		Revision as of 03:21, 27 August 2025
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	This provides 29ed12 with a comparatively larger, more diverse palette of [[consonance]]s than pure-octaves 8edo.		This provides 29ed12 with a comparatively larger, more diverse palette of [[consonance]]s than pure-octaves 8edo.

	A nice subset of 29ed12 is the subset: 1 1 2 2 1 1, which allows all of its intervals to occur at least once between degrees of the scale, but removes the most dissonant intervals above the root, making consonances more easily accessible. It has a vaguely similar feel to the whole tone scale - [[6edo]].

	The nearest [[zeta peak index]] tunings to 8edo don't have an interval within 20 cents of [[2/1]], making them unrecognisable as stretched or compressed 8edo but instead more like entirely new scales in their own right.		The nearest [[zeta peak index]] tunings to 8edo don't have an interval within 20 cents of [[2/1]], making them unrecognisable as stretched or compressed 8edo but instead more like entirely new scales in their own right.

BudjarnLambeth: /* Octave stretch and compression */

2025-08-27T03:01:24Z

Octave stretch and compression

← Older revision		Revision as of 03:01, 27 August 2025
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	This provides 29ed12 with a comparatively larger, more diverse palette of [[consonance]]s than pure-octaves 8edo.		This provides 29ed12 with a comparatively larger, more diverse palette of [[consonance]]s than pure-octaves 8edo.

			A nice subset of 29ed12 is the subset: 1 1 2 2 1 1, which allows all of its intervals to occur at least once between degrees of the scale, but removes the most dissonant intervals above the root, making consonances more easily accessible. It has a vaguely similar feel to the whole tone scale - [[6edo]].

	The nearest [[zeta peak index]] tunings to 8edo don't have an interval within 20 cents of [[2/1]], making them unrecognisable as stretched or compressed 8edo but instead more like entirely new scales in their own right.		The nearest [[zeta peak index]] tunings to 8edo don't have an interval within 20 cents of [[2/1]], making them unrecognisable as stretched or compressed 8edo but instead more like entirely new scales in their own right.