3edo - Revision history

Overthink: Undo revision 230819 by MisterShafXen (talk) High harmonics require accuracy, meaning probably none besides harmonic 5 (and maybe odd 51) are usable. I haven't seen anyone use 3edo for high-limit stuff.

2026-05-24T17:45:08Z

Undo revision 230819 by MisterShafXen (talk) High harmonics require accuracy, meaning probably none besides harmonic 5 (and maybe odd 51) are usable. I haven't seen anyone use 3edo for high-limit stuff.

← Older revision		Revision as of 17:45, 24 May 2026
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	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|3}}~~Prime harmonics 3-31 ordered from greatest to least error: 23, 29, 11, 7, 17, 19, 3, 31, 13, 5~~		{{Harmonics in equal\|3}}
	~~{{Harmonics in equal\|steps=3\|intervals=prime\|columns=9\|start=12\|collapsed=yes}}~~
	~~Prime harmonics 37-71 sorted from greatest to least error: 71, 37, 59, 47, 43, 61, 67, 53, 41~~
	~~{{Harmonics in equal\|steps=3\|intervals=prime\|start=21\|columns=9\|collapsed=yes}}~~
	~~Prime harmonics 73-109 sorted by error: 73, 89, 109, 107, 97, 83, 79, 103, 101~~

	~~Best-tuned prime harmonic <113: 101~~

	== Intervals ==		== Intervals ==

MisterShafXen: added more harmonics

2026-05-24T11:37:57Z

added more harmonics

← Older revision		Revision as of 11:37, 24 May 2026
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	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|3}}		{{Harmonics in equal\|3}}Prime harmonics 3-31 ordered from greatest to least error: 23, 29, 11, 7, 17, 19, 3, 31, 13, 5
			{{Harmonics in equal\|steps=3\|intervals=prime\|columns=9\|start=12\|collapsed=yes}}
			Prime harmonics 37-71 sorted from greatest to least error: 71, 37, 59, 47, 43, 61, 67, 53, 41
			{{Harmonics in equal\|steps=3\|intervals=prime\|start=21\|columns=9\|collapsed=yes}}
			Prime harmonics 73-109 sorted by error: 73, 89, 109, 107, 97, 83, 79, 103, 101

			Best-tuned prime harmonic <113: 101

	== Intervals ==		== Intervals ==

Overthink: grammar

2026-02-12T04:29:09Z

grammar

← Older revision		Revision as of 04:29, 12 February 2026
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	== Theory ==		== Theory ==
	Some would argue that 3edo is more chord than scale on the basis that as it is identical for what [[12edo]] uses as an augmented triad, it is familiar in that capacity. However, it is nonetheless possible to make music in it. Specifically, tonality is actually established with only two notes at a time as opposed to three—aside from octave reduplications of the Tonic that is—and the two notes directly above the Tonic are used together as a contrast chord. However, it also has real significance in the theoretical realm as well. Its associated 5-limit [[mapping]], or [[val]], is {{val\| 3 5 7 }}, meaning octaves are mapped to 3 steps (hence 3edo), the 3rd harmonic to 5 steps and the fifth harmonic to 7 steps. It represents the [[5-limit]] [[consistent]]ly and modulo 3, the mappings of 2, 3, and 5 are distinct, and both of [[4:5:6\|1–5/4–3/2]] and [[10:12:15\|1–6/5–3/2]] are mapped to 0–1–2. 3edo therefore erases the distinction between major and minor, and between the root, third and fifth of the chord, while keeping the basic outline of the triad.		Some would argue that 3edo is more chord than scale on the basis that as it is identical for what [[12edo]] uses as an augmented triad, it is familiar in that capacity. However, it is nonetheless possible to make music in it. Specifically, tonality is actually established with only two notes at a time as opposed to three—aside from octave reduplications of the Tonic that is—and the two notes directly above the Tonic are used together as a contrast chord. However, it also has real significance in the theoretical realm as well. Its associated 5-limit [[mapping]], or [[val]], is {{val\| 3 5 7 }}, meaning octaves are mapped to 3 steps (hence 3edo), the 3rd harmonic to 5 steps and the fifth harmonic to 7 steps. It represents the [[5-limit]] [[consistent]]ly, and modulo 3, the mappings of 2, 3, and 5 are distinct, and both of [[4:5:6\|1–5/4–3/2]] and [[10:12:15\|1–6/5–3/2]] are mapped to 0–1–2. 3edo therefore erases the distinction between major and minor, and between the root, third and fifth of the chord, while keeping the basic outline of the triad.

	It also erases leading tones in the sense that 10/9, 16/15 and 25/24 are all mapped to the unison, or in other words to 0 steps. It can therefore be seen as related to 19th century musical theories such as those of {{w\|Carl Friedrich Weitzmann}}, who classified triads in terms of the associated augmented triad of 12edo as semitonal displacements. Moreover if we encode the kind of triad (major or minor, or even including augmented and diminished) along with the 3edo note, we can reconstruct a 5-limit or even 7-limit version of the music. Changing the code will change the music, but preserve its 3edo skeleton. And because of the way 3edo relates to 10/9, 16/15 and 25/24, there is a tendency for voice-leading to work itself out correctly when we do.		It also erases leading tones in the sense that 10/9, 16/15 and 25/24 are all mapped to the unison, or in other words to 0 steps. It can therefore be seen as related to 19th century musical theories such as those of {{w\|Carl Friedrich Weitzmann}}, who classified triads in terms of the associated augmented triad of 12edo as semitonal displacements. Moreover if we encode the kind of triad (major or minor, or even including augmented and diminished) along with the 3edo note, we can reconstruct a 5-limit or even 7-limit version of the music. Changing the code will change the music, but preserve its 3edo skeleton. And because of the way 3edo relates to 10/9, 16/15 and 25/24, there is a tendency for voice-leading to work itself out correctly when we do.

FloraC: Recategorize

2025-06-05T04:08:51Z

Recategorize

← Older revision		Revision as of 04:08, 5 June 2025
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	* "Neainaz Antithetica, Variation I", from [https://soundcloud.com/sexytoadsandfrogsfriendcircle/sets/staffcirc-vol-7-terra-octava ''STAFFcirc vol. 7''] (2021) – [https://soundcloud.com/sexytoadsandfrogsfriendcircle/3-stc-s1002-neainaz SoundCloud] \| [https://sexytoadsandfrogsfriendcircle.bandcamp.com/track/3-neainaz-antithetica-variation-i Bandcamp]		* "Neainaz Antithetica, Variation I", from [https://soundcloud.com/sexytoadsandfrogsfriendcircle/sets/staffcirc-vol-7-terra-octava ''STAFFcirc vol. 7''] (2021) – [https://soundcloud.com/sexytoadsandfrogsfriendcircle/3-stc-s1002-neainaz SoundCloud] \| [https://sexytoadsandfrogsfriendcircle.bandcamp.com/track/3-neainaz-antithetica-variation-i Bandcamp]

	[[Category:5-limit]]		[[Category:3-limit record edos\|#]] <!-- 1-digit number -->
	[[Category:Listen]]		[[Category:Listen]]

FloraC: -redundant category

2025-02-23T13:25:34Z

-redundant category

← Older revision		Revision as of 13:25, 23 February 2025
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	[[Category:5-limit]]		[[Category:5-limit]]
	~~[[Category:Macrotonal]]~~
	[[Category:Listen]]		[[Category:Listen]]

FloraC: /* Theory */ +1

2025-02-20T15:47:35Z

Theory: +1

← Older revision		Revision as of 15:47, 20 February 2025
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	3edo can be seen as a trivial tuning of the [[augmented (temperament)\|augmented]] temperament, since it tempers [[128/125]] (the diesis) by equating three major thirds to an octave.		3edo can be seen as a trivial tuning of the [[augmented (temperament)\|augmented]] temperament, since it tempers [[128/125]] (the diesis) by equating three major thirds to an octave.

	=== ~~Harmonics~~ ===		=== Prime harmonics ===
	{{Harmonics in equal\|3}}		{{Harmonics in equal\|3}}

FloraC: /* Theory */ cleanup

2025-02-20T15:46:22Z

Theory: cleanup

← Older revision		Revision as of 15:46, 20 February 2025
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	== Theory ==		== Theory ==
	Some would argue that 3edo is more chord than scale on the basis that as it is identical for what [[12edo]] uses as an augmented triad, it is familiar in that capacity. However, it is nonetheless possible to make music in it. Specifically, tonality is actually established with only two notes at a time as opposed to three—aside from octave reduplications of the Tonic that is—and the two notes directly above the Tonic are used together as a contrast chord. However, it also has real significance in the theoretical realm as well. Its associated 5-limit [[mapping]], or [[~~Vals and Tuning Space\|~~val]], is {{val\|3 5 7}}, meaning octaves are mapped to 3 steps (hence 3edo), the 3rd harmonic to 5 steps and the fifth harmonic to 7 steps. It represents the [[5-limit]] [[consistent]]ly and modulo ~~three~~, the mappings of 2, 3, and 5 are distinct, and both of 1-5/~~4-3~~/2 and ~~1-6~~/~~5-3~~/2 are mapped to ~~0-1-2~~. 3edo therefore erases the distinction between major and minor, and between the root, third and fifth of the chord, while keeping the basic outline of the triad.		Some would argue that 3edo is more chord than scale on the basis that as it is identical for what [[12edo]] uses as an augmented triad, it is familiar in that capacity. However, it is nonetheless possible to make music in it. Specifically, tonality is actually established with only two notes at a time as opposed to three—aside from octave reduplications of the Tonic that is—and the two notes directly above the Tonic are used together as a contrast chord. However, it also has real significance in the theoretical realm as well. Its associated 5-limit [[mapping]], or [[val]], is {{val\| 3 5 7 }}, meaning octaves are mapped to 3 steps (hence 3edo), the 3rd harmonic to 5 steps and the fifth harmonic to 7 steps. It represents the [[5-limit]] [[consistent]]ly and modulo 3, the mappings of 2, 3, and 5 are distinct, and both of [[4:5:6\|1–5/4–3/2]] and [[10:12:15\|1–6/5–3/2]] are mapped to 0–1–2. 3edo therefore erases the distinction between major and minor, and between the root, third and fifth of the chord, while keeping the basic outline of the triad.

	It also erases leading tones in the sense that 10/9, 16/15 and 25/24 are all mapped to the unison, or in other words to 0 steps. It can therefore be seen as related to 19th century musical theories such as those of Carl Friedrich Weitzmann, who classified triads in terms of the associated augmented triad of ~~12EDO~~ as semitonal displacements. Moreover if we encode the kind of triad (major or minor, or even including augmented and diminished) along with the 3edo note, we can reconstruct a 5-limit or even 7-limit version of the music. Changing the code will change the music, but preserve its 3edo skeleton. And because of the way 3edo relates to 10/9, 16/15 and 25/24, there is a tendency for voice-leading to work itself out correctly when we do.		It also erases leading tones in the sense that 10/9, 16/15 and 25/24 are all mapped to the unison, or in other words to 0 steps. It can therefore be seen as related to 19th century musical theories such as those of {{w\|Carl Friedrich Weitzmann}}, who classified triads in terms of the associated augmented triad of 12edo as semitonal displacements. Moreover if we encode the kind of triad (major or minor, or even including augmented and diminished) along with the 3edo note, we can reconstruct a 5-limit or even 7-limit version of the music. Changing the code will change the music, but preserve its 3edo skeleton. And because of the way 3edo relates to 10/9, 16/15 and 25/24, there is a tendency for voice-leading to work itself out correctly when we do.

	~~When viewed from a [[regular temperament]] perspective,~~ 3edo can be seen as a tuning of the [[~~Augmented_family#Augmented~~\|augmented ~~temperament~~]], since it tempers [[128/125]] (the diesis) by equating three major thirds to an octave.		3edo can be seen as a trivial tuning of the [[augmented (temperament)\|augmented]] temperament, since it tempers [[128/125]] (the diesis) by equating three major thirds to an octave.

	=== Harmonics ===		=== Harmonics ===

ArrowHead294 at 14:52, 20 February 2025

2025-02-20T14:52:26Z

← Older revision		Revision as of 14:52, 20 February 2025
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	}}		}}
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro\|3}}		{{ED intro}}

	== Theory ==		== Theory ==
	Some would argue that 3edo is more chord than scale on the basis that as it is identical for what [[12edo]] uses as an augmented triad, it is familiar in that capacity. However, it is nonetheless possible to make music in it. Specifically, tonality is actually established with only two notes at a time as opposed to ~~three — aside~~ from octave reduplications of the Tonic that ~~is — and~~ the two notes directly above the Tonic are used together as a contrast chord. However, it also has real significance in the theoretical realm as well. Its associated 5-limit [[mapping]], or [[Vals and Tuning Space\|val]], is {{val\|3 5 7}}, meaning octaves are mapped to 3 steps (hence 3edo), the 3rd harmonic to 5 steps and the fifth harmonic to 7 steps. It represents the [[5-limit]] [[consistent]]ly and modulo three, the mappings of 2, 3, and 5 are distinct, and both of 1-5/4-3/2 and 1-6/5-3/2 are mapped to 0-1-2. 3edo therefore erases the distinction between major and minor, and between the root, third and fifth of the chord, while keeping the basic outline of the triad.		Some would argue that 3edo is more chord than scale on the basis that as it is identical for what [[12edo]] uses as an augmented triad, it is familiar in that capacity. However, it is nonetheless possible to make music in it. Specifically, tonality is actually established with only two notes at a time as opposed to three—aside from octave reduplications of the Tonic that is—and the two notes directly above the Tonic are used together as a contrast chord. However, it also has real significance in the theoretical realm as well. Its associated 5-limit [[mapping]], or [[Vals and Tuning Space\|val]], is {{val\|3 5 7}}, meaning octaves are mapped to 3 steps (hence 3edo), the 3rd harmonic to 5 steps and the fifth harmonic to 7 steps. It represents the [[5-limit]] [[consistent]]ly and modulo three, the mappings of 2, 3, and 5 are distinct, and both of 1-5/4-3/2 and 1-6/5-3/2 are mapped to 0-1-2. 3edo therefore erases the distinction between major and minor, and between the root, third and fifth of the chord, while keeping the basic outline of the triad.

	It also erases leading tones in the sense that 10/9, 16/15 and 25/24 are all mapped to the unison, or in other words to 0 steps. It can therefore be seen as related to 19th century musical theories such as those of Carl Friedrich Weitzmann, who classified triads in terms of the associated augmented triad of 12EDO as semitonal displacements. Moreover if we encode the kind of triad (major or minor, or even including augmented and diminished) along with the 3edo note, we can reconstruct a 5-limit or even 7-limit version of the music. Changing the code will change the music, but preserve its 3edo skeleton. And because of the way 3edo relates to 10/9, 16/15 and 25/24, there is a tendency for voice-leading to work itself out correctly when we do.		It also erases leading tones in the sense that 10/9, 16/15 and 25/24 are all mapped to the unison, or in other words to 0 steps. It can therefore be seen as related to 19th century musical theories such as those of Carl Friedrich Weitzmann, who classified triads in terms of the associated augmented triad of 12EDO as semitonal displacements. Moreover if we encode the kind of triad (major or minor, or even including augmented and diminished) along with the 3edo note, we can reconstruct a 5-limit or even 7-limit version of the music. Changing the code will change the music, but preserve its 3edo skeleton. And because of the way 3edo relates to 10/9, 16/15 and 25/24, there is a tendency for voice-leading to work itself out correctly when we do.
Line 21:		Line 21:
	== Intervals ==		== Intervals ==
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+ Intervals of 3edo		\|+ style="font-size: 105%;" \| Intervals of 3edo
			\|-
	! rowspan="2" \| [[Degree]]		! rowspan="2" \| [[Degree]]
	! rowspan="2" \| [[Cent]]s		! rowspan="2" \| [[Cent]]s
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	\| [[81/64]] (-7.820)		\| [[81/64]] (-7.820)
	\| [[5/4]] (+13.686)		\| [[5/4]] (+13.686)
	\| [[63/50]] (-0.108)<br>[[9/7]] (-35.084)		\| [[63/50]] (-0.108)<br />[[9/7]] (-35.084)
	\| [[34/27]] (+0.910)		\| [[34/27]] (+0.910)
	\| [[File:piano_1_3edo.mp3]]		\| [[File:piano_1_3edo.mp3]]
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	\| [[128/81]] (+7.820)		\| [[128/81]] (+7.820)
	\| [[8/5]] (-13.686)		\| [[8/5]] (-13.686)
	\| [[14/9]] (+35.084)<br>[[100/63]] (+0.108)		\| [[14/9]] (+35.084)<br />[[100/63]] (+0.108)
	\| [[27/17]] (-0.910)		\| [[27/17]] (-0.910)
	\| [[File:piano_2_3edo.mp3]]		\| [[File:piano_2_3edo.mp3]]
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	\| [[File:piano_1_1edo.mp3]]		\| [[File:piano_1_1edo.mp3]]
	\|}		\|}
			<nowiki />* Based on treating 3edo as a subset of [[12edo]], itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.
	<nowiki>*~~</nowiki> based~~ on treating 3edo as a subset of [[12edo]], itself treated as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

	== Notation ==		== Notation ==
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+ Notation of 3edo		\|+ style="font-size: 105%;" \| Notation of 3edo
			\|-
	! rowspan="2" \| [[Degree]]		! rowspan="2" \| [[Degree]]
	! rowspan="2" \| [[Cent]]s		! rowspan="2" \| [[Cent]]s
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	\| 1		\| 1
	\| 400		\| 400
	\| Major third (M3)<br>Diminished fourth (d4)		\| Major third (M3)<br />Diminished fourth (d4)
	\| F#<br>Gb		\| F#<br />Gb
	\|-		\|-
	\| 2		\| 2
	\| 800		\| 800
	\| Augmented fifth (A5)<br>Minor sixth (m6)		\| Augmented fifth (A5)<br />Minor sixth (m6)
	\| A#<br>Bb		\| A#<br />Bb
	\|-		\|-
	\| 3		\| 3
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	== Solfege ==		== Solfege ==
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|+ Solfege of 3edo		\|+ style="font-size: 105%;" \| Solfege of 3edo
			\|-
	! [[Degree]]		! [[Degree]]
	! [[Cents]]		! [[Cents]]
	! 12edo subset<br>standard [[solfege]]<br>(movable do)		! 12edo subset<br />standard [[solfege]]<br />(movable do)
	! 12edo subset<br>[[uniform solfege]]<br>(2-3 vowels)		! 12edo subset<br />[[uniform solfege]]<br />(2-3 vowels)
	\|-		\|-
	\| 0		\| 0
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	\| 400		\| 400
	\| Mi (M3)		\| Mi (M3)
	\| Ma (M3)<br>Fo (d4)		\| Ma (M3)<br />Fo (d4)
	\|-		\|-
	\| 2		\| 2
	\| 800		\| 800
	\| Si (A5)<br>Le (m6)		\| Si (A5)<br />Le (m6)
	\| Su (A5)<br>Fla (m6)		\| Su (A5)<br />Fla (m6)
	\|-		\|-
	\| 3		\| 3

AraMax: Added a new language

2024-12-19T07:17:03Z

Added a new language

← Older revision		Revision as of 07:17, 19 December 2024
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	\| es =		\| es =
	\| ja = 3平均律		\| ja = 3平均律
			\| ro = 3DEO
	}}		}}
	{{Infobox ET}}		{{Infobox ET}}

FloraC: /* Music */ -bold, +italic; misc. cleanup

2024-10-16T09:36:05Z

Music: -bold, +italic; misc. cleanup

← Older revision		Revision as of 09:36, 16 October 2024
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	; [[User:Phanomium\|Phanomium]]		; [[User:Phanomium\|Phanomium]]
	; [https://www.youtube.com/watch?v=6fmcrcQ-lGA Augmented] (2024)		* [https://www.youtube.com/watch?v=6fmcrcQ-lGA ''Augmented''] (2024)

	; [[STC_1002]]		; [[STC_1002]]
	* ~~[https://soundcloud.com/sexytoadsandfrogsfriendcircle/3-stc-s1002-neainaz~~ "Neainaz Antithetica, Variation I"], from [https://soundcloud.com/sexytoadsandfrogsfriendcircle/sets/staffcirc-vol-7-terra-octava ''STAFFcirc vol. 7''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/track/3-neainaz-antithetica-variation-i Bandcamp])		* "Neainaz Antithetica, Variation I", from [https://soundcloud.com/sexytoadsandfrogsfriendcircle/sets/staffcirc-vol-7-terra-octava ''STAFFcirc vol. 7''] (2021) – [https://soundcloud.com/sexytoadsandfrogsfriendcircle/3-stc-s1002-neainaz SoundCloud] \| [https://sexytoadsandfrogsfriendcircle.bandcamp.com/track/3-neainaz-antithetica-variation-i Bandcamp]

	~~[[Category:Equal divisions of the octave\|#]] <!-- 1-digit number -->~~
	[[Category:5-limit]]		[[Category:5-limit]]
	[[Category:Macrotonal]]		[[Category:Macrotonal]]
	[[Category:Listen]]		[[Category:Listen]]