26edo - Revision history

Lucius Chiaraviglio: /* 21st century */ Bryan Deister's''My Violet - 26edo'' (2026): Add full version

2026-06-06T02:58:51Z

21st century: Bryan Deister's''My Violet - 26edo'' (2026): Add full version

← Older revision		Revision as of 02:58, 6 June 2026
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	* [https://www.youtube.com/shorts/1aCl6tuVS0c ''Happy Together - The Turtles (microtonal cover in 26edo)''] (2026)		* [https://www.youtube.com/shorts/1aCl6tuVS0c ''Happy Together - The Turtles (microtonal cover in 26edo)''] (2026)
	* ''My Violet - 26edo'' (2026)		* ''My Violet - 26edo'' (2026)
	** [https://www.youtube.com/shorts/m76bQWxg_CA <nowiki>[short 1]</nowiki>'']		** [https://www.youtube.com/shorts/m76bQWxg_CA <nowiki>[short 1]</nowiki>] (Lumatone view)
	** [https://www.youtube.com/shorts/L2JzCNj6jak <nowiki>[short 2]</nowiki>'']		** [https://www.youtube.com/shorts/L2JzCNj6jak <nowiki>[short 2]</nowiki>] (Lumatone view)
			** [https://www.youtube.com/watch?v=XplpKE_Tc38 <nowiki>[full version[</nowiki>] (with animation by WIRED0006)
	* [https://www.youtube.com/shorts/wHGLOaeAkt8 ''26edo groove''] (2026)		* [https://www.youtube.com/shorts/wHGLOaeAkt8 ''26edo groove''] (2026)

FloraC: Move Fynn's comma to subsets and supersets section

2026-04-11T07:44:13Z

Move Fynn's comma to subsets and supersets section

← Older revision		Revision as of 07:44, 11 April 2026
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	{{~~interwiki~~		{{Interwiki
			\| en = 26edo
	\| de = 26-EDO		\| de = 26-EDO
	~~\| en = 26edo~~
	\| es =		\| es =
	\| ja =		\| ja =
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	26edo has a [[3/2\|perfect fifth]] of about 692 cents and [[tempering out\|tempers out]] [[81/80]] in the [[5-limit]], making it a very flat [[meantone]] tuning (0.088957{{c}} flat of the [[4/9-comma meantone]] fifth) with a very soft [[5L 2s\|diatonic scale]].		26edo has a [[3/2\|perfect fifth]] of about 692 cents and [[tempering out\|tempers out]] [[81/80]] in the [[5-limit]], making it a very flat [[meantone]] tuning (0.088957{{c}} flat of the [[4/9-comma meantone]] fifth) with a very soft [[5L 2s\|diatonic scale]].

	In the [[7-limit]], it tempers out [[50/49]], [[525/512]], [[875/864~~]], and [[Fynn's comma~~]], and [[support]]s temperaments like [[injera]], [[flattone]], [[lemba]], and [[doublewide]]. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13-odd-limit]] [[consistent]]ly. 26edo has a very good approximation of the harmonic seventh ([[7/4]]), as it is the denominator of a convergent to log<sub>2</sub>7.		In the [[7-limit]], it tempers out [[50/49]], [[525/512]], and [[875/864]], and [[support]]s temperaments like [[injera]], [[flattone]], [[lemba]], and [[doublewide]]. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13-odd-limit]] [[consistent]]ly. 26edo has a very good approximation of the harmonic seventh ([[7/4]]), as it is the denominator of a convergent to log<sub>2</sub>7.

	26edo's minor sixth (1.6158) is very close to {{nowrap\|''φ'' ≈ 1.6180}} (i.e. the golden ratio).		26edo's minor sixth (1.6158) is very close to {{nowrap\|''φ'' ≈ 1.6180}} (i.e. the golden ratio).
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	26edo has [[2edo]] and [[13edo]] as subsets, of which 13edo is non-trivial, sharing the 2.9.5.21.11.13.17.19-subgroup with 26edo. Multiplying 26edo by 3 yields [[78edo]], which corrects several harmonics. [[104edo]] is a notable dual-5s system. [[130edo]], [[364edo]], [[494edo]], and [[624edo]] do well in approximating JI, though they are more complex.		26edo has [[2edo]] and [[13edo]] as subsets, of which 13edo is non-trivial, sharing the 2.9.5.21.11.13.17.19-subgroup with 26edo.

			26edo tempers out [[Fynn's comma]], which sets ~7/4 to 21\26. This is shared by several notable superset edos. Multiplying 26edo by 3 yields [[78edo]], which corrects several harmonics. [[104edo]] is a notable dual-5's system. [[130edo]], [[364edo]], [[494edo]], and [[624edo]] do well in approximating JI, though they are more complex.

	== Intervals ==		== Intervals ==

Fynn Cerulean: Added note that 26EDO tempers out Fynn's comma

2026-04-11T03:24:56Z

Added note that 26EDO tempers out Fynn's comma

← Older revision		Revision as of 03:24, 11 April 2026
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	26edo has a [[3/2\|perfect fifth]] of about 692 cents and [[tempering out\|tempers out]] [[81/80]] in the [[5-limit]], making it a very flat [[meantone]] tuning (0.088957{{c}} flat of the [[4/9-comma meantone]] fifth) with a very soft [[5L 2s\|diatonic scale]].		26edo has a [[3/2\|perfect fifth]] of about 692 cents and [[tempering out\|tempers out]] [[81/80]] in the [[5-limit]], making it a very flat [[meantone]] tuning (0.088957{{c}} flat of the [[4/9-comma meantone]] fifth) with a very soft [[5L 2s\|diatonic scale]].

	In the [[7-limit]], it tempers out [[50/49]], [[525/512]], ~~and~~ [[875/864]], and [[support]]s temperaments like [[injera]], [[flattone]], [[lemba]], and [[doublewide]]. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13-odd-limit]] [[consistent]]ly. 26edo has a very good approximation of the harmonic seventh ([[7/4]]), as it is the denominator of a convergent to log<sub>2</sub>7.		In the [[7-limit]], it tempers out [[50/49]], [[525/512]], [[875/864]], and [[Fynn's comma]], and [[support]]s temperaments like [[injera]], [[flattone]], [[lemba]], and [[doublewide]]. It really comes into its own as a higher-limit temperament, being the smallest equal division which represents the [[13-odd-limit]] [[consistent]]ly. 26edo has a very good approximation of the harmonic seventh ([[7/4]]), as it is the denominator of a convergent to log<sub>2</sub>7.

	26edo's minor sixth (1.6158) is very close to {{nowrap\|''φ'' ≈ 1.6180}} (i.e. the golden ratio).		26edo's minor sixth (1.6158) is very close to {{nowrap\|''φ'' ≈ 1.6180}} (i.e. the golden ratio).

Overthink: /* Approximation to irrational intervals */ clarify

2026-03-27T22:06:34Z

Approximation to irrational intervals: clarify

← Older revision		Revision as of 22:06, 27 March 2026
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	== Approximation to irrational intervals ==		== Approximation to irrational intervals ==
	26edo approximates both [[acoustic phi]] (the [[golden ratio]]) and [[pi]] quite accurately. Not until 1076edo do we find a better edo in terms of relative error on these intervals.		26edo approximates both [[acoustic phi]] (the [[golden ratio]]) and [[pi]] quite accurately. Not until 1076edo do we find a better edo in terms of relative error on these intervals{{Clarify}}.

	{\| class="wikitable center-all"		{\| class="wikitable center-all"

Overthink: /* Intervals */ add 26/21, 64/49, and complements

2026-03-27T22:05:56Z

Intervals: add 26/21, 64/49, and complements

← Older revision		Revision as of 22:05, 27 March 2026
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	\| 8		\| 8
	\| 369.23		\| 369.23
	\| [[5/4]], [[11/9]], [[16/13]]		\| [[5/4]], [[11/9]], [[16/13]], [[26/21]]
	\| M3		\| M3
	\| F#		\| F#
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	\| 10		\| 10
	\| 461.54		\| 461.54
	\| [[21/16]], [[13/10]]		\| [[21/16]], [[13/10]], [[64/49]]
	\| d4		\| d4
	\| Gb		\| Gb
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	\| 16		\| 16
	\| 738.46		\| 738.46
	\| [[32/21]], [[20/13]]		\| [[32/21]], [[20/13]], [[49/32]]
	\| A5		\| A5
	\| A#		\| A#
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	\| 18		\| 18
	\| 830.77		\| 830.77
	\| [[13/8]], [[8/5]]		\| [[8/5]], [[13/8]], [[21/13]]
	\| m6		\| m6
	\| Bb		\| Bb

Lucius Chiaraviglio: /* 21st century */ Add Bryan Deister's ''26edo groove'' (2026)

2026-03-27T06:37:36Z

21st century: Add Bryan Deister's ''26edo groove'' (2026)

← Older revision		Revision as of 06:37, 27 March 2026
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	** [https://www.youtube.com/shorts/m76bQWxg_CA <nowiki>[short 1]</nowiki>'']		** [https://www.youtube.com/shorts/m76bQWxg_CA <nowiki>[short 1]</nowiki>'']
	** [https://www.youtube.com/shorts/L2JzCNj6jak <nowiki>[short 2]</nowiki>'']		** [https://www.youtube.com/shorts/L2JzCNj6jak <nowiki>[short 2]</nowiki>'']
			* [https://www.youtube.com/shorts/wHGLOaeAkt8 ''26edo groove''] (2026)

	; [[User:Eboone\|Ebooone]]		; [[User:Eboone\|Ebooone]]

Lucius Chiaraviglio: /* Music */ Split into 26edo proper and unequal derivatives of 26edo, starting with Bryan Deister's Daisy Bell - Harry Dacre (microtonal cover in unequal 26ish tone [displaced from 26edo in dozens]) (2026)

2026-02-25T08:36:57Z

Music: Split into 26edo proper and unequal derivatives of 26edo, starting with Bryan Deister's Daisy Bell - Harry Dacre (microtonal cover in unequal 26ish tone [displaced from 26edo in dozens]) (2026)

← Older revision		Revision as of 08:36, 25 February 2026
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	{{Catrel\|26edo tracks}}		{{Catrel\|26edo tracks}}

	=== Modern renderings ===		=== 26 equal divisions of the octave (26edo proper) ===
			==== Modern renderings ====
	; {{W\|Johann Sebastian Bach}}		; {{W\|Johann Sebastian Bach}}
	* [https://www.youtube.com/watch?v=LUNOFjiyZ0Y ''Contrapunctus 4'' from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)		* [https://www.youtube.com/watch?v=LUNOFjiyZ0Y ''Contrapunctus 4'' from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
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	* [https://www.youtube.com/watch?v=-EVO5ntuoSM ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)		* [https://www.youtube.com/watch?v=-EVO5ntuoSM ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)

	=== 21st century ===		==== 21st century ====
	; [[Abnormality]]		; [[Abnormality]]
	* [https://www.youtube.com/watch?v=Tl-AN2zQeAI ''Break''] (2024)		* [https://www.youtube.com/watch?v=Tl-AN2zQeAI ''Break''] (2024)
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	; [[YoVariable]]		; [[YoVariable]]
	* [https://www.youtube.com/watch?v=01w70PbbT3o ''Jingle Bells (26edo microtonal Lumatone cover + Mystery Song)''] (2025)		* [https://www.youtube.com/watch?v=01w70PbbT3o ''Jingle Bells (26edo microtonal Lumatone cover + Mystery Song)''] (2025)

			=== Unequal Derivatives of 26edo ===
			; [[Bryan Deister]]
			* [https://www.youtube.com/shorts/mzUGcki6H0Y ''<nowiki>Daisy Bell - Harry Dacre (microtonal cover in unequal 26ish tone [displaced from 26edo in dozens])</nowiki>''] (2026) — from Bryan Deister's video comments, "displacement in cents roughly: 0, -8, -3, -12, 14, 5, 18, 3, -12, 13, -15, 10" (these repeat every 12 notes, NOT every 13 note semi-octave, thus causing each octave to be different)

	[[Category:Listen]]		[[Category:Listen]]

Squib: /* Theory */

2026-02-21T13:22:15Z

Theory

← Older revision		Revision as of 13:22, 21 February 2026
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	# In terms of more traditional chord types we have flattone, a variant of meantone with flat fifths, which provides an interesting structure but unsatisfying intonation due mainly to the poorly tuned thirds. Extending meantone harmony to the 7-limit is quite intuitive; for example, augmented becomes supermajor, and diminished becomes subminor. Simple mappings for harmonics up to 13 are also achieved.		# In terms of more traditional chord types we have flattone, a variant of meantone with flat fifths, which provides an interesting structure but unsatisfying intonation due mainly to the poorly tuned thirds. Extending meantone harmony to the 7-limit is quite intuitive; for example, augmented becomes supermajor, and diminished becomes subminor. Simple mappings for harmonics up to 13 are also achieved.
	# As two chains of meantone fifths half an octave apart, it supports injera temperament. The generator for this is an interval which can be called either 21/20 or 15/14, and which represents two steps of 26, and hence one step of 13. Hence in 26edo (as opposed to, for instance, [[38edo]]) it can be viewed as two parallel 13edo scales, and from that point of view we can consider it as supporting the 13b&26 temperament, allowing the two chains be shifted slightly and which can be used for more atonal melodies. In this way its internal dynamics resemble those of [[14edo]].		# As two chains of meantone fifths half an octave apart, it supports injera temperament. The generator for this is an interval which can be called either 21/20 or 15/14, and which represents two steps of 26, and hence one step of 13. Hence in 26edo (as opposed to, for instance, [[38edo]]) it can be viewed as two parallel 13edo scales, and from that point of view we can consider it as supporting the 13b&26 temperament, allowing the two chains be shifted slightly and which can be used for more atonal melodies. In this way its internal dynamics resemble those of [[14edo]].
	# 26edo nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. In terms of subgroups, this is the 2.7.11 subgroup, and on this 26 tempers out the pair of commas [[65536/65219]] and ~~{{monzo\| -3 0 0 6 -4 }}~~. The 65536/65219 comma, the orgonisma, leads to the [[Orgonia\|orgone temperament]] with an approximate 77/64 generator of 7\26, with mos scales of size 7, 11 and 15. The ~~{{monzo\| -3 0 0 6 -4 }}~~ comma leads to a half-octave period and an approximate [[49/44]] generator of 4\26, leading to mos of size 8 and 14.		# 26edo nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. In terms of subgroups, this is the 2.7.11 subgroup, and on this 26 tempers out the pair of commas [[65536/65219]] and [[117649/117128]]. The 65536/65219 comma, the orgonisma, leads to the [[Orgonia\|orgone temperament]] with an approximate 77/64 generator of 7\26, with mos scales of size 7, 11 and 15. The 117649/117128 comma leads to a half-octave period and an approximate [[49/44]] generator of 4\26, leading to mos of size 8 and 14.
	# We can also treat 26edo as a full 13-limit temperament, since it is consistent on the 13-odd-limit (unlike all lower edos).		# We can also treat 26edo as a full 13-limit temperament, since it is consistent on the 13-odd-limit (unlike all lower edos).
	# It also has a pretty good 17th harmonic and tempers out the comma 459:448, thus three fourths give a 17:14 and four fifths give a 21:17; "mushtone". Mushtone is high in badness, but 26edo does it pretty well (and [[33edo]] even better). Because 26edo also tempers out 85:84, the septendecimal major and minor thirds are equivalent to their pental counterparts, making mushtone the same as flattone.		# It also has a pretty good 17th harmonic and tempers out the comma 459:448, thus three fourths give a 17:14 and four fifths give a 21:17; "mushtone". Mushtone is high in badness, but 26edo does it pretty well (and [[33edo]] even better). Because 26edo also tempers out 85:84, the septendecimal major and minor thirds are equivalent to their pental counterparts, making mushtone the same as flattone.

Hotcrystal0: standard notation

2026-02-10T14:40:16Z

standard notation

← Older revision		Revision as of 14:40, 10 February 2026
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	== Notation ==		== Notation ==

			=== Standard notation ===
			Because the chromatic semitone is 1 step, only sharps and flats are needed to notate 26edo.

			{{sharpness-sharp1}}

	=== Sagittal notation ===		=== Sagittal notation ===
	This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation\|5]], [[12edo#Sagittal notation\|12]], and [[19edo#Sagittal notation\|19]], is a subset of the notation for [[52edo#Sagittal notation\|52-EDO]], and is a superset of the notation for [[13edo#Sagittal notation\|13-EDO]].		This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation\|5]], [[12edo#Sagittal notation\|12]], and [[19edo#Sagittal notation\|19]], is a subset of the notation for [[52edo#Sagittal notation\|52-EDO]], and is a superset of the notation for [[13edo#Sagittal notation\|13-EDO]].

Lucius Chiaraviglio: Move notes to be under the corresponding tables

2026-01-14T09:14:03Z

Move notes to be under the corresponding tables

← Older revision		Revision as of 09:14, 14 January 2026
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	\| do		\| do
	\|}		\|}
			<references group="note" />

	=== Interval quality and chord names in color notation ===		=== Interval quality and chord names in color notation ===
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	\| Animist comma		\| Animist comma
	\|}		\|}
			<references group="note" />

	== Octave stretch or compression ==		== Octave stretch or compression ==
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	; [[YoVariable]]		; [[YoVariable]]
	* [https://www.youtube.com/watch?v=01w70PbbT3o ''Jingle Bells (26edo microtonal Lumatone cover + Mystery Song)''] (2025)		* [https://www.youtube.com/watch?v=01w70PbbT3o ''Jingle Bells (26edo microtonal Lumatone cover + Mystery Song)''] (2025)

	~~== Notes ==~~
	~~<references group="note" />~~

	[[Category:Listen]]		[[Category:Listen]]