22edo - Revision history

Fredg999: /* Scordatura piano */ Fix heading level

2026-07-17T04:00:20Z

Scordatura piano: Fix heading level

← Older revision		Revision as of 04:00, 17 July 2026
Line 1,547:		Line 1,547:

	== Instruments ==		== Instruments ==
	== Scordatura piano ==		=== Scordatura piano ===
	Although it does not allow for much in the way of modulation, it is possible to make some music using a piano tuned to a 12 note subset of 22edo, as shown by [[Juhani Nuorvala]]'s [https://www.youtube.com/watch?v=raRiTvogBBA ''Improvisations on a piano tuned to 22edo''] (2026).		Although it does not allow for much in the way of modulation, it is possible to make some music using a piano tuned to a 12 note subset of 22edo, as shown by [[Juhani Nuorvala]]'s [https://www.youtube.com/watch?v=raRiTvogBBA ''Improvisations on a piano tuned to 22edo''] (2026).

Fredg999: Interwiki

2026-07-17T03:59:36Z

Interwiki

← Older revision		Revision as of 03:59, 17 July 2026
Line 2:		Line 2:
	\| en = 22edo		\| en = 22edo
	\| de = 22-EDO		\| de = 22-EDO
	\| es = ~~22 EDO~~		\| es =
	\| ja = 22平均律		\| ja = 22平均律
	}}		}}

FloraC at 20:16, 1 July 2026

2026-07-01T20:16:32Z

← Older revision		Revision as of 20:16, 1 July 2026
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	! Generator		! Generator
	! Temperaments		! Temperaments
	~~!5-limit commas~~
	\|-		\|-
	\| 1		\| 1
	\| 1\22		\| 1\22
	\| [[Escapade]] / [[escaped]]<br>[[Chromo]]<br>[[Ceratitid]]		\| [[Escapade]] / [[escaped]]<br>[[Chromo]]<br>[[Ceratitid]]
	~~\|[[1220703125/1146617856]]<br>[[4294967296/4271484375]]~~
	\|-		\|-
	\| 1		\| 1
	\| 3\22		\| 3\22
	\| [[Porcupine]]		\| [[Porcupine]]
	~~\|[[250/243]]~~
	\|-		\|-
	\| 1		\| 1
	\| 5\22		\| 5\22
	\| [[Orwell]] (22) / blair (22) / winston (22f)		\| [[Orwell]] (22) / blair (22) / winston (22f)
	~~\|[[2109375/2097152]]~~
	\|-		\|-
	\| 1		\| 1
	\| 7\22		\| 7\22
	\| [[Magic]] / telepathy		\| [[Magic]] / telepathy
	~~\|[[3125/3072]]~~
	\|-		\|-
	\| 1		\| 1
	\| 9\22		\| 9\22
	\| [[Superpyth]] / [[suprapyth]]		\| [[Superpyth]] / [[suprapyth]]
	~~\|[[20480/19683]]<br>[[8388608/7971615]]~~
	\|-		\|-
	\| 2		\| 2
	\| 1\22		\| 1\22
	\| [[Shrutar]] / hemipaj<br>[[Comic]]		\| [[Shrutar]] / hemipaj<br>[[Comic]]
	\|
	\|-		\|-
	\| 2		\| 2
	\| 2\22		\| 2\22
	\| [[Srutal]] / [[pajara]] / pajarous		\| [[Srutal]] / [[pajara]] / pajarous
	~~\|[[2048/2025]]~~
	\|-		\|-
	\| 2		\| 2
	\| 3\22		\| 3\22
	\| [[Hedgehog]] / [[echidna]]		\| [[Hedgehog]] / [[echidna]]
	\|
	\|-		\|-
	\| 2		\| 2
	\| 4\22		\| 4\22
	\| [[Astrology]]<br>[[Antikythera]]<br>[[Wizard]]		\| [[Astrology]]<br>[[Antikythera]]<br>[[Wizard]]
	~~\|[[2197265625/2147483648]]~~
	\|-		\|-
	\| 2		\| 2
	\| 5\22		\| 5\22
	\| [[Doublewide]] / fleetwood		\| [[Doublewide]] / fleetwood
	~~\|[[390625/373248]]~~
	\|-		\|-
	\| 11		\| 11
	\| 1\22		\| 1\22
	\| [[Undeka]]<br>[[Hendecatonic (temperament)\|Hendecatonic]]		\| [[Undeka]]<br>[[Hendecatonic (temperament)\|Hendecatonic]]
	\|
	\|}		\|}

Naren: Add Würschmidt Tonleitern, Tonarten und Tonsysteme reference

2026-07-01T19:59:43Z

Add Würschmidt Tonleitern, Tonarten und Tonsysteme reference

← Older revision		Revision as of 19:59, 1 July 2026
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	== History ==		== History ==
	The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}. Inspired by the supposed division of the octave into 22 unequal parts in the [[Indian music\|music theory of India]], Bosanquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy. In this he was followed in the twentieth century by theorist José Würschmidt, who noted it as a possible next step after [[19edo]], and {{w\|James Murray Barbour\|J. Murray Barbour}} in his classic survey of tuning history, ''Tuning and Temperament''.		The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist {{w\|Robert Holford Macdowall Bosanquet\|R. H. M. Bosanquet}}. Inspired by the supposed division of the octave into 22 unequal parts in the [[Indian music\|music theory of India]], Bosanquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy.<ref name="bosanquet1879hindoo" /> In this he was followed in the twentieth century by theorist José Würschmidt, who noted it as a possible next step after [[19edo]], and {{w\|James Murray Barbour\|J. Murray Barbour}} in his classic survey of tuning history, ''Tuning and Temperament''.<ref name="wurschmidt1931tonleitern" /><ref name="barbour1953tuning />

	== Theory ==		== Theory ==
Line 1,589:		Line 1,589:

	== References ==		== References ==
	# Barbour, James Murray, ''Tuning and temperament, a historical survey'', East Lansing, Michigan State College Press, 1953 [c1951]		<references>
	# Bosanquet, R.H.M. [https://www.webcitation.org/5kjJcrhEx ''On the Hindoo division of the octave, with additions to the theory of higher orders''], Proceedings of the Royal Society of London vol. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, ''Hindu Music from Various Authors'', Chowkhamba Sanskrit Series, Varanasi, India, 1965		<ref name="barbour1953tuning">
			Barbour, James Murray, ''[https://archive.org/details/tuningtemperamen00barb Tuning and temperament, a historical survey]'', East Lansing, Michigan State College Press, 1953 [c1951]
			</ref>
			<ref name="bosanquet1879hindoo">
			Bosanquet, R.H.M. [https://www.webcitation.org/5kjJcrhEx ''On the Hindoo division of the octave, with additions to the theory of higher orders''], Proceedings of the Royal Society of London vol. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, ''Hindu Music from Various Authors'', Chowkhamba Sanskrit Series, Varanasi, India, 1965
			</ref>
			<ref name="wurschmidt1931tonleitern">
			Würschmidt, J., ''[https://www.zobodat.at/pdf/Sitzber-physik-med-Soc-Erlangen_63-64_0133-0238.pdf Tonleitern, Tonarten und Tonsysteme]''. Sitzungsberichte der Physikalisch-Medizinischen Sozietät zu Erlangen – 63-64: 133 - 238., 1931-1932
			</ref>
			</references>

	[[Category:Twentuning]]		[[Category:Twentuning]]

Contribution: /* Rank-2 temperaments */

2026-07-01T18:32:02Z

Rank-2 temperaments

← Older revision		Revision as of 18:32, 1 July 2026
Line 1,404:		Line 1,404:
	\| 1\22		\| 1\22
	\| [[Escapade]] / [[escaped]]<br>[[Chromo]]<br>[[Ceratitid]]		\| [[Escapade]] / [[escaped]]<br>[[Chromo]]<br>[[Ceratitid]]
	\|[[1220703125/1146617856]],		\|[[1220703125/1146617856]]<br>[[4294967296/4271484375]]
	[[4294967296/4271484375]]
	\|-		\|-
	\| 1		\| 1
Line 1,425:		Line 1,424:
	\| 9\22		\| 9\22
	\| [[Superpyth]] / [[suprapyth]]		\| [[Superpyth]] / [[suprapyth]]
	\|[[20480/19683]],		\|[[20480/19683]]<br>[[8388608/7971615]]
	[[8388608/7971615]]
	\|-		\|-
	\| 2		\| 2

Contribution: /* Rank-2 temperaments */

2026-07-01T18:31:20Z

Rank-2 temperaments

← Older revision		Revision as of 18:31, 1 July 2026
Line 1,404:		Line 1,404:
	\| 1\22		\| 1\22
	\| [[Escapade]] / [[escaped]]<br>[[Chromo]]<br>[[Ceratitid]]		\| [[Escapade]] / [[escaped]]<br>[[Chromo]]<br>[[Ceratitid]]
	\|[[1220703125/1146617856]], [[4294967296/4271484375]]		\|[[1220703125/1146617856]],
			[[4294967296/4271484375]]
	\|-		\|-
	\| 1		\| 1
Line 1,424:		Line 1,425:
	\| 9\22		\| 9\22
	\| [[Superpyth]] / [[suprapyth]]		\| [[Superpyth]] / [[suprapyth]]
	\|[[20480/19683]], [[8388608/7971615]]		\|[[20480/19683]],
			[[8388608/7971615]]
	\|-		\|-
	\| 2		\| 2

Contribution: /* Rank-2 temperaments */

2026-07-01T18:30:18Z

Rank-2 temperaments

← Older revision		Revision as of 18:30, 1 July 2026
Line 1,399:		Line 1,399:
	! Generator		! Generator
	! Temperaments		! Temperaments
			!5-limit commas
	\|-		\|-
	\| 1		\| 1
	\| 1\22		\| 1\22
	\| [[Escapade]] / [[escaped]]<br>[[Chromo]]<br>[[Ceratitid]]		\| [[Escapade]] / [[escaped]]<br>[[Chromo]]<br>[[Ceratitid]]
			\|[[1220703125/1146617856]], [[4294967296/4271484375]]
	\|-		\|-
	\| 1		\| 1
	\| 3\22		\| 3\22
	\| [[Porcupine]]		\| [[Porcupine]]
			\|[[250/243]]
	\|-		\|-
	\| 1		\| 1
	\| 5\22		\| 5\22
	\| [[Orwell]] (22) / blair (22) / winston (22f)		\| [[Orwell]] (22) / blair (22) / winston (22f)
			\|[[2109375/2097152]]
	\|-		\|-
	\| 1		\| 1
	\| 7\22		\| 7\22
	\| [[Magic]] / telepathy		\| [[Magic]] / telepathy
			\|[[3125/3072]]
	\|-		\|-
	\| 1		\| 1
	\| 9\22		\| 9\22
	\| [[Superpyth]] / [[suprapyth]]		\| [[Superpyth]] / [[suprapyth]]
			\|[[20480/19683]], [[8388608/7971615]]
	\|-		\|-
	\| 2		\| 2
	\| 1\22		\| 1\22
	\| [[Shrutar]] / hemipaj<br>[[Comic]]		\| [[Shrutar]] / hemipaj<br>[[Comic]]
			\|
	\|-		\|-
	\| 2		\| 2
	\| 2\22		\| 2\22
	\| [[Srutal]] / [[pajara]] / pajarous		\| [[Srutal]] / [[pajara]] / pajarous
			\|[[2048/2025]]
	\|-		\|-
	\| 2		\| 2
	\| 3\22		\| 3\22
	\| [[Hedgehog]] / [[echidna]]		\| [[Hedgehog]] / [[echidna]]
			\|
	\|-		\|-
	\| 2		\| 2
	\| 4\22		\| 4\22
	\| [[Astrology]]<br>[[Antikythera]]<br>[[Wizard]]		\| [[Astrology]]<br>[[Antikythera]]<br>[[Wizard]]
			\|[[2197265625/2147483648]]
	\|-		\|-
	\| 2		\| 2
	\| 5\22		\| 5\22
	\| [[Doublewide]] / fleetwood		\| [[Doublewide]] / fleetwood
			\|[[390625/373248]]
	\|-		\|-
	\| 11		\| 11
	\| 1\22		\| 1\22
	\| [[Undeka]]<br>[[Hendecatonic (temperament)\|Hendecatonic]]		\| [[Undeka]]<br>[[Hendecatonic (temperament)\|Hendecatonic]]
			\|
	\|}		\|}

Lucius Chiaraviglio: /* Instruments */ Add scordatura piano

2026-06-01T01:15:06Z

Instruments: Add scordatura piano

← Older revision		Revision as of 01:15, 1 June 2026
Line 1,547:		Line 1,547:

	== Instruments ==		== Instruments ==
			== Scordatura piano ==
			Although it does not allow for much in the way of modulation, it is possible to make some music using a piano tuned to a 12 note subset of 22edo, as shown by [[Juhani Nuorvala]]'s [https://www.youtube.com/watch?v=raRiTvogBBA ''Improvisations on a piano tuned to 22edo''] (2026).

	=== Keyboards ===		=== Keyboards ===
	[[File:22-tone halberstadt layout.png\|alt=\|frameless]]		[[File:22-tone halberstadt layout.png\|alt=\|frameless]]

FloraC: /* Notation */ SZG notation

2026-05-12T14:37:25Z

Notation: SZG notation

Show changes

Overthink: /* Theory */ 22edo is close to 24edo

2026-05-06T04:07:58Z

Theory: 22edo is close to 24edo

← Older revision		Revision as of 04:07, 6 May 2026
Line 13:		Line 13:

	== Theory ==		== Theory ==
	22edo is the third edo, after 12 and 19, which is capable of approximating the [[5-limit]] to within a [[Tenney–Euclidean temperament measures #TE error\|Tenney–Euclidean error]] of 4 cents. Moreover, it does well beyond just the 5-limit; unlike 12 or 19, it is able to approximate the [[7-limit\|7-]] and [[11-limit]] to within 3 cents of error, and in fact 22 is the smallest edo to represent the [[11-odd-limit]] [[consistent]]ly, though [[31edo]] is more accurate.		22edo is the third edo, after 12 and 19, which is capable of approximating the [[5-limit]] to within a [[Tenney–Euclidean temperament measures #TE error\|Tenney–Euclidean error]] of 4 cents. Moreover, it does well beyond just the 5-limit; unlike 12 or 19, it is able to approximate the [[7-limit\|7-]] and [[11-limit]] to within 3 cents of error, and in fact 22 is the smallest edo to represent the [[11-odd-limit]] [[consistent]]ly, though [[31edo]] is considerably more accurate.

	Possibly the most striking characteristic of 22edo to those not used to it is that it does ''not'' [[tempering out\|temper out]] [[81/80]] (the syntonic comma), and instead maps it to one step. Additionally, it is a superset of 11edo, and thus behaves like [[11edo]] and [[13edo]] in that melodic movements similar to 12edo can quickly arrive at an unfamiliar place. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory; yet it is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.		Possibly the most striking characteristic of 22edo to those not used to it is that it does ''not'' [[tempering out\|temper out]] [[81/80]] (the syntonic comma), and instead maps it to one step. Additionally, it is a superset of 11edo and is close to [[24edo]], having only 2 fewer steps than it, and thus behaves like [[11edo]] and [[13edo]] in that melodic movements similar to 12edo can quickly arrive at an unfamiliar place. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory; yet it is small enough that it can be used in live performances with suitably designed instruments, like 22-tone guitars.

	22edo's approximation to the [[7/1\|7th harmonic]] is about 13 cents sharp, somewhat similar to 12edo's approximation to the [[5/1\|5th harmonic]]. Because of this and the sharp fifth, 22edo tempers out [[64/63]], equating the pythagorean minor seventh with [[7/4]], and [[support]]ing [[superpyth]]. In that manner, 22edo can be thought of as widening the gap of [[49/48]] between septimal intervals like [[7/6]] and [[8/7]] to a full quarter-tone. However, the opposite effect consequentially occurs in the 5-limit: while 5/4 and 6/5 are closer to JI than in 12edo, 5/4 is flat and 6/5 is sharp, resulting in [[25/24]] being narrowed to a quarter tone. An important reason for this contrast is that 22edo tempers out [[50/49]], so the [[7/5]] and [[10/7]] are equated to the 600{{c}} half-octave tritone, and 5/4 and 7/4 are separated by a semioctave, as well as 6/5 and [[12/7]]. Reasonably, [[36/35]] is also tempered to 1 step just like 25/24 and 49/48.		22edo's approximation to the [[7/1\|7th harmonic]] is about 13 cents sharp, somewhat similar to 12edo's approximation to the [[5/1\|5th harmonic]]. Because of this and the sharp fifth, 22edo tempers out [[64/63]], equating the pythagorean minor seventh with [[7/4]], and [[support]]ing [[superpyth]]. In that manner, 22edo can be thought of as widening the gap of [[49/48]] between septimal intervals like [[7/6]] and [[8/7]] to a full quarter-tone. However, the opposite effect consequentially occurs in the 5-limit: while 5/4 and 6/5 are closer to JI than in 12edo, 5/4 is flat and 6/5 is sharp, resulting in [[25/24]] being narrowed to a quarter tone. An important reason for this contrast is that 22edo tempers out [[50/49]], so the [[7/5]] and [[10/7]] are equated to the 600{{c}} half-octave tritone, and 5/4 and 7/4 are separated by a semioctave, as well as 6/5 and [[12/7]]. Reasonably, [[36/35]] is also tempered to 1 step just like 25/24 and 49/48.