23edo and octave stretching - Revision history

Hstraub: Audio example for superdiatonic scale in stretched 23edo

2026-03-14T22:47:34Z

Audio example for superdiatonic scale in stretched 23edo

← Older revision		Revision as of 22:47, 14 March 2026
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	Stretched-23 is one of the best tunings to use for exploring the antidiatonic scale (and its 9-note extension, the [[superdiatonic]] scale), since its fifth is more [[consonant]] and less "[[Wolf interval\|wolfish]]" than fifths in other [[pelogic family]] temperaments.		Stretched-23 is one of the best tunings to use for exploring the antidiatonic scale (and its 9-note extension, the [[superdiatonic]] scale), since its fifth is more [[consonant]] and less "[[Wolf interval\|wolfish]]" than fifths in other [[pelogic family]] temperaments.


			[[File:Stretched23edo MavilaSuperdiatonic.mp3\|Mavila superdiatonioc scale in stretched 23edo]]

			Mavila superdiatonic scale in stretched 23edo

	== Table of intervals ==		== Table of intervals ==

Hstraub: superdiatonic, not superantidiatonic

2026-03-14T22:40:38Z

superdiatonic, not superantidiatonic

← Older revision		Revision as of 22:40, 14 March 2026
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	The perfect fifth is sharper than it is in [[7edo]], and thus the width of the perfect fifth falls within the [[syntonic temperament]]'s tuning range. However, stretched-23 is ''not'' a syntonic temperament; using the perfect fifth as [[generator]] results in an [[antidiatonic]] scale, like those of the [[mavila]] and [[pelogic]] temperaments. Because of this, stretched-23 is not an extension of or replacement for 12edo, but rather an alternative to it; its strengths tend to be 12edo's weaknesses and vice versa, so they complement each other.		The perfect fifth is sharper than it is in [[7edo]], and thus the width of the perfect fifth falls within the [[syntonic temperament]]'s tuning range. However, stretched-23 is ''not'' a syntonic temperament; using the perfect fifth as [[generator]] results in an [[antidiatonic]] scale, like those of the [[mavila]] and [[pelogic]] temperaments. Because of this, stretched-23 is not an extension of or replacement for 12edo, but rather an alternative to it; its strengths tend to be 12edo's weaknesses and vice versa, so they complement each other.

	Stretched-23 is one of the best tunings to use for exploring the antidiatonic scale (and its 9-note extension, the [[~~superantidiatonic~~]] scale), since its fifth is more [[consonant]] and less "[[Wolf interval\|wolfish]]" than fifths in other [[pelogic family]] temperaments.		Stretched-23 is one of the best tunings to use for exploring the antidiatonic scale (and its 9-note extension, the [[superdiatonic]] scale), since its fifth is more [[consonant]] and less "[[Wolf interval\|wolfish]]" than fifths in other [[pelogic family]] temperaments.

	== Table of intervals ==		== Table of intervals ==

Lucius Chiaraviglio: /* Table of intervals */ sed -E "s#([0-9]+):([0-9]+)#\\[\\[\\1/\\2\\]\\]#g" 23EDOintervals.txt > 23EDOintervals.new.txt

2026-01-02T10:28:08Z

Table of intervals: sed -E "s#([0-9]+):([0-9]+)#\\[\\[\\1/\\2\\]\\]#g" 23EDOintervals.txt > 23EDOintervals.new.txt

← Older revision		Revision as of 10:28, 2 January 2026
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	! Interval !! Width in steps !! Width in cents !! Approximations		! Interval !! Width in steps !! Width in cents !! Approximations
	\|-		\|-
	\| Quarter-tone \|\| 1 \|\| 52.45 \|\| 33:32		\| Quarter-tone \|\| 1 \|\| 52.45 \|\| [[33/32]]
	\|-		\|-
	\| Semitone \|\| 2 \|\| 105.89 \|\| 16:15, 15:14		\| Semitone \|\| 2 \|\| 105.89 \|\| [[16/15]], [[15/14]]
	\|-		\|-
	\| 3/4-tone \|\| 3 \|\| 157.34 \|\| 12:11, 11:10, 10:9		\| 3/4-tone \|\| 3 \|\| 157.34 \|\| [[12/11]], [[11/10]], [[10/9]]
	\|-		\|-
	\| Whole tone \|\| 4 \|\| 209.79 \|\| 9:8		\| Whole tone \|\| 4 \|\| 209.79 \|\| [[9/8]]
	\|-		\|-
	\| Septimal subminor third \|\| 5 \|\| 262.23 \|\| 7:6		\| Septimal subminor third \|\| 5 \|\| 262.23 \|\| [[7/6]]
	\|-		\|-
	\| Minor third \|\| 6 \|\| 314.68 \|\| 6:5		\| Minor third \|\| 6 \|\| 314.68 \|\| [[6/5]]
	\|-		\|-
	\| Major third \|\| 7 \|\| 367.13 \|\| 5:4		\| Major third \|\| 7 \|\| 367.13 \|\| [[5/4]]
	\|-		\|-
	\| Septimal supermajor third \|\| 8 \|\| 418.58 \|\| 14:11, 9:7		\| Septimal supermajor third \|\| 8 \|\| 418.58 \|\| [[14/11]], [[9/7]]
	\|-		\|-
	\| Minor fourth \|\| 9 \|\| 472.92 \|\| 13:10		\| Minor fourth \|\| 9 \|\| 472.92 \|\| [[13/10]]
	\|-		\|-
	\| Major fourth \|\| 10 \|\| 524.47 \|\| 15:11		\| Major fourth \|\| 10 \|\| 524.47 \|\| [[15/11]]
	\|-		\|-
	\| Septimal tritone \|\| 11 \|\| 576.92 \|\| 7:5		\| Septimal tritone \|\| 11 \|\| 576.92 \|\| [[7/5]]
	\|-		\|-
	\| Tridecimal tritone \|\| 12 \|\| 629.36 \|\| 10:7, 13:9		\| Tridecimal tritone \|\| 12 \|\| 629.36 \|\| [[10/7]], [[13/9]]
	\|-		\|-
	\| Natural fifth \|\| 13 \|\| 681.81 \|\| 3:2		\| Natural fifth \|\| 13 \|\| 681.81 \|\| [[3/2]]
	\|-		\|-
	\| Augmented fifth \|\| 14 \|\| 734.26 \|\| 20:13		\| Augmented fifth \|\| 14 \|\| 734.26 \|\| [[20/13]]
	\|-		\|-
	\| Undecimal minor sixth \|\| 15 \|\| 786.70 \|\| 11:7		\| Undecimal minor sixth \|\| 15 \|\| 786.70 \|\| [[11/7]]
	\|-		\|-
	\| Tridecimal neutral sixth \|\| 16 \|\| 839.15 \|\| 13:8		\| Tridecimal neutral sixth \|\| 16 \|\| 839.15 \|\| [[13/8]]
	\|-		\|-
	\| Major sixth \|\| 17 \|\| 891.60 \|\| 5:3		\| Major sixth \|\| 17 \|\| 891.60 \|\| [[5/3]]
	\|-		\|-
	\| Septimal subminor seventh;<br />septimal supermajor sixth \|\| 18 \|\| 944.04 \|\| 12:7		\| Septimal subminor seventh;<br />septimal supermajor sixth \|\| 18 \|\| 944.04 \|\| [[12/7]]
	\|-		\|-
	\| Minor seventh \|\| 19 \|\| 996.49 \|\| 7:4, 9:5		\| Minor seventh \|\| 19 \|\| 996.49 \|\| [[7/4]], [[9/5]]
	\|-		\|-
	\| Neutral seventh \|\| 20 \|\| 1048.94 \|\| 11:6		\| Neutral seventh \|\| 20 \|\| 1048.94 \|\| [[11/6]]
	\|-		\|-
	\| Major seventh \|\| 21 \|\| 1101.38 \|\| 15:8		\| Major seventh \|\| 21 \|\| 1101.38 \|\| [[15/8]]
	\|-		\|-
	\| Diminished octave \|\| 22 \|\| 1153.83 \|\|		\| Diminished octave \|\| 22 \|\| 1153.83 \|\|
	\|-		\|-
	\| Natural (stretched) octave \|\| 23 \|\| 1206.28 \|\| 2:1		\| Natural (stretched) octave \|\| 23 \|\| 1206.28 \|\| [[2/1]]
	\|}		\|}

BudjarnLambeth: /* Comparison of tunings */

2025-08-29T20:36:31Z

Comparison of tunings

← Older revision		Revision as of 20:36, 29 August 2025
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	What follows is a comparison of compressed- and stretched-octave 23edo tunings.		What follows is a comparison of compressed- and stretched-octave 23edo tunings.

	=== Compressed ===		=== (Compressed) ===

	; [[zpi\|86zpi]]		; [[zpi\|86zpi]]
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	{{Harmonics in cet\|52.114\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 85zpi (continued)}}		{{Harmonics in cet\|52.114\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 85zpi (continued)}}

	=== Pure octaves ===		=== (Pure octaves) ===

	; 23edo		; 23edo
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	{{Harmonics in equal\|23\|2\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 23edo (continued)}}		{{Harmonics in equal\|23\|2\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 23edo (continued)}}

	=== Stretched ===		=== (Stretched) ===

	; [[WE\|23et, 13-limit WE tuning]]		; [[WE\|23et, 13-limit WE tuning]]

BudjarnLambeth: /* Comparison of stretched tunings */

2025-08-29T20:31:20Z

Comparison of stretched tunings

← Older revision		Revision as of 20:31, 29 August 2025
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	[[Octave shrinking]] may also be used instead of octave stretching, which results in a scale with very different properties. 23edo’s approximations of harmonics are still improved upon, but by shifting them in the opposite direction. This is a less-studied option with fertile ground to explore.		[[Octave shrinking]] may also be used instead of octave stretching, which results in a scale with very different properties. 23edo’s approximations of harmonics are still improved upon, but by shifting them in the opposite direction. This is a less-studied option with fertile ground to explore.

	== Comparison of ~~stretched~~ tunings ==		== Comparison of tunings ==
	What follows is a comparison of compressed- and stretched-octave 23edo tunings.		What follows is a comparison of compressed- and stretched-octave 23edo tunings.

			=== Compressed ===

	; [[zpi\|86zpi]]		; [[zpi\|86zpi]]
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	{{Harmonics in cet\|52.114\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 85zpi}}		{{Harmonics in cet\|52.114\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 85zpi}}
	{{Harmonics in cet\|52.114\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 85zpi (continued)}}		{{Harmonics in cet\|52.114\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 85zpi (continued)}}

			=== Pure octaves ===

	; 23edo		; 23edo
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	{{Harmonics in equal\|23\|2\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 23edo}}		{{Harmonics in equal\|23\|2\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 23edo}}
	{{Harmonics in equal\|23\|2\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 23edo (continued)}}		{{Harmonics in equal\|23\|2\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 23edo (continued)}}

			=== Stretched ===

	; [[WE\|23et, 13-limit WE tuning]]		; [[WE\|23et, 13-limit WE tuning]]

BudjarnLambeth: /* Table of intervals */

2025-08-28T04:16:30Z

Table of intervals

BudjarnLambeth: Changed from 1216 to 1206 as the statement in the synopsis is incorrect about 1216, but true about 1206. Read through the full article and changed anything that no longer held true in 1206 (it was not too much, it's mostly the same). Changed the lists of tunings to more reasonable ones.

2025-08-28T04:08:12Z

Changed from 1216 to 1206 as the statement in the synopsis is incorrect about 1216, but true about 1206. Read through the full article and changed anything that no longer held true in 1206 (it was not too much, it's mostly the same). Changed the lists of tunings to more reasonable ones.

Show changes

Contribution: /* List of stretched 23 tunings */

2025-06-09T08:34:09Z

List of stretched 23 tunings

← Older revision		Revision as of 08:34, 9 June 2025
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	== List of stretched 23 tunings ==		== List of stretched 23 tunings ==
	These are tunings which may be used as a stretched 23edo.		These are tunings which may be used as a stretched 23edo.
			* [[11ed7/5]]
	* [[36edt]]		* [[36edt]]
	* [[59ed6]]		* [[59ed6]]

BudjarnLambeth: Added links and slightly reworded some of the stuff about pelog

2025-01-07T07:45:28Z

Added links and slightly reworded some of the stuff about pelog

BudjarnLambeth: Add 2 subheadings to break up very big introduction, add 3 sections at the end expanding the scope of the page and making navigation between pages easier

2025-01-07T07:32:08Z

Add 2 subheadings to break up very big introduction, add 3 sections at the end expanding the scope of the page and making navigation between pages easier

← Older revision		Revision as of 07:32, 7 January 2025
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	Stretched-23 is one of the best tunings to use for exploring the antidiatonic scale (and its 9-note extension, the superantidiatonic scale), since its fifth is more consonant and less "wolfish" than fifths in other pelogic-family temperaments.		Stretched-23 is one of the best tunings to use for exploring the antidiatonic scale (and its 9-note extension, the superantidiatonic scale), since its fifth is more consonant and less "wolfish" than fifths in other pelogic-family temperaments.

			== Table of intervals ==
	The table below gives the intervals and error values assuming a stretched octave of ''exactly'' 1216 cents.		The table below gives the intervals and error values assuming a stretched octave of ''exactly'' 1216 cents.

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	\|}		\|}

			== Xenharmonic and xenmelodic properties ==
	Stretching the octave by this much weakens (but does not eliminate) the sense of octave equivalency. It also yields some odd results; stacking two perfect fifths results in a (stretched) octave plus 3 steps. However the 9:8 whole tone is approximated by 4 rather than 3 steps. This is because the triple octave (8:1) is stretched by nearly a quarter tone, and thus this version of stretched 23edo is not consistent for intervals involving 8.		Stretching the octave by this much weakens (but does not eliminate) the sense of octave equivalency. It also yields some odd results; stacking two perfect fifths results in a (stretched) octave plus 3 steps. However the 9:8 whole tone is approximated by 4 rather than 3 steps. This is because the triple octave (8:1) is stretched by nearly a quarter tone, and thus this version of stretched 23edo is not consistent for intervals involving 8.

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	The octave-stretching also results is various other intervals "inverting" in unexpected ways; for instance the septimal minor seventh (7:4) is the inversion of the septimal minor third (7:6), not of the septimal whole tone (8:7)! Because of this property, this interval could also be considered a sixth rather than a seventh (depending on the context in which it occurs).		The octave-stretching also results is various other intervals "inverting" in unexpected ways; for instance the septimal minor seventh (7:4) is the inversion of the septimal minor third (7:6), not of the septimal whole tone (8:7)! Because of this property, this interval could also be considered a sixth rather than a seventh (depending on the context in which it occurs).

	~~Lookalikes: [[36edt]], [[68edo\|68ed8]], [[159edo\|159ed128]], [[227edo\|227ed1024]]~~

	== Stretched 23edo and pianos ==		== Stretched 23edo and pianos ==
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	Many non-Western scales do not have octave equivalence to begin with, so stretching or squashing the octave does not present much of a problem.		Many non-Western scales do not have octave equivalence to begin with, so stretching or squashing the octave does not present much of a problem.

			== List of stretched 23 tunings ==
			These are tunings which may be used as a stretched 23edo.
			* [[36edt]]
			* [[59ed6]]
			* [[68ed8]]
			* [[159ed128]]
			* [[227ed1024]]
			* [[1ed33/32]]
			* [[zeta peak index\|84zpi]]
			* [[Quartkeenlig]] temperament

			== Compressed 23edo ==
			[[Octave shrinking]] may also be used instead of octave stretching, which results in a scale with very different properties. 23edo’s approximations of harmonics are still improved upon, but by shifting them in the opposite direction. This is a less-studied option with fertile ground to explore.

			== List of compressed 23 tunings ==
			These are tunings which may be used as a compressed 23edo.
			* [[60ed6]]
			* [[zeta peak index\|85zpi]]
			* [[zeta peak index\|86zpi]]

	== See also ==		== See also ==
	* [[Quartkeenlig]] - a regular temperament interpretation of 23edo octave stretching.		* [[Quartkeenlig]] - a regular temperament interpretation of 23edo octave stretching.
	* [[33/32 equal step tuning\|33/32 equal-step tuning]] - can be considered a stretched 23edo.
	* [[59ed6]] - can be considered a stretched 23edo.

	[[Category:23edo]]		[[Category:23edo]]

← Older revision		Revision as of 04:16, 28 August 2025
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	! Interval !! Width in steps !! Width in cents !! Approximations		! Interval !! Width in steps !! Width in cents !! Approximations
	\|-		\|-
	\| Quarter-tone \|\| 1 \|\| 52.45 \|\|		\| Quarter-tone \|\| 1 \|\| 52.45 \|\| 33:32
	\|-		\|-
	\| Semitone \|\| 2 \|\| 105.89 \|\| 16:1, 15:14		\| Semitone \|\| 2 \|\| 105.89 \|\| 16:15, 15:14
	\|-		\|-
	\| 3/4-tone \|\| 3 \|\| 157.34 \|\| 12:11, 11:10, 10:9		\| 3/4-tone \|\| 3 \|\| 157.34 \|\| 12:11, 11:10, 10:9
	\|-		\|-
	\| Whole tone \|\| 4 \|\| 209.79 \|\| 9:8~~, 8:7~~		\| Whole tone \|\| 4 \|\| 209.79 \|\| 9:8
	\|-		\|-
	\| Septimal subminor third \|\| 5 \|\| 262.23 \|\| 7:6		\| Septimal subminor third \|\| 5 \|\| 262.23 \|\| 7:6
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	\| Major third \|\| 7 \|\| 367.13 \|\| 5:4		\| Major third \|\| 7 \|\| 367.13 \|\| 5:4
	\|-		\|-
	\| Septimal supermajor third \|\| 8 \|\| 418.58 \|\| 9:7		\| Septimal supermajor third \|\| 8 \|\| 418.58 \|\| 14:11, 9:7
	\|-		\|-
	\| Minor fourth \|\| 9 \|\| 472.92 \|\| 4:3*		\| Minor fourth \|\| 9 \|\| 472.92 \|\| 13:10
	\|-		\|-
	\| Major fourth \|\| 10 \|\| 524.47 \|\| 4:3*		\| Major fourth \|\| 10 \|\| 524.47 \|\| 15:11
	\|-		\|-
	\| Septimal tritone \|\| 11 \|\| 576.92 \|\| 7:5		\| Septimal tritone \|\| 11 \|\| 576.92 \|\| 7:5
	\|-		\|-
	\| Tridecimal tritone \|\| 12 \|\| 629.36 \|\| 13:9		\| Tridecimal tritone \|\| 12 \|\| 629.36 \|\| 10:7, 13:9
	\|-		\|-
	\| Natural fifth \|\| 13 \|\| 681.81 \|\| 3:2		\| Natural fifth \|\| 13 \|\| 681.81 \|\| 3:2
	\|-		\|-
	\| Augmented fifth \|\| 14 \|\| 734.26 \|\|		\| Augmented fifth \|\| 14 \|\| 734.26 \|\| 20:13
	\|-		\|-
	\| Undecimal minor sixth \|\| 15 \|\| 786.70 \|\| 11:7		\| Undecimal minor sixth \|\| 15 \|\| 786.70 \|\| 11:7
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	\| Major sixth \|\| 17 \|\| 891.60 \|\| 5:3		\| Major sixth \|\| 17 \|\| 891.60 \|\| 5:3
	\|-		\|-
	\| Septimal subminor seventh;<br />septimal supermajor sixth \|\| 18 \|\| 944.04 \|\| 7:4		\| Septimal subminor seventh;<br />septimal supermajor sixth \|\| 18 \|\| 944.04 \|\| 12:7
	\|-		\|-
	\| Minor seventh \|\| 19 \|\| 996.49 \|\| 9:5		\| Minor seventh \|\| 19 \|\| 996.49 \|\| 7:4, 9:5
	\|-		\|-
	\| Neutral seventh \|\| 20 \|\| 1048.94 \|\| 11:6~~, 13:7~~		\| Neutral seventh \|\| 20 \|\| 1048.94 \|\| 11:6
	\|-		\|-
	\| Major seventh \|\| 21 \|\| 1101.38 \|\|		\| Major seventh \|\| 21 \|\| 1101.38 \|\| 15:8
	\|-		\|-
	\| Diminished octave \|\| 22 \|\| 1153.83 \|\|		\| Diminished octave \|\| 22 \|\| 1153.83 \|\|

← Older revision		Revision as of 07:45, 7 January 2025
Line 1:		Line 1:
	[[23edo\|23edo]] is not typically taken seriously as a tuning except by those interested in extreme xenharmony. Its fifths are significantly flat, and is neighbors 22edo and 24edo generally get more attention.		[[23edo\|23edo]] is not typically taken seriously as a tuning except by those interested in extreme [[xenharmony]]. Its fifths are significantly flat, and is neighbors [[22edo]] and [[24edo]] generally get more attention.

	However, when using a slightly [[stretched tuning\|stretched octave]] of around 1216 cents, 23edo looks much better, and it approximates the perfect fifth (and various other ~~intervals~~ involving the 5th, 7th, 11th, and 13th ~~harmonics~~) to within 18 cents or so. If we can tolerate errors around this size in 12edo, we can probably tolerate them in stretched-~~23edo~~ as well.		However, when using a slightly [[stretched tuning\|stretched octave]] of around 1216 [[cents]], 23edo looks much better, and it approximates the [[perfect fifth]] (and various other [[interval]]s involving the 5th, 7th, 11th, and 13th [[harmonic]]s) to within 18 cents or so. If we can tolerate errors around this size in [[12edo]], we can probably tolerate them in stretched-23 as well.

	The perfect fifth is sharper than it is in 7edo, and thus the width of the perfect fifth falls within the syntonic temperament's tuning range. However, stretched-23 is ''not'' a syntonic temperament; using the perfect fifth as generator results in a [[~~Pelogic_family\|~~pelogic]] ~~("mavila" or "antidiatonic") scale~~. Because of this, stretched-23 is not an extension of or replacement for 12edo, but rather an alternative to it; its strengths tend to be 12edo's weaknesses and vice versa, so they complement each other.		The perfect fifth is sharper than it is in [[7edo]], and thus the width of the perfect fifth falls within the [[syntonic temperament]]'s tuning range. However, stretched-23 is ''not'' a syntonic temperament; using the perfect fifth as [[generator]] results in an [[antidiatonic]] scale, like those of the [[mavila]] and [[pelogic]] temperaments. Because of this, stretched-23 is not an extension of or replacement for 12edo, but rather an alternative to it; its strengths tend to be 12edo's weaknesses and vice versa, so they complement each other.

	Stretched-23 is one of the best tunings to use for exploring the antidiatonic scale (and its 9-note extension, the superantidiatonic scale), since its fifth is more consonant and less "wolfish" than fifths in other pelogic-family temperaments.		Stretched-23 is one of the best tunings to use for exploring the antidiatonic scale (and its 9-note extension, the [[superantidiatonic]] scale), since its fifth is more [[consonant]] and less "[[Wolf interval\|wolfish]]" than fifths in other [[pelogic family]] temperaments.

	== Table of intervals ==		== Table of intervals ==
Line 62:		Line 62:

	== Xenharmonic and xenmelodic properties ==		== Xenharmonic and xenmelodic properties ==
	Stretching the octave by this much weakens (but does not eliminate) the sense of octave ~~equivalency~~. It also yields some odd results; stacking two perfect fifths results in a (stretched) octave plus 3 steps. However the 9:8 whole tone is approximated by 4 rather than 3 steps. This is because the triple octave (8:1) is stretched by nearly a quarter tone, and thus this version of stretched 23edo is not consistent for intervals involving 8.		Stretching the octave by this much weakens (but does not eliminate) the sense of [[octave equivalence]]. It also yields some odd results; stacking two perfect fifths results in a (stretched) octave plus 3 steps. However the [[9/8]] whole tone is approximated by 4 rather than 3 steps. This is because the triple octave [[8/1]] is stretched by nearly a [[quarter tone]], and thus this version of stretched 23edo is not [[consistent]] for intervals involving 8.

	Another odd feature of this scale is that the perfect fourth (4:3) is sandwiched almost exactly between two scale degrees, thus resulting in two fourths (a major and a minor one). This might not actually be a bad thing. In common-practice music, the perfect fourth, despite having low harmonic entropy, was often classified as a dissonant interval for reasons relating to [https://en.wikipedia.org/wiki/Lipps%E2%80%93Meyer_law Lipps-Meyer's law]. It was considered a dissonance even after thirds and sixths began to be reclassified as consonances. Thus, by splitting the fourth in two we might actually be reducing this dissonance. Because of the split fourth, 23edo is also not consistent for intervals involving the factor 4.		Another odd feature of this scale is that the perfect fourth ([[4/3]]) is sandwiched almost exactly between two scale degrees, thus resulting in two fourths (a major and a minor one). This might not actually be a bad thing. In common-practice music, the perfect fourth, despite having low [[harmonic entropy]], was often classified as a [[dissonant]] interval for reasons relating to [https://en.wikipedia.org/wiki/Lipps%E2%80%93Meyer_law Lipps-Meyer's law]. It was considered a dissonance even after thirds and sixths began to be reclassified as consonances. Thus, by splitting the fourth in two we might actually be reducing this dissonance. Because of the split fourth, 23edo is also not consistent for intervals involving the factor 4.

	The octave-stretching also results is various other intervals "inverting" in unexpected ways; for instance the septimal minor seventh (7:4) is the inversion of the septimal minor third (7:6), not of the septimal whole tone (8:7)! Because of this property, this interval could also be considered a sixth rather than a seventh (depending on the context in which it occurs).		The octave-stretching also results is various other intervals "inverting" in unexpected ways; for instance the septimal minor seventh ([[7/4]]) is the inversion of the septimal minor third ([[7/6]]), not of the septimal whole tone ([[8/7]]) as it would be in a consistent scale. Because of this property, this interval could also be considered a sixth rather than a seventh (depending on the context in which it occurs).

	== Stretched 23edo and pianos ==		== Stretched 23edo and pianos ==
	Pianos typically have stretched octaves due to the inharmonicity of the strings. This stretch is concentrated at the low and high octaves of the piano. As a result, while the standard 88-key piano covers over seven octaves, only the middle four octaves or so are commonly used, because the treble and bass registers have so much stretch and do not sound as good. Ordinary pianos are tuned with 12 keys per octave, and 12edo is ideal for perfect octaves, but does not work as well for stretched ones. The total amount of stretch depends on the size of the piano but typically is around 70 cents (across the tuning range of the whole piano), although this stretch is unequally distributed as stated before.		[[:Category:Piano\|Pianos]] typically have stretched octaves due to the [[timbral tuning\|inharmonicity]] of the strings. This stretch is concentrated at the low and high octaves of the piano. As a result, while the standard 88-key piano covers over seven octaves, only the middle four octaves or so are commonly used, because the treble and bass registers have so much stretch and do not sound as good. Ordinary pianos are tuned with 12 keys per octave, and 12edo is ideal for perfect octaves, but does not work as well for stretched ones. The total amount of stretch depends on the size of the piano but typically is around 70 cents (across the tuning range of the whole piano), although this stretch is unequally distributed as stated before.

	Alternatively, it might be possible to design a piano in which the stretch is evenly distributed. Stretched 23edo would be ideal for such a piano. A piano covering between four and five octaves would be ideal; such a piano would have around 100 keys (versus 88 for a standard piano) and the total amount of stretch across the entire tuning range would be about the same as on a standard piano. While the range is narrower (4-5 octaves rather than 7-8), the effective ''usable'' range is about the same (since the low and high ranges of a standard piano are usually avoided due to the extreme stretch and dissonance in those regions, whereas in a stretched-23 piano, the stretch is evenly distributed and the entire range sounds equally well).		Alternatively, it might be possible to design a piano in which the stretch is evenly distributed. Stretched 23edo would be ideal for such a piano. A piano covering between four and five octaves would be ideal; such a piano would have around 100 keys (versus 88 for a standard piano) and the total amount of stretch across the entire tuning range would be about the same as on a standard piano. While the range is narrower (4-5 octaves rather than 7-8), the effective ''usable'' range is about the same (since the low and high ranges of a standard piano are usually avoided due to the extreme stretch and dissonance in those regions, whereas in a stretched-23 piano, the stretch is evenly distributed and the entire range sounds equally well).

	== Stretched 23edo and guitars ==		== Stretched 23edo and guitars ==
	~~Guitars~~, unlike pianos, typically do not have much stretch since the strings are not as stiff. Thus, stretched-23 is not as natural an option for them. However, it might be possible to design a guitar-like instrument using stiffer wire (more akin to piano wire). Such a "guitar" could only be played with a stiff plectrum; attempting to play it with just the fingers would be very painful. Other families of stringed instruments could be adapted in a similar manner.		[[Guitar]]s, unlike pianos, typically do not have much stretch since the strings are not as stiff. Thus, stretched-23 is not as natural an option for them. However, it might be possible to design a guitar-like instrument using stiffer wire (more akin to piano wire). Such a "guitar" could only be played with a stiff plectrum; attempting to play it with just the fingers would be very painful. Other families of stringed instruments could be adapted in a similar manner.

	== Stretched 23edo and other Western instruments ==		== Stretched 23edo and other Western instruments ==
	Bowed violins do ''not'' adapt well to stretched-23, since the bowing action results in mode-locking (and as a result their spectrum must be perfectly harmonic rather than stretched).		Bowed violins do ''not'' adapt well to stretched-23, since the bowing action results in mode-locking (and as a result their spectrum must be perfectly harmonic rather than stretched).

	Brass and wind ~~instruments~~ should work well if the air column is suitably shaped.		Brass and wind [[instrument]]s should work well if the air column is suitably shaped.

	== Stretched 23edo and inharmonic instruments ==		== Stretched 23edo and inharmonic instruments ==
	Stretched 23edo provides a good option for those seeking to combine Western-style instruments like piano and guitar (which have nearly harmonic spectra) with more obviously inharmonic ones (idiophones) from other cultures. In particular, the natural fifth in stretched-23 is almost halfway between the very flat fifth of 9edo ~~(which is used, for example, in~~ Indonesian pelog) and the harmonic perfect fifth. ~~As for~~ slendro~~, it~~ may be approximated as 5-5-4-5-4, or (if we use the ''diminished'' octave as our repeat unit instead) 5-4-5-4-4.		Stretched 23edo provides a good option for those seeking to combine Western-style instruments like piano and guitar (which have nearly harmonic spectra) with more obviously inharmonic ones (idiophones) from other cultures. In particular, the natural fifth in stretched-23 is almost halfway between the very flat fifth of [[9edo]] and many Indonesian [[pelog]] scales, and the harmonic perfect fifth (3/2). Indonesian [[slendro]] scales may be approximated in stretched-23 as 5-5-4-5-4, or (if we use the ''diminished'' octave as our repeat unit instead) 5-4-5-4-4.

	~~Many~~ non-Western scales do not have octave equivalence to begin with, so stretching or squashing the octave does not ~~present much~~ of a ~~problem~~.		Some non-Western scales do not have octave equivalence to begin with, so stretching or squashing the octave does not always presemt a problem. Even many of those which do - like the pelog and slendro scales above - still slightly stretch or compress the octave anyway for various purposes. So this is not too unusual of a feature to introduce.

	== List of stretched 23 tunings ==		== List of stretched 23 tunings ==