56edo - Revision history

FloraC: /* Music */ identify the original works of Bryan Deister's arrangements

2026-05-09T16:15:31Z

Music: identify the original works of Bryan Deister's arrangements

@@ Line 349: / Line 349: @@
 == Music ==
 ; [[Bryan Deister]]
 * [https://www.youtube.com/shorts/o0imqFPDh9k ''56edo''] (2023)
 * [https://www.youtube.com/watch?v=qzMOnS-lgWs ''Waltz in 56edo''] (2025)
 ; [[Budjarn Lambeth]]

Lucius Chiaraviglio: /* Music */ Add Bryan Deister's ''I Will - The Beatles (microtonal cover in 56edo)'' (2026)

2026-05-09T10:32:36Z

Music: Add Bryan Deister's ''I Will - The Beatles (microtonal cover in 56edo)'' (2026)

← Older revision		Revision as of 10:32, 9 May 2026
Line 356:		Line 356:
	** [https://www.youtube.com/shorts/M1HitTFYVu0 <nowiki>[Short]</nowiki>] (with Lumatone view)		** [https://www.youtube.com/shorts/M1HitTFYVu0 <nowiki>[Short]</nowiki>] (with Lumatone view)
	** [https://www.youtube.com/watch?v=mGcPxb-ESAQ <nowiki>[Full version]</nowiki>]		** [https://www.youtube.com/watch?v=mGcPxb-ESAQ <nowiki>[Full version]</nowiki>]
			* [https://www.youtube.com/shorts/WsvSVp3xyr8 ''I Will - The Beatles (microtonal cover in 56edo)''] (2026)

	; [[Budjarn Lambeth]]		; [[Budjarn Lambeth]]

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:32:35Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:32, 13 March 2026
Line 333:		Line 333:
	\| [[Sevond]]		\| [[Sevond]]
	\|}		\|}
	<nowiki/>* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	== Scales ==		== Scales ==

Overthink: /* Theory */ a few minor changes

2026-02-05T03:58:21Z

Theory: a few minor changes

← Older revision		Revision as of 03:58, 5 February 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	56edo shares its near perfect quality of the [[5/4\|classical major third]] with [[28edo]], which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of [[17edo]] and [[22edo]]. It has decent approximations to [[prime harmonic]]s up to [[19/1\|19]], but due to the sharpness of its harmonic [[3/1\|3]], several intervals of [[9/1\|9]] are [[consistency\|inconsistent]]. Therefore, 56edo is not very popular compared to edos like [[53edo]] or [[58edo]].		56edo shares its near perfect quality of the [[5/4\|classical major third]] with [[28edo]], which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of [[17edo]] and [[22edo]]. It has decent approximations of [[prime harmonic]]s up to [[19/1\|19]], but due to the sharpness of its harmonic [[3/1\|3]], several intervals of [[9/1\|9]] are [[consistency\|inconsistent]]. Therefore, 56edo is not very popular compared to edos like [[53edo]] or [[58edo]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 9:		Line 9:

	=== As a tuning of other temperaments ===		=== As a tuning of other temperaments ===
	In the 5-limit, 56et most notably tempers out the [[diaschisma]], as well as the [[shibboleth comma]]. Using the [[patent val]], it tempers out [[686/675]], [[875/864]], and [[1029/1024]] in the [[7-limit]], [[100/99]], [[245/242]], and [[385/384]] in the [[11-limit]], and [[91/90]] and [[169/168]] in the 13-limit. It supports the diaschismic extension [[keen]] in the 7- and 11-limit, and its 13- and 17-limit extension [[keenic]]. It also supports [[hemithirds]], [[superkleismic]], and [[sycamore]] in various limits, being an especially optimal tuning for sycamore in the 11-, and 13-limits. It is also a very sharp tuning of [[slendric]], mapping 7/6 to an [[Ultramajor and inframinor\|inframinor]] third of 257.1[[Cent\|{{c}}]], and mapping 9/7 inconsistently to an ultramajor third of 450{{c}}.		In the 5-limit, 56et most notably tempers out the [[diaschisma]], as well as the [[shibboleth comma]]. Using the [[patent val]], it tempers out [[686/675]], [[875/864]], and [[1029/1024]] in the [[7-limit]], [[100/99]], [[245/242]], and [[385/384]] in the [[11-limit]], and [[91/90]] and [[169/168]] in the 13-limit. It supports the diaschismic extension [[keen]] in the 7- and 11-limit, and its 13- and 17-limit extension [[keenic]]. It also supports [[hemithirds]], [[superkleismic]], and [[sycamore]] in various limits, being an especially optimal tuning for sycamore in the 11- and 13-limits. It also supports a very sharp tuning of [[slendric]], mapping 7/6 to an [[Ultramajor and inframinor\|inframinor]] third of 257.1[[Cent\|{{c}}]], and mapping 9/7 inconsistently to an ultramajor third of 450{{c}}.

	Another interesting val to consider is 56d ({{Val\|56 89 130 '''158'''}}), which maps 7/4 sharply to around 986{{c}}. This mapping tempers out [[50/49]] and [[64/63]] in the 7-limit, providing an alternative to [[22edo]] for [[pajara]]. It improves accuracy of the 3rd harmonic and makes the 5th harmonic basically just, especially improving [[6/5]] and [[10/9]], which are quite out of tune in 22edo. Its approximated 7th harmonic is sharper than 22edo's, and combined with the fact that the 3rd harmonic is sharp, one may ~~want~~ to [[Octave stretch\|compress the octave]], using tunings such as [[145ed6]] or [[201ed12]]. It is also an excellent tuning for the 11-limit version of pajara, which additionally tempers out [[99/98]], [[100/99]], ~~and~~ [[176/175]]. Finally, it gives an excellent tuning for the no-~~fives~~ [[supra]] temperament tempering out [[64/63]] and [[99/98]].		Another interesting val to consider is 56d ({{Val\|56 89 130 '''158''' 194}}), which maps 7/4 sharply to around 986{{c}}. This mapping tempers out [[50/49]] and [[64/63]] in the 7-limit, providing an alternative to [[22edo]] for [[pajara]]. It improves accuracy of the 3rd harmonic and makes the 5th harmonic basically just, especially improving [[6/5]] and [[10/9]], which are quite out of tune in 22edo. Its approximated 7th harmonic is sharper than 22edo's, and combined with the fact that the 3rd harmonic is sharp, one may wish to [[Octave stretch\|compress the octave]], using tunings such as [[145ed6]] or [[201ed12]]. It is also an excellent tuning for the 11-limit version of pajara, which additionally tempers out [[99/98]], [[100/99]], [[176/175]], and [[896/891]]. Finally, it gives an excellent tuning for the [[2.3.7.11 subgroup\|2.3.7.11-subgroup]] [[supra]] temperament tempering out [[64/63]] and [[99/98]].

	=== Miscellaneous properties ===		=== Miscellaneous properties ===

Lucius Chiaraviglio: /* Music */ Bryan Deister's ''Parade - Susumu Hirasawa (microtonal cover in 56edo)'' (2026): Add full version

2026-02-04T07:57:56Z

Music: Bryan Deister's ''Parade - Susumu Hirasawa (microtonal cover in 56edo)'' (2026): Add full version

← Older revision		Revision as of 07:57, 4 February 2026
Line 353:		Line 353:
	* [https://www.youtube.com/watch?v=xkfao6yGKGE ''Curious Light - DOORS (microtonal cover in 56edo)''] (2025)		* [https://www.youtube.com/watch?v=xkfao6yGKGE ''Curious Light - DOORS (microtonal cover in 56edo)''] (2025)
	* [https://www.youtube.com/watch?v=qzMOnS-lgWs ''Waltz in 56edo''] (2025)		* [https://www.youtube.com/watch?v=qzMOnS-lgWs ''Waltz in 56edo''] (2025)
	* [https://www.youtube.com/shorts/M1HitTFYVu0 ~~''Parade - Susumu Hirasawa~~ (~~microtonal cover in 56edo~~)''] ~~(2026)~~		* ''Parade - Susumu Hirasawa (microtonal cover in 56edo)'' (2026)
			** [https://www.youtube.com/shorts/M1HitTFYVu0 <nowiki>[Short]</nowiki>] (with Lumatone view)
			** [https://www.youtube.com/watch?v=mGcPxb-ESAQ <nowiki>[Full version]</nowiki>]

	; [[Budjarn Lambeth]]		; [[Budjarn Lambeth]]

Lucius Chiaraviglio: /* Music */ ''Parade - Susumu Hirasawa (microtonal cover in 56edo)'' (2026)

2026-02-03T09:02:17Z

Music: ''Parade - Susumu Hirasawa (microtonal cover in 56edo)'' (2026)

← Older revision		Revision as of 09:02, 3 February 2026
Line 353:		Line 353:
	* [https://www.youtube.com/watch?v=xkfao6yGKGE ''Curious Light - DOORS (microtonal cover in 56edo)''] (2025)		* [https://www.youtube.com/watch?v=xkfao6yGKGE ''Curious Light - DOORS (microtonal cover in 56edo)''] (2025)
	* [https://www.youtube.com/watch?v=qzMOnS-lgWs ''Waltz in 56edo''] (2025)		* [https://www.youtube.com/watch?v=qzMOnS-lgWs ''Waltz in 56edo''] (2025)
			* [https://www.youtube.com/shorts/M1HitTFYVu0 ''Parade - Susumu Hirasawa (microtonal cover in 56edo)''] (2026)

	; [[Budjarn Lambeth]]		; [[Budjarn Lambeth]]

Overthink: oops

2026-01-19T03:28:37Z

oops

← Older revision		Revision as of 03:28, 19 January 2026
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{ED intro}}		{{ED intro}}

	~~== Theory ==~~
	== Theory ==		== Theory ==
	56edo shares its near perfect quality of the [[5/4\|classical major third]] with [[28edo]], which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of [[17edo]] and [[22edo]]. It has decent approximations to [[prime harmonic]]s up to [[19/1\|19]], but due to the sharpness of its harmonic [[3/1\|3]], several intervals of [[9/1\|9]] are [[consistency\|inconsistent]]. Therefore, 56edo is not very popular compared to edos like [[53edo]] or [[58edo]].		56edo shares its near perfect quality of the [[5/4\|classical major third]] with [[28edo]], which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of [[17edo]] and [[22edo]]. It has decent approximations to [[prime harmonic]]s up to [[19/1\|19]], but due to the sharpness of its harmonic [[3/1\|3]], several intervals of [[9/1\|9]] are [[consistency\|inconsistent]]. Therefore, 56edo is not very popular compared to edos like [[53edo]] or [[58edo]].

Overthink: /* Theory */ revamp section

2026-01-19T03:28:22Z

Theory: revamp section

← Older revision		Revision as of 03:28, 19 January 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th ~~with a~~ [[~~5L 2s\|diatonic~~]] [[~~step ratio~~]] ~~that is a convergent towards the~~ [[~~Metallic~~ harmonic ~~series\|bronze metallic mean~~]]~~, following~~ [[~~17edo~~]] ~~and preceding~~ [[~~185edo~~]]~~. Because it contains 28edo's major third and also has a step size very close to the syntonic comma~~, ~~56edo contains very accurate approximations~~ of ~~both the classic major third~~ [[5/4]] ~~and the Pythagorean major third~~ [[~~81/64~~]]. ~~Unfortunately~~, ~~this "Pythagorean major third"~~ is not ~~the major third as is stacked by fifths in 56edo. However, this interval represents the pythagorean major third consistently in~~ [[~~224edo~~]]~~, which is the quadruple of 56edo~~.		== Theory ==
			56edo shares its near perfect quality of the [[5/4\|classical major third]] with [[28edo]], which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of [[17edo]] and [[22edo]]. It has decent approximations to [[prime harmonic]]s up to [[19/1\|19]], but due to the sharpness of its harmonic [[3/1\|3]], several intervals of [[9/1\|9]] are [[consistency\|inconsistent]]. Therefore, 56edo is not very popular compared to edos like [[53edo]] or [[58edo]].

	~~56edo has unambiguous approximations to prime~~ harmonics ~~up to~~ [[19/~~1\|19~~]], and ~~possibly up to~~ [[29/~~1\|29~~]]. ~~However~~, the ~~harmonic~~ [[3/1\|3]] is ~~quite sharp~~, ~~leading harmonic~~ [[9/~~1\|9~~]] to ~~be even more so~~, and ~~causing intervals like~~ [[10/9]], [[9/7]], and [[13/9]] ~~to be inconsistent~~. ~~Therefore~~, ~~56edo is not very popular compared to edos like~~ [[~~53edo\|53~~]] and [[~~58edo\|58~~]].		=== Prime harmonics ===
			{{Harmonics in equal\|56}}

			=== As a tuning of other temperaments ===
			In the 5-limit, 56et most notably tempers out the [[diaschisma]], as well as the [[shibboleth comma]]. Using the [[patent val]], it tempers out [[686/675]], [[875/864]], and [[1029/1024]] in the [[7-limit]], [[100/99]], [[245/242]], and [[385/384]] in the [[11-limit]], and [[91/90]] and [[169/168]] in the 13-limit. It supports the diaschismic extension [[keen]] in the 7- and 11-limit, and its 13- and 17-limit extension [[keenic]]. It also supports [[hemithirds]], [[superkleismic]], and [[sycamore]] in various limits, being an especially optimal tuning for sycamore in the 11-, and 13-limits. It is also a very sharp tuning of [[slendric]], mapping 7/6 to an [[Ultramajor and inframinor\|inframinor]] third of 257.1[[Cent\|{{c}}]], and mapping 9/7 inconsistently to an ultramajor third of 450{{c}}.

			Another interesting val to consider is 56d ({{Val\|56 89 130 '''158'''}}), which maps 7/4 sharply to around 986{{c}}. This mapping tempers out [[50/49]] and [[64/63]] in the 7-limit, providing an alternative to [[22edo]] for [[pajara]]. It improves accuracy of the 3rd harmonic and makes the 5th harmonic basically just, especially improving [[6/5]] and [[10/9]], which are quite out of tune in 22edo. Its approximated 7th harmonic is sharper than 22edo's, and combined with the fact that the 3rd harmonic is sharp, one may want to [[Octave stretch\|compress the octave]], using tunings such as [[145ed6]] or [[201ed12]]. It is also an excellent tuning for the 11-limit version of pajara, which additionally tempers out [[99/98]], [[100/99]], and [[176/175]]. Finally, it gives an excellent tuning for the no-fives [[supra]] temperament tempering out [[64/63]] and [[99/98]].

	One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the number of directly approximated syntonic commas per octave being 55.7976. (However, note that by [[patent val]] mapping, 56edo actually maps the syntonic comma inconsistently, to two steps.) [[Barium]] temperament realizes this proximity through regular temperament theory, and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a highly composite edo.		=== Miscellaneous properties ===

	56edo ~~can be used~~ to ~~tune~~ [[~~hemithirds~~]], [[~~superkleismic~~]], [[~~sycamore~~]] and [[~~keen~~]] ~~temperaments~~, and ~~using {{val\| 56 89 130 158 }} (56d) as the equal temperament val~~, ~~for~~ [[~~pajara~~]]. ~~It provides~~ the [[~~optimal patent val~~]] ~~for 7-~~, ~~11-~~ and ~~13-limit~~ [[~~sycamore~~]], ~~and~~ the ~~11-limit 56d val~~ is ~~close~~ to ~~the~~ [[~~POTE tuning~~]] ~~for undecimal pajara~~.		One step of 56edo is the closest to the syntonic comma, [[81/80]], of any integer edo's step size by [[direct approximation]], with the number of directly approximated syntonic commas per octave being 55.7976. [[Barium]] temperament realizes this proximity through [[regular temperament theory]], and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a [[highly composite edo]]. Because it contains 28edo's major third and has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] at 18\56, and the Pythagorean major third [[81/64]] at 19\56. Unfortunately, 56edo does not map the Pythagorean major third 19\56, but instead inconsistently to 20\56, a supermajor third of 428.6{{c}}. However, the Pythagorean major third is mapped to 19\56 consistently in [[224edo]], which is the quadruple of 56edo.

	~~=== Prime harmonics ===~~		The perfect fifth generates a [[5L 2s\|diatonic]] scale with a [[step ratio]] that is a convergent towards the [[Metallic harmonic series\|bronze metallic mean]], following [[17edo]] and preceding [[185edo]].
	~~{{Harmonics in equal~~\|~~56}}~~

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 56 factors into {{nowrap\|2<sup>3</sup> × 7}}, 56edo has subset edos {{EDOs\| 2, 4, 7, 8, 14, 28 }}.		Since 56 factors into {{nowrap\|2<sup>3</sup> × 7}}, 56edo has subset edos {{EDOs\| 2, 4, 7, 8, 14, and 28}}.

	== Intervals ==		== Intervals ==

Overthink: clarify

2026-01-18T19:51:37Z

clarify

← Older revision		Revision as of 19:51, 18 January 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series\|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo. However, this interval represents the pythagorean major third consistently in [[224edo]], which is the quadruple of 56edo.		56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th with a [[5L 2s\|diatonic]] [[step ratio]] that is a convergent towards the [[Metallic harmonic series\|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo. However, this interval represents the pythagorean major third consistently in [[224edo]], which is the quadruple of 56edo.

	56edo has unambiguous approximations to prime harmonics up to [[19/1\|19]], and possibly up to [[29/1\|29]]. However, the harmonic [[3/1\|3]] is quite sharp, leading harmonic [[9/1\|9]] to be even more so, and causing intervals like [[10/9]], [[9/7]], and [[13/9]] to be inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo\|53]] and [[58edo\|58]].		56edo has unambiguous approximations to prime harmonics up to [[19/1\|19]], and possibly up to [[29/1\|29]]. However, the harmonic [[3/1\|3]] is quite sharp, leading harmonic [[9/1\|9]] to be even more so, and causing intervals like [[10/9]], [[9/7]], and [[13/9]] to be inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo\|53]] and [[58edo\|58]].

Yourmusic Productions: Undo revision 221866 by Overthink (talk) Replaced factual statement with less useful vague one, doesn't understand the difference between metallic fifths and raw metallic numbers

2026-01-18T06:42:51Z

Undo revision 221866 by Overthink (talk) Replaced factual statement with less useful vague one, doesn't understand the difference between metallic fifths and raw metallic numbers

← Older revision		Revision as of 06:42, 18 January 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th in the ~~"shrub region" between those of~~ [[17edo]] and [[~~22edo~~]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo. However, this interval represents the pythagorean major third consistently in [[224edo]], which is the quadruple of 56edo.		56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series\|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo. However, this interval represents the pythagorean major third consistently in [[224edo]], which is the quadruple of 56edo.

	56edo has unambiguous approximations to prime harmonics up to [[19/1\|19]]. However, the harmonic [[3/1\|3]] is quite sharp, leading harmonic [[9/1\|9]] to be even more so, and causing intervals like [[10/9]], [[9/7]], and [[13/9]] to be inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo\|53]] and [[58edo\|58]].		56edo has unambiguous approximations to prime harmonics up to [[19/1\|19]], and possibly up to [[29/1\|29]]. However, the harmonic [[3/1\|3]] is quite sharp, leading harmonic [[9/1\|9]] to be even more so, and causing intervals like [[10/9]], [[9/7]], and [[13/9]] to be inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo\|53]] and [[58edo\|58]].

	One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the number of directly approximated syntonic commas per octave being 55.7976. (However, note that by [[patent val]] mapping, 56edo actually maps the syntonic comma inconsistently, to two steps.) [[Barium]] temperament realizes this proximity through regular temperament theory, and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a highly composite edo.		One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the number of directly approximated syntonic commas per octave being 55.7976. (However, note that by [[patent val]] mapping, 56edo actually maps the syntonic comma inconsistently, to two steps.) [[Barium]] temperament realizes this proximity through regular temperament theory, and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a highly composite edo.