53edo - Revision history

Lucius Chiaraviglio: /* Music */ Add Bryan Deister's ''53edo improv'' (2026-09-18) and update the title of the previous one: ''53edo improv'' (2025-12-23) --> ''Fantasy in 53edo [short clip]''; fix sub-entries for full version of this; convert 2 microtonal covers by Bryan Deister to Modern Renderings format

2026-09-19T20:24:17Z

Music: Add Bryan Deister's ''53edo improv'' (2026-09-18) and update the title of the previous one: ''53edo improv'' (2025-12-23) --> ''Fantasy in 53edo [short clip]''; fix sub-entries for full version of this; convert 2 microtonal covers by Bryan Deister to Modern Renderings format

← Older revision		Revision as of 20:24, 19 September 2026
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	* ''Pavane'', op. 50 (1887) — arranged for harpsichord and rendered by Claudi Meneghin (2026)		* ''Pavane'', op. 50 (1887) — arranged for harpsichord and rendered by Claudi Meneghin (2026)
	** [https://www.youtube.com/watch?v=lTDdD2hIRx4 Near-Pythagorean version]		** [https://www.youtube.com/watch?v=lTDdD2hIRx4 Near-Pythagorean version]

			; {{W\|Toby Fox}}
			* [https://www.youtube.com/shorts/zCthwbPH2cY ''Finale''] via ''Undertale'' (2015) – microtonal cover in 53edo by [[Bryan Deister]] (2026)

	; {{W\|George Frideric Handel}}		; {{W\|George Frideric Handel}}
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	* ''Maple Leaf Rag'' (1899) – arranged for harpsichord and rendered by Claudi Meneghin ([https://www.youtube.com/watch?v=AKXMuhB3uHQ 2024 version]; [https://www.youtube.com/shorts/VsOk3az8J40 2025 version]))		* ''Maple Leaf Rag'' (1899) – arranged for harpsichord and rendered by Claudi Meneghin ([https://www.youtube.com/watch?v=AKXMuhB3uHQ 2024 version]; [https://www.youtube.com/shorts/VsOk3az8J40 2025 version]))
	* ''Maple Leaf Rag'' (1899) – with syntonic comma adjustment, arranged for harpsichord and rendered by Claudi Meneghin ([https://www.youtube.com/watch?v=t-pRqKGX0oo 2024 version]; [https://www.youtube.com/shorts/msBeUJjFlV4 (2025 version)]		* ''Maple Leaf Rag'' (1899) – with syntonic comma adjustment, arranged for harpsichord and rendered by Claudi Meneghin ([https://www.youtube.com/watch?v=t-pRqKGX0oo 2024 version]; [https://www.youtube.com/shorts/msBeUJjFlV4 (2025 version)]

			; [https://vocaloid.fandom.com/wiki/Mushi-P Mushi-P (muship)]
			* [https://www.youtube.com/shorts/8jKjvVw4tvw ''Color & Electricity''] (2016) – microtonal cover in 53edo by [[Bryan Deister]] (2025)

	; {{W\|Shirō Sagisu}}		; {{W\|Shirō Sagisu}}
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	; [[Bryan Deister]]		; [[Bryan Deister]]
	* [https://www.youtube.com/shorts/r-Tzq33OGM4 ''microtonal improvisation in 53edo''] (2025)		* [https://www.youtube.com/shorts/r-Tzq33OGM4 ''microtonal improvisation in 53edo''] (2025)
	* ~~[https://www.youtube.com/shorts/8jKjvVw4tvw~~ ''~~Color & Electricity - muship (microtonal cover~~ in 53edo)''] (2025)		* ''Fantasy in 53edo'' (2025)
	* [https://www.youtube.com/shorts/tIx3PcOyNJo ''53edo ~~improv~~''] (2025)		** [https://www.youtube.com/shorts/tIx3PcOyNJo ''<nowiki>Fantasy in 53edo [short clip]</nowiki>''] (2025) (Lumatone view; formerly titled ''53edo improv'')
	* ''~~Fantasy in~~ 53edo'' ([https://www.youtube.com/~~shorts/~~fgsT-1pBw8g abstract version]; [https://www.youtube.com/watch?v=-q-T3HuGehk visualizer version] ~~(2025)~~		** [https://www.youtube.com/watch?v=fgsT-1pBw8g ''<nowiki>[full abstract art version]</nowiki>'']
	* [https://www.youtube.com/shorts/~~zCthwbPH2cY ''Finale - Undertale (microtonal cover in 53edo)~~''] ~~(2026)~~		** [https://www.youtube.com/watch?v=-q-T3HuGehk ''<nowiki>[full visualizer version]</nowiki>'']
	* ''Waltz in 53edo'' (2026)		* ''Waltz in 53edo'' (2026)
	** [https://www.youtube.com/shorts/WtSaDQCyfVc <nowiki>[Short]</nowiki>] (~~with~~ Lumatone view))		** [https://www.youtube.com/shorts/WtSaDQCyfVc <nowiki>[Short]</nowiki>] (Lumatone view)
	** [https://www.youtube.com/watch?v=_xumSANdf-g <nowiki>[Full version]</nowiki>]		** [https://www.youtube.com/watch?v=_xumSANdf-g <nowiki>[Full version]</nowiki>]
	* [https://www.youtube.com/shorts/7YfZZ7aMvH8 ''53edo groove''] (2026)		* [https://www.youtube.com/shorts/7YfZZ7aMvH8 ''53edo groove''] (2026)
			* [https://www.youtube.com/shorts/k5lepAk19nA ''53edo improv''] (2026-09-18)

	; [[Francium]]		; [[Francium]]

BudjarnLambeth: /* Scales approximated from JI */ attribute idiosyncratic terms

2026-09-09T06:08:27Z

Scales approximated from JI: attribute idiosyncratic terms

← Older revision		Revision as of 06:08, 9 September 2026
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	; From [[AFDO]]s		; From [[AFDO]]s
	{{Idiosyncratic terms}}		{{Idiosyncratic terms\|These terms are so far only documented being used by [[Budjarn Lambeth]]}}
			* Composite Baobab (approximated from [[30afdo]]): 14 8 9 8 7 7
	* Composite Cliffedge (approximated from [[60afdo]]): 12 10 9 19 3		* Composite Cliffedge (approximated from [[60afdo]]): 12 10 9 19 3
	* Composite Deja Vu (approximated from [[101afdo]]): 14 17 5 9 8		* Composite Deja Vu (approximated from [[101afdo]]): 14 17 5 9 8

Lucius Chiaraviglio: /* Modern renderings */ Add Gabriel Fauré's ''Pavane'', op. 50 (1887) — arranged for harpsichord and rendered by Claudi Meneghin (2026, near-Pythagorean version); make formatting of some entries above more consistent

2026-09-06T06:35:02Z

Modern renderings: Add Gabriel Fauré's ''Pavane'', op. 50 (1887) — arranged for harpsichord and rendered by Claudi Meneghin (2026, near-Pythagorean version); make formatting of some entries above more consistent

← Older revision		Revision as of 06:35, 6 September 2026
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	=== Modern renderings ===		=== Modern renderings ===
	; {{W\|Johann Sebastian Bach}}		; {{W\|Johann Sebastian Bach}}
	* [https://www.youtube.com/watch?v=ax43zKpDq9o "Jesus bleibet meine Freude" from ''Herz und Mund und Tat und Leben'', BWV 147] (1723) – tuned in 53edo, rendered by [[Claudi Meneghin]] (2021)		* [https://www.youtube.com/watch?v=ax43zKpDq9o ''"Jesus bleibet meine Freude"'' from ''Herz und Mund und Tat und Leben'', BWV 147] (1723) – tuned in 53edo, rendered by [[Claudi Meneghin]] (2021)
	* ''Prelude and Fugue in C Major, No. 1'', BWV 846, from ''The Well-Tempered Clavier'', Book I (1722) – rendered by [[Mykhaylo Khramov]]		* ''Prelude and Fugue in C Major, No. 1'', BWV 846, from ''The Well-Tempered Clavier'', Book I (1722) – rendered by [[Mykhaylo Khramov]]
	** [https://web.archive.org/web/20201127013408/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3 Prelude] · [https://web.archive.org/web/20201127012701/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/fugue1-53.mp3 Fugue]		** [https://web.archive.org/web/20201127013408/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3 Prelude]
	* [https://www.youtube.com/watch?v=WyLDjrLa94Y "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2024)		** [https://web.archive.org/web/20201127012701/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/fugue1-53.mp3 Fugue]
	* ~~[https://www.youtube.com/watch?v=GK9YwSphw5Y~~ "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2025)		* [https://www.youtube.com/watch?v=WyLDjrLa94Y ''"Ricercar a 3"'' from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2024)
	* [https://www.youtube.com/watch?v=daWx5-iegW0 ~~"Ricercar a 6" from ''The Musical Offering'', BWV 1079~~] ~~(1747) – with syntonic-comma adjustment, rendered by Claudi Meneghin (2025)~~		* ''"Ricercar a 6"'' from ''The Musical Offering'', BWV 1079 (1747) – rendered by Claudi Meneghin (2025)
	* [https://www.youtube.com/watch?v=dZyrIOMEmzo "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)		** [https://www.youtube.com/watch?v=GK9YwSphw5Y Near-Pythagorean version]
	* [https://www.youtube.com/watch?v=vcinR7nUthA "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)		** [https://www.youtube.com/watch?v=daWx5-iegW0 Schismatic version]
			* [https://www.youtube.com/watch?v=dZyrIOMEmzo ''"Contrapunctus 4"'' from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
			* [https://www.youtube.com/watch?v=vcinR7nUthA ''"Contrapunctus 11"'' from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)

	; {{W\|Nicolaus Bruhns}}		; {{W\|Nicolaus Bruhns}}
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	; {{w\|Frédéric Chopin}}		; {{w\|Frédéric Chopin}}
	* Prelude Op. 28, No. 4 in E minor « Suffocation » (1839), arranged for harpsichord, tuned into 53-edo ~~—~~ rendered by [[Claudi Meneghin]] (2025)		* Prelude Op. 28, No. 4 in E minor « Suffocation » (1839), arranged for harpsichord, tuned into 53-edo — rendered by [[Claudi Meneghin]] (2025)
	** [https://www.youtube.com/watch?v=0VB1hv0-AmE Near-Pythagorean version]		** [https://www.youtube.com/watch?v=0VB1hv0-AmE Near-Pythagorean version]
	** [https://www.youtube.com/~~shorts/~~iYtZGBKHcpU Schismatic version]		** [https://www.youtube.com/watch?v=iYtZGBKHcpU Schismatic version]
	* [https://www.youtube.com/shorts/4YEHMpaO4bA ''"Waterfall" Étude from 12 Études, op. 10''] (1829–1832) ~~&mdash~~; rendered by Claudi Meneghin (~~2025~~)		* [https://www.youtube.com/shorts/4YEHMpaO4bA ''"Waterfall" Étude from 12 Études, op. 10''] (1829–1832) — rendered by Claudi Meneghin (2025)

			; {{W\|Gabriel Fauré}}
			* ''Pavane'', op. 50 (1887) — arranged for harpsichord and rendered by Claudi Meneghin (2026)
			** [https://www.youtube.com/watch?v=lTDdD2hIRx4 Near-Pythagorean version]

	; {{W\|George Frideric Handel}}		; {{W\|George Frideric Handel}}

Lucius Chiaraviglio: /* Music */ Fix section heading

2026-08-20T18:00:40Z

Music: Fix section heading

← Older revision		Revision as of 18:00, 20 August 2026
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	* [https://www.youtube.com/watch?v=DCENVnxH6bI ''Bande-announce''] – rendered by MortisTheneRd (2024)		* [https://www.youtube.com/watch?v=DCENVnxH6bI ''Bande-announce''] – rendered by MortisTheneRd (2024)

	==== 21st century ====		=== 21st century ===
	; [[ALLY195]]		; [[ALLY195]]
	* [https://www.bilibili.com/video/BV1f54y1r7XG/ ''My Soul adaptation''] (2020)		* [https://www.bilibili.com/video/BV1f54y1r7XG/ ''My Soul adaptation''] (2020)

Lucius Chiaraviglio: /* 21st century */ Add Bryan Deister's ''53edo groove'' (2026)

2026-08-18T07:42:39Z

21st century: Add Bryan Deister's ''53edo groove'' (2026)

← Older revision		Revision as of 07:42, 18 August 2026
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	** [https://www.youtube.com/shorts/WtSaDQCyfVc <nowiki>[Short]</nowiki>] (with Lumatone view))		** [https://www.youtube.com/shorts/WtSaDQCyfVc <nowiki>[Short]</nowiki>] (with Lumatone view))
	** [https://www.youtube.com/watch?v=_xumSANdf-g <nowiki>[Full version]</nowiki>]		** [https://www.youtube.com/watch?v=_xumSANdf-g <nowiki>[Full version]</nowiki>]
			* [https://www.youtube.com/shorts/7YfZZ7aMvH8 ''53edo groove''] (2026)

	; [[Francium]]		; [[Francium]]

Fredg999: Interwiki

2026-07-17T04:18:36Z

Interwiki

← Older revision		Revision as of 04:18, 17 July 2026
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	\| en = 53edo		\| en = 53edo
	\| es =		\| es =
	\| ja =		\| ja = 53平均律
	}}		}}
	{{Infobox ET}}		{{Infobox ET}}

Fredg999: Text replacement - "[[The Riemann zeta function and tuning #" to "[[Riemann zeta function #"

2026-07-16T19:11:16Z

Text replacement - "[[The Riemann zeta function and tuning #" to "[[Riemann zeta function #"

← Older revision		Revision as of 19:11, 16 July 2026
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	== Theory ==		== Theory ==
	53edo is notable as an excellent [[5-limit]] system, a fact apparently first noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3–4):223–48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>. It is the seventh [[~~The~~ Riemann zeta function ~~and tuning~~ #Zeta EDO lists\|strict zeta edo]]. In the opinion of some, 53edo is the first equal division to deal adequately with the [[13-limit]], while others award that distinction to [[41edo]] or [[46edo]]. Like 41 and 46, 53 is distinctly [[consistent]] in the [[9-odd-limit]] (and if we exclude the most damaged interval pair, 7/5 and 10/7, is [[consistent to distance]] 2), but among them, 53 is the first that finds the [[interseptimal interval]]s [[15/13]] and [[13/10]] distinctly from adjacent [[7-limit\|septimal]] intervals [[8/7]] and [[7/6]], and [[9/7]] and [[21/16]], respectively, which is essential to its 13-limit credibility. It also avoids equating [[11/9]] with [[16/13]], so that the former is tuned very flat to equate it with a slightly flat [[~]][[39/32]] – a feature shared by 46edo. It is almost consistent to the entire [[15-odd-limit]], with the only inconsistency occurring at [[14/11]] (and its octave complement), which is mapped inconsistently sharp and equated with [[9/7]], but it has the benefit of doing very well in larger prime/subgroup-limited odd-limits. It can be treated as a no-11's, no-17's tuning, on which it is consistent all the way up to the [[27-odd-limit]]. It shines however in the 2.3.5.19 and [[2.3.5.13 subgroup\|2.3.5.13]] subgroups, where it offers excellent approximations with decent complexity.		53edo is notable as an excellent [[5-limit]] system, a fact apparently first noted by {{w\|Isaac Newton}}<ref>[https://emusicology.org/index.php/EMR/article/view/7647/6030 Muzzulini, Daniel. 2021. "Isaac Newton's Microtonal Approach to Just Intonation". ''Empirical Musicology Review'' 15 (3–4):223–48. https://doi.org/10.18061/emr.v15i3-4.7647.]</ref>. It is the seventh [[Riemann zeta function #Zeta EDO lists\|strict zeta edo]]. In the opinion of some, 53edo is the first equal division to deal adequately with the [[13-limit]], while others award that distinction to [[41edo]] or [[46edo]]. Like 41 and 46, 53 is distinctly [[consistent]] in the [[9-odd-limit]] (and if we exclude the most damaged interval pair, 7/5 and 10/7, is [[consistent to distance]] 2), but among them, 53 is the first that finds the [[interseptimal interval]]s [[15/13]] and [[13/10]] distinctly from adjacent [[7-limit\|septimal]] intervals [[8/7]] and [[7/6]], and [[9/7]] and [[21/16]], respectively, which is essential to its 13-limit credibility. It also avoids equating [[11/9]] with [[16/13]], so that the former is tuned very flat to equate it with a slightly flat [[~]][[39/32]] – a feature shared by 46edo. It is almost consistent to the entire [[15-odd-limit]], with the only inconsistency occurring at [[14/11]] (and its octave complement), which is mapped inconsistently sharp and equated with [[9/7]], but it has the benefit of doing very well in larger prime/subgroup-limited odd-limits. It can be treated as a no-11's, no-17's tuning, on which it is consistent all the way up to the [[27-odd-limit]]. It shines however in the 2.3.5.19 and [[2.3.5.13 subgroup\|2.3.5.13]] subgroups, where it offers excellent approximations with decent complexity.

	53edo has also found a certain dissemination as an edo tuning for [[Arabic, Turkish, Persian\|Arabic, Turkish, and Persian music]]. It can also be used as an extended [[3-limit\|Pythagorean tuning]], since its fifths are indistinguishable from just in most contexts.		53edo has also found a certain dissemination as an edo tuning for [[Arabic, Turkish, Persian\|Arabic, Turkish, and Persian music]]. It can also be used as an extended [[3-limit\|Pythagorean tuning]], since its fifths are indistinguishable from just in most contexts.

Godtone: /* Intervals */ 36/25 and 25/18 were missing, but also, 53edo is a model of 2.3.5.7.13 JI so it helps to have more accurate descriptors, but also, the cent values feel wrong and this is amplified by stacking them

2026-06-08T10:36:18Z

Intervals: 36/25 and 25/18 were missing, but also, 53edo is a model of 2.3.5.7.13 JI so it helps to have more accurate descriptors, but also, the cent values feel wrong and this is amplified by stacking them

Show changes

Dave Keenan: /* Sagittal notation */ In the table, swapped the order of sagittal and conventional to agree with the staff notation below it.

2026-06-02T03:10:22Z

Sagittal notation: In the table, swapped the order of sagittal and conventional to agree with the staff notation below it.

← Older revision		Revision as of 03:10, 2 June 2026
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	=== Sagittal notation ===		=== Sagittal notation ===
	53edo can be notated in [[Sagittal notation\|Sagittal]] using the [[Sagittal notation#Spartan ~~extension single-shaft~~\|Spartan ~~extension~~]], with the apotome equal to 5 edosteps and the limma to 4 edosteps. Here is a simplified table:		53edo can be notated in [[Sagittal notation\|Sagittal]] using the [[Sagittal notation#Spartan\|Spartan set]], with the apotome equal to 5 edosteps and the limma to 4 edosteps. Here is a simplified table:

	{\| class="wikitable" style="text-align: center;"		{\| class="wikitable" style="text-align: center;"
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	\| rowspan="2" \| <big>{{sagittal\|/\|}}</big>		\| rowspan="2" \| <big>{{sagittal\|/\|}}</big>
	\| rowspan="2" \| <big>{{sagittal\|//\|}}</big>		\| rowspan="2" \| <big>{{sagittal\|//\|}}</big>
	\| ~~<small>~~{{sagittal\|#}}{{sagittal\|~~\\!~~}}~~</small>~~		\| {{sagittal\|\\!}}{{sagittal\|#}}
	\| ~~<small>~~{{sagittal\|#}}{{sagittal\|\!}}~~</small>~~		\| {{sagittal\|\!}}{{sagittal\|#}}
	\| <big>{{sagittal\|#}}</big>		\| <big>{{sagittal\|#}}</big>
	\|-		\|-

FloraC: /* Notation */ SZG notation

2026-05-12T13:36:07Z

Notation: SZG notation

← Older revision		Revision as of 13:36, 12 May 2026
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	== Notation ==		== Notation ==
	=== ~~Ups~~ and downs notation ===		=== Stein–Zimmermann–Gould notation ===
	53edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).		[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
			{{Sharpness-sharp5-szg}}

			=== Kite's ups and downs notation ===
			53edo can also be notated with [[Kite's ups and downs notation\|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
	{{Ups and downs sharpness}}		{{Ups and downs sharpness}}

	Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5\|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
	~~{{Sharpness-sharp5}}~~

	=== Sagittal notation ===		=== Sagittal notation ===
	53edo can be notated in [[Sagittal notation\|Sagittal]] using the [[Sagittal notation#Spartan extension single-shaft\|Spartan extension]], with the apotome equal to 5 edosteps and the limma to 4 edosteps. Here is a simplified table:		53edo can be notated in [[Sagittal notation\|Sagittal]] using the [[Sagittal notation#Spartan extension single-shaft\|Spartan extension]], with the apotome equal to 5 edosteps and the limma to 4 edosteps. Here is a simplified table:

	{\| class="wikitable" style="text-align: center;"		{\| class="wikitable" style="text-align: center;"
	! colspan="2" \|Steps		! colspan="2" \| Steps
	!'''0'''		!'''0'''
	! 1		! 1
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	!'''5'''		!'''5'''
	\|-		\|-
	! rowspan="2" \|Symbol		! rowspan="2" \| Symbol
	!Evo		! Evo
	\| rowspan="2" \|<big>{{sagittal\|\|//\|}}</big>		\| rowspan="2" \| <big>{{sagittal\|\|//\|}}</big>
	\| rowspan="2" \|<big>{{sagittal\|/\|}}</big>		\| rowspan="2" \| <big>{{sagittal\|/\|}}</big>
	\| rowspan="2" \|<big>{{sagittal\|//\|}}</big>		\| rowspan="2" \| <big>{{sagittal\|//\|}}</big>
	\|<small>{{sagittal\|#}}{{sagittal\|\\!}}</small>		\| <small>{{sagittal\|#}}{{sagittal\|\\!}}</small>
	\|<small>{{sagittal\|#}}{{sagittal\|\!}}</small>		\| <small>{{sagittal\|#}}{{sagittal\|\!}}</small>
	\|<big>{{sagittal\|#}}</big>		\| <big>{{sagittal\|#}}</big>
	\|-		\|-
	!Revo		! Revo
	\|<big>{{sagittal\|)\|\|(}}</big>		\| <big>{{sagittal\|)\|\|(}}</big>
	\|<big>{{sagittal\|\|\|\}}</big>		\| <big>{{sagittal\|\|\|\}}</big>
	\|<big>{{sagittal\|/\|\|\}}</big>		\| <big>{{sagittal\|/\|\|\}}</big>
	\|}		\|}
	The following enharmonics from the Spartan set are present (comma tempered out):		The following enharmonics from the Spartan set are present (comma tempered out):

	* {{sagittal\|//\|}} = {{Sagittal\|/\|)}} = {{Sagittal\|/\|\}} ([[325/324]], [[352/351]])		* {{sagittal\|//\|}} = {{Sagittal\|/\|)}} = {{Sagittal\|/\|\}} ([[325/324]], [[352/351]])
	* {{sagittal\|/\|}} = {{sagittal\|\|)}} ([[225/224]])		* {{sagittal\|/\|}} = {{sagittal\|\|)}} ([[225/224]])
	* {{sagittal\|\|(}} = {{sagittal\|\|//\|}} ([[5120/5103]])		* {{sagittal\|\|(}} = {{sagittal\|\|//\|}} ([[5120/5103]])

	See [[Sagittal notation#Revo\|apotome complements]] for equivalent accidental pairs.		See [[Sagittal notation #Revo\|apotome complements]] for equivalent accidental pairs.

	Featured below is the 53edo gamut notated using the best accidental approximants; in this case, pai/pao and phai/phao.		Featured below is the 53edo gamut notated using the best accidental approximants; in this case, pai/pao and phai/phao.
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	{{Sagittal chart}}		{{Sagittal chart}}

	In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma\|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this ~~EDO~~.		In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma\|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.

	== Relationship to 12edo ==		== Relationship to 12edo ==

← Older revision		Revision as of 03:10, 2 June 2026
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	=== Sagittal notation ===		=== Sagittal notation ===
	53edo can be notated in [[Sagittal notation\|Sagittal]] using the [[Sagittal notation#Spartan ~~extension single-shaft~~\|Spartan ~~extension~~]], with the apotome equal to 5 edosteps and the limma to 4 edosteps. Here is a simplified table:		53edo can be notated in [[Sagittal notation\|Sagittal]] using the [[Sagittal notation#Spartan\|Spartan set]], with the apotome equal to 5 edosteps and the limma to 4 edosteps. Here is a simplified table:

	{\| class="wikitable" style="text-align: center;"		{\| class="wikitable" style="text-align: center;"
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	\| rowspan="2" \| <big>{{sagittal\|/\|}}</big>		\| rowspan="2" \| <big>{{sagittal\|/\|}}</big>
	\| rowspan="2" \| <big>{{sagittal\|//\|}}</big>		\| rowspan="2" \| <big>{{sagittal\|//\|}}</big>
	\| ~~<small>~~{{sagittal\|#}}{{sagittal\|~~\\!~~}}~~</small>~~		\| {{sagittal\|\\!}}{{sagittal\|#}}
	\| ~~<small>~~{{sagittal\|#}}{{sagittal\|\!}}~~</small>~~		\| {{sagittal\|\!}}{{sagittal\|#}}
	\| <big>{{sagittal\|#}}</big>		\| <big>{{sagittal\|#}}</big>
	\|-		\|-