53edo - Revision history

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 ==

FloraC: /* Music */ - 5-limit JI work

2026-05-10T18:42:58Z

Music: - 5-limit JI work

← Older revision		Revision as of 18:42, 10 May 2026
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	{{Catrel\| 53edo tracks }}		{{Catrel\| 53edo tracks }}

	~~=== 53 equal divisions of the octave (53edo proper) ===~~		=== 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)
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	* "Taking Flight" from ''Nano Particular'' (2019) – [https://open.spotify.com/track/2zp6oM57m6BvQgyOZ5kmuZ Spotify] \| [https://xotla.bandcamp.com/track/taking-flight-53edo Bandcamp] \| [https://www.youtube.com/watch?v=sIsfYQATouc YouTube]		* "Taking Flight" from ''Nano Particular'' (2019) – [https://open.spotify.com/track/2zp6oM57m6BvQgyOZ5kmuZ Spotify] \| [https://xotla.bandcamp.com/track/taking-flight-53edo Bandcamp] \| [https://www.youtube.com/watch?v=sIsfYQATouc YouTube]
	* "Detective Duckweed" from ''Jazzbeetle'' (2023) – [https://open.spotify.com/track/77iDGy7hRx8az3ODrDm5Kl Spotify] \| [https://xotla.bandcamp.com/track/detective-duckweed-53edo Bandcamp] \| [https://youtu.be/FNXEPB4Gm54 YouTube] – jazzy big band electronic hybrid		* "Detective Duckweed" from ''Jazzbeetle'' (2023) – [https://open.spotify.com/track/77iDGy7hRx8az3ODrDm5Kl Spotify] \| [https://xotla.bandcamp.com/track/detective-duckweed-53edo Bandcamp] \| [https://youtu.be/FNXEPB4Gm54 YouTube] – jazzy big band electronic hybrid

	~~=== 53 well-tempered divisions of the octave ===~~
	~~; [https://www.youtube.com/@SocraticSwansongs John Robert Bagby] (Socratic Swansongs)~~
	* [https://www.youtube.com/watch?v=F1vYmdaNHHE ''53 unequal divisions of the octave Just Intonation lumatone experiment''] (2024) (uses [https://sevish.com/scaleworkshop/?n=Harmonic+series+12+tone&l=29F28_3kF3h_pFo_3rF3k_gFf_rFp_m8Fk9_aF9_9F8_74F69_40F3h_23F1s_wFr_6F5_6rF5k_2sF29_5F4_29F1s_wFp_4iF3h_8wF6r_4F3_rFk_e8Faf_pFi_19Fw_1sF19_10Fp_afF74_14Fr_3F2_6rF4g_3hF29_pFg_3kF29_8F5_29F1e_b4F6r_5F3_rFg_3kF23_3hF20_69F3k_gF9_9F5_k9Fb4_1eFr_fF8_74F3r_1cFp_3hF1s_4gF29_2F1&version=2.4.1 this tuning])

	== Notes ==		== Notes ==

Lucius Chiaraviglio: /* Music */ Add section for 53wt, starting with John Robert Bagby's ''53 unequal divisions of the octave Just Intonation lumatone experiment'' (2024); going to need to make 53edo/Music a separate page pretty soon

2026-05-10T14:39:21Z

Music: Add section for 53wt, starting with John Robert Bagby's ''53 unequal divisions of the octave Just Intonation lumatone experiment'' (2024); going to need to make 53edo/Music a separate page pretty soon

← Older revision		Revision as of 14:39, 10 May 2026
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	{{Catrel\| 53edo tracks }}		{{Catrel\| 53edo tracks }}

	=== Modern renderings ===		=== 53 equal divisions of the octave (53edo proper) ===

			==== 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)
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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)
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	* "Taking Flight" from ''Nano Particular'' (2019) – [https://open.spotify.com/track/2zp6oM57m6BvQgyOZ5kmuZ Spotify] \| [https://xotla.bandcamp.com/track/taking-flight-53edo Bandcamp] \| [https://www.youtube.com/watch?v=sIsfYQATouc YouTube]		* "Taking Flight" from ''Nano Particular'' (2019) – [https://open.spotify.com/track/2zp6oM57m6BvQgyOZ5kmuZ Spotify] \| [https://xotla.bandcamp.com/track/taking-flight-53edo Bandcamp] \| [https://www.youtube.com/watch?v=sIsfYQATouc YouTube]
	* "Detective Duckweed" from ''Jazzbeetle'' (2023) – [https://open.spotify.com/track/77iDGy7hRx8az3ODrDm5Kl Spotify] \| [https://xotla.bandcamp.com/track/detective-duckweed-53edo Bandcamp] \| [https://youtu.be/FNXEPB4Gm54 YouTube] – jazzy big band electronic hybrid		* "Detective Duckweed" from ''Jazzbeetle'' (2023) – [https://open.spotify.com/track/77iDGy7hRx8az3ODrDm5Kl Spotify] \| [https://xotla.bandcamp.com/track/detective-duckweed-53edo Bandcamp] \| [https://youtu.be/FNXEPB4Gm54 YouTube] – jazzy big band electronic hybrid

			=== 53 well-tempered divisions of the octave ===
			; [https://www.youtube.com/@SocraticSwansongs John Robert Bagby] (Socratic Swansongs)
			* [https://www.youtube.com/watch?v=F1vYmdaNHHE ''53 unequal divisions of the octave Just Intonation lumatone experiment''] (2024) (uses [https://sevish.com/scaleworkshop/?n=Harmonic+series+12+tone&l=29F28_3kF3h_pFo_3rF3k_gFf_rFp_m8Fk9_aF9_9F8_74F69_40F3h_23F1s_wFr_6F5_6rF5k_2sF29_5F4_29F1s_wFp_4iF3h_8wF6r_4F3_rFk_e8Faf_pFi_19Fw_1sF19_10Fp_afF74_14Fr_3F2_6rF4g_3hF29_pFg_3kF29_8F5_29F1e_b4F6r_5F3_rFg_3kF23_3hF20_69F3k_gF9_9F5_k9Fb4_1eFr_fF8_74F3r_1cFp_3hF1s_4gF29_2F1&version=2.4.1 this tuning])

	== Notes ==		== Notes ==

FloraC: /* Music */ cleanup

2026-03-24T13:33:36Z

Music: cleanup

@@ Line 1,185: / Line 1,185: @@
 === 21st century ===
 ; [[Bryan Deister]]
 * [https://www.youtube.com/shorts/r-Tzq33OGM4 ''microtonal improvisation in 53edo''] (2025)
@@ Line 1,217: / Line 1,223: @@
 ; [[Aaron Krister Johnson]] ([http://www.akjmusic.com site]{{dead link}})
 * [http://www.akjmusic.com/audio/desert_prayer.mp3 ''Desert Prayer'']{{dead link}}
 ; [[Claudi Meneghin]]
@@ Line 1,275: / Line 1,284: @@
 * "Taking Flight" from ''Nano Particular'' (2019) – [https://open.spotify.com/track/2zp6oM57m6BvQgyOZ5kmuZ Spotify] | [https://xotla.bandcamp.com/track/taking-flight-53edo Bandcamp] | [https://www.youtube.com/watch?v=sIsfYQATouc YouTube]
 * "Detective Duckweed" from ''Jazzbeetle'' (2023) – [https://open.spotify.com/track/77iDGy7hRx8az3ODrDm5Kl Spotify] | [https://xotla.bandcamp.com/track/detective-duckweed-53edo Bandcamp] | [https://youtu.be/FNXEPB4Gm54 YouTube] – jazzy big band electronic hybrid
 == Notes ==

Haidilao at 08:01, 24 March 2026

2026-03-24T08:01:30Z

@@ Line 1,275: / Line 1,275: @@
 * "Taking Flight" from ''Nano Particular'' (2019) – [https://open.spotify.com/track/2zp6oM57m6BvQgyOZ5kmuZ Spotify] | [https://xotla.bandcamp.com/track/taking-flight-53edo Bandcamp] | [https://www.youtube.com/watch?v=sIsfYQATouc YouTube]
 * "Detective Duckweed" from ''Jazzbeetle'' (2023) – [https://open.spotify.com/track/77iDGy7hRx8az3ODrDm5Kl Spotify] | [https://xotla.bandcamp.com/track/detective-duckweed-53edo Bandcamp] | [https://youtu.be/FNXEPB4Gm54 YouTube] – jazzy big band electronic hybrid
 == Notes ==

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave" to "Octave-reduced form, reduced to the first half-octave"

2026-03-13T13:44:49Z

Text replacement - "Octave-reduced form, reduced to the first half-octave" to "Octave-reduced form, reduced to the first half-octave"

← Older revision		Revision as of 13:44, 13 March 2026
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	\| [[Untriton]] / [[aufo]]		\| [[Untriton]] / [[aufo]]
	\|}		\|}
	<nowiki/>* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave		<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave

	== Scales ==		== Scales ==

Eufalesio: Standardize format Sagittal

2026-03-07T12:04:34Z

Standardize format Sagittal

← Older revision		Revision as of 12:04, 7 March 2026
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	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;"
	!~~Degree~~		! colspan="2" \|Steps
	\|'''0'''		!'''0'''
	~~\| +~~1		! 1
	~~\| +~~2		! 2
	~~\| +~~3		! 3
	~~\| +~~4		! 4
	\|'''+5'''		!'''5'''
	\|-		\|-
			! rowspan="2" \|Symbol
	!Evo		!Evo
	\| rowspan="2" \|<big>{{sagittal\|\|//\|}}</big>		\| rowspan="2" \|<big>{{sagittal\|\|//\|}}</big>

Lucius Chiaraviglio: /* 21st century */ Bryan Deister's''Waltz in 53edo'' (2026): Add full version

2026-03-05T06:29:26Z

21st century: Bryan Deister's''Waltz in 53edo'' (2026): Add full version

← Older revision		Revision as of 06:29, 5 March 2026
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	* ''Fantasy in 53edo'' ([https://www.youtube.com/shorts/fgsT-1pBw8g abstract version]; [https://www.youtube.com/watch?v=-q-T3HuGehk visualizer version] (2025)		* ''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/shorts/zCthwbPH2cY ''Finale - Undertale (microtonal cover in 53edo)''] (2026)		* [https://www.youtube.com/shorts/zCthwbPH2cY ''Finale - Undertale (microtonal cover in 53edo)''] (2026)
	* [https://www.youtube.com/shorts/WtSaDQCyfVc ~~''Waltz in 53edo''~~] (~~2026~~)		* ''Waltz in 53edo'' (2026)
			** [https://www.youtube.com/shorts/WtSaDQCyfVc <nowiki>[Short]</nowiki>] (with Lumatone view))
			** [https://www.youtube.com/watch?v=_xumSANdf-g <nowiki>[Full version]</nowiki>]

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

← 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>
	\|-		\|-