53edo: Difference between revisions

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=== Prime harmonics ===

{{Harmonics in equal|53|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 53edo (continued)}}

{{Harmonics in equal|53|columns=12|start=12|collapsed=true|title=Approximation of prime harmonics in 53edo (continued)}}

See [[#Approximation to JI]] for details and a more in-depth discussion.

See [[#Approximation to JI]] for details and a more in-depth discussion on the higher harmonics.

=== As a tuning of other temperaments ===

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53edo is the 16th [[prime edo]], following [[47edo]] and coming before [[59edo]].

Many of its multiples such as [[159edo]], [[212edo]], [[742edo]], [[901edo]] and [[954edo]] have good consistency limits and are each of their own interest. The [[mercator family]] comprises rank-2 temperaments with 1/53-octave periods.

Many of its multiples such as [[159edo]], [[212edo]], [[742edo]], [[901edo]] and the zeta [[954edo]] have good consistency limits and are each of their own interest. The [[mercator family]] comprises rank-2 temperaments with 1/53-octave periods.

== Intervals ==

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== 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:

=== 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).

~~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~~]]:

=== Sagittal notation ===

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;"

! colspan="2" | Steps

!'''0'''

! 1

! 2

! 3

! 4

!'''5'''

|-

! rowspan="2" | Symbol

! Evo

| rowspan="2" | <big>{{sagittal||//|}}</big>

| rowspan="2" | <big>{{sagittal|/|}}</big>

| rowspan="2" | <big>{{sagittal|//|}}</big>

| {{sagittal|\\!}}{{sagittal|#}}

| {{sagittal|\!}}{{sagittal|#}}

| <big>{{sagittal|#}}</big>

|-

! Revo

| <big>{{sagittal|)||(}}</big>

| <big>{{sagittal|||\}}</big>

| <big>{{sagittal|/||\}}</big>

|}

The following enharmonics from the Spartan set are present (comma tempered out):

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

* {{sagittal|/|}} = {{sagittal||)}} ([[225/224]])

* {{sagittal||(}} = {{sagittal||//|}} ([[5120/5103]])

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.

~~=== Sagittal notation ===~~

==== Evo flavor ====

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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 ==

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|}

Because the 5th is so accurate, 53edo also offers ~~good approximations for~~ Pythagorean ~~thirds~~. In addition, the 43\53 interval is only 4.8 cents wider than 7/4, so 53edo can also be used for 7-limit harmony, in which it tempers out the [[septimal kleisma]], 225/224.

Because the 5th is so incredibly accurate (next edo with a more accurate fifth is [[200edo]]), 53edo also offers a great approximation to Pythagorean tuning. In addition, the 43\53 interval is only 4.8 cents wider than 7/4, so 53edo can also be used for 7-limit harmony, in which it tempers out the [[septimal kleisma]], 225/224.

=== 15-odd-limit interval mappings ===

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| 0.42

| Sathurugu

| ~~Schismina~~

| Minisma

|}

Line 1,026:

Line 1,061:

| 362.3

| 16/13

| [[~~Submajor~~]]

| [[Demibuzzard]] / submajor / interpental

|-

| 1

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| [[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 ==

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; {{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/shorts/iYtZGBKHcpU Schismatic version]

* [https://www.youtube.com/shorts/4YEHMpaO4bA ''"Waterfall" Étude from 12 Études, op. 10''] (1829–1832) — rendered by Claudi Meneghin (2025)

; {{W|George Frideric Handel}}

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; {{W|Scott Joplin}}

* ~~[https://www.youtube.com/watch?v=AKXMuhB3uHQ~~ ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)

* ''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

* ''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://www.youtube.com/watch?v=t-pRqKGX0oo (2024 version)]

** [https://www.youtube.com/shorts/msBeUJjFlV4 (2025 version)]

; {{W|Shirō Sagisu}}

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* [https://www.youtube.com/watch?v=DCENVnxH6bI ''Bande-announce''] – rendered by MortisTheneRd (2024)

=== 21st century ===

==== 21st century ====

; [[ALLY195]]

* [https://www.bilibili.com/video/BV1f54y1r7XG/ ''My Soul adaptation''] (2020)

; [[Alxeusxiao]]

* [https://www.bilibili.com/video/BV1zM4m1m7Gz/ ''53edo exploration''] (2024)

; [[Bryan Deister]]

* [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)

* [https://www.youtube.com/shorts/tIx3PcOyNJo ''53edo improv''] (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)

* ''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]]

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* "Husband Head Void" from ''Void'' (2025) – [https://open.spotify.com/track/4yvyDZv8dBjOiurzoTjpBj Spotify] | [https://francium223.bandcamp.com/track/husband-head-void Bandcamp] | [https://www.youtube.com/watch?v=HMnklwjEdF0 YouTube]

* "Lasagna Cat" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/6sJil69QOqxNWrWrkgm3rl Spotify] | [https://francium223.bandcamp.com/track/lasagna-cat Bandcamp] | [https://www.youtube.com/watch?v=Ay2zhVnTlxw YouTube]

* [https://www.youtube.com/watch?v=efGrW8uSGuE ''Opunish Bathomet''] (2025)

; [[Andrew Heathwaite]]

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; [[Aaron Krister Johnson]] ([http://www.akjmusic.com site]{{dead link}})

* [http://www.akjmusic.com/audio/desert_prayer.mp3 ''Desert Prayer'']{{dead link}}

; [[Logan02A4]]

* [https://www.bilibili.com/video/BV1mBCRYmEhg/ ''53edo try''] (2024)

; [[Claudi Meneghin]]

* [https://www.youtube.com/shorts/g7C2OrFd-nk ''Orwell Micro Trio, for Organ (Just: 7 Orwells = 1 Twelfth)''] (2025) — actually in open-ended Orwell tuning, but with the generator adjusted to be extremely close to 12\53, at 271.71{{c}}

; [[Merct]]

Line 1,221:

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; [[Valeriana of the Night]]

* [https://www.youtube.com/watch?v=eMPQDRTHGhg ''Hero''] (2023)

; [[VitaminCD]]

* [https://www.youtube.com/watch?v=KCWhecfwlMw ''<nowiki>Orwellian in Nature (Orwell [9] Microtonal Lament)</nowiki>''] (2025)

; [[Randy Wells]]

@@ Line 18: / Line 18: @@
 === Prime harmonics ===
 {{Harmonics in equal|53|columns=11}}
-{{Harmonics in equal|53|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 53edo (continued)}}
+{{Harmonics in equal|53|columns=12|start=12|collapsed=true|title=Approximation of prime harmonics in 53edo (continued)}}
-See [[#Approximation to JI]] for details and a more in-depth discussion.
+See [[#Approximation to JI]] for details and a more in-depth discussion on the higher harmonics.
 === As a tuning of other temperaments ===
@@ Line 28: / Line 28: @@
 edo is the 16th [[prime edo]], following [[47edo]] and coming before [[59edo]].
-Many of its multiples such as [[159edo]], [[212edo]], [[742edo]], [[901edo]] and [[954edo]] have good consistency limits and are each of their own interest. The [[mercator family]] comprises rank-2 temperaments with 1/53-octave periods.
+Many of its multiples such as [[159edo]], [[212edo]], [[742edo]], [[901edo]] and the zeta [[954edo]] have good consistency limits and are each of their own interest. The [[mercator family]] comprises rank-2 temperaments with 1/53-octave periods.
 == Intervals ==
@@ Line 633: / Line 633: @@
 == Notation ==
-=== Ups and downs notation ===
+=== Stein–Zimmermann–Gould notation ===
-edo 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 ===
+edo 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}}
-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]]:
+=== Sagittal notation ===
-{{Sharpness-sharp5}}
+edo 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;"
+! colspan="2" | Steps
+!'''0'''
+! 1
+! 2
+! 3
+! 4
+!'''5'''
+|-
+! rowspan="2" | Symbol
+! Evo
+| rowspan="2" | <big>{{sagittal||//|}}</big>
+| rowspan="2" | <big>{{sagittal|/|}}</big>
+| rowspan="2" | <big>{{sagittal|//|}}</big>
+| {{sagittal|\\!}}{{sagittal|#}}
+| {{sagittal|\!}}{{sagittal|#}}
+| <big>{{sagittal|#}}</big>
+|-
+! Revo
+| <big>{{sagittal|)||(}}</big>
+| <big>{{sagittal|||\}}</big>
+| <big>{{sagittal|/||\}}</big>
+|}
+The following enharmonics from the Spartan set are present (comma tempered out):
+* {{sagittal|//|}} = {{Sagittal|/|)}} = {{Sagittal|/|\}} ([[325/324]], [[352/351]])
+* {{sagittal|/|}} = {{sagittal||)}} ([[225/224]])
+* {{sagittal||(}} = {{sagittal||//|}} ([[5120/5103]])
+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.
-=== Sagittal notation ===
 ==== Evo flavor ====
 {{Sagittal chart|Evo}}
@@ Line 647: / Line 682: @@
 {{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 ==
@@ Line 698: / Line 733: @@
 |}
-Because the 5th is so accurate, 53edo also offers good approximations for Pythagorean thirds. In addition, the 43\53 interval is only 4.8 cents wider than 7/4, so 53edo can also be used for 7-limit harmony, in which it tempers out the [[septimal kleisma]], 225/224.
+Because the 5th is so incredibly accurate (next edo with a more accurate fifth is [[200edo]]), 53edo also offers a great approximation to Pythagorean tuning. In addition, the 43\53 interval is only 4.8 cents wider than 7/4, so 53edo can also be used for 7-limit harmony, in which it tempers out the [[septimal kleisma]], 225/224.
 === 15-odd-limit interval mappings ===
@@ Line 940: / Line 975: @@
 | 0.42
 | Sathurugu
-| Schismina
+| Minisma
 |}
@@ Line 1,026: / Line 1,061: @@
 | 362.3
 | 16/13
-| [[Submajor]]
+| [[Demibuzzard]] / submajor / interpental
 |-
 | 1
@@ Line 1,076: / Line 1,111: @@
 | [[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 ==
@@ Line 1,134: / Line 1,169: @@
 ; {{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 &mdash; rendered by [[Claudi Meneghin]] (2025)
 ** [https://www.youtube.com/watch?v=0VB1hv0-AmE Near-Pythagorean version]
 ** [https://www.youtube.com/shorts/iYtZGBKHcpU Schismatic version]
+* [https://www.youtube.com/shorts/4YEHMpaO4bA ''"Waterfall" Étude from 12 Études, op. 10''] (1829–1832) &mdash; rendered by Claudi Meneghin (2025)
 ; {{W|George Frideric Handel}}
@@ Line 1,142: / Line 1,178: @@
 ; {{W|Scott Joplin}}
-* [https://www.youtube.com/watch?v=AKXMuhB3uHQ ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)
+* ''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
+* ''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://www.youtube.com/watch?v=t-pRqKGX0oo (2024 version)]
-** [https://www.youtube.com/shorts/msBeUJjFlV4 (2025 version)]
 ; {{W|Shirō Sagisu}}
@@ Line 1,151: / Line 1,185: @@
 * [https://www.youtube.com/watch?v=DCENVnxH6bI ''Bande-announce''] – rendered by MortisTheneRd (2024)
-=== 21st century ===
+==== 21st century ====
+; [[ALLY195]]
+* [https://www.bilibili.com/video/BV1f54y1r7XG/ ''My Soul adaptation''] (2020)
+; [[Alxeusxiao]]
+* [https://www.bilibili.com/video/BV1zM4m1m7Gz/ ''53edo exploration''] (2024)
 ; [[Bryan Deister]]
 * [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)
+* [https://www.youtube.com/shorts/tIx3PcOyNJo ''53edo improv''] (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)
+* ''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]]
@@ Line 1,163: / Line 1,210: @@
 * "Husband Head Void" from ''Void'' (2025) – [https://open.spotify.com/track/4yvyDZv8dBjOiurzoTjpBj Spotify] | [https://francium223.bandcamp.com/track/husband-head-void Bandcamp] | [https://www.youtube.com/watch?v=HMnklwjEdF0 YouTube]
 * "Lasagna Cat" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/6sJil69QOqxNWrWrkgm3rl Spotify] | [https://francium223.bandcamp.com/track/lasagna-cat Bandcamp] | [https://www.youtube.com/watch?v=Ay2zhVnTlxw YouTube]
+* [https://www.youtube.com/watch?v=efGrW8uSGuE ''Opunish Bathomet''] (2025)
 ; [[Andrew Heathwaite]]
@@ Line 1,176: / Line 1,224: @@
 ; [[Aaron Krister Johnson]] ([http://www.akjmusic.com site]{{dead link}})
 * [http://www.akjmusic.com/audio/desert_prayer.mp3 ''Desert Prayer'']{{dead link}}
+; [[Logan02A4]]
+* [https://www.bilibili.com/video/BV1mBCRYmEhg/ ''53edo try''] (2024)
+; [[Claudi Meneghin]]
+* [https://www.youtube.com/shorts/g7C2OrFd-nk ''Orwell Micro Trio, for Organ (Just: 7 Orwells = 1 Twelfth)''] (2025) &mdash; actually in open-ended Orwell tuning, but with the generator adjusted to be extremely close to 12\53, at 271.71{{c}}
 ; [[Merct]]
@@ Line 1,221: / Line 1,275: @@
 ; [[Valeriana of the Night]]
 * [https://www.youtube.com/watch?v=eMPQDRTHGhg ''Hero''] (2023)
+; [[VitaminCD]]
+* [https://www.youtube.com/watch?v=KCWhecfwlMw ''<nowiki>Orwellian in Nature (Orwell [9] Microtonal Lament)</nowiki>''] (2025)
 ; [[Randy Wells]]