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53edo: Difference between revisions

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m →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
 
(20 intermediate revisions by 5 users not shown)
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53edo is the 16th [[prime edo]], following [[47edo]] and coming before [[59edo]].
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 ==
== Intervals ==
Line 49: Line 49:
|-
|-
| 1
| 1
| 22.6
| 22.64
| ''[[50/49]]'', [[64/63]], [[81/80]]
| ''[[50/49]]'', [[64/63]], [[81/80]]
| ^1
| ^1
Line 58: Line 58:
|-
|-
| 2
| 2
| 45.3
| 45.28
| [[33/32]], [[36/35]], [[49/48]], [[128/125]]
| [[33/32]], [[36/35]], [[49/48]], [[128/125]]
| ^^1, vvm2
| ^^1, vvm2
Line 67: Line 67:
|-
|-
| 3
| 3
| 67.9
| 67.92
| ''[[22/21]]'', [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| ''[[22/21]]'', [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| vvA1, vm2
| vvA1, vm2
Line 76: Line 76:
|-
|-
| 4
| 4
| 90.6
| 90.57
| [[19/18]], [[20/19]], [[21/20]], [[256/243]]
| [[19/18]], [[20/19]], [[21/20]], [[256/243]]
| vA1, m2
| vA1, m2
Line 85: Line 85:
|-
|-
| 5
| 5
| 113.2
| 113.21
| [[15/14]], [[16/15]]
| [[15/14]], [[16/15]]
| A1, ^m2
| A1, ^m2
Line 94: Line 94:
|-
|-
| 6
| 6
| 135.8
| 135.85
| [[13/12]], [[14/13]], [[27/25]]
| [[13/12]], [[14/13]], [[27/25]]
| ^^m2
| ^^m2
Line 103: Line 103:
|-
|-
| 7
| 7
| 158.5
| 158.49
| [[11/10]], [[12/11]], [[35/32]], [[57/52]], [[800/729]]
| [[11/10]], [[12/11]], [[35/32]], [[57/52]], [[800/729]]
| vvM2
| vvM2
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|-
|-
| 8
| 8
| 181.1
| 181.13
| [[10/9]]
| [[10/9]]
| vM2
| vM2
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|-
|-
| 9
| 9
| 203.8
| 203.77
| [[9/8]]
| [[9/8]]
| M2
| M2
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|-
|-
| 10
| 10
| 226.4
| 226.42
| [[8/7]], [[256/225]]
| [[8/7]], [[256/225]]
| ^M2
| ^M2
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|-
|-
| 11
| 11
| 249.1
| 249.06
| [[15/13]], [[22/19]], [[125/108]], [[144/125]]
| [[15/13]], [[22/19]], [[125/108]], [[144/125]]
| ^^M2, vvm3
| ^^M2, vvm3
Line 148: Line 148:
|-
|-
| 12
| 12
| 271.7
| 271.70
| [[7/6]], [[75/64]]
| [[7/6]], [[75/64]]
| vm3
| vm3
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|-
|-
| 13
| 13
| 294.3
| 294.34
| [[13/11]], [[19/16]], [[32/27]]
| [[13/11]], [[19/16]], [[32/27]]
| m3
| m3
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|-
|-
| 14
| 14
| 317.0
| 316.98
| [[6/5]]
| [[6/5]]
| ^m3
| ^m3
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|-
|-
| 15
| 15
| 339.6
| 339.62
| [[11/9]], [[243/200]]
| [[11/9]], [[243/200]]
| ^^m3
| ^^m3
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|-
|-
| 16
| 16
| 362.3
| 362.26
| [[16/13]], [[100/81]]
| [[16/13]], [[100/81]]
| vvM3
| vvM3
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|-
|-
| 17
| 17
| 384.9
| 384.91
| [[5/4]]
| [[5/4]]
| vM3
| vM3
Line 202: Line 202:
|-
|-
| 18
| 18
| 407.5
| 407.55
| [[19/15]], [[24/19]], [[81/64]]
| [[19/15]], [[24/19]], [[81/64]]
| M3
| M3
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|-
|-
| 19
| 19
| 430.2
| 430.19
| [[9/7]], ''[[14/11]]''
| [[9/7]], ''[[14/11]]''
| ^M3
| ^M3
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|-
|-
| 20
| 20
| 452.8
| 452.83
| [[13/10]], [[125/96]], [[162/125]]
| [[13/10]], [[125/96]], [[162/125]]
| ^^M3, vv4
| ^^M3, vv4
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|-
|-
| 21
| 21
| 475.5
| 475.47
| [[21/16]], [[25/19]], [[320/243]], [[675/512]]
| [[21/16]], [[25/19]], [[320/243]], [[675/512]]
| v4
| v4
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|-
|-
| 22
| 22
| 498.1
| 498.11
| [[4/3]]
| [[4/3]]
| P4
| P4
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|-
|-
| 23
| 23
| 520.8
| 520.75
| [[19/14]], [[27/20]]
| [[19/14]], [[27/20]]
| ^4
| ^4
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|-
|-
| 24
| 24
| 543.4
| 543.40
| [[11/8]], [[15/11]], [[26/19]]
| [[11/8]], [[15/11]], [[26/19]]
| ^^4
| ^^4
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|-
|-
| 25
| 25
| 566.0
| 566.04
| [[18/13]]
| [[18/13]], [[25/18]]
| vvA4, vd5
| vvA4, vd5
| dudaug 4th, downdim 5th
| dudaug 4th, downdim 5th
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|-
|-
| 26
| 26
| 588.7
| 588.68
| [[7/5]], [[45/32]]
| [[7/5]], [[45/32]]
| vA4, d5
| vA4, d5
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|-
|-
| 27
| 27
| 611.3
| 611.32
| [[10/7]], [[64/45]]
| [[10/7]], [[64/45]]
| A4, ^d5
| A4, ^d5
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|-
|-
| 28
| 28
| 634.0
| 633.96
| [[13/9]]
| [[13/9]], [[36/25]]
| ^A4, ^^d5
| ^A4, ^^d5
| upaug 4th, dupdim 5th
| upaug 4th, dupdim 5th
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|-
|-
| 29
| 29
| 656.6
| 656.60
| [[16/11]], [[19/13]], [[22/15]]
| [[16/11]], [[19/13]], [[22/15]]
| vv5
| vv5
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|-
|-
| 30
| 30
| 679.2
| 679.25
| [[28/19]], [[40/27]]
| [[28/19]], [[40/27]]
| v5
| v5
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|-
|-
| 31
| 31
| 701.9
| 701.89
| [[3/2]]
| [[3/2]]
| P5
| P5
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|-
|-
| 32
| 32
| 724.5
| 724.53
| [[32/21]], [[38/25]], [[243/160]], [[1024/675]]
| [[32/21]], [[38/25]], [[243/160]], [[1024/675]]
| ^5
| ^5
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|-
|-
| 33
| 33
| 747.2
| 747.17
| [[20/13]], [[125/81]], [[192/125]]
| [[20/13]], [[125/81]], [[192/125]]
| ^^5, vvm6
| ^^5, vvm6
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|-
|-
| 34
| 34
| 769.8
| 769.81
| ''[[11/7]]'', [[14/9]], [[25/16]]
| ''[[11/7]]'', [[14/9]], [[25/16]]
| vm6
| vm6
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|-
|-
| 35
| 35
| 792.5
| 792.45
| [[19/12]], [[30/19]], [[128/81]]
| [[19/12]], [[30/19]], [[128/81]]
| m6
| m6
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|-
|-
| 36
| 36
| 815.1
| 815.09
| [[8/5]]
| [[8/5]]
| ^m6
| ^m6
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|-
|-
| 37
| 37
| 837.7
| 837.74
| [[13/8]], [[81/50]]
| [[13/8]], [[81/50]]
| ^^m6
| ^^m6
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|-
|-
| 38
| 38
| 860.4
| 860.38
| [[18/11]], [[400/243]]
| [[18/11]], [[400/243]]
| vvM6
| vvM6
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|-
|-
| 39
| 39
| 883.0
| 883.02
| [[5/3]]
| [[5/3]]
| vM6
| vM6
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|-
|-
| 40
| 40
| 905.7
| 905.66
| [[22/13]], [[27/16]], [[32/19]]
| [[22/13]], [[27/16]], [[32/19]]
| M6
| M6
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|-
|-
| 41
| 41
| 928.3
| 928.30
| [[12/7]]
| [[12/7]]
| ^M6
| ^M6
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|-
|-
| 42
| 42
| 950.9
| 950.94
| [[19/11]], [[26/15]], [[125/72]], [[216/125]]
| [[19/11]], [[26/15]], [[125/72]], [[216/125]]
| ^^M6, vvm7
| ^^M6, vvm7
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|-
|-
| 43
| 43
| 973.6
| 973.58
| [[7/4]]
| [[7/4]]
| vm7
| vm7
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|-
|-
| 44
| 44
| 996.2
| 996.23
| [[16/9]]
| [[16/9]]
| m7
| m7
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|-
|-
| 45
| 45
| 1018.9
| 1018.87
| [[9/5]]
| [[9/5]]
| ^m7
| ^m7
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|-
|-
| 46
| 46
| 1041.5
| 1041.51
| [[11/6]], [[20/11]], [[64/35]], [[729/400]]
| [[11/6]], [[20/11]], [[64/35]], [[729/400]]
| ^^m7
| ^^m7
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|-
|-
| 47
| 47
| 1064.2
| 1064.15
| [[13/7]], [[24/13]], [[50/27]]
| [[13/7]], [[24/13]], [[50/27]]
| vvM7
| vvM7
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|-
|-
| 48
| 48
| 1086.8
| 1086.79
| [[15/8]]
| [[15/8]]
| vM7
| vM7
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|-
|-
| 49
| 49
| 1109.4
| 1109.43
| [[19/10]], [[36/19]], [[40/21]], [[243/128]]
| [[19/10]], [[36/19]], [[40/21]], [[243/128]]
| M7
| M7
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|-
|-
| 50
| 50
| 1132.1
| 1132.08
| ''[[21/11]]'', [[25/13]], [[27/14]], [[52/27]], [[48/25]]
| ''[[21/11]]'', [[25/13]], [[27/14]], [[52/27]], [[48/25]]
| ^M7
| ^M7
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|-
|-
| 51
| 51
| 1154.7
| 1154.72
| [[35/18]], [[64/33]], [[96/49]], [[125/64]]
| [[35/18]], [[64/33]], [[96/49]], [[125/64]]
| ^^M7, vv8
| ^^M7, vv8
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|-
|-
| 52
| 52
| 1177.4
| 1177.36
| ''[[49/25]]'', [[63/32]], [[160/81]]
| ''[[49/25]]'', [[63/32]], [[160/81]]
| v8
| v8
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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]]:
=== Sagittal notation ===
{{Sharpness-sharp5}}
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 ====
==== Evo flavor ====
{{Sagittal chart|Evo}}
{{Sagittal chart|Evo}}
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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 ==
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 ===
=== 15-odd-limit interval mappings ===
Line 940: Line 975:
| 0.42
| 0.42
| Sathurugu
| Sathurugu
| Schismina
| Minisma
|}
|}


Line 1,026: Line 1,061:
| 362.3
| 362.3
| 16/13
| 16/13
| [[Submajor]]
| [[Demibuzzard]] / submajor / interpental
|-
|-
| 1
| 1
Line 1,076: Line 1,111:
| [[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 ==
Line 1,143: Line 1,178:


; {{W|Scott Joplin}}
; {{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}}
; {{W|Shirō Sagisu}}
Line 1,152: Line 1,185:
* [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]]
* [https://www.bilibili.com/video/BV1f54y1r7XG/ ''My Soul adaptation''] (2020)
 
; [[Alxeusxiao]]
* [https://www.bilibili.com/video/BV1zM4m1m7Gz/ ''53edo exploration''] (2024)
 
; [[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)
* [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]]
; [[Francium]]
Line 1,179: Line 1,224:
; [[Aaron Krister Johnson]] ([http://www.akjmusic.com site]{{dead link}})
; [[Aaron Krister Johnson]] ([http://www.akjmusic.com site]{{dead link}})
* [http://www.akjmusic.com/audio/desert_prayer.mp3 ''Desert Prayer'']{{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]]
; [[Claudi Meneghin]]
Line 1,227: Line 1,275:
; [[Valeriana of the Night]]
; [[Valeriana of the Night]]
* [https://www.youtube.com/watch?v=eMPQDRTHGhg ''Hero''] (2023)
* [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]]
; [[Randy Wells]]
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