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Overhaul on the interval table in favor of highlighting simpler ratios
→Rank-2 temperaments: 81/80 can't be the associated ratio for 1\63 in Sevond, because 63edo maps 81/80 inconsistently to 2\63; I think 64/63 fits the bill, but better check on that
 
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== Theory ==
== Theory ==
63edo [[tempering out|tempers out]] [[3125/3072]] in the 5-limit and [[225/224]], [[245/243]], and [[875/864]] in the 7-limit, so that it [[support]]s [[magic]] temperament. In the 11-limit it tempers out [[100/99]], supporting 11-limit magic, plus [[385/384]] and [[540/539]], [[896/891]]. In the 13-limit it tempers out [[169/168]], [[275/273]], [[640/637]], [[352/351]], [[364/363]] and [[676/675]]. It provides the [[optimal patent val]] for [[immune]], the {{nowrap| 29 & 34d }} temperament in the 7-, 11- and 13-limit.  
63edo is almost [[consistent]] to the [[15-odd-limit]]; the only inconsistency is that [[10/9]] is mapped to 9\63 (1\7, the same as what [[11/10]] is mapped to consistently) so that it is almost 11{{cent}} out of tune. This corresponds to 63edo exaggerating the syntonic comma, [[81/80]], to two steps, so that it finds a somewhat flat mean-tone between ~10/9 and ~9/8.
 
As an equal temperament, it [[tempering out|tempers out]] [[3125/3072]] in the 5-limit and [[225/224]], [[245/243]], and [[875/864]] in the 7-limit, so that it [[support]]s [[magic]] temperament. In the 11-limit it tempers out [[100/99]], supporting 11-limit magic, plus [[385/384]] and [[540/539]], [[896/891]]. In the 13-limit it tempers out [[169/168]], [[275/273]], [[640/637]], [[352/351]], [[364/363]] and [[676/675]]. It provides the [[optimal patent val]] for [[immune]], the {{nowrap| 29 & 34d }} temperament in the 7-, 11- and 13-limit.  


63 is also a fascinating division to look at in the [[47-limit]]. Although it does not deal as well with primes 5, 17, 19, 37 and 41, it excels in the 2.3.7.11.13.23.29.31.43.47 [[subgroup]], and is a great candidate for a [[gentle]] tuning. Its regular augmented fourth (+6 fifths) is less than 0.3 cents sharp of [[23/16]], therefore tempering out [[736/729]]. Its diesis (+12 fifths) can represent [[33/32]], [[32/31]], [[30/29]], [[29/28]], [[28/27]], as well as [[91/88]], and more, so it is very versatile, making chains of fifths of 12 tones or longer very useful in covering harmonic and melodic ground while providing a lot of different colour in different keys. We can take advantage of the representation of 27:28:29:30:31:32:33, which splits [[11/9]] into six "small dieses" as a result; here it can be seen more clearly why these are not regular quarter-tones so are best distinguished from such with the qualifier "large", as otherwise we would expect to see some flavour of minor third after six of them.  
63 is also a fascinating division to look at in the [[47-limit]]. Although it does not deal as well with primes 5, 17, 19, 37 and 41, it excels in the 2.3.7.11.13.23.29.31.43.47 [[subgroup]], and is a great candidate for a [[gentle]] tuning. Its regular augmented fourth (+6 fifths) is less than 0.3 cents sharp of [[23/16]], therefore tempering out [[736/729]]. Its diesis (+12 fifths) can represent [[33/32]], [[32/31]], [[30/29]], [[29/28]], [[28/27]], as well as [[91/88]], and more, so it is very versatile, making chains of fifths of 12 tones or longer very useful in covering harmonic and melodic ground while providing a lot of different colour in different keys. We can take advantage of the representation of 27:28:29:30:31:32:33, which splits [[11/9]] into six "small dieses" as a result; here it can be seen more clearly why these are not regular quarter-tones so are best distinguished from such with the qualifier "large", as otherwise we would expect to see some flavour of minor third after six of them.  
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| 10
| 10
| 190.5
| 190.5
| [[29/26]], [[39/35]], [[49/44]]
| [[19/17]], [[29/26]], [[39/35]], [[49/44]]
|-
|-
| 11
| 11
Line 242: Line 244:
| 53
| 53
| 1009.5
| 1009.5
| [[52/29]], [[70/39]], [[88/49]]
| [[34/19]], [[52/29]], [[70/39]], [[88/49]]
|-
|-
| 54
| 54
Line 286: Line 288:
<nowiki>*</nowiki> As a 2.3.5.7.11.13.23.29.31-subgroup (no-17 no-19 31-limit) temperament, inconsistent intervals in ''italics''
<nowiki>*</nowiki> As a 2.3.5.7.11.13.23.29.31-subgroup (no-17 no-19 31-limit) temperament, inconsistent intervals in ''italics''


See also [[63edo/Godtone's approach]] for some higher-limit ratios.  
See the below section for a machine-generated table including higher-limit ratios selected with a mind towards higher accuracy.
 
=== Higher-accuracy interpretations ===
 
The following table was created using [[User:Godtone#My python 3 code|Godtone's code]] with the command <code><nowiki>interpret_edo(63,ol=53,no=[5,17,19,25,27,37,41,51],add=[73,75,87,89,91,93,105],dec="''",wiki=23)</nowiki></code> (run in a Python 3 interactive console) plus manual correction of the order of some inconsistent intervals, removal of unsimplified intervals of 75, and adding of (the inconsistent but simple) 10/9, 21/20 and their octave-complements.
 
As the command and description indicates, it is a(n accurate) "no-5's"* no-17's no-19's no-25's no-27's no-37's no-41's 49-odd-limit add-53 add-63 add-73 add-87 add-89 add-91 add-93 add-105 interpretation, tuned to the strengths of 63edo. * Note that because of the cancellation of factors, some odd harmonics of 5 (the simpler/more relevant ones) are present, EG {{nowrap|75/3 {{=}} 25}}, {{nowrap|45/3 {{=}} 15}}, {{nowrap|105/75 {{=}} 7/5}}, {{nowrap| 75/35/2 {{=}} 15/14}}, and {{nowrap|45/9 {{=}} 5}}, so it isn't really "no-5's", just has a de-emphasized focus.
 
Intervals are listed in order of size, so that one can know their relative order at a glance and deem the value of the interpretation for a harmonic context, and [[23-limit]] intervals are highlighted for navigability as [[13-limit]] intervals are more likely to already have pages, and as we are excluding primes 17 and 19, we are only adding prime 23 to the 13-limit.
 
Inconsistent intervals are ''in italics''.
 
{| class="wikitable center-all right-2 left-3 mw-collapsible mw-collapsed"
|-
! Degree
! Cents
! Approximate ratios<ref group="note">{{sg|limit=2.3.5.7.11.13.23.29.31.43.47.53.73.89-subgroup (no-17's no-19's no-37's no-41's 53-limit add-73 add-89 add-105)}} Note that due to the error on 5, only low-complexity intervals involving 5 are included here.</ref>
|-
| 0
| 0.0
| [[1/1]]
|-
| 1
| 19.05
| 106/105, [[105/104]], 94/93, 93/92, [[92/91]], [[91/90]], 90/89, 89/88, 88/87, 87/86, 73/72, [[65/64]], [[64/63]]
|-
| 2
| 38.1
| ''[[66/65]]'', 53/52, [[49/48]], 48/47, 47/46, 93/91, [[46/45]], 91/89, [[45/44]], 89/87, 44/43, 43/42
|-
| 3
| 57.14
| [[36/35]], [[33/32]], 32/31, 94/91, 31/30, 92/89, [[91/88]], 30/29, 89/86, 29/28, ''[[25/24]]''
|-
| 4
| 76.19
| [[26/25]], 49/47, 73/70, [[24/23]], 47/45, 93/89, [[23/22]], 91/87, 45/43, [[22/21]]
|-
| 5
| 95.24
| ''[[21/20]]'', 98/93, [[96/91]], 94/89, 56/53, 93/88, 92/87, 91/86, 89/84, [[35/33]], [[52/49]]
|-
| 6
| 114.29
| 33/31, [[49/46]], [[16/15]], 47/44, 78/73, 31/29, 46/43, [[15/14]]
|-
| 7
| 133.33
| [[14/13]], 96/89, 94/87, 93/86, 53/49, [[13/12]]
|-
| 8
| 152.38
| [[49/45]], [[12/11]], 47/43, [[35/32]], 58/53, [[23/21]]
|-
| 9
| 171.43
| [[11/10]], 98/89, 43/39, 32/29, 53/48, 116/105, 73/66, 52/47, 31/28, ''[[10/9]]''
|-
| 10
| 190.48
| [[49/44]], [[39/35]], 29/26, 48/43, 105/94, 104/93, 47/42
|-
| 11
| 209.52
| ''[[28/25]]'', [[9/8]], 98/87, 53/47, [[44/39]], 35/31, [[26/23]], 60/53, ''[[25/22]]''
|-
| 12
| 228.57
| 33/29, 49/43, 106/93, 73/64, 89/78, [[105/92]], [[8/7]]
|-
| 13
| 247.62
| 84/73, 53/46, [[15/13]], [[52/45]]
|-
| 14
| 266.67
| ''29/25'', 36/31, 106/91, [[7/6]], 104/89, 62/53
|-
| 15
| 285.71
| 73/62, 53/45, 86/73, [[33/28]], [[46/39]], 105/89, 124/105, [[13/11]], 58/49
|-
| 16
| 304.76
| 106/89, 56/47, 87/73, 31/26, [[105/88]], 43/36, 104/87
|-
| 17
| 323.81
| [[6/5]], 112/93, 53/44, 47/39, 88/73, 35/29, 64/53, 29/24, 52/43
|-
| 18
| 342.86
| 73/60, [[28/23]], 106/87, [[39/32]], [[128/105]], 89/73, 105/86, [[11/9]], [[60/49]]
|-
| 19
| 361.9
| 43/35, [[16/13]], 53/43, 90/73, 58/47, 89/72, [[26/21]]
|-
| 20
| 380.95
| 31/25, 36/29, 87/70, [[56/45]], 66/53, 91/73, 116/93, [[5/4]]
|-
| 21
| 400.0
| [[49/39]], [[44/35]], 39/31, 112/89, 73/58, 92/73, 29/23, 53/42, [[91/72]], 62/49
|-
| 22
| 419.05
| [[33/26]], 89/70, [[14/11]], 93/73, 116/91, 60/47, [[23/18]]
|-
| 23
| 438.1
| ''[[32/25]]'', [[9/7]], 112/87, 94/73, 58/45, 40/31, 31/24
|-
| 24
| 457.14
| [[13/10]], 56/43, 43/33, 116/89, 73/56, [[30/23]], 47/36, [[64/49]]
|-
| 25
| 476.19
| [[21/16]], [[46/35]], 96/73, 29/22, [[120/91]], 62/47, 70/53
|-
| 26
| 495.24
| 93/70, [[4/3]]
|-
| 27
| 514.29
| 98/73, 47/35, 43/32, 39/29, [[35/26]], [[66/49]], 31/23, 120/89, 89/66, 58/43
|-
| 28
| 533.33
| 42/31, 72/53, 53/39, 87/64, [[49/36]], 64/47, 124/91, [[15/11]]
|-
| 29
| 552.38
| [[48/35]], [[11/8]], 128/93, 73/53, 62/45, [[91/66]], 40/29, 29/21
|-
| 30
| 571.43
| [[18/13]], 43/31, 146/105, 89/64, [[32/23]], [[39/28]], 124/89, [[46/33]], 60/43
|-
| 31
| 590.48
| [[7/5]], 87/62, 73/52, 66/47, [[45/32]], [[128/91]], 31/22
|-
| 32
| 609.52
| 44/31, [[91/64]], [[64/45]], 47/33, 104/73, 124/87, [[10/7]]
|-
| 33
| 628.57
| 43/30, [[33/23]], 89/62, [[56/39]], [[23/16]], 128/89, 105/73, 62/43, [[13/9]]
|-
| 34
| 647.62
| 42/29, 29/20, [[132/91]], 45/31, 106/73, 93/64, [[16/11]], [[35/24]]
|-
| 35
| 666.67
| [[22/15]], 91/62, 47/32, [[72/49]], 128/87, 78/53, 53/36, 31/21
|-
| 36
| 685.71
| 43/29, 132/89, 89/60, 46/31, [[49/33]], [[52/35]], 58/39, 64/43, 70/47, 73/49
|-
| 37
| 704.76
| [[3/2]], 140/93
|-
| 38
| 723.81
| 53/35, 47/31, [[91/60]], 44/29, 73/48, [[35/23]], [[32/21]]
|-
| 39
| 742.86
| [[49/32]], 72/47, [[23/15]], 112/73, 89/58, 66/43, 43/28, [[20/13]]
|-
| 40
| 761.9
| 48/31, 31/20, 45/29, 73/47, 87/56, [[14/9]], ''[[25/16]]''
|-
| 41
| 780.95
| [[36/23]], 47/30, 91/58, 146/93, [[11/7]], 140/89, [[52/33]]
|-
| 42
| 800.0
| 49/31, [[144/91]], 84/53, 46/29, 73/46, 116/73, 89/56, 62/39, [[35/22]], [[78/49]]
|-
| 43
| 819.05
| [[8/5]], 93/58, 146/91, 53/33, [[45/28]], 140/87, 29/18, 50/31
|-
| 44
| 838.1
| [[21/13]], 144/89, 47/29, 73/45, 86/53, [[13/8]], 70/43
|-
| 45
| 857.14
| [[49/30]], [[18/11]], 172/105, 146/89, [[105/64]], [[64/39]], 87/53, [[23/14]], 120/73
|-
| 46
| 876.19
| 43/26, 48/29, 53/32, 58/35, 73/44, 78/47, 88/53, 93/56, [[5/3]]
|-
| 47
| 895.24
| 87/52, 72/43, [[176/105]], 52/31, 146/87, 47/28, 89/53
|-
| 48
| 914.29
| 49/29, [[22/13]], 105/62, 178/105, [[39/23]], [[56/33]], 73/43, 90/53, 124/73
|-
| 49
| 933.33
| 53/31, 89/52, [[12/7]], 91/53, 31/18, ''50/29''
|-
| 50
| 952.38
| [[45/26]], [[26/15]], 92/53, 73/42
|-
| 51
| 971.43
| [[7/4]], [[184/105]], 156/89, 128/73, 93/53, 86/49, 58/33
|-
| 52
| 990.48
| ''[[44/25]]'', 53/30, [[23/13]], 62/35, [[39/22]], 94/53, 87/49, [[16/9]], ''[[25/14]]''
|-
| 53
| 1009.52
| 84/47, 93/52, 188/105, 43/24, 52/29, [[70/39]], [[88/49]]
|-
| 54
| 1028.57
| ''[[9/5]]', 56/31, 47/26, 132/73, 105/58, 96/53, 29/16, 78/43, 89/49, [[20/11]]
|-
| 55
| 1047.62
| [[42/23]], 53/29, [[64/35]], 86/47, [[11/6]], [[90/49]]
|-
| 56
| 1066.67
| [[24/13]], 98/53, 172/93, 87/47, 89/48, [[13/7]]
|-
| 57
| 1085.71
| [[28/15]], 43/23, 58/31, 73/39, 88/47, [[15/8]], [[92/49]], 62/33
|-
| 58
| 1104.76
| [[49/26]], [[66/35]], 168/89, 172/91, 87/46, 176/93, 53/28, 89/47, [[91/48]], 93/49, ''[[40/21]]''
|-
| 59
| 1123.81
| [[21/11]], 86/45, 174/91, [[44/23]], 178/93, 90/47, [[23/12]], 140/73, 94/49, [[25/13]]
|-
| 60
| 1142.86
| ''[[48/25]]'', 56/29, 172/89, 29/15, [[176/91]], 89/46, 60/31, 91/47, 31/16, [[64/33]], [[35/18]]
|-
| 61
| 1161.9
| 84/43, 43/22, 174/89, [[88/45]], 178/91, [[45/23]], 182/93, 92/47, 47/24, [[96/49]], 104/53
|-
| 62
| 1180.95
| [[63/32]], 144/73, 172/87, 87/44, 176/89, 89/45, [[180/91]], [[91/46]], 184/93, 93/47, [[208/105]], 105/53
|-
| 63
| 1200.0
| [[2/1]]
|}
<references group="note" />


== Notation ==
== Notation ==
=== Ups and downs notation ===
63edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
{{Sharpness-sharp7a}}
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7}}
=== Sagittal notation ===
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[56edo #Sagittal notation|56edo]].
This notation uses the same sagittal sequence as [[56edo #Sagittal notation|56edo]].
Line 316: Line 601:
</imagemap>
</imagemap>


=== Ups and downs notation ===
== Approximation to JI ==
Using [[Helmholtz–Ellis]] accidentals, 63edo can be notated using [[ups and downs notation]]:
=== Interval mappings ===
{{Sharpness-sharp7}}
{{Q-odd-limit intervals}}


== Approximation to JI ==
=== Zeta peak index ===
=== Zeta peak index ===
{| class="wikitable center-all"
{{ZPI
| zpi = 321
| steps = 63.0192885705350
| step size = 19.0417890652143
| tempered height = 6.768662
| pure height = 6.534208
| integral = 1.049023
| gap = 15.412920
| octave = 1199.63271110850
| consistent = 8
| distinct = 8
}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
! colspan="3" | Tuning
| 2.3
! colspan="3" | Strength
| {{Monzo| 100 -63 }}
! colspan="2" | Closest edo
| {{Mapping| 63 100 }}
! colspan="2" | Integer limit
| -0.885
| 0.885
| 4.65
|-
|-
! ZPI
| 2.3.5
! Steps per octave
| 3125/3072, 1638400/1594323
! Step size (cents)
| {{Mapping| 63 100 146 }}
! Height
| +0.177
! Integral
| 1.67
! Gap
| 8.77
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
|-
| [[321zpi]]
| 2.3.5.7
| 63.0192885705350
| 225/224, 245/243, 51200/50421
| 19.0417890652143
| {{Mapping| 63 100 146 177 }}
| 6.768662
| -0.099
| 1.049023
| 1.52
| 15.412920
| 8.00
| 63edo
|-
| 1199.63271110850
| 2.3.5.7.11
| 8
| 100/99, 225/224, 245/243, 1331/1323
| 8
| {{mapping| 63 100 146 177 218 }}
| -0.141
| 1.37
| 7.17
|-
| 2.3.5.7.11.13
| 100/99, 169/168, 225/224, 245/243, 275/273
| {{mapping| 63 100 146 177 218 233 }}
| -0.008
| 1.28
| 6.73
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 2\63
| 38.10
| 49/48
| [[Slender]]
|-
| 1
| 13\63
| 247.62
| 15/13
| [[Immune]]
|-
| 1
| 19\63
| 361.90
| 16/13
| [[Submajor]]
|-
| 1
| 20\63
| 380.95
| 5/4
| [[Magic]]
|-
| 1
| 25\63
| 476.19
| 21/16
| [[Subfourth]]
|-
| 3
| 26\63<br>(5\63)
| 495.24<br>(95.24)
| 4/3<br>(21/20)
| [[Fog]]
|-
| 7
| 26\63<br>(1\63)
| 495.24<br>(19.05)
| 4/3<br>(64/63)
| [[Sevond]]
|-
| 9
| 13\63<br>(1\63)
| 247.62<br>(19.05)
| 15/13<br>(99/98)
| [[Enneaportent]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
Line 356: Line 731:
* Timeywimey (original/default tuning): 16 10 7 4 11 5 10
* Timeywimey (original/default tuning): 16 10 7 4 11 5 10
* Sandcastle (original/default tuning): 8 10 8 11 8 8 10
* Sandcastle (original/default tuning): 8 10 8 11 8 8 10
== Instruments ==
* [[Lumatone mapping for 63edo]]
* [[Skip fretting system 63 3 17]]


== Music ==
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/IYLzF4ogl_w ''microtonal improvisation in 63edo''] (2025)
; [[Cam Taylor]]
; [[Cam Taylor]]
* [https://soundcloud.com/cam-taylor-2-1/12tone63edo1 ''Improvisation in 12-tone fifths chain''] (2015)
* [https://soundcloud.com/cam-taylor-2-1/12tone63edo1 ''Improvisation in 12-tone fifths chain''] (2015)
Retrieved from "https://en.xen.wiki/w/63edo"