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manually remove intervals of 75 as always being too damaged for their complexity, as evidenced by always appearing at the extreme ends of the spectrum of interpretations, if not as the most extreme entry then as the second-most, next to a vastly simpler interval that can justify the damage
→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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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 63 factors into {{factorization|63}}, 63edo has subset edos {{EDOs| 3, 7, 9, and 21 }}.
Since 63 factors into primes as {{nowrap| 3<sup>2</sup> × 7 }}, 63edo has subset edos {{EDOs| 3, 7, 9, and 21 }}.


Its representation of the 2.3.5.7.13 subgroup (no-11's 13-limit) can uniquely be described in terms of accurate approximations contained in its main subsets of [[7edo]] and [[9edo]]:
Its representation of the 2.3.5.7.13 subgroup (no-11's 13-limit) can uniquely be described in terms of accurate approximations contained in its main subsets of [[7edo]] and [[9edo]]:
* 1\9 = ~[[14/13]]~[[13/12]], implying (the much more accurate) 2\9 = ~[[7/6]] ([[septiennealic]])
* 1\9 = [[14/13]]~[[13/12]], implying the much more accurate 2\9 = ~[[7/6]] ([[septiennealic]])
* 2\7 = ~[[39/32]]~[[128/105]], via [[4096/4095]] and the [[akjaysma]] (which are naturally paired)
* 2\7 = [[39/32]]~[[128/105]], via [[4096/4095]] and the [[akjaysma]] (which are naturally paired)
If we avoid equating 14/13 and 13/12 (which is by far the highest damage equivalence) so that we achieve 7/6 = 2\9 directly, we get the 63 & 441 microtemperament in the same subgroup.
If we avoid equating 14/13 and 13/12 (which is by far the highest damage equivalence) so that we achieve {{nowrap| 7/6 {{=}} 2\9 }} directly, we get the {{nowrap| 63 & 441 }} microtemperament in the same subgroup.


== Intervals ==
== Intervals ==
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 and removal of unsimplified intervals of 75.
{| class="wikitable center-all right-2 left-3"
|-
! Degree
! Cents
! Approximate ratios*
|-
| 0
| 0.0
| [[1/1]]
|-
| 1
| 19.0
| ''[[50/49]]'', ''[[55/54]]'', [[64/63]], [[65/64]], [[91/90]], [[105/104]]
|-
| 2
| 38.1
| [[45/44]], [[46/45]], [[49/48]], ''[[56/55]]'', ''[[66/65]]'', ''[[81/80]]''
|-
| 3
| 57.1
| ''[[25/24]]'', [[28/27]], [[29/28]], [[30/29]], [[31/30]], [[32/31]], [[33/32]], [[36/35]]
|-
| 4
| 76.2
| [[22/21]], [[23/22]], [[24/23]], [[26/25]], ''[[27/26]]''
|-
| 5
| 95.2
| ''[[21/20]]'', [[35/33]]
|-
| 6
| 114.3
| [[15/14]], [[16/15]]
|-
| 7
| 133.3
| [[13/12]], [[14/13]]
|-
| 8
| 152.4
| [[12/11]]
|-
| 9
| 171.4
| ''[[10/9]]'', [[11/10]], [[31/28]], [[32/29]]
|-
| 10
| 190.5
| [[19/17]], [[29/26]], [[39/35]], [[49/44]]
|-
| 11
| 209.5
| [[9/8]]
|-
| 12
| 228.6
| [[8/7]]
|-
| 13
| 247.6
| [[15/13]]
|-
| 14
| 266.7
| [[7/6]]
|-
| 15
| 285.7
| [[13/11]]
|-
| 16
| 304.8
| [[31/26]]
|-
| 17
| 323.8
| [[6/5]]
|-
| 18
| 342.9
| [[11/9]], [[28/23]], [[39/32]]
|-
| 19
| 361.9
| [[16/13]], [[26/21]], [[27/22]]
|-
| 20
| 381.0
| [[5/4]]
|-
| 21
| 400.0
| [[29/23]], [[44/35]], [[49/39]]
|-
| 22
| 419.0
| [[14/11]]
|-
| 23
| 438.1
| [[9/7]]
|-
| 24
| 457.1
| [[13/10]]
|-
| 25
| 476.2
| [[21/16]]
|-
| 26
| 495.2
| [[4/3]]
|-
| 27
| 514.3
| [[35/26]]
|-
| 28
| 533.3
| [[15/11]], ''[[27/20]]''
|-
| 29
| 552.4
| [[11/8]]
|-
| 30
| 571.4
| [[18/13]], [[32/23]]
|-
| 31
| 590.5
| [[7/5]]
|-
| 32
| 609.5
| [[10/7]]
|-
| 33
| 628.6
| [[13/9]], [[23/16]]
|-
| 34
| 647.6
| [[16/11]]
|-
| 35
| 666.7
| [[22/15]]
|-
| 36
| 685.7
| [[52/35]]
|-
| 37
| 704.8
| [[3/2]]
|-
| 38
| 723.8
| [[32/21]]
|-
| 39
| 742.9
| [[20/13]]
|-
| 40
| 761.9
| [[14/9]]
|-
| 41
| 781.0
| [[11/7]]
|-
| 42
| 800.0
| [[35/22]], [[46/29]]
|-
| 43
| 819.0
| [[8/5]]
|-
| 44
| 838.1
| [[13/8]], [[21/13]], [[44/27]]
|-
| 45
| 857.1
| [[18/11]], [[23/14]], [[64/39]]
|-
| 46
| 876.2
| [[5/3]]
|-
| 47
| 895.2
| [[52/31]]
|-
| 48
| 914.3
| [[22/13]]
|-
| 49
| 933.3
| [[12/7]]
|-
| 50
| 952.4
| [[26/15]]
|-
| 51
| 971.4
| [[7/4]]
|-
| 52
| 990.5
| [[16/9]]
|-
| 53
| 1009.5
| [[34/19]], [[52/29]], [[70/39]], [[88/49]]
|-
| 54
| 1028.6
| ''[[9/5]]'', [[20/11]], [[29/16]], [[56/31]]
|-
| 55
| 1047.6
| [[11/6]]
|-
| 56
| 1066.7
| [[13/7]], [[24/13]]
|-
| 57
| 1085.7
| [[15/8]], [[28/15]]
|-
| 58
| 1104.8
| ''[[40/21]]'', [[66/35]]
|-
| 59
| 1123.8
| [[21/11]], [[23/12]], [[25/13]], [[44/23]], ''[[52/27]]''
|-
| 60
| 1142.9
| [[27/14]], [[29/15]], [[31/16]], [[35/18]], ''[[48/25]]'', [[56/29]], [[60/31]], [[64/33]]
|-
| 61
| 1161.9
| [[45/23]], ''[[55/28]]'', [[88/45]], [[96/49]], ''[[160/81]]''
|-
| 62
| 1181.0
| ''[[49/25]]'', [[63/32]], [[65/33]], ''[[108/55]]'', [[180/91]], [[208/105]]
|-
| 63
| 1200.0
| [[2/1]]
|}
<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 the below section for a machine-generated table including higher-limit ratios selected with a mind towards higher accuracy.
 
=== Higher-accuracy interpretations ===


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.
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.
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.
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Inconsistent intervals are ''in italics''.
Inconsistent intervals are ''in italics''.


{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3 mw-collapsible mw-collapsed"
|-
|-
! Degree
! Degree
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| 5
| 5
| 95.24
| 95.24
| 98/93, [[96/91]], 94/89, 56/53, 93/88, 92/87, 91/86, 89/84, [[35/33]], [[52/49]]
| ''[[21/20]]'', 98/93, [[96/91]], 94/89, 56/53, 93/88, 92/87, 91/86, 89/84, [[35/33]], [[52/49]]
|-
|-
| 6
| 6
Line 74: Line 344:
| 9
| 9
| 171.43
| 171.43
| [[11/10]], 98/89, 43/39, 32/29, 53/48, 116/105, 73/66, 52/47, 31/28
| [[11/10]], 98/89, 43/39, 32/29, 53/48, 116/105, 73/66, 52/47, 31/28, ''[[10/9]]''
|-
|-
| 10
| 10
Line 218: Line 488:
| 45
| 45
| 857.14
| 857.14
| [[49/30]], [[18/11]], 172/105, 146/89, [[105/64]], [[64/39]], 87/53, [[23/14]], 120/73, [[150/91]]
| [[49/30]], [[18/11]], 172/105, 146/89, [[105/64]], [[64/39]], 87/53, [[23/14]], 120/73
|-
|-
| 46
| 46
Line 226: Line 496:
| 47
| 47
| 895.24
| 895.24
| 87/52, 72/43, [[176/105]], 52/31, 146/87, 47/28, 89/53, 150/89
| 87/52, 72/43, [[176/105]], 52/31, 146/87, 47/28, 89/53
|-
|-
| 48
| 48
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| 54
| 54
| 1028.57
| 1028.57
| 56/31, 47/26, 132/73, 105/58, 96/53, 29/16, 78/43, 89/49, [[20/11]]
| ''[[9/5]]', 56/31, 47/26, 132/73, 105/58, 96/53, 29/16, 78/43, 89/49, [[20/11]]
|-
|-
| 55
| 55
Line 270: Line 540:
| 58
| 58
| 1104.76
| 1104.76
| [[49/26]], [[66/35]], 168/89, 172/91, 87/46, 176/93, 53/28, 89/47, [[91/48]], 93/49
| [[49/26]], [[66/35]], 168/89, 172/91, 87/46, 176/93, 53/28, 89/47, [[91/48]], 93/49, ''[[40/21]]''
|-
|-
| 59
| 59
Line 295: Line 565:


== 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|56-EDO]].
This notation uses the same sagittal sequence as [[56edo #Sagittal notation|56edo]].


==== Evo flavor ====
==== Evo flavor ====
Line 322: 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]] (%)
|-
| 2.3
| {{Monzo| 100 -63 }}
| {{Mapping| 63 100 }}
| -0.885
| 0.885
| 4.65
|-
| 2.3.5
| 3125/3072, 1638400/1594323
| {{Mapping| 63 100 146 }}
| +0.177
| 1.67
| 8.77
|-
|-
! colspan="3" | Tuning
| 2.3.5.7
! colspan="3" | Strength
| 225/224, 245/243, 51200/50421
! colspan="2" | Closest edo
| {{Mapping| 63 100 146 177 }}
! colspan="2" | Integer limit
| -0.099
| 1.52
| 8.00
|-
|-
! ZPI
| 2.3.5.7.11
! Steps per octave
| 100/99, 225/224, 245/243, 1331/1323
! Step size (cents)
| {{mapping| 63 100 146 177 218 }}
! Height
| -0.141
! Integral
| 1.37
! Gap
| 7.17
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
|-
| [[321zpi]]
| 2.3.5.7.11.13
| 63.0192885705350
| 100/99, 169/168, 225/224, 245/243, 275/273
| 19.0417890652143
| {{mapping| 63 100 146 177 218 233 }}
| 6.768662
| -0.008
| 1.049023
| 1.28
| 15.412920
| 6.73
| 63edo
|}
| 1199.63271110850
 
| 8
=== Rank-2 temperaments ===
| 8
{| 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 ==
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* 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"