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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
Overhaul on the interval table in favor of highlighting simpler ratios
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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 [[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.  
Line 14: Line 14:


=== 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.
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"
{| class="wikitable center-all right-2 left-3"
|-
|-
! Degree
! Degree
! Cents
! 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>
! Approximate ratios*
|-
|-
| 0
| 0
Line 41: Line 33:
|-
|-
| 1
| 1
| 19.05
| 19.0
| 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]]
| ''[[50/49]]'', ''[[55/54]]'', [[64/63]], [[65/64]], [[91/90]], [[105/104]]
|-
|-
| 2
| 2
| 38.1
| 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
| [[45/44]], [[46/45]], [[49/48]], ''[[56/55]]'', ''[[66/65]]'', ''[[81/80]]''
|-
|-
| 3
| 3
| 57.14
| 57.1
| [[36/35]], [[33/32]], 32/31, 94/91, 31/30, 92/89, [[91/88]], 30/29, 89/86, 29/28, ''[[25/24]]''
| ''[[25/24]]'', [[28/27]], [[29/28]], [[30/29]], [[31/30]], [[32/31]], [[33/32]], [[36/35]]
|-
|-
| 4
| 4
| 76.19
| 76.2
| [[26/25]], 49/47, 73/70, [[24/23]], 47/45, 93/89, [[23/22]], 91/87, 45/43, [[22/21]]
| [[22/21]], [[23/22]], [[24/23]], [[26/25]], ''[[27/26]]''
|-
|-
| 5
| 5
| 95.24
| 95.2
| 98/93, [[96/91]], 94/89, 56/53, 93/88, 92/87, 91/86, 89/84, [[35/33]], [[52/49]]
| ''[[21/20]]'', [[35/33]]
|-
|-
| 6
| 6
| 114.29
| 114.3
| 33/31, [[49/46]], [[16/15]], 47/44, 78/73, 31/29, 46/43, [[15/14]]
| [[15/14]], [[16/15]]
|-
|-
| 7
| 7
| 133.33
| 133.3
| [[14/13]], 96/89, 94/87, 93/86, 53/49, [[13/12]]
| [[13/12]], [[14/13]]
|-
|-
| 8
| 8
| 152.38
| 152.4
| [[49/45]], [[12/11]], 47/43, [[35/32]], 58/53, [[23/21]]
| [[12/11]]
|-
|-
| 9
| 9
| 171.43
| 171.4
| [[11/10]], 98/89, 43/39, 32/29, 53/48, 116/105, 73/66, 52/47, 31/28
| ''[[10/9]]'', [[11/10]], [[31/28]], [[32/29]]
|-
|-
| 10
| 10
| 190.48
| 190.5
| [[49/44]], [[39/35]], 29/26, 48/43, 105/94, 104/93, 47/42
| [[29/26]], [[39/35]], [[49/44]]
|-
|-
| 11
| 11
| 209.52
| 209.5
| ''[[28/25]]'', [[9/8]], 98/87, 53/47, [[44/39]], 35/31, [[26/23]], 60/53, ''[[25/22]]''
| [[9/8]]
|-
|-
| 12
| 12
| 228.57
| 228.6
| 33/29, 49/43, 106/93, 73/64, 89/78, [[105/92]], [[8/7]]
| [[8/7]]
|-
|-
| 13
| 13
| 247.62
| 247.6
| 84/73, 53/46, [[15/13]], [[52/45]]
| [[15/13]]
|-
|-
| 14
| 14
| 266.67
| 266.7
| ''29/25'', 36/31, 106/91, [[7/6]], 104/89, 62/53
| [[7/6]]
|-
|-
| 15
| 15
| 285.71
| 285.7
| 73/62, 53/45, 86/73, [[33/28]], [[46/39]], 105/89, 124/105, [[13/11]], 58/49
| [[13/11]]
|-
|-
| 16
| 16
| 304.76
| 304.8
| 106/89, 56/47, 87/73, 31/26, [[105/88]], 43/36, 104/87
| [[31/26]]
|-
|-
| 17
| 17
| 323.81
| 323.8
| [[6/5]], 112/93, 53/44, 47/39, 88/73, 35/29, 64/53, 29/24, 52/43
| [[6/5]]
|-
|-
| 18
| 18
| 342.86
| 342.9
| 73/60, [[28/23]], 106/87, [[39/32]], [[128/105]], 89/73, 105/86, [[11/9]], [[60/49]]
| [[11/9]], [[28/23]], [[39/32]]
|-
|-
| 19
| 19
| 361.9
| 361.9
| 43/35, [[16/13]], 53/43, 90/73, 58/47, 89/72, [[26/21]]
| [[16/13]], [[26/21]], [[27/22]]
|-
|-
| 20
| 20
| 380.95
| 381.0
| 31/25, 36/29, 87/70, [[56/45]], 66/53, 91/73, 116/93, [[5/4]]
| [[5/4]]
|-
|-
| 21
| 21
| 400.0
| 400.0
| [[49/39]], [[44/35]], 39/31, 112/89, 73/58, 92/73, 29/23, 53/42, [[91/72]], 62/49
| [[29/23]], [[44/35]], [[49/39]]
|-
|-
| 22
| 22
| 419.05
| 419.0
| [[33/26]], 89/70, [[14/11]], 93/73, 116/91, 60/47, [[23/18]]
| [[14/11]]
|-
|-
| 23
| 23
| 438.1
| 438.1
| ''[[32/25]]'', [[9/7]], 112/87, 94/73, 58/45, 40/31, 31/24
| [[9/7]]
|-
|-
| 24
| 24
| 457.14
| 457.1
| [[13/10]], 56/43, 43/33, 116/89, 73/56, [[30/23]], 47/36, [[64/49]]
| [[13/10]]
|-
|-
| 25
| 25
| 476.19
| 476.2
| [[21/16]], [[46/35]], 96/73, 29/22, [[120/91]], 62/47, 70/53
| [[21/16]]
|-
|-
| 26
| 26
| 495.24
| 495.2
| 93/70, [[4/3]]
| [[4/3]]
|-
|-
| 27
| 27
| 514.29
| 514.3
| 98/73, 47/35, 43/32, 39/29, [[35/26]], [[66/49]], 31/23, 120/89, 89/66, 58/43
| [[35/26]]
|-
|-
| 28
| 28
| 533.33
| 533.3
| 42/31, 72/53, 53/39, 87/64, [[49/36]], 64/47, 124/91, [[15/11]]
| [[15/11]], ''[[27/20]]''
|-
|-
| 29
| 29
| 552.38
| 552.4
| [[48/35]], [[11/8]], 128/93, 73/53, 62/45, [[91/66]], 40/29, 29/21
| [[11/8]]
|-
|-
| 30
| 30
| 571.43
| 571.4
| [[18/13]], 43/31, 146/105, 89/64, [[32/23]], [[39/28]], 124/89, [[46/33]], 60/43
| [[18/13]], [[32/23]]
|-
|-
| 31
| 31
| 590.48
| 590.5
| [[7/5]], 87/62, 73/52, 66/47, [[45/32]], [[128/91]], 31/22
| [[7/5]]
|-
|-
| 32
| 32
| 609.52
| 609.5
| 44/31, [[91/64]], [[64/45]], 47/33, 104/73, 124/87, [[10/7]]
| [[10/7]]
|-
|-
| 33
| 33
| 628.57
| 628.6
| 43/30, [[33/23]], 89/62, [[56/39]], [[23/16]], 128/89, 105/73, 62/43, [[13/9]]
| [[13/9]], [[23/16]]
|-
|-
| 34
| 34
| 647.62
| 647.6
| 42/29, 29/20, [[132/91]], 45/31, 106/73, 93/64, [[16/11]], [[35/24]]
| [[16/11]]
|-
|-
| 35
| 35
| 666.67
| 666.7
| [[22/15]], 91/62, 47/32, [[72/49]], 128/87, 78/53, 53/36, 31/21
| [[22/15]]
|-
|-
| 36
| 36
| 685.71
| 685.7
| 43/29, 132/89, 89/60, 46/31, [[49/33]], [[52/35]], 58/39, 64/43, 70/47, 73/49
| [[52/35]]
|-
|-
| 37
| 37
| 704.76
| 704.8
| [[3/2]], 140/93
| [[3/2]]
|-
|-
| 38
| 38
| 723.81
| 723.8
| 53/35, 47/31, [[91/60]], 44/29, 73/48, [[35/23]], [[32/21]]
| [[32/21]]
|-
|-
| 39
| 39
| 742.86
| 742.9
| [[49/32]], 72/47, [[23/15]], 112/73, 89/58, 66/43, 43/28, [[20/13]]
| [[20/13]]
|-
|-
| 40
| 40
| 761.9
| 761.9
| 48/31, 31/20, 45/29, 73/47, 87/56, [[14/9]], ''[[25/16]]''
| [[14/9]]
|-
|-
| 41
| 41
| 780.95
| 781.0
| [[36/23]], 47/30, 91/58, 146/93, [[11/7]], 140/89, [[52/33]]
| [[11/7]]
|-
|-
| 42
| 42
| 800.0
| 800.0
| 49/31, [[144/91]], 84/53, 46/29, 73/46, 116/73, 89/56, 62/39, [[35/22]], [[78/49]]
| [[35/22]], [[46/29]]
|-
|-
| 43
| 43
| 819.05
| 819.0
| [[8/5]], 93/58, 146/91, 53/33, [[45/28]], 140/87, 29/18, 50/31
| [[8/5]]
|-
|-
| 44
| 44
| 838.1
| 838.1
| [[21/13]], 144/89, 47/29, 73/45, 86/53, [[13/8]], 70/43
| [[13/8]], [[21/13]], [[44/27]]
|-
|-
| 45
| 45
| 857.14
| 857.1
| [[49/30]], [[18/11]], 172/105, 146/89, [[105/64]], [[64/39]], 87/53, [[23/14]], 120/73, [[150/91]]
| [[18/11]], [[23/14]], [[64/39]]
|-
|-
| 46
| 46
| 876.19
| 876.2
| 43/26, 48/29, 53/32, 58/35, 73/44, 78/47, 88/53, 93/56, [[5/3]]
| [[5/3]]
|-
|-
| 47
| 47
| 895.24
| 895.2
| 87/52, 72/43, [[176/105]], 52/31, 146/87, 47/28, 89/53, 150/89
| [[52/31]]
|-
|-
| 48
| 48
| 914.29
| 914.3
| 49/29, [[22/13]], 105/62, 178/105, [[39/23]], [[56/33]], 73/43, 90/53, 124/73
| [[22/13]]
|-
|-
| 49
| 49
| 933.33
| 933.3
| 53/31, 89/52, [[12/7]], 91/53, 31/18, ''50/29''
| [[12/7]]
|-
|-
| 50
| 50
| 952.38
| 952.4
| [[45/26]], [[26/15]], 92/53, 73/42
| [[26/15]]
|-
|-
| 51
| 51
| 971.43
| 971.4
| [[7/4]], [[184/105]], 156/89, 128/73, 93/53, 86/49, 58/33
| [[7/4]]
|-
|-
| 52
| 52
| 990.48
| 990.5
| ''[[44/25]]'', 53/30, [[23/13]], 62/35, [[39/22]], 94/53, 87/49, [[16/9]], ''[[25/14]]''
| [[16/9]]
|-
|-
| 53
| 53
| 1009.52
| 1009.5
| 84/47, 93/52, 188/105, 43/24, 52/29, [[70/39]], [[88/49]]
| [[52/29]], [[70/39]], [[88/49]]
|-
|-
| 54
| 54
| 1028.57
| 1028.6
| 56/31, 47/26, 132/73, 105/58, 96/53, 29/16, 78/43, 89/49, [[20/11]]
| ''[[9/5]]'', [[20/11]], [[29/16]], [[56/31]]
|-
|-
| 55
| 55
| 1047.62
| 1047.6
| [[42/23]], 53/29, [[64/35]], 86/47, [[11/6]], [[90/49]]
| [[11/6]]
|-
|-
| 56
| 56
| 1066.67
| 1066.7
| [[24/13]], 98/53, 172/93, 87/47, 89/48, [[13/7]]
| [[13/7]], [[24/13]]
|-
|-
| 57
| 57
| 1085.71
| 1085.7
| [[28/15]], 43/23, 58/31, 73/39, 88/47, [[15/8]], [[92/49]], 62/33
| [[15/8]], [[28/15]]
|-
|-
| 58
| 58
| 1104.76
| 1104.8
| [[49/26]], [[66/35]], 168/89, 172/91, 87/46, 176/93, 53/28, 89/47, [[91/48]], 93/49
| ''[[40/21]]'', [[66/35]]
|-
|-
| 59
| 59
| 1123.81
| 1123.8
| [[21/11]], 86/45, 174/91, [[44/23]], 178/93, 90/47, [[23/12]], 140/73, 94/49, [[25/13]]
| [[21/11]], [[23/12]], [[25/13]], [[44/23]], ''[[52/27]]''
|-
|-
| 60
| 60
| 1142.86
| 1142.9
| ''[[48/25]]'', 56/29, 172/89, 29/15, [[176/91]], 89/46, 60/31, 91/47, 31/16, [[64/33]], [[35/18]]
| [[27/14]], [[29/15]], [[31/16]], [[35/18]], ''[[48/25]]'', [[56/29]], [[60/31]], [[64/33]]
|-
|-
| 61
| 61
| 1161.9
| 1161.9
| 84/43, 43/22, 174/89, [[88/45]], 178/91, [[45/23]], 182/93, 92/47, 47/24, [[96/49]], 104/53
| [[45/23]], ''[[55/28]]'', [[88/45]], [[96/49]], ''[[160/81]]''
|-
|-
| 62
| 62
| 1180.95
| 1181.0
| [[63/32]], 144/73, 172/87, 87/44, 176/89, 89/45, [[180/91]], [[91/46]], 184/93, 93/47, [[208/105]], 105/53
| ''[[49/25]]'', [[63/32]], [[65/33]], ''[[108/55]]'', [[180/91]], [[208/105]]
|-
|-
| 63
| 63
Line 292: Line 284:
| [[2/1]]
| [[2/1]]
|}
|}
<references group="note" />
<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.


== Notation ==
== Notation ==
=== 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 ====

Revision as of 10:07, 12 March 2025

← 62edo 63edo 64edo →
Prime factorization 32 × 7
Step size 19.0476 ¢ 
Fifth 37\63 (704.762 ¢)
Semitones (A1:m2) 7:4 (133.3 ¢ : 76.19 ¢)
Consistency limit 7
Distinct consistency limit 7

63 equal divisions of the octave (abbreviated 63edo or 63ed2), also called 63-tone equal temperament (63tet) or 63 equal temperament (63et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 63 equal parts of about 19 ¢ each. Each step represents a frequency ratio of 21/63, or the 63rd root of 2.

Theory

63edo tempers out 3125/3072 in the 5-limit and 225/224, 245/243, and 875/864 in the 7-limit, so that it supports 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 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.

A 17-tone fifths chain looks on the surface a little similar to 17edo, but as −17 fifths gets us to 64/63, observing the comma becomes an essential part in progressions favouring prime 7. Furthermore, its prime 5 is far from unusable; although 25/16 is barely inconsistent, this affords the tuning supporting 7-limit magic, which may be considered interesting or desirable in of itself. And if this was not enough, if you really want to, it offers reasonable approximations to some yet higher primes too; namely 43/32, 47/32, and 53/32; see the tables below.

Prime harmonics

Approximation of prime harmonics in 63edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.00 +2.81 -5.36 +2.60 +1.06 -2.43 +9.33 +7.25 +0.30 -1.01 -2.18 -3.72
Relative (%) +0.0 +14.7 -28.1 +13.7 +5.6 -12.8 +49.0 +38.1 +1.6 -5.3 -11.4 -19.6
Steps
(reduced) 		63
(0) 		100
(37) 		146
(20) 		177
(51) 		218
(29) 		233
(44) 		258
(6) 		268
(16) 		285
(33) 		306
(54) 		312
(60) 		328
(13)
Approximation of prime harmonics in 63edo (continued)
Harmonic 41 43 47 53 59 61 67 71 73 79 83 89
Error Absolute (¢) +9.03 +2.77 +1.16 +2.69 +7.50 +6.92 -3.12 -8.27 +0.78 -2.63 +7.10 +0.55
Relative (%) +47.4 +14.5 +6.1 +14.1 +39.3 +36.4 -16.4 -43.4 +4.1 -13.8 +37.3 +2.9
Steps
(reduced) 		338
(23) 		342
(27) 		350
(35) 		361
(46) 		371
(56) 		374
(59) 		382
(4) 		387
(9) 		390
(12) 		397
(19) 		402
(24) 		408
(30)

Subsets and supersets

Since 63 factors into primes as 32 × 7, 63edo has subset 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:

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.

Intervals

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 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 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

* 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.

Notation

Sagittal notation

This notation uses the same sagittal sequence as 56edo.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/6381/8033/32

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/6381/8033/32

Ups and downs notation

Using Helmholtz–Ellis accidentals, 63edo can be notated using ups and downs notation:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Sharp symbol  
Flat symbol

Approximation to JI

Zeta peak index

Tuning Strength Closest edo Integer limit
ZPI Steps per octave Step size (cents) Height Integral Gap Edo Octave (cents) Consistent Distinct
321zpi 63.0192885705350 19.0417890652143 6.768662 1.049023 15.412920 63edo 1199.63271110850 8 8

Scales

  • Approximation of Pelog lima: 6 9 21 6 21
  • Timeywimey (original/default tuning): 16 10 7 4 11 5 10
  • Sandcastle (original/default tuning): 8 10 8 11 8 8 10

Music

Cam Taylor
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