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→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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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|63}}
{{ED intro}}


== Theory ==
== Theory ==
The equal temperament [[tempering out|tempers out]] [[3125/3072]] in the 5-limit and [[225/224]], [[245/243]], [[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 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.  


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.
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 ===
=== Prime harmonics ===
Line 14: Line 16:


=== 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]]:
* 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)
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=47,no=[5,17,19,25,27,37,41],add=[63,73,75,87,89,91,93],dec="''",wiki=23)</nowiki></code> (run in a Python 3 interactive console) plus manual correction of the order of some inconsistent intervals.
{| 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 ===
 
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 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 47-odd-limit add-63 add-73 add-75 add-87 add-89 add-91 add-93 interpretation, tuned to the strengths of [[63edo]]. Note that because of the cancellation of factors, some odd harmonics of 5 (the more relevant ones) are present, specifically 75/3 = 25, 45/3 = 15 and 45/9 = 5.
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.
Line 25: Line 300:
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
! Cents
! Cents
! Approximate Ratios*
! 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
Line 37: Line 312:
| 1
| 1
| 19.05
| 19.05
| 94/93, 93/92, [[92/91]], [[91/90]], 90/89, 89/88, 88/87, 87/86, 73/72, [[64/63]]
| 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
| 2
| 38.1
| 38.1
| ''63/62'', 48/47, 47/46, 93/91, [[46/45]], 91/89, [[45/44]], 89/87, 44/43, 43/42, 75/73
| ''[[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
| 3
Line 49: Line 324:
| 4
| 4
| 76.19
| 76.19
| [[26/25]], 73/70, [[24/23]], 47/45, 93/89, [[23/22]], 91/87, 45/43, [[22/21]]
| [[26/25]], 49/47, 73/70, [[24/23]], 47/45, 93/89, [[23/22]], 91/87, 45/43, [[22/21]]
|-
|-
| 5
| 5
| 95.24
| 95.24
| ''[[21/20]]'', [[96/91]], 94/89, 93/88, 92/87, 91/86, 89/84, [[35/33]]
| ''[[21/20]]'', 98/93, [[96/91]], 94/89, 56/53, 93/88, 92/87, 91/86, 89/84, [[35/33]], [[52/49]]
|-
|-
| 6
| 6
| 114.29
| 114.29
| 33/31, [[16/15]], 47/44, 78/73, 31/29, 46/43, [[15/14]]
| 33/31, [[49/46]], [[16/15]], 47/44, 78/73, 31/29, 46/43, [[15/14]]
|-
|-
| 7
| 7
| 133.33
| 133.33
| [[14/13]], 96/89, 94/87, 93/86, [[13/12]]
| [[14/13]], 96/89, 94/87, 93/86, 53/49, [[13/12]]
|-
|-
| 8
| 8
| 152.38
| 152.38
| 63/58, [[12/11]], 47/43, [[35/32]], [[23/21]]
| [[49/45]], [[12/11]], 47/43, [[35/32]], 58/53, [[23/21]]
|-
|-
| 9
| 9
| 171.43
| 171.43
| [[11/10]], 43/39, 32/29, 73/66, 52/47, 31/28, ''[[10/9]]''
| [[11/10]], 98/89, 43/39, 32/29, 53/48, 116/105, 73/66, 52/47, 31/28, ''[[10/9]]''
|-
|-
| 10
| 10
| 190.48
| 190.48
| [[39/35]], 29/26, 48/43, 104/93, 47/42
| [[49/44]], [[39/35]], 29/26, 48/43, 105/94, 104/93, 47/42
|-
|-
| 11
| 11
| 209.52
| 209.52
| ''[[28/25]]'', [[9/8]], [[44/39]], 35/31, [[26/23]], ''[[25/22]]''
| ''[[28/25]]'', [[9/8]], 98/87, 53/47, [[44/39]], 35/31, [[26/23]], 60/53, ''[[25/22]]''
|-
|-
| 12
| 12
| 228.57
| 228.57
| 33/29, 73/64, 89/78, [[8/7]]
| 33/29, 49/43, 106/93, 73/64, 89/78, [[105/92]], [[8/7]]
|-
|-
| 13
| 13
| 247.62
| 247.62
| ''86/75'', 84/73, [[15/13]], [[52/45]], 73/63
| 84/73, 53/46, [[15/13]], [[52/45]]
|-
|-
| 14
| 14
| 266.67
| 266.67
| ''29/25'', 36/31, [[7/6]], 104/89, [[75/64]]
| ''29/25'', 36/31, 106/91, [[7/6]], 104/89, 62/53
|-
|-
| 15
| 15
| 285.71
| 285.71
| [[88/75]], 73/62, 86/73, [[33/28]], [[46/39]], [[13/11]], ''[[25/21]]''
| 73/62, 53/45, 86/73, [[33/28]], [[46/39]], 105/89, 124/105, [[13/11]], 58/49
|-
|-
| 16
| 16
| 304.76
| 304.76
| 89/75, 56/47, 87/73, 31/26, 43/36, 104/87
| 106/89, 56/47, 87/73, 31/26, [[105/88]], 43/36, 104/87
|-
|-
| 17
| 17
| 323.81
| 323.81
| [[6/5]], 112/93, 47/39, 88/73, 35/29, 29/24, 52/43, 75/62
| [[6/5]], 112/93, 53/44, 47/39, 88/73, 35/29, 64/53, 29/24, 52/43
|-
|-
| 18
| 18
| 342.86
| 342.86
| ''[[63/52]]'', [[91/75]], 73/60, [[28/23]], [[39/32]], 89/73, [[11/9]]
| 73/60, [[28/23]], 106/87, [[39/32]], [[128/105]], 89/73, 105/86, [[11/9]], [[60/49]]
|-
|-
| 19
| 19
| 361.9
| 361.9
| [[92/75]], 43/35, [[16/13]], 90/73, 58/47, 89/72, [[26/21]]
| 43/35, [[16/13]], 53/43, 90/73, 58/47, 89/72, [[26/21]]
|-
|-
| 20
| 20
| 380.95
| 380.95
| 31/25, 36/29, 87/70, [[56/45]], 91/73, 116/93, [[5/4]]
| 31/25, 36/29, 87/70, [[56/45]], 66/53, 91/73, 116/93, [[5/4]]
|-
|-
| 21
| 21
| 400.0
| 400.0
| 94/75, [[44/35]], 39/31, 112/89, 73/58, 92/73, 29/23, [[91/72]]
| [[49/39]], [[44/35]], 39/31, 112/89, 73/58, 92/73, 29/23, 53/42, [[91/72]], 62/49
|-
|-
| 22
| 22
Line 125: Line 400:
| 23
| 23
| 438.1
| 438.1
| ''[[32/25]]'', [[9/7]], 112/87, 94/73, 58/45, 40/31, 31/24, 75/58
| ''[[32/25]]'', [[9/7]], 112/87, 94/73, 58/45, 40/31, 31/24
|-
|-
| 24
| 24
| 457.14
| 457.14
| [[13/10]], 56/43, 43/33, 116/89, 73/56, [[30/23]], 47/36
| [[13/10]], 56/43, 43/33, 116/89, 73/56, [[30/23]], 47/36, [[64/49]]
|-
|-
| 25
| 25
| 476.19
| 476.19
| [[21/16]], [[46/35]], 96/73, 29/22, [[120/91]], 62/47
| [[21/16]], [[46/35]], 96/73, 29/22, [[120/91]], 62/47, 70/53
|-
|-
| 26
| 26
| 495.24
| 495.24
| 93/70, [[4/3]], ''[[75/56]]''
| 93/70, [[4/3]]
|-
|-
| 27
| 27
| 514.29
| 514.29
| 63/47, 47/35, 43/32, 39/29, [[35/26]], 31/23, 120/89, 89/66, 58/43
| 98/73, 47/35, 43/32, 39/29, [[35/26]], [[66/49]], 31/23, 120/89, 89/66, 58/43
|-
|-
| 28
| 28
| 533.33
| 533.33
| 42/31, 87/64, 64/47, 124/91, [[15/11]], 86/63
| 42/31, 72/53, 53/39, 87/64, [[49/36]], 64/47, 124/91, [[15/11]]
|-
|-
| 29
| 29
| 552.38
| 552.38
| [[63/46]], [[48/35]], [[11/8]], 128/93, 62/45, [[91/66]], 40/29, 29/21
| [[48/35]], [[11/8]], 128/93, 73/53, 62/45, [[91/66]], 40/29, 29/21
|-
|-
| 30
| 30
| 571.43
| 571.43
| [[18/13]], [[104/75]], 43/31, 89/64, [[32/23]], [[39/28]], 124/89, [[46/33]], 60/43, [[88/63]]
| [[18/13]], 43/31, 146/105, 89/64, [[32/23]], [[39/28]], 124/89, [[46/33]], 60/43
|-
|-
| 31
| 31
| 590.48
| 590.48
| [[7/5]], 87/62, 73/52, 66/47, [[45/32]], [[128/91]], 31/22, 89/63
| [[7/5]], 87/62, 73/52, 66/47, [[45/32]], [[128/91]], 31/22
|-
|-
| 32
| 32
| 609.52
| 609.52
| 126/89, 44/31, [[91/64]], [[64/45]], 47/33, 104/73, 124/87, [[10/7]]
| 44/31, [[91/64]], [[64/45]], 47/33, 104/73, 124/87, [[10/7]]
|-
|-
| 33
| 33
| 628.57
| 628.57
| [[63/44]], 43/30, [[33/23]], 89/62, [[56/39]], [[23/16]], 128/89, 62/43, [[75/52]], [[13/9]]
| 43/30, [[33/23]], 89/62, [[56/39]], [[23/16]], 128/89, 105/73, 62/43, [[13/9]]
|-
|-
| 34
| 34
| 647.62
| 647.62
| 42/29, 29/20, [[132/91]], 45/31, 93/64, [[16/11]], [[35/24]], [[92/63]]
| 42/29, 29/20, [[132/91]], 45/31, 106/73, 93/64, [[16/11]], [[35/24]]
|-
|-
| 35
| 35
| 666.67
| 666.67
| 63/43, [[22/15]], 91/62, 47/32, 128/87, 31/21
| [[22/15]], 91/62, 47/32, [[72/49]], 128/87, 78/53, 53/36, 31/21
|-
|-
| 36
| 36
| 685.71
| 685.71
| 43/29, 132/89, 89/60, 46/31, [[52/35]], 58/39, 64/43, 70/47, 94/63
| 43/29, 132/89, 89/60, 46/31, [[49/33]], [[52/35]], 58/39, 64/43, 70/47, 73/49
|-
|-
| 37
| 37
| 704.76
| 704.76
| ''[[112/75]]'', [[3/2]], 140/93
| [[3/2]], 140/93
|-
|-
| 38
| 38
| 723.81
| 723.81
| 47/31, [[91/60]], 44/29, 73/48, [[35/23]], [[32/21]]
| 53/35, 47/31, [[91/60]], 44/29, 73/48, [[35/23]], [[32/21]]
|-
|-
| 39
| 39
| 742.86
| 742.86
| 72/47, [[23/15]], 112/73, 89/58, 66/43, 43/28, [[20/13]]
| [[49/32]], 72/47, [[23/15]], 112/73, 89/58, 66/43, 43/28, [[20/13]]
|-
|-
| 40
| 40
| 761.9
| 761.9
| 116/75, 48/31, 31/20, 45/29, 73/47, 87/56, [[14/9]], ''[[25/16]]''
| 48/31, 31/20, 45/29, 73/47, 87/56, [[14/9]], ''[[25/16]]''
|-
|-
| 41
| 41
Line 201: Line 476:
| 42
| 42
| 800.0
| 800.0
| [[144/91]], 46/29, 73/46, 116/73, 89/56, 62/39, [[35/22]], 75/47
| 49/31, [[144/91]], 84/53, 46/29, 73/46, 116/73, 89/56, 62/39, [[35/22]], [[78/49]]
|-
|-
| 43
| 43
| 819.05
| 819.05
| [[8/5]], 93/58, 146/91, [[45/28]], 140/87, 29/18, 50/31
| [[8/5]], 93/58, 146/91, 53/33, [[45/28]], 140/87, 29/18, 50/31
|-
|-
| 44
| 44
| 838.1
| 838.1
| [[21/13]], 144/89, 47/29, 73/45, [[13/8]], 70/43, [[75/46]]
| [[21/13]], 144/89, 47/29, 73/45, 86/53, [[13/8]], 70/43
|-
|-
| 45
| 45
| 857.14
| 857.14
| [[18/11]], 146/89, [[64/39]], [[23/14]], 120/73, [[150/91]], ''[[104/63]]''
| [[49/30]], [[18/11]], 172/105, 146/89, [[105/64]], [[64/39]], 87/53, [[23/14]], 120/73
|-
|-
| 46
| 46
| 876.19
| 876.19
| 124/75, 43/26, 48/29, 58/35, 73/44, 78/47, 93/56, [[5/3]]
| 43/26, 48/29, 53/32, 58/35, 73/44, 78/47, 88/53, 93/56, [[5/3]]
|-
|-
| 47
| 47
| 895.24
| 895.24
| 87/52, 72/43, 52/31, 146/87, 47/28, 150/89
| 87/52, 72/43, [[176/105]], 52/31, 146/87, 47/28, 89/53
|-
|-
| 48
| 48
| 914.29
| 914.29
| ''[[42/25]]'', [[22/13]], [[39/23]], [[56/33]], 73/43, 124/73, [[75/44]]
| 49/29, [[22/13]], 105/62, 178/105, [[39/23]], [[56/33]], 73/43, 90/53, 124/73
|-
|-
| 49
| 49
| 933.33
| 933.33
| [[128/75]], 89/52, [[12/7]], 31/18, ''50/29''
| 53/31, 89/52, [[12/7]], 91/53, 31/18, ''50/29''
|-
|-
| 50
| 50
| 952.38
| 952.38
| 126/73, [[45/26]], [[26/15]], 73/42, ''75/43''
| [[45/26]], [[26/15]], 92/53, 73/42
|-
|-
| 51
| 51
| 971.43
| 971.43
| [[7/4]], 156/89, 128/73, 58/33
| [[7/4]], [[184/105]], 156/89, 128/73, 93/53, 86/49, 58/33
|-
|-
| 52
| 52
| 990.48
| 990.48
| ''[[44/25]]'', [[23/13]], 62/35, [[39/22]], [[16/9]], ''[[25/14]]''
| ''[[44/25]]'', 53/30, [[23/13]], 62/35, [[39/22]], 94/53, 87/49, [[16/9]], ''[[25/14]]''
|-
|-
| 53
| 53
| 1009.52
| 1009.52
| 84/47, 93/52, 43/24, 52/29, [[70/39]]
| 84/47, 93/52, 188/105, 43/24, 52/29, [[70/39]], [[88/49]]
|-
|-
| 54
| 54
| 1028.57
| 1028.57
| ''[[9/5]]'', 56/31, 47/26, 132/73, 29/16, 78/43, [[20/11]]
| ''[[9/5]]', 56/31, 47/26, 132/73, 105/58, 96/53, 29/16, 78/43, 89/49, [[20/11]]
|-
|-
| 55
| 55
| 1047.62
| 1047.62
| [[42/23]], [[64/35]], 86/47, [[11/6]], 116/63
| [[42/23]], 53/29, [[64/35]], 86/47, [[11/6]], [[90/49]]
|-
|-
| 56
| 56
| 1066.67
| 1066.67
| [[24/13]], 172/93, 87/47, 89/48, [[13/7]]
| [[24/13]], 98/53, 172/93, 87/47, 89/48, [[13/7]]
|-
|-
| 57
| 57
| 1085.71
| 1085.71
| [[28/15]], 43/23, 58/31, 73/39, 88/47, [[15/8]], 62/33
| [[28/15]], 43/23, 58/31, 73/39, 88/47, [[15/8]], [[92/49]], 62/33
|-
|-
| 58
| 58
| 1104.76
| 1104.76
| [[66/35]], 168/89, 172/91, 87/46, 176/93, 89/47, [[91/48]], ''[[40/21]]''
| [[49/26]], [[66/35]], 168/89, 172/91, 87/46, 176/93, 53/28, 89/47, [[91/48]], 93/49, ''[[40/21]]''
|-
|-
| 59
| 59
| 1123.81
| 1123.81
| [[21/11]], 86/45, 174/91, [[44/23]], 178/93, 90/47, [[23/12]], 140/73, [[25/13]]
| [[21/11]], 86/45, 174/91, [[44/23]], 178/93, 90/47, [[23/12]], 140/73, 94/49, [[25/13]]
|-
|-
| 60
| 60
Line 277: Line 552:
| 61
| 61
| 1161.9
| 1161.9
| 146/75, 84/43, 43/22, 174/89, [[88/45]], 178/91, [[45/23]], 182/93, 92/47, 47/24, ''124/63''
| 84/43, 43/22, 174/89, [[88/45]], 178/91, [[45/23]], 182/93, 92/47, 47/24, [[96/49]], 104/53
|-
|-
| 62
| 62
| 1180.95
| 1180.95
| [[63/32]], 144/73, 172/87, 87/44, 176/89, 89/45, [[180/91]], [[91/46]], 184/93, 93/47
| [[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
| 63
Line 287: Line 562:
| [[2/1]]
| [[2/1]]
|}
|}
<nowiki>*</nowiki> as a 2.3.5.7.11.13.23.29.31.43.47.73-subgroup (no-17's no-19's no-37's no-41's 47-limit add-73 add-89) temperament with accurate or low-complexity intervals involving 5 added.
<references group="note" />


== Notation ==
== Notation ==


===Sagittal notation===
=== Ups and downs notation ===
This notation uses the same sagittal sequence as [[56edo#Sagittal notation|56-EDO]].
====Evo flavor====


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 ===
This notation uses the same sagittal sequence as [[56edo #Sagittal notation|56edo]].
==== Evo flavor ====
<imagemap>
<imagemap>
File:63-EDO_Evo_Sagittal.svg
File:63-EDO_Evo_Sagittal.svg
Line 306: Line 589:
</imagemap>
</imagemap>


====Revo flavor====
==== Revo flavor ====
 
<imagemap>
<imagemap>
File:63-EDO_Revo_Sagittal.svg
File:63-EDO_Revo_Sagittal.svg
Line 319: Line 601:
</imagemap>
</imagemap>


== Zeta properties ==
== Approximation to JI ==
===Zeta peak index===
=== Interval mappings ===
{| class="wikitable"
{{Q-odd-limit intervals}}
! colspan="3" |Tuning
 
! colspan="3" |Strength
=== Zeta peak index ===
! colspan="2" |Closest EDO
{{ZPI
! colspan="2" |Integer limit
| 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
|-
| 2.3.5.7
| 225/224, 245/243, 51200/50421
| {{Mapping| 63 100 146 177 }}
| -0.099
| 1.52
| 8.00
|-
| 2.3.5.7.11
| 100/99, 225/224, 245/243, 1331/1323
| {{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]]
|-
|-
!ZPI
| 1
!Steps per octave
| 20\63
!Step size (cents)
| 380.95
!Height
| 5/4
!Integral
| [[Magic]]
!Gap
!EDO
!Octave (cents)
!Consistent
!Distinct
|-
|-
|[[321zpi]]
| 1
|63.0192885705350
| 25\63
|19.0417890652143
| 476.19
|6.768662
| 21/16
|1.049023
| [[Subfourth]]
|15.412920
|-
|63edo
| 3
|1199.63271110850
| 26\63<br>(5\63)
|8
| 495.24<br>(95.24)
|8
| 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 ==
* Approximation of ''[[Pelog]] lima'': 6 9 21 6 21
* Approximation of ''[[Pelog]] lima'': 6 9 21 6 21
* 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"