Xenharmonic Wiki

63edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
Godtone (talk | contribs)
autopatrolled
2,289 edits
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
Line 3: Line 3:


== 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 ====
Retrieved from "https://en.xen.wiki/w/63edo"