63edo: Difference between revisions
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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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== 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 589: | Line 600: | ||
default [[File:63-EDO_Revo_Sagittal.svg]] | default [[File:63-EDO_Revo_Sagittal.svg]] | ||
</imagemap> | </imagemap> | ||
== Approximation to JI == | == Approximation to JI == | ||
=== Interval mappings === | === Interval mappings === | ||
{{Q-odd-limit intervals}} | {{Q-odd-limit intervals}} | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 664: | Line 640: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 100/99, 225/224, 245/243, 1331/1323 | | 100/99, 225/224, 245/243, 1331/1323 | ||
| {{ | | {{Mapping| 63 100 146 177 218 }} | ||
| -0.141 | | -0.141 | ||
| 1.37 | | 1.37 | ||
| Line 671: | Line 647: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 100/99, 169/168, 225/224, 245/243, 275/273 | | 100/99, 169/168, 225/224, 245/243, 275/273 | ||
| {{ | | {{Mapping| 63 100 146 177 218 233 }} | ||
| -0.008 | | -0.008 | ||
| 1.28 | | 1.28 | ||
| Line 689: | Line 665: | ||
| 1 | | 1 | ||
| 2\63 | | 2\63 | ||
| 38. | | 38.1 | ||
| 49/48 | | 49/48 | ||
| [[Slender]] | | [[Slender]] | ||
| Line 695: | Line 671: | ||
| 1 | | 1 | ||
| 13\63 | | 13\63 | ||
| 247. | | 247.6 | ||
| 15/13 | | 15/13 | ||
| [[Immune]] | | [[Immune]] | ||
| Line 701: | Line 677: | ||
| 1 | | 1 | ||
| 19\63 | | 19\63 | ||
| 361. | | 361.9 | ||
| 16/13 | | 16/13 | ||
| [[ | | [[Demibuzzard]] / submajor | ||
|- | |- | ||
| 1 | | 1 | ||
| 20\63 | | 20\63 | ||
| | | 381.0 | ||
| 5/4 | | 5/4 | ||
| [[Magic]] | | [[Magic]] | ||
| Line 713: | Line 689: | ||
| 1 | | 1 | ||
| 25\63 | | 25\63 | ||
| 476. | | 476.2 | ||
| 21/16 | | 21/16 | ||
| [[Subfourth]] | | [[Subfourth]] / [[lemongrass]] | ||
|- | |||
| 1 | |||
| 26\63 | |||
| 495.2 | |||
| 4/3 | |||
| [[Leapweek]] / [[leapmonth]] | |||
|- | |- | ||
| 3 | | 3 | ||
| 26\63<br>(5\63) | | 26\63<br>(5\63) | ||
| 495. | | 495.2<br>(95.2) | ||
| 4/3<br>(21/20) | | 4/3<br>(21/20) | ||
| [[Fog]] | | [[Fog]] | ||
| Line 725: | Line 707: | ||
| 7 | | 7 | ||
| 26\63<br>(1\63) | | 26\63<br>(1\63) | ||
| 495. | | 495.2<br>(19.0) | ||
| 4/3<br>( | | 4/3<br>(64/63) | ||
| [[Sevond]] | | [[Sevond]] | ||
|- | |- | ||
| 9 | | 9 | ||
| 13\63<br>(1\63) | | 13\63<br>(1\63) | ||
| 247. | | 247.6<br>(19.0) | ||
| 15/13<br>(99/98) | | 15/13<br>(99/98) | ||
| [[Enneaportent]] | | [[Enneaportent]] | ||
|} | |} | ||
<nowiki/>* [[Normal | <nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|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 | ||
* Sandcastle (original/default tuning): 8 10 8 11 8 8 10 (is quasi-[[equiheptatonic]]) | |||
* 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) | |||
* [https://www.youtube.com/shorts/EEAJ0AaWoJM ''63edo improv''] (2025) | |||
* [https://www.youtube.com/shorts/l-M2FZn2nC0 ''Venus as a Boy - Björk (microtonal cover in 63edo)''] (2025) | |||
* [https://www.youtube.com/shorts/qSFURozTf5A ''Hartmann's Youkai Girl - ZUN (microtonal cover in 63edo)''] (2026) | |||
; [[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) | ||