63edo: Difference between revisions
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{{EDO intro|63}} | {{EDO intro|63}} | ||
== 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. | |||
63 is also a fascinating division to look at in the 31-limit, as its regular augmented fourth (+6 fifths) is less than 0.3c sharp of 23/16, therefore tempering out 736/729, and its "large quarter-tone", or diesis, is only 2.2c off of [[32/31]] which is equated with [[31/30]] (even more accurate) and [[30/29]] on the other side, hence tempering [[961/960|S31]] and [[900/899|S30]], but also completing a streak of large quartertones/small dieses of [[superparticular interval]]s in the harmonic series by continuing to equate them on the large side with [[29/28]] and [[28/27]] (tempering [[841/840|S29]] and [[784/783|S28]]) and on the small side with [[33/32]] (tempering [[961/960|S31]]). Although it doesn't deal as well with primes 5, 17, and 19, it excels in the 2.3.7.11.13.23.29.31 group, and is a great candidate for a rank-1 or rank-2 gentle tuning. As a fifths-system, the diesis after 12 fifths can represent 32:33, 27:28, any superparticular interval imbetween those, 88:91, and more, so it is very versatile, making chains of fifths 12 or longer very useful in covering harmonic and melodic ground while providing a lot of different colour in different keys. 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. Alternatively, using the quarter-tone interval 3\63 = 1\21, 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. 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 wasn't 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 table below. | 63 is also a fascinating division to look at in the 31-limit, as its regular augmented fourth (+6 fifths) is less than 0.3c sharp of 23/16, therefore tempering out 736/729, and its "large quarter-tone", or diesis, is only 2.2c off of [[32/31]] which is equated with [[31/30]] (even more accurate) and [[30/29]] on the other side, hence tempering [[961/960|S31]] and [[900/899|S30]], but also completing a streak of large quartertones/small dieses of [[superparticular interval]]s in the harmonic series by continuing to equate them on the large side with [[29/28]] and [[28/27]] (tempering [[841/840|S29]] and [[784/783|S28]]) and on the small side with [[33/32]] (tempering [[961/960|S31]]). Although it doesn't deal as well with primes 5, 17, and 19, it excels in the 2.3.7.11.13.23.29.31 group, and is a great candidate for a rank-1 or rank-2 gentle tuning. As a fifths-system, the diesis after 12 fifths can represent 32:33, 27:28, any superparticular interval imbetween those, 88:91, and more, so it is very versatile, making chains of fifths 12 or longer very useful in covering harmonic and melodic ground while providing a lot of different colour in different keys. 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. Alternatively, using the quarter-tone interval 3\63 = 1\21, 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. 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 wasn't 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 table below. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|63|columns=16}} | {{Harmonics in equal|63|columns=16}} | ||
== | === Subsets and supersets === | ||
It is divisible by 3, 7, 9 and 21. | |||
== Intervals == | |||
{{Interval table}} | {{Interval table}} | ||
== Scales == | |||
* [[5- to 10-tone scales in 63edo]] | |||
== Music == | == Music == | ||
; [[Cam Taylor]] | |||
* [https://soundcloud.com/camtaylor-1/63edobosanquetaxis-8thjuly2016-237111323-seconds-and-otonal-shifts Seconds and Otonal Shifts] | * [https://soundcloud.com/camtaylor-1/63edobosanquetaxis-8thjuly2016-237111323-seconds-and-otonal-shifts Seconds and Otonal Shifts] | ||
* [https://soundcloud.com/cam-taylor-2-1/17-out-of-63edo-wurly-those-early-dreams those early dreams] | * [https://soundcloud.com/cam-taylor-2-1/17-out-of-63edo-wurly-those-early-dreams those early dreams] | ||
* [https://archive.org/details/17_63EDOEarlyDreamsTwo Early Dreams 2] | * [https://archive.org/details/17_63EDOEarlyDreamsTwo Early Dreams 2] | ||
* [https://soundcloud.com/cam-taylor-2-1/12tone63edo1 Improvisation in 12-tone fifths chain in 63EDO] | * [https://soundcloud.com/cam-taylor-2-1/12tone63edo1 Improvisation in 12-tone fifths chain in 63EDO] | ||