63edo: Difference between revisions

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It tempers out [[3125/3072]] in the 5-limit and [[875/864]], [[225/224]] and [[245/243]] 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 [[896/891]], [[385/384]] and [[540/539]]. In the 13-limit it tempers out 275/273, 169/168, 640/637, [[352/351]], [[364/363]] and [[676/675]]. It provides the optimal patent val for the 29&63 temperament in the 7-, 11- and 13-limit. It is divisible by 3, 7, 9 and 21.

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 [[43/16]], [[47/16]] and [[53/16]]; 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.

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 It tempers out [[3125/3072]] in the 5-limit and [[875/864]], [[225/224]] and [[245/243]] 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 [[896/891]], [[385/384]] and [[540/539]]. In the 13-limit it tempers out 275/273, 169/168, 640/637, [[352/351]], [[364/363]] and [[676/675]]. It provides the optimal patent val for the 29&amp;63 temperament in the 7-, 11- and 13-limit. It is divisible by 3, 7, 9 and 21.
-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 [[43/16]], [[47/16]] and [[53/16]]; see the table below.
+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.
 {{Harmonics in equal|63|columns=16}}