63edo: Difference between revisions

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== 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. It is almost [[consistent]] in the [[15-odd-limit]]; the only inconsistency is that [[10/9]] is mapped to 1\7 = 9\63 (the same as what [[11/10]] is mapped to consistently) so that it's almost 11{{cent}} out of tune. This corresponds to 63edo exaggerating 81/80 to 2\63, so that it finds a somewhat flat mean-tone between ~10/9 and ~9/8.

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.

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 == Theory ==
-edo [[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. It is almost [[consistent]] in the [[15-odd-limit]]; the only inconsistency is that [[10/9]] is mapped to 1\7 = 9\63 (the same as what [[11/10]] is mapped to consistently) so that it's almost 11{{cent}} out of tune. This corresponds to 63edo exaggerating 81/80 to 2\63, so that it finds a somewhat flat mean-tone between ~10/9 and ~9/8.
+edo 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.
 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.