202edo: Difference between revisions

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== Theory ==

202edo is [[consistent]] to the [[9-odd-limit]] with a flat tendency in harmonics 3, 5, and 7. It also has a decent harmonic [[11/1|11]], though it is sharp unlike the previous harmonics, with [[11/9]] barely exceeding 50% relative error. Despite this, it is most notable in the 11-limit, providing the optimal patent val for many temperaments tempering out [[243/242]].

202edo is [[consistent]] to the [[9-odd-limit]] with a flat tendency in harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]]. It also has a decent harmonic [[11/1|11]], though it is sharp unlike the previous harmonics, with [[11/9]] barely exceeding 50% [[relative interval error|relative error]]. Despite this, it is most notable in the [[11-limit]], providing the [[optimal patent val]] for many temperaments tempering out [[243/242]].

Using the patent val, 202et [[tempering out|tempers out]] [[2401/2400]], [[19683/19600]] and [[65625/65536]] in the [[7-limit]], and [[243/242]], [[441/440]], [[4000/3993]] in the [[11-limit]]. It also notably tempers out the [[quartisma]], equating a stack of five [[33/32]] quartertones with [[7/6]]. It is the [[optimal patent val]] for the 11-limit rank-2 temperaments [[harry]] and [[tertiaseptal]], the rank-3 temperament [[jove]] tempering out 243/242 and 441/440, which also tempers out [[540/539]], and the rank-4 [[rastmic]] temperament, which tempers out 243/242.

Using the patent val, 202et [[tempering out|tempers out]] [[2401/2400]], [[19683/19600]] and [[65625/65536]] in the [[7-limit]], and [[243/242]], [[441/440]], [[4000/3993]] in the 11-limit. It also notably tempers out the [[quartisma]], equating a stack of five [[33/32]] quartertones with [[7/6]]. It is the [[optimal patent val]] for the 11-limit rank-2 temperaments [[harry]] and [[tertiaseptal]], the rank-3 temperament [[jove]] tempering out 243/242 and 441/440, which also tempers out [[540/539]], and the rank-4 [[rastmic]] temperament, which tempers out 243/242.

It extends less well to the [[13-limit]], with harmonic [[13/1|13]] being about halfway between its steps. Nonetheless, the patent val tempers out [[351/350]], [[364/363]], [[676/675]], [[729/728]], and [[2080/2079]], supporting [[breed family #Jovial|jovial]] and [[breed family #Jovis|jovis]], as well as 13-limit harry. Primes [[17/1|17]] and [[23/1|23]] are quite sharp, but prime [[19/1|19]] is accurate. 202edo can thus be considered a 2.3.5.7.11.13.19-subgroup temperament with a mostly flat tendency, with the exception of prime 11. The intervals [[11/9]], [[13/11]], and their octave complements are the only inconsistencies in the no-17 [[21-odd-limit]], and the no-11 no-17 21-odd limit is completely consistent, though one may also want to exclude prime 13 given its inaccuracy, giving us the 2.3.5.7.19 subgroup.

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 == Theory ==
-edo is [[consistent]] to the [[9-odd-limit]] with a flat tendency in harmonics 3, 5, and 7. It also has a decent harmonic [[11/1|11]], though it is sharp unlike the previous harmonics, with [[11/9]] barely exceeding 50% relative error. Despite this, it is most notable in the 11-limit, providing the optimal patent val for many temperaments tempering out [[243/242]].
+edo is [[consistent]] to the [[9-odd-limit]] with a flat tendency in harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]]. It also has a decent harmonic [[11/1|11]], though it is sharp unlike the previous harmonics, with [[11/9]] barely exceeding 50% [[relative interval error|relative error]]. Despite this, it is most notable in the [[11-limit]], providing the [[optimal patent val]] for many temperaments tempering out [[243/242]].
-Using the patent val, 202et [[tempering out|tempers out]] [[2401/2400]], [[19683/19600]] and [[65625/65536]] in the [[7-limit]], and [[243/242]], [[441/440]], [[4000/3993]] in the [[11-limit]]. It also notably tempers out the [[quartisma]], equating a stack of five [[33/32]] quartertones with [[7/6]]. It is the [[optimal patent val]] for the 11-limit rank-2 temperaments [[harry]] and [[tertiaseptal]], the rank-3 temperament [[jove]] tempering out 243/242 and 441/440, which also tempers out [[540/539]], and the rank-4 [[rastmic]] temperament, which tempers out 243/242.
+Using the patent val, 202et [[tempering out|tempers out]] [[2401/2400]], [[19683/19600]] and [[65625/65536]] in the [[7-limit]], and [[243/242]], [[441/440]], [[4000/3993]] in the 11-limit. It also notably tempers out the [[quartisma]], equating a stack of five [[33/32]] quartertones with [[7/6]]. It is the [[optimal patent val]] for the 11-limit rank-2 temperaments [[harry]] and [[tertiaseptal]], the rank-3 temperament [[jove]] tempering out 243/242 and 441/440, which also tempers out [[540/539]], and the rank-4 [[rastmic]] temperament, which tempers out 243/242.
 It extends less well to the [[13-limit]], with harmonic [[13/1|13]] being about halfway between its steps. Nonetheless, the patent val tempers out [[351/350]], [[364/363]], [[676/675]], [[729/728]], and [[2080/2079]], supporting [[breed family #Jovial|jovial]] and [[breed family #Jovis|jovis]], as well as 13-limit harry. Primes [[17/1|17]] and [[23/1|23]] are quite sharp, but prime [[19/1|19]] is accurate. 202edo can thus be considered a 2.3.5.7.11.13.19-subgroup temperament with a mostly flat tendency, with the exception of prime 11. The intervals [[11/9]], [[13/11]], and their octave complements are the only inconsistencies in the no-17 [[21-odd-limit]], and the no-11 no-17 21-odd limit is completely consistent, though one may also want to exclude prime 13 given its inaccuracy, giving us the 2.3.5.7.19 subgroup.