202edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | |||
==Theory== | == Theory == | ||
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]]. | |||
==Regular temperament properties== | 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. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|202}} | |||
=== Subsets and supersets === | |||
Since 202 factors into {{nowrap| 2 × 101 }}, 202edo contains [[2edo]] and [[101edo]] as subset edos. | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2 | ! rowspan="2" | [[Subgroup]] | ||
| | ! rowspan="2" | [[Comma list]] | ||
| | ! rowspan="2" | [[Mapping]] | ||
| | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
| | ! colspan="2" | Tuning error | ||
| | |- | ||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{ | | {{Monzo| -13 17 -6 }}, {{monzo| 23 6 -14 }} | ||
|{{ | | {{Mapping| 202 320 469 }} | ||
| 0.2280 | | +0.2280 | ||
| 0.2710 | | 0.2710 | ||
| 4.56 | | 4.56 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|2401/2400, 19683/19600, 65625/65536 | | 2401/2400, 19683/19600, 65625/65536 | ||
|{{ | | {{Mapping| 202 320 469 567 }} | ||
| 0.2164 | | +0.2164 | ||
| 0.2356 | | 0.2356 | ||
| 3.97 | | 3.97 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|243/242, 441/440, | | 243/242, 441/440, 4000/3993, 65625/65536 | ||
|{{ | | {{Mapping| 202 320 469 567 699 }} | ||
| 0.1061 | | +0.1061 | ||
| 0.3049 | | 0.3049 | ||
| 5.13 | | 5.13 | ||
| Line 48: | Line 51: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>ratio | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|13\202 | | 13\202 | ||
|77.23 | | 77.23 | ||
|256/245 | | 256/245 | ||
|[[Tertiaseptal]] | | [[Tertiaseptal]] | ||
|- | |- | ||
|1 | | 1 | ||
|51\202 | | 51\202 | ||
|302.97 | | 302.97 | ||
|25/21 | | 25/21 | ||
|[[Quinmite]] | | [[Quinmite]] | ||
|- | |- | ||
|1 | | 1 | ||
|85\202 | | 85\202 | ||
|504.95 | | 504.95 | ||
|104976/78125 | | 104976/78125 | ||
|[[Countermeantone]] | | [[Countermeantone]] | ||
|- | |- | ||
|1 | | 1 | ||
|87\202 | | 87\202 | ||
|516.83 | | 516.83 | ||
|27/20 | | 27/20 | ||
|[[ | | [[Larry]] | ||
|- | |- | ||
|2 | | 2 | ||
|12\202 | | 12\202 | ||
|71.29 | | 71.29 | ||
|25/24 | | 25/24 | ||
|[[ | | [[Narayana]] | ||
|- | |- | ||
|2 | | 2 | ||
|87\202<br>(14\202) | | 87\202<br>(14\202) | ||
|516.83<br>(83.17) | | 516.83<br>(83.17) | ||
|27/20<br>(21/20) | | 27/20<br>(21/20) | ||
|[[Harry]] | | [[Harry]] | ||
|} | |} | ||
<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 == | ||
* [[Jove1]] | * [[Jove1]], [[jove2]], [[jove3]], [[jove4]], [[jove5]], [[jove6]] | ||
* [[ | * [[Elfjove7]], [[elfjove8d]], [[elfjove10]], [[elfjove11c]], [[elfjove12]] | ||
* [[Oktone]] | |||
* [[ | |||
== Music == | == Music == | ||
[https://www.youtube.com/watch?v=_bNbb2o5K80 Home Planet Nostalgia] | ; [[Mundoworld]] | ||
* [https://www.youtube.com/watch?v=_bNbb2o5K80 ''Home Planet Nostalgia''] – in Oktone scale | |||
[[Category:Harry]] | [[Category:Harry]] | ||
[[Category:Tertiaseptal]] | [[Category:Tertiaseptal]] | ||
[[Category:Jove]] | [[Category:Jove]] | ||
[[Category:Rastmic]] | [[Category:Rastmic]] | ||
[[Category: | [[Category:Listen]] | ||