202edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
202et 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]]. 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, and the rank-4 rastmic temperament | 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. | |||
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 === | === Prime harmonics === | ||
{{Harmonics in equal|202}} | {{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 == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[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 | ||
| Line 35: | Line 36: | ||
| 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 50: | 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> | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 79: | Line 81: | ||
| 516.83 | | 516.83 | ||
| 27/20 | | 27/20 | ||
| [[ | | [[Larry]] | ||
|- | |- | ||
| 2 | | 2 | ||
| Line 85: | Line 87: | ||
| 71.29 | | 71.29 | ||
| 25/24 | | 25/24 | ||
| [[ | | [[Narayana]] | ||
|- | |- | ||
| 2 | | 2 | ||
| Line 93: | Line 95: | ||
| [[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 == | ||
; [[Mundoworld]] | ; [[Mundoworld]] | ||
* [https://www.youtube.com/watch?v=_bNbb2o5K80 ''Home Planet Nostalgia''] | * [https://www.youtube.com/watch?v=_bNbb2o5K80 ''Home Planet Nostalgia''] – in Oktone scale | ||
[[Category:Harry]] | [[Category:Harry]] | ||
| Line 110: | Line 110: | ||
[[Category:Jove]] | [[Category:Jove]] | ||
[[Category:Rastmic]] | [[Category:Rastmic]] | ||
[[Category: | [[Category:Listen]] | ||