250edo: Difference between revisions
"7\10" is clearer than "0.7 octaves". Harmonics -> subgroup. Add the missing 2 in 2.11.13. Resolve edo vs et |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | |||
250edo is [[enfactoring|enfactored]] in the 7-limit, with the same tuning as 125edo, but provides a closer approximation to the harmonics 11 and 13, where the 13/8 derives from [[10edo]] (7\10). Even so, there are a number of mappings to be considered, in particular, a less flat-tending [[patent val]] {{val| 250 396 580 '''702''' '''865''' '''925''' … }} and a more flat-tending 250deff… val {{val| 250 396 580 '''701''' '''864''' '''924''' … }}. | 250edo is [[enfactoring|enfactored]] in the 7-limit, with the same tuning as 125edo, but provides a closer approximation to the harmonics 11 and 13, where the 13/8 derives from [[10edo]] (7\10). Even so, there are a number of mappings to be considered, in particular, a less flat-tending [[patent val]] {{val| 250 396 580 '''702''' '''865''' '''925''' … }} and a more flat-tending 250deff… val {{val| 250 396 580 '''701''' '''864''' '''924''' … }}. | ||
The patent val tempers out [[243/242]], [[3025/3024]], 4375/4356, [[9801/9800]], 14700/14641 in the 11-limit and [[1716/1715]], [[2080/2079]], and [[2200/2197]] in the 13-limit. It [[support]]s the [[Minortonic family #Seminar|seminar]] temperament. | |||
The 250deff… val tempers out [[441/440]], 4125/4096, [[8019/8000]], 9801/9800, 12005/11979, [[14641/14580]] in the 11-limit and [[325/324]], [[676/675]], and 1287/1280 in the 13-limit. | |||
=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|250}} | {{Harmonics in equal|250}} | ||
=== | === Subsets and supersets === | ||
250edo has subset edos {{EDOs| 2, 5, 10, 25, 50, 125 }}. | 250edo has subset edos {{EDOs| 2, 5, 10, 25, 50, 125 }}. | ||
Since the 2.3.5.7 subgroup in the patent val comes from 125et, and the 2.11.13 subgroup in the patent val comes from 50et, this system is worthy of being considered as a superset of these two temperaments. | Since the 2.3.5.7 subgroup in the patent val comes from 125et, and the 2.11.13 subgroup in the patent val comes from 50et, this system is worthy of being considered as a superset of these two temperaments. | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
|- | |||
! 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]] (%) | |||
|- | |||
| 2.3.5.7.11 | |||
| 441/440, 4125/4096, 14641/14580, 15625/15552 | |||
| {{mapping| 250 396 580 701 864 }} (250de) | |||
| +0.8703 | |||
| 0.4930 | |||
| 10.3 | |||
|- | |||
| 2.3.5.7.11 | |||
| 225/224, 243/242, 4375/4356, 589824/588245 | |||
| {{mapping| 250 396 580 701 864 }} (250) | |||
| +0.2503 | |||
| 0.5149 | |||
| 10.7 | |||
|} | |||