250edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|250}} | {{EDO intro|250}} | ||
==Theory== | |||
== 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 | 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. | 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. | ||
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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== | == 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|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3.5.7.11 | ||
| | | 441/440, 4125/4096, 14641/14580, 15625/15552 | ||
|{{ | | {{mapping| 250 396 580 701 864 }} (250de) | ||
| 0. | | +0.8703 | ||
| 0. | | 0.4930 | ||
| | | 10.3 | ||
|- | |- | ||
|2.3.5 | | 2.3.5.7.11 | ||
| | | 225/224, 243/242, 4375/4356, 589824/588245 | ||
|{{ | | {{mapping| 250 396 580 701 864 }} (250) | ||
| 0. | | +0.2503 | ||
| 0. | | 0.5149 | ||
| | | 10.7 | ||
|} | |} | ||
Revision as of 15:08, 22 March 2024
| ← 249edo | 250edo | 251edo → |
Theory
250edo is 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 ⟨250 396 580 702 865 925 …] and a more flat-tending 250deff… 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 supports the 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
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.16 | -2.31 | +0.77 | -2.31 | +0.68 | -0.53 | +1.33 | +0.64 | +0.09 | -0.38 | +0.53 |
| Relative (%) | -24.1 | -48.2 | +16.1 | -48.1 | +14.2 | -11.0 | +27.7 | +13.4 | +1.8 | -7.9 | +11.0 | |
| Steps (reduced) |
396 (146) |
580 (80) |
702 (202) |
792 (42) |
865 (115) |
925 (175) |
977 (227) |
1022 (22) |
1062 (62) |
1098 (98) |
1131 (131) | |
Subsets and supersets
250edo has subset 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.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7.11 | 441/440, 4125/4096, 14641/14580, 15625/15552 | [⟨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 | [⟨250 396 580 701 864]] (250) | +0.2503 | 0.5149 | 10.7 |