250edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 249edo250edo251edo →
Prime factorization 2 × 53
Step size 4.8¢ 
Fifth 146\250 (700.8¢) (→73\125)
Semitones (A1:m2) 22:20 (105.6¢ : 96¢)
Consistency limit 5
Distinct consistency limit 5

250 equal divisions of the octave (abbreviated 250edo or 250ed2), also called 250-tone equal temperament (250tet) or 250 equal temperament (250et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 250 equal parts of exactly 4.8 ¢ each. Each step represents a frequency ratio of 21/250, or the 250th root of 2.

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

Approximation of odd harmonics in 250edo
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 -48 +16 -48 +14 -11 +28 +13 +2 -8 +11
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