201edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 200edo201edo202edo →
Prime factorization 3 × 67
Step size 5.97015¢ 
Fifth 118\201 (704.478¢)
Semitones (A1:m2) 22:13 (131.3¢ : 77.61¢)
Dual sharp fifth 118\201 (704.478¢)
Dual flat fifth 117\201 (698.507¢) (→39\67)
Dual major 2nd 34\201 (202.985¢)
Consistency limit 5
Distinct consistency limit 5

210 equal divisions of the octave (abbreviated 210edo or 210ed2), also called 210-tone equal temperament (210tet) or 210 equal temperament (210et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 210 equal parts of about 5.71 ¢ each. Each step represents a frequency ratio of 21/210, or the 210th root of 2.

201edo is only consistent to the 5-odd-limit, and harmonic 3 is about halfway between its steps.

Using the patent val, it tempers out 393216/390625 (würschmidt comma) and [25 -26 7 in the 5-limit; 245/243, 50421/50000, and 2100875/2097152 in the 7-limit; 385/384, 896/891, 1331/1323, and 47432/46875 in the 11-limit; 196/195, 325/324, 2080/2079, 2200/2197, and 3146/3125 in the 13-limit.

Using the 201e val, it tempers out 441/440, 2200/2187, 3388/3375, and 65536/65219 in the 11-limit; 196/195, 325/324, 352/351, 1001/1000, and 106496/105875 in the 13-limit.

Using the 201de val, it tempers out 4000/3969, 10976/10935, and 4194304/4134375 in the 7-limit; 540/539, 896/891, 1375/1372, and 234375/234256 in the 11-limit; 325/324, 352/351, 364/363, 640/637, and 4394/4375 in the 13-limit (supporting the pluto temperament).

Using the 201b val, it tempers out 1990656/1953125 (valentine comma) and [-31 24 -3 in the 5-limit; 126/125, 1029/1024, and [-2 19 0 -10 in the 7-limit; 540/539, 1944/1925, 2835/2816, and 483153/480200 in the 11-limit; 1287/1280, 1575/1573, 1716/1715, 2200/2197, and 3146/3125 in the 13-limit.

Using the 201bcf val, it tempers out 15625/15552 (kleisma) and [-56 31 3 in the 5-limit; 1029/1024, 250047/250000, and 273375/268912 in the 7-limit; 385/384, 441/440, 4000/3993, and 295245/290521 in the 11-limit; 351/350, 975/968, 1287/1280, 1573/1568, and 10935/10816 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 201edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +2.52 +1.75 -1.66 -0.92 -2.06 +1.26 -1.70 +2.51 +0.99 +0.86 -1.41
Relative (%) +42.3 +29.2 -27.8 -15.5 -34.6 +21.2 -28.5 +42.0 +16.7 +14.4 -23.6
Steps
(reduced)
319
(118)
467
(65)
564
(162)
637
(34)
695
(92)
744
(141)
785
(182)
822
(18)
854
(50)
883
(79)
909
(105)

Subsets and supersets

Since 201 factors into 3 × 67, 201edo contains 3edo and 67edo as its subsets. 402edo, which doubles it, provides a good correction to the approximation of harmonic 3.