# 263edo

 ← 262edo 263edo 264edo →
Prime factorization 263 (prime)
Step size 4.56274¢
Fifth 154\263 (702.662¢)
Semitones (A1:m2) 26:19 (118.6¢ : 86.69¢)
Consistency limit 5
Distinct consistency limit 5

263 equal divisions of the octave (abbreviated 263edo or 263ed2), also called 263-tone equal temperament (263tet) or 263 equal temperament (263et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 263 equal parts of about 4.56 ¢ each. Each step represents a frequency ratio of 21/263, or the 263rd root of 2.

## Theory

263et tempers out 393216/390625 (würschmidt comma) and [50 -33 1 in the 5-limit. Using the patent val, it tempers out 4375/4374, 50421/50000, and 458752/455625 in the 7-limit; 441/440, 3388/3375, 16384/16335, and 26411/26244 in the 11-limit; 364/363, 2080/2079, 2197/2187, and 3584/3575 in the 13-limit; 595/594, 833/832, 936/935, and 1156/1155 in the 17-limit.

Using the 263d val, it tempers out 5120/5103, 16875/16807, and 1959552/1953125 in the 7-limit; 540/539, 1375/1372, 16384/16335, and 43923/43750 in the 11-limit; 351/350, 1001/1000, 1573/1568, 2197/2187, and 4225/4224 in the 13-limit.

Using the 263df val, it tempers out 352/351, 640/637, 729/728, and 3584/3575 in the 13-limit.

Finally, it is accurate for the 17th harmonic, as the denominator of a convergent to log217, after 80 and before 343.

### Prime harmonics

Approximation of prime harmonics in 263edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.71 +1.52 -1.53 +0.77 -0.98 -0.01 -0.94 +1.38 +1.60 +0.21
Relative (%) +0.0 +15.5 +33.3 -33.4 +16.9 -21.6 -0.3 -20.5 +30.3 +35.1 +4.6
Steps
(reduced)
263
(0)
417
(154)
611
(85)
738
(212)
910
(121)
973
(184)
1075
(23)
1117
(65)
1190
(138)
1278
(226)
1303
(251)

### Subsets and supersets

263edo is the 56th prime edo.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [417 -263 263 417] -0.2229 0.2229 4.89
2.3.5 393216/390625, [50 -33 1 263 417 611] -0.3666 0.2728 5.98

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 40\263 182.51 10/9 Minortone
1 85\263 387.83 5/4 Würschmidt

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct