# 248edo

 ← 247edo 248edo 249edo →
Prime factorization 23 × 31
Step size 4.83871¢
Fifth 145\248 (701.613¢)
Semitones (A1:m2) 23:19 (111.3¢ : 91.94¢)
Consistency limit 11
Distinct consistency limit 11

248 equal divisions of the octave (abbreviated 248edo or 248ed2), also called 248-tone equal temperament (248tet) or 248 equal temperament (248et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 248 equal parts of about 4.84 ¢ each. Each step represents a frequency ratio of 21/248, or the 248th root of 2.

## Theory

248 = 8 × 31, and 248edo shares the mapping of harmonics 5 and 7 with 31edo. It has a decent 13-limit interpretation despite not being consistent. The equal temperament tempers out 32805/32768 in the 5-limit; 3136/3125 and 420175/419904 in the 7-limit; 441/440, 8019/8000 in the 11-limit; 729/728, 847/845, 1001/1000, 1575/1573 and 2200/2197 in the 13-limit. It also notably tempers out the quartisma.

It supports the bischismic temperament, providing the optimal patent val for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for the essence temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.

### Prime harmonics

Approximation of prime harmonics in 248edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.34 +0.78 -1.08 +0.29 +1.41 +1.50 -2.35 +0.76 +1.07 +1.74
Relative (%) +0.0 -7.1 +16.2 -22.4 +6.1 +29.1 +30.9 -48.6 +15.7 +22.1 +35.9
Steps
(reduced)
248
(0)
393
(145)
576
(80)
696
(200)
858
(114)
918
(174)
1014
(22)
1053
(61)
1122
(130)
1205
(213)
1229
(237)

### Subsets and supersets

Since 248 factors into 23 × 31, 248edo has subset edos 2, 4, 8, 31, 62, and 124.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [287 -181 [248 393]] +0.108 0.108 2.23
2.3.5 32805/32768, [12 32 -27 [248 393 576]] -0.041 0.228 4.70
2.3.5.7 3136/3125, 32805/32768, 420175/419904 [248 393 576 696]] +0.066 0.270 5.58
2.3.5.7.11 441/440, 3136/3125, 8019/8000, 41503/41472 [248 393 576 696 858]] +0.036 0.249 5.15
2.3.5.7.11.13 441/440, 729/728, 847/845, 1001/1000, 3136/3125 [248 393 576 696 858 918]] +0.079 0.275 5.69

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 5\248 24.19 686/675 Sengagen
1 103\248 498.39 4/3 Helmholtz
2 77\248
(47\248)
372.58
(227.42)
26/21
(154/135)
Essence
2 103\248 498.39 4/3 Bischismic
8 117\248
(7\248)
566.13
(33.87)
104/75
(49/48)
Octowerck
31 103\248
(1\248)
498.39
(4.84)
4/3
(385/384)
Birds

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