# 239edo

 ← 238edo 239edo 240edo →
Prime factorization 239 (prime)
Step size 5.02092¢
Fifth 140\239 (702.929¢)
Semitones (A1:m2) 24:17 (120.5¢ : 85.36¢)
Consistency limit 11
Distinct consistency limit 11

239 equal divisions of the octave (abbreviated 239edo or 239ed2), also called 239-tone equal temperament (239tet) or 239 equal temperament (239et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 239 equal parts of about 5.02 ¢ each. Each step represents a frequency ratio of 21/239, or the 239th root of 2.

## Theory

239edo has a sharp tendency, with prime harmonics 3 through 11 all tuned sharp. The equal temperament tempers out 2401/2400, 5120/5103, and 29360128/29296875 in the 7-limit, supporting the hemififths temperament and providing an excellent tuning. It also supports and provides a good tuning for quasiorwell and alphaquarter. In the 11-limit, it tempers out 3025/3024, 4000/3993, 5632/5625, and 12005/11979.

### Prime harmonics

Approximation of prime harmonics in 239edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.97 +0.30 +0.21 +0.98 -2.03 +0.48 -1.28 -0.66 -0.29 -0.27
Relative (%) +0.0 +19.4 +5.9 +4.2 +19.6 -40.5 +9.6 -25.5 -13.1 -5.7 -5.3
Steps
(reduced)
239
(0)
379
(140)
555
(77)
671
(193)
827
(110)
884
(167)
977
(21)
1015
(59)
1081
(125)
1161
(205)
1184
(228)

### Subsets and supersets

239edo is the 52nd prime edo.

## Regular temperament properties

Subgroup Comma List Mapping Optimal 8ve
Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [379 -239 [239 379]] -0.307 0.307 6.12
2.3.5 [3 -18 11, [32 -7 -9 [239 379 555]] -0.247 0.265 5.27
2.3.5.7 2401/2400, 5120/5103, 29360128/29296875 [239 379 555 671]] -0.204 0.241 4.80
2.3.5.7.11 2401/2400, 3025/3024, 4000/3993, 5120/5103 [239 379 555 671 827]] -0.220 0.218 4.34

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 3\239 15.06 121/120 Yarman I (239)
1 11\239 35.15 1990656/1953125 Gammic (5-limit)
1 7\239 55.23 33/32 Escapade / alphaquarter
1 35\239 175.73 72/65 Quadrafifths (239f)
1 54\239 271.13 90/77 Quasiorwell (239)
1 70\239 351.46 49/40 Hemififths (7-limit)
1 79\239 396.65 44/35 Squarschmidt
1 83\239 416.74 14/11 Unthirds (239f)
1 116\239 582.43 7/5 Neptune (7-limit)

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

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