# 299edo

 ← 298edo 299edo 300edo →
Prime factorization 13 × 23
Step size 4.01338¢
Fifth 175\299 (702.341¢)
Semitones (A1:m2) 29:22 (116.4¢ : 88.29¢)
Consistency limit 7
Distinct consistency limit 7

299 equal divisions of the octave (abbreviated 299edo or 299ed2), also called 299-tone equal temperament (299tet) or 299 equal temperament (299et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 299 equal parts of about 4.01 ¢ each. Each step represents a frequency ratio of 21/299, or the 299th root of 2.

## Theory

In the 5-limit, 299et tempers out the kleisma, 15625/15552, in the 7-limit 10976/10935, in the 11-limit 385/384; and in the 13-limit 325/324, 625/624 and 676/675. It provides the optimal patent val for the 13-limit rank-3 enlil temperament, and the rank-4 temperament tempering out 325/324 and 385/384.

### Prime harmonics

Approximation of prime harmonics in 299edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.39 -1.03 -1.60 -1.49 -1.73 -0.61 -0.52 +1.83 +1.86 -1.22
Relative (%) +0.0 +9.6 -25.7 -39.9 -37.0 -43.1 -15.1 -13.0 +45.5 +46.4 -30.5
Steps
(reduced)
299
(0)
474
(175)
694
(96)
839
(241)
1034
(137)
1106
(209)
1222
(26)
1270
(74)
1353
(157)
1453
(257)
1481
(285)

### Subsets and supersets

Since 299 factors into 13 × 23, 299edo contains 13edo and 23edo as subsets.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [474 -299 [299 474]] -0.1218 0.1218 3.04
2.3.5 15625/15552, [80 -49 -1 [299 474 694]] +0.0665 0.2844 7.09
2.3.5.7 10976/10935, 15625/15552, 823543/819200 [299 474 694 839]] +0.1925 0.3291 8.20
2.3.5.7.11 385/384, 6250/6237, 10976/10935, 12005/11979 [299 474 694 839 1034]] +0.2399 0.3092 7.70
2.3.5.7.11.13 325/324, 385/384, 625/624, 10648/10647, 10976/10935 [299 474 694 839 1034 1106]] +0.2779 0.2948 7.34
2.3.5.7.11.13.17 325/324, 385/384, 595/594, 625/624, 2058/2057, 8624/8619 [299 474 694 839 1034 1106 1222]] +0.2595 0.2767 6.89
2.3.5.7.11.13.17.19 325/324, 343/342, 385/384, 595/594, 625/624, 1216/1215, 1445/1444 [299 474 694 839 1034 1106 1222 1270]] +0.2424 0.2627 6.54

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 79\299 317.06 6/5 Hanson
1 124\299 497.66 4/3 Cotoneum (7-limit)
1 124\299 505.69 75/56 Marfifths

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