299edo

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← 298edo299edo300edo →
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