298edo

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← 297edo298edo299edo →
Prime factorization 2 × 149
Step size 4.02685¢
Fifth 174\298 (700.671¢) (→87\149)
Semitones (A1:m2) 26:24 (104.7¢ : 96.64¢)
Consistency limit 5
Distinct consistency limit 5

298 equal divisions of the octave (abbreviated 298edo), or 298-tone equal temperament (298tet), 298 equal temperament (298et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 298 equal parts of about 4.03 ¢ each. Each step of 298edo represents a frequency ratio of 21/298, or the 298th root of 2.

Theory

298edo is consistent in the 5-odd-limit, where it is enfactored, with the same tuning as 149edo. Since 149edo is notable for being the smallest edo distinctly consistent in the 17-odd-limit, 298edo is related to 149edo - it retains the mapping for 2.3.5.17 but differs on the mapping for harmonics 7, 11, 13. 298edo tempers out the rastma, splitting 3/2 inherited from 149edo into two steps representing 11/9. It also tempers out the ratwolfsma. The patent val supports the bison temperament and the rank-3 temperament hemimage. In the 2.5.11.13 subgroup, 298edo supports emka. In the full 13-limit, 298edo supports an unnamed 77 & 298 temperament with 13/8 as its generator.

Aside from the patent val, there is a number of mappings to be considered. One can approach 298edo's vals as a double of 149edo again, by simply viewing its prime harmonics as variations from 149edo by its own half-step. The 298d val, 298 472 692 836 1031], which includes 149edo's 7-limit tuning, is better tuned than the patent val in the 11-limit (though not in the 17-limit). It supports hagrid, in addition to the 31 & 298d variant and the 118 & 298d variant of hemithirds. Some of the commas it tempers out make for much more interesting temperaments than the patent val - for example it still tempers out 243/242, but now it adds 1029/1024, 3136/3125, and 9801/9800.

The 298cd val, 298 472 691 836 1031] supports miracle.

In higher limits, 298edo has excellent representation of the 2.5.11.17.23.43.53.59 subgroup, with all the harmonics having errors of less than 10 r¢. A comma basis for the 2.5.11.17.23.43.53.59 subgroup is {1376/1375, 3128/3127, 4301/4300, 25075/25069, 38743/38720, 58351/58300, 973360/972961}.

Prime harmonics

Approximation of odd harmonics in 298edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error absolute (¢) -1.28 +0.26 +1.64 +1.46 +0.36 +1.08 -1.02 -0.26 +0.47 +0.36 -0.09
relative (%) -32 +7 +41 +36 +9 +27 -25 -6 +12 +9 -2
Steps
(reduced)
472
(174)
692
(96)
837
(241)
945
(51)
1031
(137)
1103
(209)
1164
(270)
1218
(26)
1266
(74)
1309
(117)
1348
(156)

Regular temperament properties

Subgroup Comma list Mapping Optimal

8ve stretch (¢)

Tuning error
Absolute (¢) Relative (%)
2.3.5.7 6144/6125, 321489/320000, 3796875/3764768 [298 472 692 837]] 0.0275 0.5022 ?
2.3.5.7.11 243/242, 1375/1372, 6144/6125, 72171/71680 [298 472 692 837 1031]] 0.0012 0.4523 ?
2.3.5.7.11 243/242, 1029/1024, 3136/3125, 9801/9800 [298 472 692 836 1031]] (298d) 0.2882 0.4439 ?
2.3.5.7.11.13 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925 [298 472 692 837 1031 1103]] ?
2.3.5.7.11.13.17 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925 [298 472 692 837 1031 1103 1218]] ?

Rank-2 temperaments

Note: 5-limit temperaments represented by 149edo are not included.

Periods
per Octave
Generator
(Reduced)
Cents
(Reduced)
Associated
Ratio
Temperaments
1 113\298 455.033 13/10 Petrtri (2.11/5.13/5)
1 137\298 551.67 11/8 Emka
2 39\298 157.04 35/32 Bison

Scales

The concoctic scale for 298edo is a scale produced by a generator of 105 steps (paraconcoctic), and the associated rank-2 temperament is 105 & 298.