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 (298edo), or 298-tone equal temperament (298tet), 298 equal temperament (298et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 298 equal parts of about 4.03 ¢ each.

Theory

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¢. It is a double of 149edo, the smallest uniquely consistent EDO in the 17-limit. In the 2.5.11.17.23.43.53.59 subgroup, 298edo tempers out 3176/3175, 3128/3125, 3128/3127, 32906/32065 and 76585/76582.

Patent val

298edo's patent val is the lowest error val in the 17-limit among 298edo vals, but they differ on the mapping of the 7th, 11th, and 13th harmonics. Thus it can be viewed as a "spicy 149edo" as a result.

The patent val in 298edo 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.

298edo tempers out the rastma and the ratwolfsma, meaning it splits its perfect fifth which it inherits from 149edo, into two steps representing 11/9, and also supports the ratwolf triad.

Other vals

Different temperaments can be extracted from 298edo by simply viewing its prime harmonics as variations from 149edo by its own half-step, although it is important to note that these vals are not better tuned than the patent val.

The 298d val in 11-limit (149edo with 298edo 11/8) is better tuned than the patent val (although not in the 17-limit) and 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. It still tempers out 243/242, but now it adds 1029/1024, 3136/3125, and 9801/9800.

The 298cd val supports miracle.

Prime harmonics

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

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
1 137\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.