298edo
| ← 297edo | 298edo | 299edo → |
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
| 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 | 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.