298edo: Difference between revisions
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== Theory == | == 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]], but | 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]] and its 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. | ||
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. | |||
298edo supports unconventional extensions of [[sensi]] to higher dimensions. The 298d val in 11-limit (149edo with 298edo 11/8) supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. The 298cd val supports [[miracle]]. | 298edo supports unconventional extensions of [[sensi]] to higher dimensions. The 298d val in 11-limit (149edo with 298edo 11/8) supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. The 298cd val supports [[miracle]]. | ||
Revision as of 18:16, 28 September 2022
| ← 297edo | 298edo | 299edo → |
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 and its 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.
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.
298edo supports unconventional extensions of sensi to higher dimensions. The 298d val in 11-limit (149edo with 298edo 11/8) supports hagrid, in addition to the 31 & 298d variant and the 118 & 298d variant of hemithirds. The 298cd val supports miracle.
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.
In the 2.5.11.17.23.43.53.59, 298edo tempers out 3176/3175, 3128/3125, 3128/3127, 32906/32065 and 76585/76582.
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 (%) | -31.9 | +6.5 | +40.8 | +36.2 | +8.9 | +26.9 | -25.3 | -6.4 | +11.8 | +8.9 | -2.1 | |
| Steps (reduced) |
472 (174) |
692 (96) |
837 (241) |
945 (51) |
1031 (137) |
1103 (209) |
1164 (270) |
1218 (26) |
1266 (74) |
1309 (117) |
1348 (156) | |
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.