298edo: Difference between revisions
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== Theory == | == Theory == | ||
298edo | 298edo is [[consistent]] in the 5-odd-limit, where it is [[enfactoring|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 [[7/4|7]], [[11/8|11]], [[13/8|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, {{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, {{val|298 472 '''691''' '''836''' 1031}} supports [[miracle]]. | |||
The | |||
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 === | === Prime harmonics === | ||
{{Harmonics in equal|298}} | {{Harmonics in equal|298}} | ||
Revision as of 14:54, 3 February 2024
| ← 297edo | 298edo | 299edo → |
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
| 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) | |
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.