298edo: Difference between revisions
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== Theory == | == Theory == | ||
298edo is [[enfactoring|enfactored]] in the [[5-limit]] and only [[consistent]] in the [[5-odd-limit]], 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 | 298edo is [[enfactoring|enfactored]] in the [[5-limit]] and only [[consistent]] in the [[5-odd-limit]], 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 [[harmonic]]s [[2/1|2]], [[3/1|3]], [[5/1|5]], and [[17/1|17]] but differs on the mapping for [[7/4|7]], [[11/8|11]], [[13/8|13]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] the [[rastma]] in the 11-limit, splitting [[3/2]] inherited from 149edo into two steps representing [[11/9]]. It also tempers out the [[ratwolfsma]] in the 13-limit. It [[support]]s 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 | 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 298cd val, {{val| 298 472 '''691''' '''836''' 1031 }} supports [[miracle]]. | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| Line 49: | Line 41: | ||
| 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925 | | 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925 | ||
| {{mapping| 298 472 692 837 1031 1103 }} (298) | | {{mapping| 298 472 692 837 1031 1103 }} (298) | ||
| | | −0.0478 | ||
| 0.4271 | | 0.4271 | ||
| 10.6 | | 10.6 | ||
| Line 56: | Line 48: | ||
| 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925 | | 243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925 | ||
| {{mapping| 298 472 692 837 1031 1103 1218 }} (298) | | {{mapping| 298 472 692 837 1031 1103 1218 }} (298) | ||
| | | −0.0320 | ||
| 0.3974 | | 0.3974 | ||
| 9.87 | | 9.87 | ||
{{comma basis end}} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
Note: 5-limit temperaments supported by 149et are not listed. | Note: 5-limit temperaments supported by 149et are not listed. | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 88: | Line 74: | ||
| 35/32 | | 35/32 | ||
| [[Bison]] | | [[Bison]] | ||
{{rank-2 end}} | |||
{{orf}} | |||
== Scales == | == Scales == | ||
Revision as of 05:10, 16 November 2024
| ← 297edo | 298edo | 299edo → |
Theory
298edo is enfactored in the 5-limit and only consistent in the 5-odd-limit, 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 harmonics 2, 3, 5, and 17 but differs on the mapping for 7, 11, 13. Using the patent val, the equal temperament tempers out the rastma in the 11-limit, splitting 3/2 inherited from 149edo into two steps representing 11/9. It also tempers out the ratwolfsma in the 13-limit. It 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}.
Odd 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
Template:Comma basis begin |- | 2.3.5.7 | 6144/6125, 78732/78125, 3796875/3764768 | [⟨298 472 692 837]] (298) | +0.0275 | 0.5022 | 12.5 |- | 2.3.5.7.11 | 243/242, 1375/1372, 6144/6125, 72171/71680 | [⟨298 472 692 837 1031]] (298) | +0.0012 | 0.4523 | 11.2 |- | 2.3.5.7.11 | 243/242, 1029/1024, 3136/3125, 9801/9800 | [⟨298 472 692 836 1031]] (298d) | +0.2882 | 0.4439 | 11.0 |- | 2.3.5.7.11.13 | 243/242, 351/350, 1375/1372, 4096/4095, 16038/15925 | [⟨298 472 692 837 1031 1103]] (298) | −0.0478 | 0.4271 | 10.6 |- | 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]] (298) | −0.0320 | 0.3974 | 9.87 Template:Comma basis end
Rank-2 temperaments
Note: 5-limit temperaments supported by 149et are not listed. Template:Rank-2 begin |- | 1 | 113\298 | 455.033 | 13/10 | Petrtri |- | 1 | 137\298 | 551.67 | 11/8 | Emka |- | 2 | 39\298 | 157.04 | 35/32 | Bison Template:Rank-2 end Template:Orf
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.