298edo: Difference between revisions
Created page with "'''298 equal division''' divides the octave into steps of 4.027 cents each. == Theory == {{primes in edo|298|columns=18}}" |
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{{Infobox ET}} | |||
{{ED intro}} | |||
== 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—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]] and is a strong tuning for [[barton]]. In the full 13-limit, 298edo supports an unnamed {{nowrap|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 {{nowrap|31 & 298d}} variant and the {{nowrap|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]]. | |||
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 === | |||
{{Harmonics in equal|298}} | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3.5.7 | |||
| 6144/6125, 78732/78125, 3796875/3764768 | |||
| {{mapping| 298 472 692 837 }} (298) | |||
| +0.0275 | |||
| 0.5022 | |||
| 12.5 | |||
|- | |||
| 2.3.5.7.11 | |||
| 243/242, 1375/1372, 6144/6125, 72171/71680 | |||
| {{mapping| 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 | |||
| {{mapping| 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 | |||
| {{mapping| 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 | |||
| {{mapping| 298 472 692 837 1031 1103 1218 }} (298) | |||
| −0.0320 | |||
| 0.3974 | |||
| 9.87 | |||
|} | |||
=== Rank-2 temperaments === | |||
Note: 5-limit temperaments supported by 149et are not listed. | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 113\298 | |||
| 455.033 | |||
| 13/10 | |||
| [[Petrtri]] | |||
|- | |||
| 1 | |||
| 137\298 | |||
| 551.67 | |||
| 11/8 | |||
| [[Emka]] | |||
|- | |||
| 2 | |||
| 39\298 | |||
| 157.04 | |||
| 35/32 | |||
| [[Bison]] | |||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Scales == | |||
The [[concoctic]] scale for 298edo is a scale produced by a generator of 105 steps (paraconcoctic), and the associated rank-2 temperament is {{nowrap|105 & 298}}. | |||
[[Category:Bison]] | |||
[[Category:Emka]] | |||