298edo: Difference between revisions
→Rank-2 temperaments: petrtri |
→Theory: +barton |
||
| (9 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
298edo | 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]]. | |||
298edo's | |||
The | The 298cd val, {{val| 298 472 '''691''' '''836''' 1031 }} supports [[miracle]]. | ||
298edo | 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}} | {{Harmonics in equal|298}} | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! rowspan="2" | [[Subgroup]] | ||
![[TE simple badness|Relative]] (%) | ! 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 | | 2.3.5.7 | ||
|6144/6125, | | 6144/6125, 78732/78125, 3796875/3764768 | ||
| | | {{mapping| 298 472 692 837 }} (298) | ||
|0.0275 | | +0.0275 | ||
|0.5022 | | 0.5022 | ||
| | | 12.5 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|243/242, 1375/1372, 6144/6125, 72171/71680 | | 243/242, 1375/1372, 6144/6125, 72171/71680 | ||
| | | {{mapping| 298 472 692 837 1031 }} (298) | ||
|0.0012 | | +0.0012 | ||
|0.4523 | | 0.4523 | ||
| | | 11.2 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|243/242, 1029/1024, 3136/3125, 9801/9800 | | 243/242, 1029/1024, 3136/3125, 9801/9800 | ||
| | | {{mapping| 298 472 692 836 1031 }} (298d) | ||
|0.2882 | | +0.2882 | ||
|0.4439 | | 0.4439 | ||
| | | 11.0 | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
|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) | ||
| | | −0.0478 | ||
| | | 0.4271 | ||
| | | 10.6 | ||
|- | |- | ||
|2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
|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) | ||
| | | −0.0320 | ||
| | | 0.3974 | ||
| | | 9.87 | ||
|} | |} | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
Note: 5-limit temperaments | Note: 5-limit temperaments supported by 149et are not listed. | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
! Periods<br>per | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Generator | |- | ||
! Cents | ! Periods<br />per 8ve | ||
! Associated<br> | ! Generator* | ||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|113\298 | | 113\298 | ||
|455.033 | | 455.033 | ||
|13/10 | | 13/10 | ||
|[[Petrtri]] | | [[Petrtri]] | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 97: | Line 91: | ||
| [[Bison]] | | [[Bison]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Scales == | == 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. | 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:Bison]] | ||
[[Category:Emka | [[Category:Emka]] | ||