298edo: Difference between revisions
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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 the 2.5.11.17.23.43.53.59, | 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}. | ||
The [[concoctic]] scale for 298edo is a generator of 105 steps (paraconcoctic). | === 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]] | |||
Latest revision as of 05:25, 14 August 2025
| ← 297edo | 298edo | 299edo → |
298 equal divisions of the octave (abbreviated 298edo or 298ed2), also called 298-tone equal temperament (298tet) or 298 equal temperament (298et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 298 equal parts of about 4.03 ¢ each. Each step represents a frequency ratio of 21/298, or the 298th root of 2.
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 and is a strong tuning for barton. 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
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 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 |
Rank-2 temperaments
Note: 5-limit temperaments supported by 149et are not listed.
| Periods per 8ve |
Generator* | Cents* | Associated 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 |
* Octave-reduced form, reduced to the first half-octave, and 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 105 & 298.