298edo: Difference between revisions
Cleanup and remove misconceptions |
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{{Infobox ET | {{Infobox ET | ||
| Prime factorization = 2 × 149 | | Prime factorization = 2 × 149 | ||
| Fifth = 174\298 ([[149edo|87\149]]) | | Step size = 4.0268¢ | ||
|Semitones=26:24 (104. | | Fifth = 174\298 (→ [[149edo|87\149]]) | ||
| Semitones = 26:24 (104.70¢:96.64¢) | |||
}} | |||
{{EDO intro|298}} | {{EDO intro|298}} | ||
== Theory == | == Theory == | ||
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¢. It is a double of [[149edo]], but the patent vals differ on the mapping of the 7th, 11th, and 13th harmonics. Thus it can be viewed as a "spicy 149edo" as a result, and different temperaments can be extracted from 298edo by simply viewing its prime harmonics as variations from 149edo by its own half-step. | |||
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 | |||
298edo supports unconventional extensions of [[ | 298edo supports unconventional extensions of [[sensi]] to higher dimensions. The 298d val in 11-limit (149edo with 298edo 11/8) supports [[hagrid]], in addition to the 31 & 298d variant and the 118 & 298d variant of [[hemithirds]]. The 298cd val supports [[miracle]]. | ||
The patent val in 298edo | The patent val in 298edo 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. | ||
In the 2.5.11.17.23.43.53.59, 298edo tempers out 3176/3175, 3128/3125, 3128/3127, 32906/32065 and 76585/76582. | In the 2.5.11.17.23.43.53.59, 298edo tempers out 3176/3175, 3128/3125, 3128/3127, 32906/32065 and 76585/76582. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|298}} | |||
== Rank | == Rank-2 temperaments == | ||
Note: | Note: 5-limit temperaments represented by 149edo are not included. | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
!Periods | ! Periods<br>per Octave | ||
per | ! Generator<br>(Reduced) | ||
!Generator | ! Cents<br>(Reduced) | ||
( | ! Associated<br>Ratio | ||
!Cents | ! Temperaments | ||
( | |||
!Associated | |||
!Temperaments | |||
|- | |- | ||
|1 | | 1 | ||
|137\298 | | 137\298 | ||
|551.67 | | 551.67 | ||
|11/8 | | 11/8 | ||
|[[Emka]] | | [[Emka]] | ||
|- | |- | ||
|2 | | 2 | ||
|39\298 | | 39\298 | ||
|157.04 | | 157.04 | ||
|35/32 | | 35/32 | ||
|[[Bison]] | | [[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. | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Bison]] | [[Category:Bison]] | ||
[[Category:Emka family]] | [[Category:Emka family]] | ||
Revision as of 14:45, 10 August 2022
| ← 297edo | 298edo | 299edo → |
Theory
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¢. It is a double of 149edo, but the patent vals differ on the mapping of the 7th, 11th, and 13th harmonics. Thus it can be viewed as a "spicy 149edo" as a result, and different temperaments can be extracted from 298edo by simply viewing its prime harmonics as variations from 149edo by its own half-step.
298edo supports unconventional extensions of sensi to higher dimensions. The 298d val in 11-limit (149edo with 298edo 11/8) supports hagrid, in addition to the 31 & 298d variant and the 118 & 298d variant of hemithirds. The 298cd val supports miracle.
The patent val in 298edo 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.
In the 2.5.11.17.23.43.53.59, 298edo tempers out 3176/3175, 3128/3125, 3128/3127, 32906/32065 and 76585/76582.
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) | |
Rank-2 temperaments
Note: 5-limit temperaments represented by 149edo are not included.
| Periods per Octave |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 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.