298edo: Difference between revisions
→Theory: 298 patent is lowest in badness |
expanded, added information, will calculate errors later as I have a lot of college |
||
| Line 8: | Line 8: | ||
== 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]] | 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]], the smallest uniquely consistent EDO in the 17-limit. In the 2.5.11.17.23.43.53.59 subgroup, 298edo tempers out 3176/3175, 3128/3125, 3128/3127, 32906/32065 and 76585/76582. | ||
=== Patent val === | |||
298edo's patent val is the lowest error val in the 17-limit among 298edo vals, but they differ on the mapping of the 7th, 11th, and 13th harmonics. Thus it can be viewed as a "spicy 149edo" as a result. | |||
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. | |||
298edo tempers out the [[rastma]] and the [[ratwolfsma]], meaning it splits its perfect fifth which it inherits from 149edo, into two steps representing 11/9, and also supports the [[ratwolf triad]]. | |||
=== Other vals === | |||
Different temperaments can be extracted from 298edo by simply viewing its prime harmonics as variations from 149edo by its own half-step, although it is important to note that these vals are not better tuned than the patent val. | Different temperaments can be extracted from 298edo by simply viewing its prime harmonics as variations from 149edo by its own half-step, although it is important to note that these vals are not better tuned than the patent val. | ||
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]]. | 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]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|298}} | {{Harmonics in equal|298}} | ||
== Rank-2 temperaments == | == Regular temperament properties == | ||
Note: this assumes the patent val. | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" |Subgroup | |||
! rowspan="2" |[[Comma list]] | |||
! rowspan="2" |[[Mapping]] | |||
! rowspan="2" |Optimal | |||
8ve stretch (¢) | |||
! colspan="2" |Tuning error | |||
|- | |||
![[TE error|Absolute]] (¢) | |||
![[TE simple badness|Relative]] (%) | |||
|- | |||
|2.3.5.7 | |||
|6144/6125, 321489/320000, 3796875/3764768 | |||
|[{{val|298 472 692 837}}] | |||
|0.0275 | |||
|0.5022 | |||
|? | |||
|- | |||
|2.3.5.7.11 | |||
|243/242, 1375/1372, 6144/6125, 72171/71680 | |||
|[{{val|298 472 692 837 1031}}] | |||
|0.2881 | |||
|0.4439 | |||
|? | |||
|- | |||
|2.3.5.7.11.13 | |||
|243/242, 351/350, 1375/1372, 4096/4095, 16038/15925 | |||
|[{{val|298 472 692 837 1031 1103}}] | |||
| | |||
| | |||
|? | |||
|- | |||
|2.3.5.7.11.13.17 | |||
|243/242, 351/350, 561/560, 1375/1372, 14175/14144, 16038/15925 | |||
|[{{val|298 472 692 837 1031 1103 1218}}] | |||
| | |||
| | |||
| ? | |||
|} | |||
=== Rank-2 temperaments === | |||
Note: 5-limit temperaments represented by 149edo are not included. | Note: 5-limit temperaments represented by 149edo are not included. | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||