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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2 × 149
{{ED intro}}
| Step size = 4.0268¢
| Fifth = 174\298 (→ [[149edo|87\149]])
| Semitones = 26:24 (104.70¢:96.64¢)
|Consistency=5}}
{{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 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.   


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]].  
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 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.
The 298cd val, {{val| 298 472 '''691''' '''836''' 1031 }} supports [[miracle]].  


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 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}.


=== Prime harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|298}}
{{Harmonics in equal|298}}


== Rank-2 temperaments ==
== Regular temperament properties ==
Note: 5-limit temperaments represented by 149edo are not included.
{| 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"
{| class="wikitable center-all left-5"
! Periods<br>per Octave
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Generator<br>(Reduced)
|-
! Cents<br>(Reduced)
! Periods<br />per 8ve
! Associated<br>Ratio
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
| 1
| 113\298
| 455.033
| 13/10
| [[Petrtri]]
|-
|-
| 1
| 1
Line 40: 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 &amp; 298}}.


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Bison]]
[[Category:Bison]]
[[Category:Emka family]]
[[Category:Emka]]
Retrieved from "https://en.xen.wiki/w/298edo"