298edo: Difference between revisions

← Older edit

VisualWikitext

@@ Line 1: / Line 1: @@
-'''298 equal division''' divides the octave into steps of 4.027 cents each.
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
-{{primes in edo|298|columns=17}}
+edo 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 &amp; 298}} temperament with [[13/8]] as its generator.
-edo has excellent representation of the 2.5.11.17.23.43.53.59 subgroup, with all the harmonics having errors of less than 10 rc. It is a double of [[149edo]], which is the smallest edo that is uniquely consistent within the [[17-odd-limit]]. It supports a 17-limit extension of [[Sensi]], 111 & 103 & 298. However, compared to 149edo, 298edo's patent val differs on the mapping of 7, 11, and 13th harmonics.
+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 &amp; 298d}} variant and the {{nowrap|118 &amp; 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]].
-In the patent val, 298edo tempers out 351/350, 561/560, 936/935, and 1156/1155 in the full 17-limit. In the 2.5.11.13.17 subgroup, it tempers out [[2200/2197]] and [[6656/6655]].
+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}.
-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 &amp; 298}}.
+[[Category:Bison]]
+[[Category:Emka]]