118th-octave temperaments: Difference between revisions

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+{{Infobox fractional-octave|118}}
 [[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets. Furthermore, one step of 118edo is in direct proximity to essential tempering commas [[169/168]] and [[170/169]].
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 [[Badness]]: 0.145166
-==== 11-limit ====
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
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 Badness: 0.049316
-=== Centenniamajor ===
+=== Peithoian ===
-Centenniamajor is an extension of parakleischis which retains the 5-limit mapping of 118edo and provides the correction for 13th harmonic. 13-limit is the first prime limit that 118edo does not tune consistently, and the goal of centenniamajor temperament is to expand on that. Named after the fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) = 118.
+Peithoian is an extension of parakleischis which retains the 5-limit mapping of 118edo and provides the correction for 13th harmonic. 13-limit is the first prime limit that 118edo does not tune consistently, and the goal of peithoian temperament is to expand on that. Named after the minor planet [[wikipedia:118 Peitho|118 Peitho]].
 Subgroup: 2.3.5.7.11
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 Badness: 0.357
-===== 13-limit =====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
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 [[Optimal tuning]] ([[CTE]]): ~1953125/1354752 = 634.0068
-{{Optimal ET sequence|legend=1| 354, 2360, 2714, 3068, 3442, 3776, 7198cd, 10974bccdd }}
+{{Optimal ET sequence|legend=1| 354, 2360, 2714, 3068, 3422, 3776, 7198cd, 10974bccdd }}
 [[Badness]]: 2.66
-==== 11-limit ====
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
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 Optimal tuning (CTE): ~1953125/1354752 = 634.0085
-{{Optimal ET sequence|legend=1| 354, 3068e, 3442, 3776, 11682ccdde }}
+{{Optimal ET sequence|legend=1| 354, 3068e, 3422, 3776, 11682ccdde }}
 Badness: 0.568
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
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 Badness: 0.172
-==== 17-limit ====
+=== 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
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 {{Optimal ET sequence|legend=1| 354, 3068e, 3422, 3776 }}
-Badness: 0.105
+=== 19-limit ===
+The closest superparticular to one step of 118edo is [[171/170]], so 19-limit extension for oganesson is prescribed.
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 4096/4095, 6175/6174, 9801/9800, 14365/14364, 28900/28899, 3438981/3437500
+Mapping: [{{val| 118 0 274 643 1094 499 607 1000}}, {{val| 0 3 0 -5 -11 -1 2 -8}}]
+: mapping generators: ~171/170, ~238/165
+Optimal tuning (CTE): ~238/165 = 634.006
+{{Optimal ET sequence|legend=1| 354, ..., 3422, 3776 }}
+{{Navbox fractional-octave}}
+[[Category:118edo]]