Fractional-octave temperaments: Difference between revisions

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 * [[Tritrizo clan #Schisennealimmal|Schisennealimmal]] (1\171 period)
 * [[Tritrizo clan #Lunennealimmal|Lunennealimmal]] (1\441 period)
-== 14th-octave temperaments ==
-While 14edo is poor in simple harmonics, some of its multiples (such as [[224edo]] and [[742edo]]) are members of zeta edo list.
-=== Silicon ===
-The name of silicon temperament comes from the 14th element. Defined upwards to the 13-limit. When tuned in 742edo, it is generated by a [[53edo]] fifth intermingled with [[14edo]] periods.
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 14348907/14336000, 56358560858112/56296884765625
-[[Mapping]]: [{{val| 14 0 -145 239 }}, {{val| 0 1 8 -9 }}]
-Mapping generators: ~6125/5832, ~3
-[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.870
-{{Val list|legend=1| 224, 518, 742, 966, 1708, 2674 }}
-[[Badness]]: 0.196
-==== 11-limit ====
-Subgroup: 2.3.5.7.11
-Comma list: 9801/9800, 1240029/1239040, 2359296/2358125
-Mapping: [{{val| 14 0 -145 239 115 }}, {{val| 0 1 8 -9 -3 }}]
-Optimal tuning (CTE): ~3/2 = 701.872
-Optimal GPV sequence: {{val list| 224, 518, 742, 966, 1708, 4382e }}
-Badness: 0.0450
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 4096/4095, 6656/6655, 9801/9800, 24192/24167
-Mapping: [{{val| 14 0 -145 239 115 74 }}, {{val| 0 1 8 -9 -3 -1 }}]
-Optimal tuning (CTE): ~3/2 = 701.8733
-Optimal GPV sequence: {{val list| 224, 518, 742, 966, 1708f }}
-Badness: 0.0269
@@ Line 218: / Line 171: @@
 Badness: 0.0345
+== 61st-octave temperaments ==
+=== Promethium ===
+Promethium tempers out the [[dipromethia]] and can be described as the 183 & 2684 temperament. By tempering out 4100625/4100096 promethium identifies the diaschisma with [[2025/2002]] in the 13-limit and also in the 17-limit.
+Subgroup: 2.3.5.7.11.13
+Comma list: 10648/10647, 196625/196608, 4100625/4100096, 204800000/204788493
+Mapping: [{{val|61 0 335 703 66 -161}}, {{val|0 2 -4 -11 3 8}}]
+Mapping generators: ~2025/2002 = 1\61, ~6875/3969 = 950.970
+Optimal tuning (CTE): ~11907/6875 = 950.970
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 14400/14399, 37180/37179, 121875/121856, 140800/140777, 3536379/3536000
+Mapping: [{{val|61 0 335 703 66 -161 201}}, {{val|0 2 -4 -11 3 8 1}}]
+Mapping generators: ~2025/2002 = 1\61, ~11907/6875 = 950.970
+Optimal tuning (CTE): ~11907/6875 = 950.970
 == 65th-octave temperaments ==
 [[65edo]] is accurate for harmonics 3, 5, and 11, so various 65th-octave temperaments actually make sense.