Fractional-octave temperaments: Difference between revisions

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 == 14th-octave temperaments ==
-While 14edo is poor in LCJI harmonics, some of its multiples (such as [[224edo]] and [[742edo]]) are members of zeta edo list.
+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. In 742edo, what's also unique is that it is generated by a [[53edo]] fifth intermingled with [[14edo]] periods.
+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.
-==== 5-limit ====
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5
-Comma list: {{Monzo|-145 112 -14}}
+[[Comma list]]: 14348907/14336000, 56358560858112/56296884765625
-Mapping generators: ~282429536481/268435456000, ~3/2
+[[Mapping]]: [{{val| 14 0 -145 239 }}, {{val| 0 1 8 -9 }}]
-Mapping: [{{val|14 14 -33}}, {{val|0 1 8}}]
+Mapping generators: ~6125/5832, ~3
-Optimal tuning (CTE): ~3/2 = 701.864
+[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.870
-==== 7-limit ====
+{{Val list|legend=1| 224, 518, 742, 966, 1708, 2674 }}
-Subgroup: 2.3.5.7
-Comma list: 14348907/14336000, 56358560858112/56296884765625
+[[Badness]]: 0.196
-Mapping generators: ~6125/5832, ~3/2
-Mapping: [{{val|14 14 -33 113}}, {{val|0 1 8 -9}}]
-Optimal tuning (CTE): ~3/2 = 701.870
 ==== 11-limit ====
@@ Line 110: / Line 102: @@
 Comma list: 9801/9800, 1240029/1239040, 2359296/2358125
-Mapping generators: ~605/576, ~3/2
+Mapping: [{{val| 14 0 -145 239 115 }}, {{val| 0 1 8 -9 -3 }}]
+Optimal tuning (CTE): ~3/2 = 701.872
-Mapping: [{{val|14 14 -33 113 73}}, {{val|0 1 8 -9 -3}}]
+Optimal GPV sequence: {{val list| 224, 518, 742, 966, 1708, 4382e }}
-Optimal tuning (CTE): ~3/2 = 701.872
+Badness: 0.0450
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 4096/4095, 9801/9800, 67392/67375, 59535/59488
+Comma list: 4096/4095, 6656/6655, 9801/9800, 24192/24167
-Mapping generators: ~104/99, ~3/2
+Mapping: [{{val| 14 0 -145 239 115 74 }}, {{val| 0 1 8 -9 -3 -1 }}]
-Mapping: [{{val|14 14 -33 113 73 60}}, {{val|0 1 8 -9 -3 -1}}]
+Optimal tuning (CTE): ~3/2 = 701.8733
-Optimal tuning (CTE): ~3/2 = 701.873
+Optimal GPV sequence: {{val list| 224, 518, 742, 966, 1708f }}
-Vals: {{EDOs|70d, 224, 294, 448, 518, 672, 742, 966, 1190, 1260}}
+Badness: 0.0269
 == 37th-octave temperaments ==
@@ Line 230: / Line 224: @@
 == 56th-octave temperaments ==
-One step of 56edo is close to a syntonic comma.
+=== Barium ===
+One step of 56edo is close to a syntonic comma. Named after the 56th element, barium tempers out the {{monzo| -225 224 -56 }} comma, which sets 56 syntonic commas equal to the octave. It can be expressed as the 224 & 2072 temperament.
-=== Barium ===
+[[Subgroup]]: 2.3.5
-Named after the 56th element, and tempers out the [-225 224 -56] comma, which sets 56 syntonic commas equal to the octave. Best expressed as 224 & 2072 temperament.
+[[Comma list]]: {{monzo| -225 24 -56 }}
-Subgroup: 2.3.5
+[[Mapping]]: [{{val| 56 0 -225 }}, {{val| 0 1 4 }}]
-Comma list: {{monzo|-225 24 -56}}
+Mapping generators: ~81/80, ~3
-Mapping: [{{val|56 89 131}}, {{val|0 -1 -4}}]
+[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.9379
-Mapping generators: ~81/80, ~3/2
+{{Val list|legend=1| 224, 1176, 1400, 1624, 1848, 2072, 5992, 8064, 26264, 34328b, 42392b }}
-Optimal tuning (CTE): ~3/2 = 701.938
+[[Badness]]: 4.70
-Vals: {{EDOs|56, 224, 2072, 1848, 2296, 1624, 448, 3920, 4144, 3696, 2520}}
+==== 7-limit ====
-=== 7-limit===
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7
+[[Comma list]]: {{monzo| -12 29 -11 -3 }}, {{monzo| 47 -7 -7 -7 }}
-Comma list: {{monzo|-12 29 -11 -3}}, {{monzo|47 -7  -7 -7}}
+[[Mapping]]: [{{val| 56 0 -225 601 }}, {{val| 0 1 4 -5 }}]
-Mapping: [{{val|56 56 -1 321}}, {{val|0 1 4 -5}}]
+[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.9433
-Mapping generators: ~81/80, ~3/2
+{{Val list|legend=1| 224, 1176, 1400, 1624, 1848, 2072, 5768, 7616, 17080, 24696cd }}
-Optimal tuning (CTE): ~3/2 = 701.943
+[[Badness]]: 0.227
-Vals: 224, 1848, 2072
+==== 11-limit ====
-===11-limit===
 Subgroup: 2.3.5.7.11
 Comma list: 9801/9800, 1019215872/1019046875, 14765025303/14763950080
-Mapping: [{{val|56 56 -1 321 292}}, {{val|0 1 4 5 -3}}]
+Mapping: [{{val| 56 0 -225 601 460 }}, {{val| 0 1 4 5 -3 }}]
+Optimal tuning (CTE): ~3/2 = 701.9431
-Mapping generators: ~81/80, ~3/2
+Optimal GPV sequence: {{Val list| 224, 1176, 1400, 1624, 1848, 3920, 5768, 7616, 21000cd, 28616cd }}
-Optimal tuning (CTE): ~3/2 = 701.943
+Badness: 0.0345
 == 65th-octave temperaments ==