Diminished family: Difference between revisions

← Older edit

VisualWikitext

@@ Line 12: / Line 12: @@
 {{Mapping|legend=1| 4 0 3 | 0 1 1 }}
+: mapping generators: ~6/5, ~3
 [[Optimal tuning]]s:
@@ Line 26: / Line 28: @@
 {{Main| Diminished (temperament) }}
-Septimal diminished equates the 300c minor third to [[7/6]] along with 6/5, and the 600c tritone to [[7/5]].
+Septimal diminished equates the 300{{c}} minor third to [[7/6]] along with 6/5, and the 600{{c}} tritone to [[7/5]][[~]][[10/7]].
 [[Subgroup]]: 2.3.5.7
@@ Line 49: / Line 51: @@
 Comma list: 36/35, 50/49, 56/55
-Mapping: {{mapping| 4 0 3 5 14 | 0 1 1 1 0 }}
+{{Mapping|legend=0| 4 0 3 5 14 | 0 1 1 1 0 }}
 Optimal tunings:
@@ Line 64: / Line 66: @@
 Comma list: 36/35, 40/39, 50/49, 66/65
-Mapping: {{mapping| 4 0 3 5 14 15 | 0 1 1 1 0 0 }}
+{{Mapping|legend=0| 4 0 3 5 14 15 | 0 1 1 1 0 0 }}
 Optimal tunings:
@@ Line 79: / Line 81: @@
 Comma list: 36/35, 45/44, 50/49
-Mapping: {{mapping| 4 0 3 5 -5 | 0 1 1 1 3 }}
+{{Mapping|legend=0| 4 0 3 5 -5 | 0 1 1 1 3 }}
 Optimal tunings:
@@ Line 96: / Line 98: @@
 Comma list: 36/35, 50/49, 125/121
-Mapping: {{mapping| 4 1 4 6 6 | 0 2 2 2 3 }}
+{{Mapping|legend=0| 4 1 4 6 6 | 0 2 2 2 3 }}
 : mapping generators: ~6/5, ~11/7
@@ Line 113: / Line 115: @@
 Comma list: 36/35, 50/49, 66/65, 125/121
-Mapping: {{mapping| 4 1 4 6 6 7 | 0 2 2 2 3 3 }}
+{{Mapping|legend=0| 4 1 4 6 6 7 | 0 2 2 2 3 3 }}
 Optimal tunings:
@@ Line 149: / Line 151: @@
 Comma list: 49/48, 77/75, 243/242
-Mapping: {{mapping| 4 0 3 8 -2 | 0 2 2 1 5 }}
+{{Mapping|legend=0| 4 0 3 8 -2 | 0 2 2 1 5 }}
 Optimal tunings:
@@ Line 164: / Line 166: @@
 Comma list: 49/48, 66/65, 77/75, 243/242
-Mapping: {{mapping| 4 0 3 8 -2 -1 | 0 2 2 1 5 5 }}
+{{Mapping|legend=0| 4 0 3 8 -2 -1 | 0 2 2 1 5 5 }}
 Optimal tunings:
@@ Line 202: / Line 204: @@
 Comma list: 56/55, 77/75, 245/243
-Mapping: {{mapping| 8 0 6 -3 15 | 0 1 1 2 1 }}
+{{Mapping|legend=0| 8 0 6 -3 15 | 0 1 1 2 1 }}
 Optimal tunings:
@@ Line 217: / Line 219: @@
 Comma list: 56/55, 66/65, 77/75, 507/500
-Mapping: {{mapping| 8 0 6 -3 15 17 | 0 1 1 2 1 1 }}
+{{Mapping|legend=0| 8 0 6 -3 15 17 | 0 1 1 2 1 1 }}
 Optimal tunings:
@@ Line 227: / Line 229: @@
 Badness (Sintel): 1.26
-[[Category:Temperament families]]
+== Subgroup extensions ==
-[[Category:Pages with mostly numerical content]]
+Diminished works on 17- and 19-limit subgroups very well, where the 1/4-octave period stands in for ~6/5, ~19/16, and ~20/17.
+=== Diminished (2.3.5.17 subgroup) ===
+[[Subgroup]]: 2.3.5.17
+[[Comma list]]: 51/50, 289/288
+{{Mapping|legend=3| 4 0 3 10 | 0 1 1 1 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~6/5 = 299.899{{c}}, ~3/2 = 698.251{{c}} (~17/16 = 98.453{{c}})
+* [[CWE]]: ~6/5 = 300.000{{c}}, ~3/2 = 698.137{{c}} (~17/16 = 98.137{{c}})
+{{Optimal ET sequence|legend=1| 4, 8, 12 }}
+[[Badness]] (Sintel): 0.509
+=== Diminished (2.3.5.19 subgroup) ===
+[[Subgroup]]: 2.3.5.19
+[[Comma list]]: 96/95, 513/500
+{{Mapping|legend=3| 4 0 3 17 | 0 1 1 0 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~6/5 = 299.744{{c}}, ~3/2 = 698.206{{c}} (~20/19 = 98.718{{c}})
+* [[CWE]]: ~6/5 = 300.000{{c}}, ~3/2 = 698.195{{c}} (~20/19 = 98.195{{c}})
+{{Optimal ET sequence|legend=1| 4, 8, 12 }}
+[[Badness]] (Sintel): 0.681
 [[Category:Diminished family| ]] <!-- main article -->
 [[Category:Diminished| ]] <!-- key article -->
-[[Category:Rank 2]]
+[[Category:Temperament families]]
+[[Category:Catalogs of rank-2 temperaments]]