Biyatismic clan: Difference between revisions

← Older edit Newer edit →

VisualWikitext

@@ Line 215: / Line 215: @@
 Badness: 1.456 × 10<sup>-3</sup>
-==== (13-limit) [[#Eros|Eros]] ====
+==== Eros ====
-This is the [[13-limit]] restriction of [[#Eros|23-limit Eros]], documented for readability in a separate section of this page.
+Eros fairs impressively into the 23-limit as a rank 3 temperament<sup>[''how?'']</sup>.
 Subgroup: 2.3.5.7.11.13
@@ Line 230: / Line 230: @@
 Badness: 1.150 × 10<sup>-3</sup>
-==== Inanna ====
+===== 17-limit =====
-Subgroup: 2.3.5.7.11.13
+Note that this extension requires the 29g val for 29edo, which has the sizes of 17/16 and 18/17 swapped.
-Comma list: 105/104, 121/120, 275/273
+Subgroup: 2.3.5.7.11.13.17
-Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}
+Comma list: 121/120, 154/153, 196/195, 352/351
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7754, ~21/20 = 79.6096
+Mapping: {{mapping| 1 0 1 3 2 7 6 | 0 1 1 0 1 -2 -1 | 0 0 -4 -3 -2 -2 -5 }}
-{{Optimal ET sequence|legend=1| 14cf, 15, 29, 31, 45ef, 60e }}
+Optimal tunings:
+* CTE: ~2 = 1\1, ~3/2 = 701.9299, ~22/21 = 78.2539
+* CWE: ~2 = 1\1, ~3/2 = 701.7925, ~22/21 = 78.6203
-Badness: 1.077 × 10<sup>-3</sup>
+Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 77, 106de }}
-==== Ishtar ====
+Badness:
-Subgroup: 2.3.5.7.11.13
+* Smith: 0.979 × 10<sup>-3</sup>
+* Dirichlet: 0.931
-Comma list: 91/90, 121/120, 441/440
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
-Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}
+Comma list: 121/120, 154/153, 196/195, 286/285, 352/351
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.3952, ~21/20 = 78.9578
+Mapping: {{mapping| 1 0 1 3 2 7 6 9 | 0 1 1 0 1 -2 -1 -3 | 0 0 -4 -3 -2 -2 -5 0 }}
-{{Optimal ET sequence|legend=1| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
+Optimal tunings:
+* CTE: ~2 = 1\1, ~3/2 = 701.5642, ~22/21 = 78.2353
+* CWE: ~2 = 1\1, ~3/2 = 701.6963, ~22/21 = 78.6479
-Badness: 1.151 × 10<sup>-3</sup>
+Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}
-== Eros ==
+Badness:
-This temperament is an extension of [[#Aphrodite]], specifically [[#(13-limit) Eros|(13-limit) Eros]], into the 23-limit, in which it fairs impressively as a rank 3 temperament.
+* Smith: 1.13 × 10<sup>-3</sup>
+* Dirichlet: 1.159
-Due to having natural extensions to such a high prime limit and due to its unusual efficiency, it is documented here in its own section to keep other extensions of Aphrodite more readable.
+===== 23-limit =====
+Subgroup: 2.3.5.7.11.13.17.19.23
-=== 13-limit ===
+Comma list: 121/120, 154/153, 161/160, 196/195, 286/285, 352/351
-Subgroup: [[13-limit]]
-Comma list: 121/120, 441/440, 352/351
+Mapping: {{mapping| 1 0 1 3 2 7 6 9 3 | 0 1 1 0 1 -2 -1 -3 1 | 0 0 -4 -3 -2 -2 -5 0 -1 }}
-Mapping: {{mapping| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}
+Optimal tunings:
+* CTE: ~2 = 1\1, ~3 = 1901.7115, ~23/22 = 78.2054
+* CWE: ~2 = 1\1, ~3 = 1901.8010, ~23/22 = 78.7188
-Optimal tuning ([[CTE]]): 2, ~3 = 1901.722, ~22/21 = 78.110
+Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 31, 46, 60e, 75dfgh, 77, 106de }}
-[[Patent val]] EDO tunings: [[29edo|29]], [[31edo|31]], [[46edo|46]], [[77edo|77]], [[92edo|92]]
+Badness:
+* Smith: 0.939 × 10<sup>-3</sup>
+* Dirichlet: 1.084
-Dirichlet badness: 1.075
+==== Inanna ====
+Subgroup: 2.3.5.7.11.13
-=== 17-limit ===
+Comma list: 105/104, 121/120, 275/273
-Note that there is one more patent val EDO tuning in the 17-limit than in the 19- and 23-limit corresponding to using 46 EDO with a better mapping for 19. 29 disappears as it requires the 29g val.
-Subgroup: [[17-limit]]
+Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}
-Comma list: 121/120, 441/440, 352/351, 154/153
+Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7754, ~21/20 = 79.6096
-Mapping: {{mapping| 1 0 1 3 2 7 6 | 0 1 1 0 1 -2 -1 | 0 0 -4 -3 -2 -2 -5 }}
+{{Optimal ET sequence|legend=1| 14cf, 15, 29, 31, 45ef, 60e }}
-Optimal tuning ([[CTE]]): 2, ~3 = 1901.930, ~22/21 = 78.254
+Badness: 1.077 × 10<sup>-3</sup>
-[[Patent val]] EDO tunings: [[31edo|31]], [[46edo|46]], [[77edo|77]], [[92edo|92]]
+==== Ishtar ====
+Subgroup: 2.3.5.7.11.13
-Dirichlet badness: 0.931
+Comma list: 91/90, 121/120, 441/440
-=== 19-limit ===
+Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}
-Subgroup: [[19-limit]]
-Comma list: 441/440, 1617/1600, 196/195, 154/153, 286/285
+Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.3952, ~21/20 = 78.9578
-Mapping: {{mapping| 1 0 1 3 2 7 6 9 | 0 1 1 0 1 -2 -1 -3 | 0 0 -4 -3 -2 -2 -5 0 }}
+{{Optimal ET sequence|legend=1| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
-Optimal tuning ([[CTE]]): 2, ~3/2 = 1901.564, ~22/21 = 78.235
+Badness: 1.151 × 10<sup>-3</sup>
-[[Patent val]] EDO tunings: [[31edo|31]], [[46edo|46]], [[77edo|77]]
-Dirichlet badness: 1.159
-=== 23-limit ===
-Subgroup: [[23-limit]]
-Comma list: 441/440, 1617/1600, 196/195, 154/153, 286/285, 161/160
-Mapping: {{mapping| 1 0 1 3 2 7 6 9 3 | 0 1 1 0 1 -2 -1 -3 1 | 0 0 -4 -3 -2 -2 -5 0 -1 }}
-Optimal tuning ([[CTE]]): 2, ~3 = 1901.711, ~23/22 = 78.205
-[[Patent val]] EDO tunings: [[31edo|31]], [[46edo|46]], [[77edo|77]]
-Dirichlet badness: 1.084
 [[Category:Temperament clans]]