Blackwood family: Difference between revisions

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-The '''limma family''' tempers out the Pythagorean limma, [[256/243]]. As a consequence, [[3/2]] is always represented by 3\5, 720 [[cent]]s assuming pure octaves. While quite sharp, this is close enough to a just fifth to serve as one, and some people are fond of it.
+{{Technical data page}}
+The '''blackwood family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] 256/243, the [[Pythagorean limma]]. As a consequence, [[3/2]] is always represented by 3\5, 720 [[cent]]s assuming pure octaves. While quite sharp, this is close enough to a just fifth to serve as a fifth, and some people are fond of it. All temperaments shown here are pentaploid acot.
 == Blackwood ==
-Subgroup: 2.3.5
+{{Main| Blackwood }}
+Blackwood is the 5edo [[circle of fifths]] with an independent dimension for the harmonic 5. It can be described as the {{nowrap| 5 & 10 }} temperament. [[15edo]] is an obvious tuning.
+The only extension to the 7-limit that makes any sense is to map the [[7/4|harmonic seventh]] to 4\5, tempering out [[28/27]], [[49/48]], and [[64/63]]. This is known as ''blacksmith'' in earlier materials, including [[Graham Breed]]'s temperament finder.
+[[Subgroup]]: 2.3.5
 [[Comma list]]: 256/243
-[[Mapping]]: [{{val| 5 8 0 }}, {{val| 0 0 1 }}]
+{{Mapping|legend=1| 5 8 0 | 0 0 1 }}
-Mapping generators: ~9/8, ~5
+: mapping generators: ~9/8, ~5
-[[POTE generator]]: ~5/4 = 399.594
+[[Optimal tuning]]s:
+* [[WE]]: ~8/7 = 238.851{{c}}, ~5/4 = 397.681{{c}}
+: [[error map]]: {{val| -5.746 +8.852 -0.124 }}
+* [[CWE]]: ~8/7 = 240.000{{c}}, ~5/4 = 395.126{{c}}
+: error map: {{val| 0.000 +18.045 +8.812 }}
-{{Val list|legend=1| 5, 10, 15 }}
+{{Optimal ET sequence|legend=1| 5, 10, 15 }}
-[[Badness]]: 0.063760
+[[Badness]] (Sintel): 1.50
-=== Blacksmith ===
+== Septimal blackwood ==
-[[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of blacksmith]]
+{{Main| Blackwood }}
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 28/27, 49/48
-[[Mapping]]: [{{val| 5 8 0 14 }}, {{val| 0 0 1 0 }}]
+{{Mapping|legend=1| 5 8 0 14 | 0 0 1 0 }}
-Mapping generators: ~7/6, ~5
-{{Multival|legend=1| 0 5 0 8 0 -14 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~8/7 = 239.426{{c}}, ~5/4 = 391.828{{c}}
+: [[error map]]: {{val| -2.870 +13.453 -0.225 -16.861 }}
+* [[CWE]]: ~8/7 = 240.000{{c}}, ~5/4 = 391.098{{c}}
+: error map: {{val| 0.000 +18.045 +4.784 -8.826 }}
-[[POTE generator]]: ~5/4 = 392.767
+{{Optimal ET sequence|legend=1| 5, 10, 15, 40b }}
-{{Val list|legend=1| 5, 10, 15, 40b, 55b }}
+[[Badness]] (Sintel): 0.649
-[[Badness]]: 0.025640
+=== Undecimal blackwood ===
-==== 11-limit ====
 Subgroup: 2.3.5.7.11
 Comma list: 28/27, 49/48, 55/54
-Mapping: [{{val| 5 8 0 14 29 }}, {{val| 0 0 1 0 -1 }}]
+Mapping: {{mapping| 5 8 0 14 29 | 0 0 1 0 -1 }}
-POTE generator: ~5/4 = 394.948
+Optimal tunings:
+* WE: ~8/7 = 239.341{{c}}, ~5/4 = 393.864{{c}}
+* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 394.655{{c}}
-Vals: {{Val list| 5, 10, 15, 40be, 55be, 70bde, 85bcde}}
+{{Optimal ET sequence|legend=0| 5, 10, 15, 40be }}
-Badness: 0.024641
+Badness (Sintel): 0.815
-===== 13-limit =====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
 Comma list: 28/27, 40/39, 49/48, 55/54
-Mapping: [{{val| 5 8 0 14 29 7 }}, {{val| 0 0 1 0 -1 1 }}]
+Mapping: {{mapping| 5 8 0 14 29 7 | 0 0 1 0 -1 1 }}
-POTE generator: ~5/4 = 391.037
+Optimal tunings:
+* WE: ~8/7 = 239.187{{c}}, ~5/4 = 389.713{{c}}
+* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 390.282{{c}}
-Vals: {{Val list| 5, 10, 15, 25e, 40bef}}
+{{Optimal ET sequence|legend=0| 5, 10, 15, 25e }}
-Badness: 0.020498
+Badness (Sintel): 0.847
-==== Farrier ====
+=== Farrier ===
 Subgroup: 2.3.5.7.11
 Comma list: 28/27, 49/48, 77/75
-Mapping: [{{val| 5 8 0 14 -6 }}, {{val| 0 0 1 0 2 }}]
+Mapping: {{mapping| 5 8 0 14 -6 | 0 0 1 0 2 }}
-POTE generator: ~5/4 = 398.070
+Optimal tunings:
+* WE: ~8/7 = 239.389{{c}}, ~5/4 = 397.056{{c}}
+* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 396.599{{c}}
-Vals: {{Val list| 5e, 10e, 15 }}
+{{Optimal ET sequence|legend=0| 5e, 10e, 15 }}
-Badness: 0.029200
+Badness (Sintel): 0.965
-===== 13-limit =====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
 Comma list: 28/27, 40/39, 49/48, 66/65
-Mapping: [{{val| 5 8 0 14 -6 7 }}, {{val| 0 0 1 0 2 1 }}]
+Mapping: {{mapping| 5 8 0 14 -6 7 | 0 0 1 0 2 1 }}
-POTE generator: ~5/4 = 396.812
+Optimal tunings:
+* WE: ~8/7 = 239.196{{c}}, ~5/4 = 395.483{{c}}
+* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 394.759{{c}}
-Vals: {{Val list| 5e, 10e, 15 }}
+{{Optimal ET sequence|legend=0| 5e, 10e, 15 }}
-Badness: 0.022325
+Badness (Sintel): 0.922
-==== Ferrum ====
+=== Ferrum ===
 Subgroup: 2.3.5.7.11
 Comma list: 28/27, 35/33, 49/48
-Mapping: [{{val| 5 8 0 14 6 }}, {{val| 0 0 1 0 1 }}]
+Mapping: {{mapping| 5 8 0 14 6 | 0 0 1 0 1 }}
+Optimal tunings:
+* WE: ~8/7 = 239.058{{c}}, ~5/4 = 373.292{{c}}
+* CWE: ~8/7 = 240.000{{c}}, ~5/4 = 371.659{{c}}
+{{Optimal ET sequence|legend=0| 5e, 10 }}
+Badness (Sintel): 1.02
+== Quindecic ==
+Quindecic preserves the [[11-limit]] structure of [[15edo]], with an independent generator for [[13/1|harmonic 13]].
+[[Subgroup]]: 2.3.5.7.11.13
+[[Comma list]]: 28/27, 49/48, 55/54, 77/75
+[[Mapping]]: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }}
+: mapping generators: ~22/21, ~13
-POTE generator: ~5/4 = 374.763
+[[Optimal tuning]]s:
+* [[WE]]: ~22/21 = 79.770{{c}}, ~13/8 = 850.476{{c}} (~40/39 = 26.999{{c}})
+* [[CWE]]: ~22/21 = 80.000{{c}}, ~13/8 = 850.793{{c}} (~40/39 = 29.207{{c}})
-Vals: {{Val list| 5e, 10 }}
+{{Optimal ET sequence|legend=1| 15, 30 }}
-Badness: 0.030883
+[[Badness]] (Sintel): 1.20
-[[Category:Regular temperament theory]]
+[[Category:Blackwood family| ]] <!-- main article -->
-[[Category:Temperament family]]
+[[Category:Temperament families]]
-[[Category:Limma family]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]
-[[Category:Blackwood]]