Blackwood family: Difference between revisions

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-{{interwiki
-| en = Limmic temperaments
-| de = Blackwood-Limmisch
-| es =
-| ja =
-}}
 {{Technical data page}}
-'''Limmic temperaments''' are [[temperament]]s that [[temper 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 a fifth, and some people are fond of it. All temperaments shown here are pentaploid acot.
+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 ==
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 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.
-=== 5-limit ===
 [[Subgroup]]: 2.3.5
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 [[Badness]] (Sintel): 1.50
-=== 7-limit ===
+== Septimal blackwood ==
+{{Main| Blackwood }}
 [[Subgroup]]: 2.3.5.7
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 [[Badness]] (Sintel): 0.649
-==== Undecimal blackwood ====
+=== Undecimal blackwood ===
 Subgroup: 2.3.5.7.11
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 Badness (Sintel): 0.815
-===== 13-limit =====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
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 Badness (Sintel): 0.847
-==== Farrier ====
+=== Farrier ===
 Subgroup: 2.3.5.7.11
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 Badness (Sintel): 0.965
-===== 13-limit =====
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
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 Badness (Sintel): 0.922
-==== Ferrum ====
+=== Ferrum ===
 Subgroup: 2.3.5.7.11
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 Badness (Sintel): 1.02
-== Blackweed ==
+== Quindecic ==
-Blackweed is a [[restriction]] of undecimal blackwood as it tempers out 256/243 alike but in the 2.3.11/7 [[subgroup]]. 20edo is close to the optimum, which has 4\20 as the period and 420{{c}} as the generator.
+Quindecic preserves the [[11-limit]] structure of [[15edo]], with an independent generator for [[13/1|harmonic 13]].
-[[Subgroup]]: 2.3.11/7
+[[Subgroup]]: 2.3.5.7.11.13
-[[Comma list]]: {{monzo| 8 -5 }} (256/243)
+[[Comma list]]: 28/27, 49/48, 55/54, 77/75
-{{Mapping|legend=2| 5 8 0 | 0 0 1 }}
+[[Mapping]]: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }}
+: mapping generators: ~22/21, ~13
-: sval mapping generators: ~9/8, ~11/7
+[[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}})
-[[Optimal tuning]]s:
+{{Optimal ET sequence|legend=1| 15, 30 }}
-* [[Tp tuning|subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}
-: [[error map]]: {{val| -5.746 +8.852 -0.035 }}
-* [[Tp tuning|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}
-: error map: {{val| 0.000 +18.045 +2.475 }}
-{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}
+[[Badness]] (Sintel): 1.20
-[[Category:Temperament collections]]
+[[Category:Blackwood family| ]] <!-- main article -->
-[[Category:Pages with mostly numerical content]]
+[[Category:Temperament families]]
-[[Category:Limmic temperaments]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]
-[[Category:Blackwood]]