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Blackwood family: Difference between revisions

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{{interwiki
| en = Limmic temperaments
| de = Blackwood-Limmisch
| es =
| ja =
}}
{{Technical data page}}
{{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.
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 ==
== 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.  
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
[[Subgroup]]: 2.3.5


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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~9/8 = 240.000, ~5/4 = 386.314
* [[WE]]: ~8/7 = 238.851{{c}}, ~5/4 = 397.681{{c}}
: [[error map]]: {{val| 0.000 +18.045 0.000 }}
: [[error map]]: {{val| -5.746 +8.852 -0.124 }}
* [[POTE]]: ~9/8 = 240.000, ~5/4 = 399.594
* [[CWE]]: ~8/7 = 240.000{{c}}, ~5/4 = 395.126{{c}}
: error map: {{val| 0.000 +18.045 +13.280 }}
: error map: {{val| 0.000 +18.045 +8.812 }}


{{Optimal ET sequence|legend=1| 5, 10, 15 }}
{{Optimal ET sequence|legend=1| 5, 10, 15 }}


[[Badness]] (Smith): 0.063760
[[Badness]] (Sintel): 1.50
 
== Septimal blackwood ==
{{Main| Blackwood }}


=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~8/7 = 240.000, ~5/4 = 386.313
* [[WE]]: ~8/7 = 239.426{{c}}, ~5/4 = 391.828{{c}}
: [[error map]]: {{val| 0.000 +18.045 0.000 -8.826 }}
: [[error map]]: {{val| -2.870 +13.453 -0.225 -16.861 }}
* [[POTE]]: ~8/7 = 240.000, ~5/4 = 392.767
* [[CWE]]: ~8/7 = 240.000{{c}}, ~5/4 = 391.098{{c}}
: error map: {{val| 0.000 +18.045 +6.454 -8.826 }}
: error map: {{val| 0.000 +18.045 +4.784 -8.826 }}


{{Optimal ET sequence|legend=1| 5, 10, 15, 40b }}
{{Optimal ET sequence|legend=1| 5, 10, 15, 40b }}


[[Badness]] (Smith): 0.025640
[[Badness]] (Sintel): 0.649


==== Undecimal blackwood ====
=== Undecimal blackwood ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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Mapping: {{mapping| 5 8 0 14 29 | 0 0 1 0 -1 }}
Mapping: {{mapping| 5 8 0 14 29 | 0 0 1 0 -1 }}


Optimal tuning (POTE): ~8/7 = 240.000, ~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}}


{{Optimal ET sequence|legend=0| 5, 10, 15, 40be, 55be, 70bde, 85bcde }}
{{Optimal ET sequence|legend=0| 5, 10, 15, 40be }}


Badness (Smith): 0.024641
Badness (Sintel): 0.815


===== 13-limit =====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Mapping: {{mapping| 5 8 0 14 29 7 | 0 0 1 0 -1 1 }}
Mapping: {{mapping| 5 8 0 14 29 7 | 0 0 1 0 -1 1 }}


Optimal tuning (POTE): ~8/7 = 240.000, ~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}}


{{Optimal ET sequence|legend=0| 5, 10, 15, 25e, 40bef }}
{{Optimal ET sequence|legend=0| 5, 10, 15, 25e }}


Badness (Smith): 0.020498
Badness (Sintel): 0.847


==== Farrier ====
=== Farrier ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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Mapping: {{mapping| 5 8 0 14 -6 | 0 0 1 0 2 }}
Mapping: {{mapping| 5 8 0 14 -6 | 0 0 1 0 2 }}


Optimal tuning (POTE): ~8/7 = 240.000, ~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}}


{{Optimal ET sequence|legend=0| 5e, 10e, 15 }}
{{Optimal ET sequence|legend=0| 5e, 10e, 15 }}


Badness (Smith): 0.029200
Badness (Sintel): 0.965


===== 13-limit =====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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Mapping: {{mapping| 5 8 0 14 -6 7 | 0 0 1 0 2 1 }}
Mapping: {{mapping| 5 8 0 14 -6 7 | 0 0 1 0 2 1 }}


Optimal tuning (POTE): ~8/7 = 240.000, ~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}}


{{Optimal ET sequence|legend=0| 5e, 10e, 15 }}
{{Optimal ET sequence|legend=0| 5e, 10e, 15 }}


Badness (Smith): 0.022325
Badness (Sintel): 0.922


==== Ferrum ====
=== Ferrum ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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Mapping: {{mapping| 5 8 0 14 6 | 0 0 1 0 1 }}
Mapping: {{mapping| 5 8 0 14 6 | 0 0 1 0 1 }}


Optimal tuning (POTE): ~8/7 = 240.000, ~5/4 = 374.763
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 }}
{{Optimal ET sequence|legend=0| 5e, 10 }}


Badness (Smith): 0.030883
Badness (Sintel): 1.02


== Blackweed ==
== Quindecic ==
Blackweed is a variant of 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]] ([[Tp tuning|subgroup POTE]]): ~9/8 = 240.0000, ~11/7 = 786.2215
{{Optimal ET sequence|legend=1| 15, 30 }}


{{Optimal ET sequence|legend=1| 15, 20, 35b }}
[[Badness]] (Sintel): 1.20


[[Category:Temperament collections]]
[[Category:Blackwood family| ]] <!-- main article -->
[[Category:Limmic temperaments]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Rank 2]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Blackwood]]
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