Blackwood family: Difference between revisions
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''' | 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. | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
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[[Badness]] (Sintel): 1.50 | [[Badness]] (Sintel): 1.50 | ||
== | == Septimal blackwood == | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Mapping|legend=2| 5 8 0 | 0 0 1 }} | {{Mapping|legend=2| 5 8 0 | 0 0 1 }} | ||
: mapping generators: ~9/8, ~11/7 | |||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }} | {{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }} | ||
[[Category: | [[Category:Blackwood family| ]] <!-- main article --> | ||
[[Category:Temperaments families]] | |||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
Revision as of 10:44, 28 June 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The blackwood family of temperaments tempers out 256/243, the Pythagorean limma. As a consequence, 3/2 is always represented by 3\5, 720 cents 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 is the 5edo circle of fifths with an independent dimension for the harmonic 5. It can be described as the 5 & 10 temperament. 15edo is an obvious tuning.
The only extension to the 7-limit that makes any sense is to map the 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: [⟨5 8 0], ⟨0 0 1]]
- mapping generators: ~9/8, ~5
- WE: ~8/7 = 238.851 ¢, ~5/4 = 397.681 ¢
- error map: ⟨-5.746 +8.852 -0.124]
- CWE: ~8/7 = 240.000 ¢, ~5/4 = 395.126 ¢
- error map: ⟨0.000 +18.045 +8.812]
Optimal ET sequence: 5, 10, 15
Badness (Sintel): 1.50
Septimal blackwood
Subgroup: 2.3.5.7
Comma list: 28/27, 49/48
Mapping: [⟨5 8 0 14], ⟨0 0 1 0]]
- WE: ~8/7 = 239.426 ¢, ~5/4 = 391.828 ¢
- error map: ⟨-2.870 +13.453 -0.225 -16.861]
- CWE: ~8/7 = 240.000 ¢, ~5/4 = 391.098 ¢
- error map: ⟨0.000 +18.045 +4.784 -8.826]
Optimal ET sequence: 5, 10, 15, 40b
Badness (Sintel): 0.649
Undecimal blackwood
Subgroup: 2.3.5.7.11
Comma list: 28/27, 49/48, 55/54
Mapping: [⟨5 8 0 14 29], ⟨0 0 1 0 -1]]
Optimal tunings:
- WE: ~8/7 = 239.341 ¢, ~5/4 = 393.864 ¢
- CWE: ~8/7 = 240.000 ¢, ~5/4 = 394.655 ¢
Optimal ET sequence: 5, 10, 15, 40be
Badness (Sintel): 0.815
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 28/27, 40/39, 49/48, 55/54
Mapping: [⟨5 8 0 14 29 7], ⟨0 0 1 0 -1 1]]
Optimal tunings:
- WE: ~8/7 = 239.187 ¢, ~5/4 = 389.713 ¢
- CWE: ~8/7 = 240.000 ¢, ~5/4 = 390.282 ¢
Optimal ET sequence: 5, 10, 15, 25e
Badness (Sintel): 0.847
Farrier
Subgroup: 2.3.5.7.11
Comma list: 28/27, 49/48, 77/75
Mapping: [⟨5 8 0 14 -6], ⟨0 0 1 0 2]]
Optimal tunings:
- WE: ~8/7 = 239.389 ¢, ~5/4 = 397.056 ¢
- CWE: ~8/7 = 240.000 ¢, ~5/4 = 396.599 ¢
Optimal ET sequence: 5e, 10e, 15
Badness (Sintel): 0.965
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 28/27, 40/39, 49/48, 66/65
Mapping: [⟨5 8 0 14 -6 7], ⟨0 0 1 0 2 1]]
Optimal tunings:
- WE: ~8/7 = 239.196 ¢, ~5/4 = 395.483 ¢
- CWE: ~8/7 = 240.000 ¢, ~5/4 = 394.759 ¢
Optimal ET sequence: 5e, 10e, 15
Badness (Sintel): 0.922
Ferrum
Subgroup: 2.3.5.7.11
Comma list: 28/27, 35/33, 49/48
Mapping: [⟨5 8 0 14 6], ⟨0 0 1 0 1]]
Optimal tunings:
- WE: ~8/7 = 239.058 ¢, ~5/4 = 373.292 ¢
- CWE: ~8/7 = 240.000 ¢, ~5/4 = 371.659 ¢
Badness (Sintel): 1.02
Blackweed
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 ¢ as the generator.
Subgroup: 2.3.11/7
Comma list: [8 -5⟩ (256/243)
Subgroup-val mapping: [⟨5 8 0], ⟨0 0 1]]
- mapping generators: ~9/8, ~11/7
- error map: ⟨-5.746 +8.852 -0.035]
- error map: ⟨0.000 +18.045 +2.475]
Optimal ET sequence: 15, 20, 35b, 55b