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 ==
== 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
[[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
[[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
[[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 }}


{{Multival|legend=1| 0 5 0 8 0 -14 }}
{{Optimal ET sequence|legend=1| 5, 10, 15, 40b }}


[[POTE generator]]: ~5/4 = 392.767
[[Badness]] (Sintel): 0.649


{{Val list|legend=1| 5, 10, 15, 40b, 55b }}
=== Undecimal blackwood ===
 
[[Badness]]: 0.025640
 
==== 11-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 28/27, 49/48, 55/54
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
Subgroup: 2.3.5.7.11.13


Comma list: 28/27, 40/39, 49/48, 55/54
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
Subgroup: 2.3.5.7.11


Comma list: 28/27, 49/48, 77/75
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
Subgroup: 2.3.5.7.11.13


Comma list: 28/27, 40/39, 49/48, 66/65
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
Subgroup: 2.3.5.7.11


Comma list: 28/27, 35/33, 49/48
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 }}


POTE generator: ~5/4 = 374.763
Badness (Sintel): 1.02


Vals: {{Val list| 5e, 10 }}
== Quindecic ==
Quindecic preserves the [[11-limit]] structure of [[15edo]], with an independent generator for [[13/1|harmonic 13]].


Badness: 0.030883
[[Subgroup]]: 2.3.5.7.11.13
== Blackweed ==
So named because it's a 5L 5s temperament and the 20edo tuning has 4\20 as the period and 420¢ as the generator.


Subgroup: 2.3.14/11
[[Comma list]]: 28/27, 49/48, 55/54, 77/75


Commas: 256/243
[[Mapping]]: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }}
: mapping generators: ~22/21, ~13


Mapping: [⟨5 8 0], ⟨0 0 1]]  
[[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}})


Mapping generators: ~9/8, ~14/11
{{Optimal ET sequence|legend=1| 15, 30 }}


POTE generator: ~14/11 = 413.7785
[[Badness]] (Sintel): 1.20


Vals: {{Vals|20, 35b}}
[[Category:Blackwood family| ]] <!-- main article -->
[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Limma family]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Blackwood]]u

Latest revision as of 11:50, 14 August 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

Main article: 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

Optimal tunings:

  • 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

Main article: Blackwood

Subgroup: 2.3.5.7

Comma list: 28/27, 49/48

Mapping: [⟨5 8 0 14], ⟨0 0 1 0]]

Optimal tunings:

  • 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 ¢

Optimal ET sequence: 5e, 10

Badness (Sintel): 1.02

Quindecic

Quindecic preserves the 11-limit structure of 15edo, with an independent generator for harmonic 13.

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 49/48, 55/54, 77/75

Mapping: [⟨15 24 35 42 52 0], ⟨0 0 0 0 0 1]]

mapping generators: ~22/21, ~13

Optimal tunings:

  • WE: ~22/21 = 79.770 ¢, ~13/8 = 850.476 ¢ (~40/39 = 26.999 ¢)
  • CWE: ~22/21 = 80.000 ¢, ~13/8 = 850.793 ¢ (~40/39 = 29.207 ¢)

Optimal ET sequence: 15, 30

Badness (Sintel): 1.20

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