Blackwood family: Difference between revisions

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This '''limmic temperaments''' page collects various temperaments tempering 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.
{{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.


== Blacksmith ==
== Blackwood ==
=== 5-limit (blackwood) ===
{{Main| Blackwood }}
Subgroup: 2.3.5
 
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


=== 7-limit ===
== 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


{{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
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}}


Optimal GPV sequence: {{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 ====
Line 54: Line 66:
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}}


Optimal GPV sequence: {{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 ===
Line 67: Line 81:
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}}


Optimal GPV sequence: {{Val list| 5e, 10e, 15 }}
{{Optimal ET sequence|legend=0| 5e, 10e, 15 }}


Badness: 0.029200
Badness (Sintel): 0.965


==== 13-limit ====
==== 13-limit ====
Line 80: Line 96:
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}}


Optimal GPV sequence: {{Val list| 5e, 10e, 15 }}
{{Optimal ET sequence|legend=0| 5e, 10e, 15 }}


Badness: 0.022325
Badness (Sintel): 0.922


=== Ferrum ===
=== Ferrum ===
Line 93: Line 111:
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 }}


POTE generator: ~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 GPV sequence: {{Val list| 5e, 10 }}
{{Optimal ET sequence|legend=0| 5e, 10 }}


Badness: 0.030883
Badness (Sintel): 1.02


== Blackweed ==
== Quindecic ==
Blackweed is so named because the 20EDO tuning has 4\20 as the period and 420¢ 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]]: 256/243
[[Comma list]]: 28/27, 49/48, 55/54, 77/75


[[Mapping]]: [{{val| 5 8 0 }}, {{val| 0 0 1 }}]
[[Mapping]]: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }}
: mapping generators: ~22/21, ~13


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}})


[[POTE generator]]: ~14/11 = 413.7785
{{Optimal ET sequence|legend=1| 15, 30 }}


{{Val list|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]]

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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