Blackwood family: Difference between revisions

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{{interwiki
{{Technical data page}}
| de = Blackwood-Limmisch
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.
| en = Limmic_temperaments
| es =
| ja =
}}


This '''limmic temperaments''' page collects various [[temperament]]s [[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.
== Blackwood ==
{{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.  


== Blacksmith ==
=== 5-limit (blackwood) ===
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 18: Line 17:
: mapping generators: ~9/8, ~5
: mapping generators: ~9/8, ~5


[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\5, ~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 }}


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


[[Badness]]: 0.063760
[[Badness]] (Sintel): 1.50


[[File:BlackwoodMajor 15edo.mp3]] [[:BlackwoodMajor 15edo.mp3|BlackwoodMajor 15edo.mp3]]
== Septimal blackwood ==
 
{{Main| Blackwood }}
Blackwood major scale in 15edo
 
=== 7-limit ===
[[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of blacksmith]]


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 37: Line 36:
{{Mapping|legend=1| 5 8 0 14 | 0 0 1 0 }}
{{Mapping|legend=1| 5 8 0 14 | 0 0 1 0 }}


{{Multival|legend=1| 0 5 0 8 0 -14 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~8/7 = 239.426{{c}}, ~5/4 = 391.828{{c}}
[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~5/4 = 392.767
: [[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 }}


{{Optimal ET sequence|legend=1| 5, 10, 15, 40b, 55b }}
{{Optimal ET sequence|legend=1| 5, 10, 15, 40b }}


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


=== 11-limit ===
=== Undecimal blackwood ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 52: Line 53:
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 = 1\5, ~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=1| 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 65: Line 68:
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 = 1\5, ~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=1| 5, 10, 15, 25e, 40bef }}
{{Optimal ET sequence|legend=0| 5, 10, 15, 25e }}


Badness: 0.020498
Badness (Sintel): 0.847


=== Farrier ===
=== Farrier ===
Line 78: Line 83:
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 = 1\5, ~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=1| 5e, 10e, 15 }}
{{Optimal ET sequence|legend=0| 5e, 10e, 15 }}


Badness: 0.029200
Badness (Sintel): 0.965


==== 13-limit ====
==== 13-limit ====
Line 91: Line 98:
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 = 1\5, ~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=1| 5e, 10e, 15 }}
{{Optimal ET sequence|legend=0| 5e, 10e, 15 }}


Badness: 0.022325
Badness (Sintel): 0.922


=== Ferrum ===
=== Ferrum ===
Line 104: Line 113:
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 = 1\5, ~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=1| 5e, 10 }}
{{Optimal ET sequence|legend=0| 5e, 10 }}


Badness: 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¢ 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]]): ~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]]

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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