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. All temperaments shown here are pentaploid acot.
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 ==
Line 15: Line 9:
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


Line 34: Line 27:
[[Badness]] (Sintel): 1.50
[[Badness]] (Sintel): 1.50


=== 7-limit ===
== Septimal blackwood ==
{{Main| Blackwood }}
 
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 51: Line 46:
[[Badness]] (Sintel): 0.649
[[Badness]] (Sintel): 0.649


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


Line 66: Line 61:
Badness (Sintel): 0.815
Badness (Sintel): 0.815


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


Line 81: Line 76:
Badness (Sintel): 0.847
Badness (Sintel): 0.847


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


Line 96: Line 91:
Badness (Sintel): 0.965
Badness (Sintel): 0.965


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


Line 111: Line 106:
Badness (Sintel): 0.922
Badness (Sintel): 0.922


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


Line 126: Line 121:
Badness (Sintel): 1.02
Badness (Sintel): 1.02


== Blackweed ==
== Quindecic ==
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{{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]]s:
{{Optimal ET sequence|legend=1| 15, 30 }}
* [[Tp tuning|subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}
: [[error map]]: {{val| -5.746 +8.852 -0.035 }}
* [[Tp tuning|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}
: error map: {{val| 0.000 +18.045 +2.475 }}


{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}
[[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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