Limmic temperaments: Difference between revisions

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{{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.
'''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.


== Blackwood ==
== Blackwood ==
Line 25: Line 25:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~9/8 = 240.000, ~5/4 = 386.314
* [[WE]]: ~8/7 = 238.851{{c}}, ~5/4 = 397.681{{c}}
: [[error map]]: {{val| 0.000 +18.045 0.000 }}
: [[error map]]: {{val| -5.746 +8.852 -0.124 }}
* [[POTE]]: ~9/8 = 240.000, ~5/4 = 399.594
* [[CWE]]: ~8/7 = 240.000{{c}}, ~5/4 = 395.126{{c}}
: error map: {{val| 0.000 +18.045 +13.280 }}
: 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]] (Smith): 0.063760
[[Badness]] (Sintel): 1.50


=== 7-limit (blacksmith) ===
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 40: Line 40:


{{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:  
[[Optimal tuning]]s:  
* [[CTE]]: ~8/7 = 240.000, ~5/4 = 386.313
* [[WE]]: ~8/7 = 239.426{{c}}, ~5/4 = 391.828{{c}}
: [[error map]]: {{val| 0.000 +18.045 0.000 -8.826 }}
: [[error map]]: {{val| -2.870 +13.453 -0.225 -16.861 }}
* [[POTE]]: ~8/7 = 240.000, ~5/4 = 392.767
* [[CWE]]: ~8/7 = 240.000{{c}}, ~5/4 = 391.098{{c}}
: error map: {{val| 0.000 +18.045 +6.454 -8.826 }}
: error map: {{val| 0.000 +18.045 +4.784 -8.826 }}


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


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


==== Undecimal blackwood ====
==== Undecimal blackwood ====
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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 = 240.000, ~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=0| 5, 10, 15, 40be, 55be, 70bde, 85bcde }}
{{Optimal ET sequence|legend=0| 5, 10, 15, 40be }}


Badness (Smith): 0.024641
Badness (Sintel): 0.815


===== 13-limit =====
===== 13-limit =====
Line 73: Line 73:
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 = 240.000, ~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=0| 5, 10, 15, 25e, 40bef }}
{{Optimal ET sequence|legend=0| 5, 10, 15, 25e }}


Badness (Smith): 0.020498
Badness (Sintel): 0.847


==== Farrier ====
==== Farrier ====
Line 86: Line 88:
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 = 240.000, ~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=0| 5e, 10e, 15 }}
{{Optimal ET sequence|legend=0| 5e, 10e, 15 }}


Badness (Smith): 0.029200
Badness (Sintel): 0.965


===== 13-limit =====
===== 13-limit =====
Line 99: Line 103:
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 = 240.000, ~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=0| 5e, 10e, 15 }}
{{Optimal ET sequence|legend=0| 5e, 10e, 15 }}


Badness (Smith): 0.022325
Badness (Sintel): 0.922


==== Ferrum ====
==== Ferrum ====
Line 112: Line 118:
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 = 240.000, ~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=0| 5e, 10 }}
{{Optimal ET sequence|legend=0| 5e, 10 }}


Badness (Smith): 0.030883
Badness (Sintel): 1.02


== Blackweed ==
== Blackweed ==
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.
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.


[[Subgroup]]: 2.3.11/7
[[Subgroup]]: 2.3.11/7


[[Comma list]]: {{monzo| 8 -5 }} = 256/243
[[Comma list]]: {{monzo| 8 -5 }} (256/243)


{{Mapping|legend=2| 5 8 0 | 0 0 1 }}
{{Mapping|legend=2| 5 8 0 | 0 0 1 }}
Line 129: Line 137:
: sval mapping generators: ~9/8, ~11/7
: sval mapping generators: ~9/8, ~11/7


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~9/8 = 240.000, ~11/7 = 786.2215
[[Optimal tuning]]s:
* [[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 }}
{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Limmic temperaments]] <!-- main article -->
[[Category:Limmic temperaments]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Blackwood]]
[[Category:Blackwood]]

Latest revision as of 10:42, 20 July 2025

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.

Limmic temperaments are temperaments that temper out the Pythagorean limma, 256/243. 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.

5-limit

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

7-limit

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

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)

Sval mapping: [⟨5 8 0], ⟨0 0 1]]

sval mapping generators: ~9/8, ~11/7

Optimal tunings:

  • subgroup WE: ~8/7 = 238.851 ¢, ~11/7 = 782.457 ¢
error map: ⟨-5.746 +8.852 -0.035]
  • subgroup CWE: ~8/7 = 240.000 ¢, ~11/7 = 784.967 ¢
error map: ⟨0.000 +18.045 +2.475]

Optimal ET sequence: 15, 20, 35b, 55b

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