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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 ==
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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


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[[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


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{{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 =====
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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 ====
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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 =====
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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 ====
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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 }}
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: 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]]
Retrieved from "https://en.xen.wiki/w/Limmic_temperaments"