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Limmic temperaments: 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.
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{{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.


== Blacksmith ==
== Blackwood ==
=== 5-limit (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.
 
=== 5-limit ===
[[Subgroup]]: 2.3.5
[[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


[[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


=== 7-limit ===
=== 7-limit ===
[[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of blacksmith]]
[[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]] ([[POTE]]): ~9/8 = 1\5, ~5/4 = 392.767
[[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 }}


{{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


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 }}


Optimal tuning (POTE): ~9/8 = 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 =====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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 }}


Optimal tuning (POTE): ~9/8 = 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 ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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 }}


Optimal tuning (POTE): ~9/8 = 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 =====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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 }}


Optimal tuning (POTE): ~9/8 = 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 ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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 }}


Optimal tuning (POTE): ~9/8 = 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 ==
== 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)


[[Sval]] [[mapping]]: [{{val| 5 8 0 }}, {{val| 0 0 1 }}]
{{Mapping|legend=2| 5 8 0 | 0 0 1 }}


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


[[Optimal tuning]] ([[subgroup POTE]]): ~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"