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Porwell temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (3/)
Switch to Sintel's badness, WE & CWE tunings (4/4)
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: mapping generators: ~1323/1280, ~3
: mapping generators: ~1323/1280, ~3


[[Optimal tuning]] ([[POTE]]): ~1323/1280 = 52.1739{{c}}, ~64/63 = 29.3586{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~1323/1280 = 52.1732{{c}}, ~3/2 = 701.0660{{c}}
: [[error map]]: {{val| -0.017 -0.906 +1.679 -0.386 }}
* [[CWE]]: ~1323/1280 = 52.1739{{c}}, ~3/2 = 701.0722{{c}}
: error map: {{val| 0.000 -0.883 +1.715 -0.333 }}


{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}


[[Badness]] (Smith): 0.196622
[[Badness]] (Sintel): 4.98


=== 11-limit ===
=== 11-limit ===
Line 533: Line 537:
Mapping: {{mapping| 23 0 17 101 116 | 0 1 1 -1 -1 }}
Mapping: {{mapping| 23 0 17 101 116 | 0 1 1 -1 -1 }}


Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.3980{{c}}
Optimal tunings:
* WE: ~33/32 = 52.1740{{c}}, ~3/2 = 701.0379{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0370{{c}}


{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


Badness (Smith): 0.064613
Badness (Sintel): 2.14


=== 13-limit ===
=== 13-limit ===
Line 546: Line 552:
Mapping: {{mapping| 23 0 17 101 116 158 | 0 1 1 -1 -1 -2 }}
Mapping: {{mapping| 23 0 17 101 116 158 | 0 1 1 -1 -1 -2 }}


Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.2830{{c}}
Optimal tunings:
* WE: ~33/32 = 52.1724{{c}}, ~3/2 = 701.1310{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.1524{{c}}


{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


Badness (Smith): 0.040484
Badness (Sintel): 1.67


=== 17-limit ===
=== 17-limit ===
Line 559: Line 567:
Mapping: {{mapping| 23 0 17 101 116 158 94 | 0 1 1 -1 -1 -2 0 }}
Mapping: {{mapping| 23 0 17 101 116 158 94 | 0 1 1 -1 -1 -2 0 }}


Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.2800{{c}}
Optimal tunings:
* WE: ~33/32 = 52.1735{{c}}, ~3/2 = 701.1493{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.1549{{c}}


{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


Badness (Smith): 0.024676
Badness (Sintel): 1.26


=== 19-limit ===
=== 19-limit ===
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Mapping: {{mapping| 23 0 17 101 116 158 94 207 | 0 1 1 -1 -1 -2 0 -3 }}
Mapping: {{mapping| 23 0 17 101 116 158 94 207 | 0 1 1 -1 -1 -2 0 -3 }}


Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.3760{{c}}
Optimal tunings:
* WE: ~33/32 = 52.1744{{c}}, ~3/2 = 701.0649{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0582{{c}}


{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}


Badness (Smith): 0.021579
Badness (Sintel): 1.31


=== 23-limit ===
=== 23-limit ===
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Mapping: {{mapping| 23 0 17 101 116 158 94 207 104 | 0 1 1 -1 -1 -2 0 -3 0 }}
Mapping: {{mapping| 23 0 17 101 116 158 94 207 104 | 0 1 1 -1 -1 -2 0 -3 0 }}


Optimal tuning (POTE): ~33/32 = 52.1739{{c}}, ~64/63 = 29.3471{{c}}
Optimal tunings:
* WE: ~33/32 = 52.1768{{c}}, ~3/2 = 701.1259{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0841{{c}}


{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368ci }}
{{Optimal ET sequence|legend=0| 46, 115, 161, 207 }}


Badness (Smith): 0.017745
Badness (Sintel): 1.27


== Absurdity ==
== Absurdity ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Absurdity (5-limit)]].''
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Absurdity (5-limit)]].''
{{See also| Fifth-chroma temperaments }}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 601: Line 616:
: mapping generators: ~972/875, ~3
: mapping generators: ~972/875, ~3


[[Optimal tuning]] ([[POTE]]): ~972/875 = 171.4286{{c}}, ~3/2 = 700.5854{{c}} (or ~10/9 = 186.2997{{c}})
[[Optimal tuning]]s:
* [[WE]]: ~972/875 = 171.4382{{c}}, ~3/2 = 700.6247{{c}}
: [[error map]]: {{val| +0.067 -1.263 +1.313 +0.450 }}
* [[CWE]]: ~972/875 = 171.4286{{c}}, ~3/2 = 700.5871{{c}}
: error map: {{val| 0.000 -1.368 +1.162 +0.254 }}


{{Optimal ET sequence|legend=1| 77, 84, 161 }}
{{Optimal ET sequence|legend=1| 77, 84, 161 }}


[[Badness]] (Smith): 0.133520
[[Badness]] (Sintel): 3.38


=== 11-limit ===
=== 11-limit ===
Line 614: Line 633:
Mapping: {{mapping| 7 0 -17 64 124 | 0 1 3 -4 -9 }}
Mapping: {{mapping| 7 0 -17 64 124 | 0 1 3 -4 -9 }}


Optimal tuning (POTE): ~495/448 = 171.4286{{c}}, ~3/2 = 700.6354{{c}} (or ~10/9 = 186.3497{{c}})
Optimal tunings:
* WE: ~495/448 = 171.4346{{c}}, ~3/2 = 700.6602{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 700.6339{{c}}


{{Optimal ET sequence|legend=0| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


Badness (Smith): 0.081564
Badness (Sintel): 2.70


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 7 0 -17 64 124 37 | 0 1 3 -4 -9 -1 }}
Mapping: {{mapping| 7 0 -17 64 124 37 | 0 1 3 -4 -9 -1 }}


Optimal tuning (POTE): ~72/65 = 171.4286{{c}}, ~3/2 = 700.6291{{c}} (or ~10/9 = 186.3434{{c}})
Optimal tunings:
* WE: ~72/65 = 171.4223{{c}}, ~3/2 = 700.6036{{c}}
* CWE: ~72/65 = 171.4286{{c}}, ~3/2 = 700.6306{{c}}


{{Optimal ET sequence|legend=0| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}


Badness (Smith): 0.041600
Badness (Sintel): 1.72


=== 17-limit ===
=== 17-limit ===
Line 640: Line 663:
Mapping: {{mapping| 7 0 -17 64 124 37 -49 | 0 1 3 -4 -9 -1 7 }}
Mapping: {{mapping| 7 0 -17 64 124 37 -49 | 0 1 3 -4 -9 -1 7 }}


Optimal tuning (POTE): ~72/65 = 171.4286{{c}}, ~3/2 = 700.6524{{c}} (or ~10/9 = 186.3667{{c}})
Optimal tunings:
* WE: ~72/65 = 171.4263{{c}}, ~3/2 = 700.6429{{c}}
* CWE: ~72/65 = 171.4286{{c}}, ~3/2 = 700.6525{{c}}


{{Optimal ET sequence|legend=0| 77, 161 }}
{{Optimal ET sequence|legend=0| 77, 161 }}


Badness (Smith): 0.031783
Badness (Sintel): 1.62


=== 19-limit ===
=== 19-limit ===
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Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 | 0 1 3 -4 -9 -1 7 -3 }}
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 | 0 1 3 -4 -9 -1 7 -3 }}


Optimal tuning (POTE): ~21/19 = 171.4286{{c}}, ~3/2 = 700.6565{{c}} (or ~10/9 = 186.3708{{c}})
Optimal tunings:
* WE: ~21/19 = 171.4244{{c}}, ~3/2 = 700.6395{{c}}
* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6568{{c}}


{{Optimal ET sequence|legend=0| 77, 161 }}
{{Optimal ET sequence|legend=0| 77, 161 }}


Badness (Smith): 0.022291
Badness (Sintel): 1.36


=== 23-limit ===
=== 23-limit ===
Line 666: Line 693:
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 | 0 1 3 -4 -9 -1 7 -3 -4 }}
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 | 0 1 3 -4 -9 -1 7 -3 -4 }}


Optimal tuning ([[CTE]]): ~21/19 = 171.429{{c}}, ~3/2 = 700.629{{c}} (or ~10/9 = 186.343{{c}})
Optimal tunings:
* WE: ~21/19 = 171.4321{{c}}, ~3/2 = 700.6475{{c}}
* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6325{{c}}


{{Optimal ET sequence|legend=0| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
Badness (Sintel): 1.34


=== 29-limit ===
=== 29-limit ===
{{See also| Fifth-chroma temperaments }}
Subgroup: 2.3.5.7.11.13.17.19.23.29
Subgroup: 2.3.5.7.11.13.17.19.23.29


Line 679: Line 708:
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 34 | 0 1 3 -4 -9 -1 7 -3 -4 0 }}
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 34 | 0 1 3 -4 -9 -1 7 -3 -4 0 }}


Optimal tuning ([[CTE]]): ~21/19 = 171.429{{c}}, ~3/2 = 700.629{{c}} (or ~10/9 = 186.343{{c}})
Optimal tunings:
* WE: ~21/19 = 171.4348{{c}}, ~3/2 = 700.6612{{c}}
* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6351{{c}}


{{Optimal ET sequence|legend=0| 77, 84, 161 }}
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
Badness (Sintel): 1.25


== Dodifo ==
== Dodifo ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Dodifo]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Dodifo]].''


Also named by [[Petr Pařízek]] in 2011, ''dodifo'' refers to the (tetraptolemaic) double-diminished fourth, which is a generator of this temperament<ref name="petr's long post"/>. The extension here is a less accurate 7-limit intepretation.  
Also named by [[Petr Pařízek]] in 2011, ''dodifo'' refers to the (tetraptolemaic) double-diminished fourth, which is a generator of this temperament<ref name="petr's long post"/>. The extension here is a less accurate 7-limit interpretation.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 692: Line 725:
[[Comma list]]: 6144/6125, 2500000/2470629
[[Comma list]]: 6144/6125, 2500000/2470629


{{Mapping|legend=1| 1 12 5 4 | 0 -35 -9 -4 }}
{{Mapping|legend=1| 1 -23 -4 0 | 0 35 9 4 }}
: mapping generators: ~2, ~80/49


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/40 = 357.070{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6429{{c}}, ~80/49 = 842.6790{{c}}
: [[error map]]: {{val| -0.357 +0.228 -0.774 +1.890 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9243{{c}}
: error map: {{val| 0.000 +0.396 +0.005 +2.871 }}


{{Optimal ET sequence|legend=1| 37, 84, 121, 205 }}
{{Optimal ET sequence|legend=1| 37, 84, 121, 205 }}


[[Badness]] (Smith): 0.179692
[[Badness]] (Sintel): 4.55


=== 11-limit ===
=== 11-limit ===
Line 705: Line 743:
Comma list: 1375/1372, 2560/2541, 4375/4356
Comma list: 1375/1372, 2560/2541, 4375/4356


Mapping: {{mapping| 1 12 5 4 -1 | 0 -35 -9 -4 15 }}
Mapping: {{mapping| 1 -23 -4 0 14 | 0 35 9 4 -15 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/40 = 357.048{{c}}
Optimal tunings:
* WE: ~2 = 1199.3401{{c}}, ~80/49 = 842.4880{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/49 = 842.9457{{c}}


{{Optimal ET sequence|legend=0| 37, 84, 121, 326dee }}
{{Optimal ET sequence|legend=0| 37, 84, 121, 326dee }}


Badness (Smith): 0.081923
Badness (Sintel): 2.71


=== 13-limit ===
=== 13-limit ===
Line 720: Line 760:
Mapping: {{mapping| 1 12 5 4 -1 4 | 0 -35 -9 -4 15 -1 }}
Mapping: {{mapping| 1 12 5 4 -1 4 | 0 -35 -9 -4 15 -1 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/13 = 357.049{{c}}
Optimal tunings:
* WE: ~2 = 1199.3410{{c}}, ~13/8 = 842.4885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9466{{c}}


{{Optimal ET sequence|legend=0| 37, 84, 121, 326deef }}
{{Optimal ET sequence|legend=0| 37, 84, 121, 326deef }}


Badness (Smith): 0.039533
Badness (Sintel): 1.63


== References ==
== References ==
Retrieved from "https://en.xen.wiki/w/Porwell_temperaments"