Xenharmonic Wiki

Porwell temperaments: Difference between revisions

← Older editNewer edit →
VisualWikitext
m Units & misc. cleanup. - redundant category
Switch to Sintel's badness, WE & CWE tunings (1/). + intro to hendecatonic and twothirdtonic
Line 28: Line 28:
: ''For the 5-limit version, see [[11th-octave temperaments #Hendecapent]].''  
: ''For the 5-limit version, see [[11th-octave temperaments #Hendecapent]].''  


The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represent [[9/7]].
The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represent [[9/7]]. It tempers out 10976/10935, the [[hemimage comma]], and may be described as the {{nowrap| 22 & 99 }} temperament, with [[99edo]] giving an almost perfect tuning.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 37: Line 37:
: mapping generators: ~16/15, ~3
: mapping generators: ~16/15, ~3


[[Optimal tuning]] ([[POTE]]): ~16/15 = 109.091{{c}}, ~3/2 = 703.054{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~16/15 = 109.0526{{c}}, ~3/2 = 702.8069{{c}}
: [[error map]]: {{val| -0.421 +0.431 +0.563 -0.265 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 702.9705{{c}}
: error map: {{val| 0.000 +1.015 +1.625 +0.751 }}


{{Optimal ET sequence|legend=1| 22, 55, 77, 99 }}
{{Optimal ET sequence|legend=1| 22, 55, 77, 99 }}


[[Badness]] (Smith): 0.041081
[[Badness]] (Sintel): 1.04


=== 11-limit ===
=== 11-limit ===
Line 50: Line 54:
Mapping: {{mapping| 11 0 43 -4 38 | 0 1 -1 2 0 }}
Mapping: {{mapping| 11 0 43 -4 38 | 0 1 -1 2 0 }}


Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 702.636{{c}}
Optimal tunings:
* WE: ~16/15 = 109.0977{{c}}, ~3/2 = 702.6801{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 702.6484{{c}}


{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176e, 275e }}
{{Optimal ET sequence|legend=0| 22, 55, 77, 99 }}


Badness (Smith): 0.046088
Badness (Sintel): 1.52


==== 13-limit ====
==== 13-limit ====
Line 63: Line 69:
Mapping: {{mapping| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}
Mapping: {{mapping| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}


Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 702.291{{c}}
Optimal tunings:
* WE: ~16/15 = 109.1092{{c}}, ~3/2 = 702.4093{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 702.2930{{c}}


{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176e }}
{{Optimal ET sequence|legend=0| 22, 55, 77, 99 }}


Badness (Smith): 0.040099
Badness (Sintel): 1.66


==== 17-limit ====
==== 17-limit ====
Line 76: Line 84:
Mapping: {{mapping| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }}
Mapping: {{mapping| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }}


Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 702.301{{c}}
Optimal tunings:
* WE: ~16/15 = 109.0933{{c}}, ~3/2 = 702.3170{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 702.3017{{c}}


{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176eg }}
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176eg }}


Badness (Smith): 0.029054
Badness (Sintel): 1.48


=== Cohendecatonic ===
=== Cohendecatonic ===
Line 89: Line 99:
Mapping: {{mapping| 11 0 43 -4 73 | 0 1 -1 2 -2 }}
Mapping: {{mapping| 11 0 43 -4 73 | 0 1 -1 2 -2 }}


Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 703.686{{c}}
Optimal tunings:
* WE: ~16/15 = 109.0237{{c}}, ~3/2 = 703.2522{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.6563{{c}}


{{Optimal ET sequence|legend=0| 22, 77e, 99e, 121, 220e }}
{{Optimal ET sequence|legend=0| 22, 77e, 99e, 121, 220e }}


Badness (Smith): 0.038042
Badness (Sintel): 1.26


==== 13-limit ====
==== 13-limit ====
Line 102: Line 114:
Mapping: {{mapping| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}
Mapping: {{mapping| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}


Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 703.888{{c}}
Optimal tunings:
* WE: ~16/15 = 109.0189{{c}}, ~3/2 = 703.4228{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.9248{{c}}


{{Optimal ET sequence|legend=0| 22, 77eff, 99ef, 121, 341bdeeff }}
{{Optimal ET sequence|legend=0| 22, 99ef, 121, 341bdeeff }}


Badness (Smith): 0.036112
Badness (Sintel): 1.49


==== 17-limit ====
==== 17-limit ====
Line 115: Line 129:
Mapping: {{mapping| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }}
Mapping: {{mapping| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }}


Optimal tuning (POTE): ~16/15 = 109.091{{c}}, ~3/2 = 703.877{{c}}
Optimal tunings:
* WE: ~16/15 = 109.0159{{c}}, ~3/2 = 703.3932{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.9110{{c}}


{{Optimal ET sequence|legend=0| 22, 77eff, 99ef, 121, 220efg, 341bdeeffgg }}
{{Optimal ET sequence|legend=0| 22, 99ef, 121, 220efg, 341bdeeffgg }}


Badness (Smith): 0.022590
Badness (Sintel): 1.15


=== Icosidillic ===
=== Icosidillic ===
Line 129: Line 145:
: mapping generators: ~33/32, ~3
: mapping generators: ~33/32, ~3


Optimal tuning (POTE): ~33/32 = 54.545{{c}}, ~3/2 = 702.914{{c}}
Optimal tunings:
* WE: ~33/32 = 54.5305{{c}}, ~3/2 = 702.7206{{c}}
* CWE: ~33/32 = 54.5455{{c}}, ~3/2 = 702.8829{{c}}


{{Optimal ET sequence|legend=0| 22, 154, 176, 198 }}
{{Optimal ET sequence|legend=0| 22, 154, 176, 198 }}


Badness (Smith): 0.057725
Badness (Sintel): 1.84


== Twothirdtonic ==
== Twothirdtonic ==
Twothirdtonic tempers out 686/675, the [[senga]], in addition to the porwell comma, and may be described as the {{nowrap| 37 & 46 }} temperament, generated by one third of a [[5/4|classical major third]] that represents [[15/14]], [[14/13]], and [[13/12]] in the [[13-limit]] interpretation. Note that in the data below, the generator is taken to be its [[octave complement]], thirteen of which [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]]; it follows that the [[ploidacot]] for this temperament is 11-sheared 13-cot. [[46edo]] may be recommended as a tuning.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 686/675, 6144/6125
[[Comma list]]: 686/675, 6144/6125


{{Mapping|legend=1| 1 3 2 4 | 0 -13 3 -11 }}
{{Mapping|legend=1| 1 -10 5 -7 | 0 13 -3 11 }}
: mapping generators: ~2, ~15/14
: mapping generators: ~2, ~28/15


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~15/14 = 130.401{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}}
: [[error map]]: {{val| -0.693 +1.736 +3.278 -5.176 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}}
: error map: {{val| 0.000 +2.515 +4.962 -3.505 }}


{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}


[[Badness]] (Smith): 0.099601
[[Badness]] (Sintel): 2.52


=== 11-limit ===
=== 11-limit ===
Line 154: Line 178:
Comma list: 121/120, 176/175, 686/675
Comma list: 121/120, 176/175, 686/675


Mapping: {{mapping| 1 3 2 4 4 | 0 -13 3 -11 -5 }}
Mapping: {{mapping| 1 -10 5 -7 -1 | 0 13 -3 11 5 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/14 = 130.430{{c}}
Optimal tunings:
* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}}


{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}


Badness (Smith): 0.040768
Badness (Sintel): 1.35


=== 13-limit ===
=== 13-limit ===
Line 167: Line 193:
Comma list: 91/90, 121/120, 169/168, 176/175
Comma list: 91/90, 121/120, 169/168, 176/175


Mapping: {{mapping| 1 3 2 4 4 5 | 0 -13 3 -11 -5 -12 }}
Mapping: {{mapping| 1 -10 5 -7 -1 -7 | 0 13 -3 11 5 12 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/13 = 130.409{{c}}
Optimal tunings:
* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}}


{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}
{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}


Badness (Smith): 0.025941
Badness (Sintel): 1.07


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