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Meantone family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus (2/)
Switch to Sintel's badness, WE & CWE tunings, per community consensus (3/)
Line 1,552: Line 1,552:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~128/105 = 348.437{{c}}
* [[WE]]: ~2 = 1200.8160{{c}}, ~128/105 = 348.6518{{c}}
: [[error map]]: {{val| +0.816 -3.835 +2.901 +0.900 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 348.4194{{c}}
: error map: {{val| 0.000 -5.116 +1.041 -1.439 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~128/105 = 348.437{{c}}
: [[error map]]: {{val| 0.000 -5.080 +1.186 -1.637 }}
: [[error map]]: {{val| 0.000 -5.080 +1.186 -1.637 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~128/105 = 348.415{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~128/105 = 348.415{{c}}
: error map: {{val| 0.000 -5.125 +1.005 -1.390 }}
: error map: {{val| 0.000 -5.125 +1.005 -1.390 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 1,571: Line 1,575:
{{Optimal ET sequence|legend=1| 7, 24, 31 }}
{{Optimal ET sequence|legend=1| 7, 24, 31 }}


[[Badness]] (Smith): 0.055714
[[Badness]] (Sintel): 1.41


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,583: Line 1,587:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 348.561{{c}}
* WE: ~2 = 1201.1562{{c}}, ~11/9 = 348.8124{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 348.477{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.4910{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/9 = 348.561{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 348.477{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 1,597: Line 1,603:
{{Optimal ET sequence|legend=0| 7, 24, 31 }}
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


Badness (Smith): 0.026064
Badness (Sintel): 0.862


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,609: Line 1,615:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 348.587{{c}}
* WE: ~2 = 1200.4256{{c}}, ~11/9 = 348.6819{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 348.558{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5622{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/9 = 348.587{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 348.558{{c}} -->


{{Optimal ET sequence|legend=0| 7, 24, 31 }}
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


Badness (Smith): 0.023388
Badness (Sintel): 0.966


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,626: Line 1,634:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 348.735{{c}}
* WE: ~2 = 1200.0382{{c}}, ~11/9 = 348.7471{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 348.736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.7360{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/9 = 348.735{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 348.736{{c}} -->


{{Optimal ET sequence|legend=0| 7, 24, 31 }}
{{Optimal ET sequence|legend=0| 7, 24, 31 }}


Badness (Smith): 0.020576
Badness (Sintel): 1.05


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,643: Line 1,653:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 348.821{{c}}
* WE: ~2 = 1199.7469{{c}}, ~11/9 = 348.7367{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 348.810{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.8117{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/9 = 348.821{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 348.810{{c}} -->


{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }}
{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }}


Badness (Smith): 0.017302
Badness (Sintel): 1.05


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,662: Line 1,674:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~25/21 = 350.352{{c}}
* [[WE]]: ~2 = 1199.0661{{c}}, ~25/21 = 350.3127{{c}}
: [[error map]]: {{val| -0.934 -2.264 +16.188 -13.827 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 350.4856{{c}}
: error map: {{val| 0.000 -0.984 +17.571 -12.513 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~25/21 = 350.352{{c}}
: [[error map]]: {{val| 0.000 -1.250 +16.505 -14.245 }}
: [[error map]]: {{val| 0.000 -1.250 +16.505 -14.245 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~25/21 = 350.586{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~25/21 = 350.586{{c}}
: error map: {{val| 0.000 -0.784 +18.370 -11.214 }}
: error map: {{val| 0.000 -0.784 +18.370 -11.214 }} -->


{{Optimal ET sequence|legend=1| 7d, 17c, 24 }}
{{Optimal ET sequence|legend=1| 7d, 17c, 24 }}


[[Badness]] (Smith): 0.077734
[[Badness]] (Sintel): 1.97


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 1,681: Line 1,697:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 350.347{{c}}
* WE: ~2 = 1199.1924{{c}}, ~11/9 = 350.3286{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 350.565{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4821{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/9 = 350.347{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 350.565{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 17c, 24 }}
{{Optimal ET sequence|legend=0| 7d, 17c, 24 }}


Badness: 0.036207
Badness (Sintel): 1.20


Scales (Smith): [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]


=== 13-limit ===
=== 13-limit ===
Line 1,698: Line 1,716:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/9 = 350.365{{c}}
* WE: ~2 = 1198.5986{{c}}, ~11/9 = 350.3353{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 350.745{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6459{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/9 = 350.365{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/9 = 350.745{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }}
{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }}


Badness: 0.028738
Badness (Sintel): 1.19


Scales (Smith): [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]


== Liese ==
== Liese ==
Line 1,721: Line 1,741:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~10/7 = 632.783{{c}}
* [[WE]]: ~2 = 1201.5548{{c}}, ~10/7 = 633.2251{{c}}
: [[error map]]: {{val| +1.555 -2.280 +6.168 -8.015 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 632.5640{{c}}
: error map: {{val| 0.000 -4.263 +4.454 -10.622 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/7 = 632.783{{c}}
: [[error map]]: {{val| 0.000 -3.606 +7.084 -8.212 }}
: [[error map]]: {{val| 0.000 -3.606 +7.084 -8.212 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/7 = 632.406{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/7 = 632.406{{c}}
: error map: {{val| 0.000 -4.738 +2.554 -12.363 }}
: error map: {{val| 0.000 -4.738 +2.554 -12.363 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 1,735: Line 1,759:
{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}
{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}


[[Badness]] (Smith): 0.046706
[[Badness]] (Sintel): 1.18


=== Liesel ===
=== Liesel ===
Line 1,745: Line 1,769:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~10/7 = 632.812{{c}}
* WE: ~2 = 1198.8507{{c}}, ~10/7 = 632.4668{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 633.073{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 632.9963{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/7 = 632.812{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 633.073{{c}} -->


{{Optimal ET sequence|legend=0| 17c, 19, 36 }}
{{Optimal ET sequence|legend=0| 17c, 19, 36 }}


Badness (Smith): 0.040721
Badness (Sintel): 1.35


==== 13-limit ====
==== 13-limit ====
Line 1,760: Line 1,786:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~10/7 = 632.925{{c}}
* WE: ~2 = 1199.4968{{c}}, ~10/7 = 632.7766{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 633.042{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.0082{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/7 = 632.925{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 633.042{{c}} -->


{{Optimal ET sequence|legend=0| 17c, 19, 36, 91ceef }}
{{Optimal ET sequence|legend=0| 17c, 19, 36 }}


Badness (Smith): 0.027304
Badness (Sintel): 1.13


=== Elisa ===
=== Elisa ===
Line 1,775: Line 1,803:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~10/7 = 633.317{{c}}
* WE: ~2 = 1201.0489{{c}}, ~10/7 = 633.6147{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 633.061{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1644{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/7 = 633.317{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 633.061{{c}} -->


{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}


Badness (Smith): 0.041592
Badness (Sintel): 1.37


==== 13-limit ====
==== 13-limit ====
Line 1,790: Line 1,820:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~10/7 = 633.370{{c}}
* WE: ~2 = 1201.4815{{c}}, ~10/7 = 633.7720{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 632.991{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1281{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/7 = 633.370{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 632.991{{c}} -->


{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}
{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}


Badness (Smith): 0.026922
Badness (Sintel): 1.11


=== Lisa ===
=== Lisa ===
Line 1,805: Line 1,837:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~10/7 = 631.981{{c}}
* WE: ~2 = 1202.6773{{c}}, ~10/7 = 632.7783{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 631.370{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.6175{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/7 = 631.981{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 631.370{{c}} -->


{{Optimal ET sequence|legend=0| 17cee, 19 }}
{{Optimal ET sequence|legend=0| 17cee, 19 }}


Badness (Smith): 0.054829
Badness (Sintel): 1.81


==== 13-limit ====
==== 13-limit ====
Line 1,820: Line 1,854:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~10/7 = 632.093{{c}}
* WE: ~2 = 1203.6086{{c}}, ~10/7 = 633.1193{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 631.221{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.5346{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~10/7 = 632.093{{c}}
* POTE: ~2 = 1200.000{{c}}, ~10/7 = 631.221{{c}} -->


{{Optimal ET sequence|legend=0| 17cee, 19 }}
{{Optimal ET sequence|legend=0| 17cee, 19 }}


Badness (Smith): 0.036144
Badness (Sintel): 1.49


== Superpine ==
== Superpine ==
{{see also|No-sevens subgroup temperaments #Superpine}}
{{See also| No-sevens subgroup temperaments #Superpine }}


The superpine temperament is generated by 1/3 of a fourth, represented by [[~]][[35/32]], which resembles [[porcupine]], but it favors flat fifths instead of sharp ones. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]]–harmonics other than 3 all require the 15-tone mos to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as [[11/10]] as in porcupine, which makes [[11/8]] high-[[complexity]] like the other harmonics, but in the 13-limit 5 generators up closely approximates [[13/8]]. [[43edo]] is a good tuning especially for the higher-limit extensions.
The superpine temperament is generated by 1/3 of a fourth, represented by [[~]][[35/32]], which resembles [[porcupine]], but it favors flat fifths instead of sharp ones. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]]–harmonics other than 3 all require the 15-tone mos to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as [[11/10]] as in porcupine, which makes [[11/8]] high-[[complexity]] like the other harmonics, but in the 13-limit 5 generators up closely approximates [[13/8]]. [[43edo]] is a good tuning especially for the higher-limit extensions.
Line 1,839: Line 1,875:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~35/32 = 167.279{{c}}
* [[WE]]: ~2 = 1199.3652{{c}}, ~35/32 = 167.1615{{c}}
: [[error map]]: {{val| 0.000 -3.793 +6.336 +5.804 }}
: [[error map]]: {{val| -0.635 -4.709 +5.209 +3.639 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~35/32 = 167.256{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 167.2561{{c}}
: error map: {{val| 0.000 -3.723 +6.613 +5.503 }}
: error map: {{val| 0.000 -3.723 +6.613 +5.503 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~35/32 = 167.279{{c}}
: [[error map]]: {{val| 0.000 -3.793 +6.336 +5.804 }} -->


{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}
{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}


[[Badness]] (Smith): 0.137
[[Badness]] (Sintel): 3.46


=== 11-limit ===
=== 11-limit ===
Line 1,856: Line 1,894:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 167.407{{c}}
* WE: ~2 = 1199.0522{{c}}, ~11/10 = 167.1904{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/10 = 167.338{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3382{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 167.407{{c}} -->


{{Optimal ET sequence|legend=0| 7, 36, 43 }}
{{Optimal ET sequence|legend=0| 7, 36, 43 }}


Badness (Smith): 0.0576
Badness (Sintel): 1.90


=== 13-limit ===
=== 13-limit ===
Line 1,871: Line 1,910:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 167.427{{c}}
* WE: ~2 = 1199.4286{{c}}, ~11/10 = 167.3105{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/10 = 167.396{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3958{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 167.427{{c}} -->


{{Optimal ET sequence|legend=0| 7, 36, 43 }}
{{Optimal ET sequence|legend=0| 7, 36, 43 }}


Badness (Smith): 0.0368
Badness (Sintel): 1.52


== Lithium ==
== Lithium ==
Line 1,890: Line 1,930:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~56/45 = 400.000{{c}}, ~3/2 = 695.827{{c}} (~15/14 = 104.173{{c}})
* [[WE]]: ~56/45 = 400.6744{{c}}, ~3/2 = 695.8474{{c}} {~15/14 = 105.5015{{c}})
: [[error map]]: {{val| 0.000 -6.128 -3.007 +6.135 }}
: [[error map]]: {{val| +2.023 -4.084 -2.924 +4.910 }}
* [[CWE]]: ~56/45 = 400.000{{c}}, ~3/2 = 695.141{{c}} {~15/14 = 104.859{{c}})
* [[CWE]]: ~56/45 = 400.0000{{c}}, ~3/2 = 695.1413{{c}} {~15/14 = 104.8587{{c}})
: error map: {{val| 0.000 -6.814 -5.748 +2.022 }}
: error map: {{val| 0.000 -6.814 -5.748 +2.022 }}
<!-- * [[CTE]]: ~56/45 = 400.000{{c}}, ~3/2 = 695.827{{c}} (~15/14 = 104.173{{c}})
: error map: {{val| 0.000 -6.128 -3.007 +6.135 }} -->


{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}
{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}


[[Badness]] (Smith): 0.0692
[[Badness]] (Sintel): 1.75


== Squares ==
== Squares ==
Retrieved from "https://en.xen.wiki/w/Meantone_family"