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Meantone family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus (1/)
Switch to Sintel's badness, WE & CWE tunings, per community consensus (2/)
Line 707: Line 707:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.153{{c}}
* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.250{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.153{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.250{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 716: Line 718:
{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}
{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}


Badness (Smith): 0.021423
Badness (Sintel): 0.708


==== 13-limit ====
==== 13-limit ====
Line 726: Line 728:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.098{{c}}
* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.146{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.098{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.146{{c}} -->


{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}
{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}


Badness (Smith): 0.021182
Badness (Sintel): 0.875


===== 17-limit =====
===== 17-limit =====
Line 741: Line 745:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.216{{c}}
* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.575{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.216{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.575{{c}} -->


{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}


Badness (Smith): 0.022980
Badness (Sintel): 1.17


===== 19-limit =====
===== 19-limit =====
Line 756: Line 762:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.277{{c}}
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.706{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.277{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.706{{c}} -->


{{Optimal ET sequence|legend=0| 12f, 19, 31e }}
{{Optimal ET sequence|legend=0| 12f, 19, 31e }}


Badness (Smith): 0.020293
Badness (Sintel): 1.23


==== Vincenzo ====
==== Vincenzo ====
Line 771: Line 779:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.790{{c}}
* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 695.060{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.790{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 695.060{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 12, 19 }}
{{Optimal ET sequence|legend=0| 7d, 12, 19 }}


Badness (Smith): 0.024763
Badness (Sintel): 1.02


===== 17-limit =====
===== 17-limit =====
Line 786: Line 796:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.011{{c}}
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 695.858{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.011{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 695.858{{c}} -->


{{Optimal ET sequence|legend=0| 12, 19 }}
{{Optimal ET sequence|legend=0| 12, 19 }}


Badness (Smith): 0.025535
Badness (Sintel): 1.30


===== 19-limit =====
===== 19-limit =====
Line 801: Line 813:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.120{{c}}
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.131{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.120{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.131{{c}} -->


{{Optimal ET sequence|legend=0| 12, 19 }}
{{Optimal ET sequence|legend=0| 12, 19 }}


Badness (Smith): 0.022302
Badness (Sintel): 1.36


==== Meanundec ====
==== Meanundec ====
Line 816: Line 830:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.620{{c}}
* WE: ~2 = 1196.0359{{c}}, ~3/2 = 694.9504{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 697.254{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7474{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.620{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 697.254{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 12f, 19f }}
{{Optimal ET sequence|legend=0| 7d, 12f, 19f }}


Badness (Smith): 0.024243
Badness (Sintel): 1.00


===== 17-limit =====
===== 17-limit =====
Line 831: Line 847:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.279{{c}}
* WE: ~2 = 1196.8604{{c}}, ~3/2 = 695.7613{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 697.586{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1744{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.279{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 697.586{{c}} -->


{{Optimal ET sequence|legend=0| 7dg, 12f }}
{{Optimal ET sequence|legend=0| 7dg, 12f }}


Badness (Smith): 0.021400
Badness (Sintel): 1.09


===== 19-limit =====
===== 19-limit =====
Line 846: Line 864:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.849{{c}}
* WE: ~2 = 1196.9296{{c}}, ~3/2 = 696.3321{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 698.118{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.7122{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.849{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 698.118{{c}} -->


{{Optimal ET sequence|legend=0| 7dgh, 12f }}
{{Optimal ET sequence|legend=0| 7dgh, 12f }}


Badness (Smith): 0.018996
Badness (Sintel): 1.16


=== Meanundeci ===
=== Meanundeci ===
Line 863: Line 883:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.702{{c}}
* WE: ~2 = 1205.7146{{c}}, ~3/2 = 697.9977{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 694.689{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.1805{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.702{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 694.689{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 12e, 19e }}
{{Optimal ET sequence|legend=0| 7d, 12e, 19e }}


Badness (Smith): 0.031539
Badness (Sintel): 1.04


==== 13-limit ====
==== 13-limit ====
Line 878: Line 900:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.241{{c}}
* WE: ~2 = 1205.5631{{c}}, ~3/2 = 697.9847{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 694.764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.0144{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.241{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 694.764{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 12e, 19e }}
{{Optimal ET sequence|legend=0| 7d, 12e, 19e }}


Badness (Smith): 0.026288
Badness (Sintel): 1.09


=== Bimeantone ===
=== Bimeantone ===
Line 897: Line 921:


Optimal tunings:  
Optimal tunings:  
* CTE: ~63/44 = 600.000{{c}}, ~3/2 = 696.520{{c}}
* WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}}
* POTE: ~63/44 = 600.000{{c}}, ~3/2 = 696.016{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}}
<!-- * CTE: ~63/44 = 600.000{{c}}, ~3/2 = 696.520{{c}}
* POTE: ~63/44 = 600.000{{c}}, ~3/2 = 696.016{{c}} -->


{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}
{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}


Badness (Smith): 0.038122
Badness (Sintel): 1.26


==== 13-limit ====
==== 13-limit ====
Line 912: Line 938:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 600.000{{c}}, ~3/2 = 696.341{{c}}
* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}
* POTE: ~55/39 = 600.000{{c}}, ~3/2 = 695.836{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}
<!-- * CTE: ~55/39 = 600.000{{c}}, ~3/2 = 696.341{{c}}
* POTE: ~55/39 = 600.000{{c}}, ~3/2 = 695.836{{c}} -->


{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}


Badness (Smith): 0.028817
Badness (Sintel): 1.19


==== 17-limit ====
==== 17-limit ====
Line 927: Line 955:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~3/2 = 696.353{{c}}
* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~3/2 = 695.783{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}
<!-- * CTE: ~17/12 = 600.000{{c}}, ~3/2 = 696.353{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~3/2 = 695.783{{c}} -->


{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}
{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}


Badness (Smith): 0.022666
Badness (Sintel): 1.15


==== 19-limit ====
==== 19-limit ====
Line 942: Line 972:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000{{c}}, ~3/2 = 696.384{{c}}
* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~3/2 = 695.752{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}
<!-- * CTE: ~17/12 = 600.000{{c}}, ~3/2 = 696.384{{c}}
* POTE: ~17/12 = 600.000{{c}}, ~3/2 = 695.752{{c}} -->


{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}
{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}


Badness (Smith): 0.017785
Badness (Sintel): 1.08


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


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 961: Line 993:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 167.707{{c}}
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 167.805{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 167.707{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 167.805{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}
{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}


Badness (Smith): 0.050729
Badness (Sintel): 1.68


==== 13-limit ====
==== 13-limit ====
Line 976: Line 1,010:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 167.712{{c}}
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 167.790{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 167.712{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 167.790{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}
{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}


Badness (Smith): 0.035445
Badness (Sintel): 1.46


==== 17-limit ====
==== 17-limit ====
Line 991: Line 1,027:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 167.705{{c}}
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 167.786{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 167.705{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 167.786{{c}} -->


{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}
{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}


Badness (Smith): 0.025221
Badness (Sintel): 1.28


== Flattone ==
== Flattone ==
Line 1,010: Line 1,048:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 693.552{{c}}
* [[WE]]: ~2 = 1203.6308{{c}}, ~3/2 = 695.8782{{c}}
: [[error map]]: {{val| +3.631 -2.446 -2.801 -2.684 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.7334{{c}}
: error map: {{val| 0.000 -8.222 -11.380 -12.426 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 693.552{{c}}
: [[error map]]: {{val| 0.000 -8.403 -12.106 -10.794 }}
: [[error map]]: {{val| 0.000 -8.403 -12.106 -10.794 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 693.779{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 693.779{{c}}
: error map: {{val| 0.000 -8.176 -11.197 -12.838 }}
: error map: {{val| 0.000 -8.176 -11.197 -12.838 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 1,032: Line 1,074:
{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}
{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}


[[Badness]] (Smith): 0.038553
[[Badness]] (Sintel): 0.976


=== 11-limit ===
=== 11-limit ===
Line 1,044: Line 1,086:


Optimal tuning:  
Optimal tuning:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 693.251{{c}}
* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 693.126{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 693.251{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 693.126{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 1,053: Line 1,097:
{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}
{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}


Badness (Smith): 0.033839
Badness (Sintel): 1.12


=== 13-limit ===
=== 13-limit ===
Line 1,063: Line 1,107:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 693.029{{c}}
* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 693.058{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 693.029{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 693.058{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 1,072: Line 1,118:
{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}
{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}


Badness (Smith): 0.022260
Badness (Sintel): 0.920


== Flattertone ==
== Flattertone ==
Line 1,086: Line 1,132:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 692.698{{c}}
* [[WE]]: ~2 = 1204.4511{{c}}, ~3/2 = 694.3258{{c}}
: [[error map]]: {{val| 0.000 -9.257 -15.520 +7.047 }}
: [[error map]]: {{val| +4.451 -3.178 -9.011 +3.554 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 692.048{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 692.0479{{c}}
: error map: {{val| 0.000 -9.907 -18.122 -4.012 }}
: error map: {{val| 0.000 -9.907 -18.122 -4.012 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 692.698{{c}}
: [[error map]]: {{val| 0.000 -9.257 -15.520 +7.047 }} -->


{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}
{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}


[[Badness]] (Smith): 0.0961
[[Badness]] (Sintel): 2.43


==== 11-limit ====
==== 11-limit ====
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 45/44, 81/80, 1375/1344
Comma list: 45/44, 81/80, 1375/1344


{{Mapping|legend=1| 1 0 -4 -24 0| 0 1 4 17 6 }}
Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }}


: mapping generators: ~2, ~3
Optimal tunings:  
* WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 692.642{{c}} -->


[[Optimal tuning]]s:
{{Optimal ET sequence|legend=0| 7d, 19d, 26 }}
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 692.642{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 692.042{{c}}


{{Optimal ET sequence|legend=0| 7d, 19d, 26, 59bcd, 85bccd }}
Badness (Sintel): 1.53


; Music
; Music
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] - [[Budjarn Lambeth]] (2024)
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)


== Dominant ==
== Dominant ==
Line 1,128: Line 1,177:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 699.622{{c}}
* [[WE]]: ~2 = 1195.3384{{c}}, ~3/2 = 698.8478{{c}}
: [[error map]]: {{val| -4.662 -7.769 +9.077 +14.832 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.1125{{c}}
: error map: {{val| 0.000 -0.842 +18.136 +28.949 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 699.622{{c}}
: [[error map]]: {{val| 0.000 -2.333 +12.173 +31.931 }}
: [[error map]]: {{val| 0.000 -2.333 +12.173 +31.931 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.573{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.573{{c}}
: error map: {{val| 0.000 -0.382 +19.979 +28.028 }}
: error map: {{val| 0.000 -0.382 +19.979 +28.028 }} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 1,140: Line 1,193:
{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}


[[Badness]] (Smith): 0.020690
[[Badness]] (Sintel): 0.524


=== 11-limit ===
=== 11-limit ===
Line 1,154: Line 1,207:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.334{{c}}
* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 703.254{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.334{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 703.254{{c}} -->


{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}
{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}


Badness (Smith): 0.024180
Badness (Sintel): 0.799


==== 13-limit ====
==== 13-limit ====
Line 1,169: Line 1,224:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 704.847{{c}}
* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 703.636{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 704.847{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 703.636{{c}} -->


Tuning ranges:  
Tuning ranges:  
Line 1,176: Line 1,233:
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]


{{Optimal ET sequence|legend=0|12f, 17c, 29cdef }}
{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }}


Badness (Smith): 0.024108
Badness (Sintel): 0.996


==== Dominion ====
==== Dominion ====
Line 1,188: Line 1,245:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 704.034{{c}}
* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.905{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 704.034{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.905{{c}} -->


{{Optimal ET sequence|legend=0| 5, 12, 17c, 46cde }}
{{Optimal ET sequence|legend=0| 5, 12, 17c }}


Badness (Smith): 0.027295
Badness (Sintel): 1.13


=== Domineering ===
=== Domineering ===
Line 1,203: Line 1,262:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.240{{c}}
* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 698.776{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.240{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 698.776{{c}} -->


{{Optimal ET sequence|legend=0| 5e, 7, 12, 19d, 43de }}
{{Optimal ET sequence|legend=0| 5e, 7, 12 }}


Badness (Smith): 0.021978
Badness (Sintel): 0.727


==== 13-limit ====
==== 13-limit ====
Line 1,218: Line 1,279:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.315{{c}}
* WE: ~2 = 1198.1958{{c}}, ~3/2 = 694.7159{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 695.762{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.6809{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.315{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 695.762{{c}} -->


{{Optimal ET sequence|legend=0| 5ef, 7, 12, 19d, 31def }}
{{Optimal ET sequence|legend=0| 7, 12 }}


Badness (Smith): 0.027039
Badness (Sintel): 1.12


===== 17-limit =====
===== 17-limit =====
Line 1,233: Line 1,296:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.894{{c}}
* WE: ~2 = 1197.7959{{c}}, ~3/2 = 694.8362{{c}}
* POTE:  ~2 = 1200.000{{c}}, ~3/2 = 696.115{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.0834{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.894{{c}}
* POTE:  ~2 = 1200.000{{c}}, ~3/2 = 696.115{{c}} -->


{{Optimal ET sequence|legend=0| 5ef, 7, 12, 19d, 31def }}
{{Optimal ET sequence|legend=0| 7, 12 }}


Badness (Smith): 0.024539
Badness (Sintel): 1.25


===== 19-limit =====
===== 19-limit =====
Line 1,248: Line 1,313:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.139{{c}}
* WE: ~2 = 1197.6198{{c}}, ~3/2 = 694.8362{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.217{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2075{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.139{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.217{{c}} -->


{{Optimal ET sequence|legend=0| 5ef, 7, 12, 19d, 31def }}
{{Optimal ET sequence|legend=0| 5ef, 7, 12, 19d, 31def }}


Badness (Smith): 0.020398
Badness (Sintel): 1.24


==== Dominatrix ====
==== Dominatrix ====
Line 1,263: Line 1,330:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 694.840{{c}}
* WE: ~2 = 1193.1574{{c}}, ~3/2 = 694.5610{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 698.544{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.7268{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 694.840{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 698.544{{c}} -->


{{Optimal ET sequence|legend=0| 5e, 7, 12f, 19df }}
{{Optimal ET sequence|legend=0| 5e, 7, 12f }}


Badness (Smith): 0.018289
Badness (Sintel): 0.756


=== Domination ===
=== Domination ===
Line 1,278: Line 1,347:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.268{{c}}
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 705.004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.268{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 705.004{{c}} -->


{{Optimal ET sequence|legend=0| 5e, 12e, 17c, 46cd }}
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


Badness (Smith): 0.036562
Badness (Sintel): 1.21


==== 13-limit ====
==== 13-limit ====
Line 1,293: Line 1,364:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.719{{c}}
* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 705.496{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.719{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 705.496{{c}} -->


{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}
{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}


Badness (Smith): 0.027435
Badness (Sintel): 1.13


=== Arnold ===
=== Arnold ===
Line 1,308: Line 1,381:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 698.546{{c}}
* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 698.491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 698.546{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 698.491{{c}} -->


{{Optimal ET sequence|legend=0| 5, 7, 12e }}
{{Optimal ET sequence|legend=0| 5, 7, 12e }}


Badness (Smith): 0.026141
Badness (Sintel): 0.864


==== 13-limit ====
==== 13-limit ====
Line 1,323: Line 1,398:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.929{{c}}
* WE: ~2 = 1197.8123{{c}}, ~3/2 = 695.4727{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.743{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.5713{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 695.929{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.743{{c}} -->


{{Optimal ET sequence|legend=0| 5, 7, 12ef, 19def }}
{{Optimal ET sequence|legend=0| 5, 7 }}


Badness (Smith): 0.023300
Badness (Sintel): 0.963


==== 17-limit ====
==== 17-limit ====
Line 1,338: Line 1,415:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.683{{c}}
* WE: ~2 = 1197.6327{{c}}, ~3/2 = 695.6030{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.978{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9316{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.683{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.978{{c}} -->


{{Optimal ET sequence|legend=0| 5, 7, 12ef, 19def }}
{{Optimal ET sequence|legend=0| 5, 7 }}


Badness (Smith): 0.024535
Badness (Sintel): 1.25


==== 19-limit ====
==== 19-limit ====
Line 1,353: Line 1,432:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.996{{c}}
* WE: ~2 = 1197.5295{{c}}, ~3/2 = 695.6325{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 697.068{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0579{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 696.996{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 697.068{{c}} -->


{{Optimal ET sequence|legend=0| 5, 7, 12ef, 19def }}
{{Optimal ET sequence|legend=0| 5, 7, 12ef, 19def }}


Badness (Smith): 0.021098
Badness (Sintel): 1.28


== Sharptone ==
== Sharptone ==
Line 1,372: Line 1,453:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.732{{c}}
* [[WE]]: ~2 = 1204.2961{{c}}, ~3/2 = 702.6463{{c}}
: [[error map]]: {{val| +4.296 +4.987 +24.271 -56.591 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4928{{c}}
: error map: {{val| 0.000 -0.462 +19.657 -64.347 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.732{{c}}
: [[error map]]: {{val| 0.000 +1.777 +28.614 -57.630 }}
: [[error map]]: {{val| 0.000 +1.777 +28.614 -57.630 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.140{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.140{{c}}
: error map: {{val| 0.000 -1.815 +14.245 -68.407 }}
: error map: {{val| 0.000 -1.815 +14.245 -68.407 }} -->


{{Optimal ET sequence|legend=1| 5, 7d, 12d }}
{{Optimal ET sequence|legend=1| 5, 7d, 12d }}


[[Badness]] (Smith): 0.024848
[[Badness]] (Sintel): 0.629


=== Meanertone ===
=== Meanertone ===
Line 1,389: Line 1,474:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.730{{c}}
* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.615{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.730{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 696.615{{c}} -->


{{Optimal ET sequence|legend=0| 5, 7d, 12de }}
{{Optimal ET sequence|legend=0| 5, 7d, 12de }}


Badness (Smith): 0.025167
Badness (Sintel): 0.832


== Supermean ==
== Supermean ==
Line 1,404: Line 1,491:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.811{{c}}
* [[WE]]: ~2 = 1195.4372{{c}}, ~3/2 = 702.2086{{c}}
: [[error map]]: {{val| -4.563 -4.309 +22.521 -8.319 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5375{{c}}
: error map: {{val| 0.000 +2.583 +31.836 -0.763 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.811{{c}}
: [[error map]]: {{val| 0.000 +1.856 +28.929 -11.665 }}
: [[error map]]: {{val| 0.000 +1.856 +28.929 -11.665 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 704.889{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 704.889{{c}}
: error map: {{val| 0.000 +2.934 +33.242 +4.507 }}
: error map: {{val| 0.000 +2.934 +33.242 +4.507 }} -->


{{Optimal ET sequence|legend=1| 5d, 12d, 17c, 29c }}
{{Optimal ET sequence|legend=1| 5d, 12d, 17c }}


[[Badness]] (Smith): 0.134204
[[Badness]] (Sintel): 3.40


=== 11-limit ===
=== 11-limit ===
Line 1,421: Line 1,512:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 704.016{{c}}
* WE: ~2 = 1195.7270{{c}}, ~3/2 = 702.5848{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 705.096{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7471{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 704.016{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 705.096{{c}} -->


{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}
{{Optimal ET sequence|legend=0| 5de, 12de, 17c }}


Badness (Smith): 0.063262
Badness (Sintel): 2.09


=== 13-limit ===
=== 13-limit ===
Line 1,436: Line 1,529:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 704.121{{c}}
* WE: ~2 = 1196.3958{{c}}, ~3/2 = 702.9766{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 705.094{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7940{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 704.121{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 705.094{{c}} -->


{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}
{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}


Badness (Smith): 0.040324
Badness (Sintel): 1.67


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