User:Recentlymaterialized/Miscellaneous rank-2 temperaments: Difference between revisions

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worsetone 19-limit mapping changed from 55fgh & 67 to 55f & 67
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Line 1: Line 1:


== Senscape (27e & 87) ==
== Senscape (27e & 87) ==


[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
Line 22: Line 22:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 400.3122{{cc}}, 220.5893{{cc}}
| 400.3122{{cent}}, 220.5893{{cent}}
| 399.994{{cc}}, 220.6882{{cc}}
| 399.994{{c}}, 220.6882{{c}}
| 399.9947{{cc}}, 220.6886{{cc}}
| 399.9947{{c}}, 220.6886{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\3, 220.729{{cc}}
| 1\3, 220.729{{c}}
| 1\3, 220.6899{{cc}}
| 1\3, 220.6899{{c}}
| 1\3, 220.6873{{cc}}
| 1\3, 220.6873{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\3, 220.4173{{cc}}
| 1\3, 220.4173{{c}}
| 1\3, 220.6915{{cc}}
| 1\3, 220.6915{{c}}
| 1\3, 220.6915{{cc}}
| 1\3, 220.6915{{c}}
|}
|}


Line 52: Line 52:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1199.982{{cc}}
| 1199.982{{c}}
| 1903.4292{{cc}}
| 1903.4292{{c}}
| 2786.1822{{cc}}
| 2786.1822{{c}}
| 3365.482{{cc}}
| 3365.482{{c}}
| 4151.6942{{cc}}
| 4151.6942{{c}}
| 4441.3165{{cc}}
| 4441.3165{{c}}
|-
|-
| [[Error]]
| [[Error]]
| -0.018{{cc}}
| -0.018{{c}}
| +1.4742{{cc}}
| +1.4742{{c}}
| −0.1315{{cc}}
| −0.1315{{c}}
| −3.3439{{cc}}
| −3.3439{{c}}
| +0.3762{{cc}}
| +0.3762{{c}}
| +0.7889{{cc}}
| +0.7889{{c}}
|}
|}


Line 74: Line 74:
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.05783
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.05783


== Tri[[BPS|pierce]] (b30 & b39) ==
== Tri[[BPS|pierce]] (b30 & b39) ==


: ''See also: [[User:Dummy_index/3rd-tritave_temperaments#BPS_III_.28b30_.26_b9.2C_b39_.26_b9.29|BPS III]]''
: ''See also: [[User:Dummy_index/3rd-tritave_temperaments#BPS_III_.28b30_.26_b9.2C_b39_.26_b9.29|BPS III]]''
Line 99: Line 99:
| ???
| ???
| ???
| ???
| 634.5206{{cc}}, 1075.4459{{cc}}
| 634.5206{{c}}, 1075.4459{{c}}
| 634.5215{{cc}}, 1075.4475{{cc}}
| 634.5215{{c}}, 1075.4475{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\b3, 1074.0886{{cc}}
| 1\b3, 1074.0886{{c}}
| 1\b3, 1074.6634{{cc}}
| 1\b3, 1074.6634{{c}}
| 1\b3, 1075.1282{{cc}}
| 1\b3, 1075.1282{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| ???
| ???
| 1\b3, 1074.5381{{cc}}
| 1\b3, 1074.5381{{c}}
| 1\b3, 1074.5382{{cc}}
| 1\b3, 1074.5382{{c}}
|}
|}


Line 126: Line 126:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1903.5617{{cc}}
| 1903.5617{{c}}
| 2785.4124{{cc}}
| 2785.4124{{c}}
| 3366.1979{{cc}}
| 3366.1979{{c}}
| 4441.6439{{cc}}
| 4441.6439{{c}}
|-
|-
| [[Error]]
| [[Error]]
| +1.6067{{cc}}
| +1.6067{{c}}
| −0.9013{{cc}}
| −0.9013{{c}}
| −2.628{{cc}}
| −2.628{{c}}
| +1.1162{{cc}}
| +1.1162{{c}}
|}
|}


Line 144: Line 144:
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.26872
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.26872


=== Trisemi[[BPS|pierce]] (b39 & b60) ===
=== Trisemi[[BPS|pierce]] (b39 & b60) ===
: ''See also: [[User:Dummy_index/3rd-tritave_temperaments#weak_extension_3.5.7.11.13_.28b39_.26_b60.29|BPS III - weak extension]]''
: ''See also: [[User:Dummy_index/3rd-tritave_temperaments#weak_extension_3.5.7.11.13_.28b39_.26_b60.29|BPS III - weak extension]]''


Line 167: Line 167:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 634.4577{{cc}}, 537.4447{{cc}}
| 634.4577{{c}}, 537.4447{{c}}
| 634.5219{{cc}}, 537.7266{{cc}}
| 634.5219{{c}}, 537.7266{{c}}
| 634.5229{{cc}}, 537.7274{{cc}}
| 634.5229{{c}}, 537.7274{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\b3, 537.0443{{cc}}
| 1\b3, 537.0443{{c}}
| 1\b3, 537.3589{{cc}}
| 1\b3, 537.3589{{c}}
| 1\b3, 537.7233{{cc}}
| 1\b3, 537.7233{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\b3, 537.0443{{cc}}
| 1\b3, 537.0443{{c}}
| 1\b3, 537.2716{{cc}}
| 1\b3, 537.2716{{c}}
| 1\b3, 537.2716{{cc}}
| 1\b3, 537.2716{{c}}
|}
|}


Line 196: Line 196:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1903.5658{{cc}}
| 1903.5658{{c}}
| 2785.4282{{cc}}
| 2785.4282{{c}}
| 3366.2003{{cc}}
| 3366.2003{{c}}
| 4151.2674{{cc}}
| 4151.2674{{c}}
| 4441.6534{{cc}}
| 4441.6534{{c}}
|-
|-
| [[Error]]
| [[Error]]
| +1.6108{{cc}}
| +1.6108{{c}}
| −0.8855{{cc}}
| −0.8855{{c}}
| −2.6256{{cc}}
| −2.6256{{c}}
| −0.0505{{cc}}
| −0.0505{{c}}
| +1.1258{{cc}}
| +1.1258{{c}}
|}
|}


Line 216: Line 216:
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.44749
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.44749


=== Tri[[dubhe]] (b69 & b78) ===
=== Tri[[dubhe]] (b69 & b78) ===


[[Subgroup]]: [[3.5.7.13.17.19_subgroup|3.5.7.13.17.19]]
[[Subgroup]]: [[3.5.7.13.17.19_subgroup|3.5.7.13.17.19]]
Line 238: Line 238:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 634.4242{{cc}}, 220.2042{{cc}}
| 634.4242{{c}}, 220.2042{{c}}
| 634.3781{{cc}}, 220.1438{{cc}}
| 634.3781{{c}}, 220.1438{{c}}
| 634.3798{{cc}}, 220.1444{{cc}}
| 634.3798{{c}}, 220.1444{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\b3, 220.0518{{cc}}
| 1\b3, 220.0518{{c}}
| 1\b3, 220.0811{{cc}}
| 1\b3, 220.0811{{c}}
| 1\b3, 220.4087{{cc}}
| 1\b3, 220.4087{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\b3, 220.0518{{cc}}
| 1\b3, 220.0518{{c}}
| 1\b3, 220.0074{{cc}}
| 1\b3, 220.0074{{c}}
| 1\b3, 220.0074{{cc}}
| 1\b3, 220.0074{{c}}
|}
|}


Line 268: Line 268:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1903.1344{{cc}}
| 1903.1344{{c}}
| 2783.7095{{cc}}
| 2783.7095{{c}}
| 3365.9811{{cc}}
| 3365.9811{{c}}
| 4440.6468{{cc}}
| 4440.6468{{c}}
| 4906.9877{{cc}}
| 4906.9877{{c}}
| 5101.0782{{cc}}
| 5101.0782{{c}}
|-
|-
| [[Error]]
| [[Error]]
| +1.1794{{cc}}
| +1.1794{{c}}
| −2.6042{{cc}}
| −2.6042{{c}}
| −2.8448{{cc}}
| −2.8448{{c}}
| +0.1192{{cc}}
| +0.1192{{c}}
| +2.0323{{cc}}
| +2.0323{{c}}
| +3.5652{{cc}}
| +3.5652{{c}}
|}
|}


Line 290: Line 290:
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.48415
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.48415


=== Trisemi[[dubhe]] (b60 & b78) ===
=== Trisemi[[dubhe]] (b60 & b78) ===


[[Subgroup]]: [[3.5.7.11.13.17.19_subgroup|3.5.7.11.13.17.19]]
[[Subgroup]]: [[3.5.7.11.13.17.19_subgroup|3.5.7.11.13.17.19]]
Line 313: Line 313:
| ???
| ???
| ???
| ???
| 317.2028{{cc}}, 537.3966{{cc}}
| 317.2028{{c}}, 537.3966{{c}}
| 317.2037{{cc}}, 537.3981{{cc}}
| 317.2037{{c}}, 537.3981{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\b6, 537.0443{{cc}}
| 1\b6, 537.0443{{c}}
| 1\b6, 537.1261{{cc}}
| 1\b6, 537.1261{{c}}
| 1\b6, 537.5291{{cc}}
| 1\b6, 537.5291{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| ???
| ???
| 1\b6, 1074.0806{{cc}}
| 1\b6, 1074.0806{{c}}
| 1\b6, 1074.0806{{cc}}
| 1\b6, 1074.0806{{c}}
|}
|}


Line 343: Line 343:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1903.2168{{cc}}
| 1903.2168{{c}}
| 2783.9919{{cc}}
| 2783.9919{{c}}
| 3366.0459{{cc}}
| 3366.0459{{c}}
| 4149.8121{{cc}}
| 4149.8121{{c}}
| 4440.8391{{cc}}
| 4440.8391{{c}}
| 4907.4024{{cc}}
| 4907.4024{{c}}
| 5101.4204{{cc}}
| 5101.4204{{c}}
|-
|-
| [[Error]]
| [[Error]]
| +1.2618{{cc}}
| +1.2618{{c}}
| −2.3218{{cc}}
| −2.3218{{c}}
| −2.780{{cc}}
| −2.780{{c}}
| −1.5059{{cc}}
| −1.5059{{c}}
| +0.3114{{cc}}
| +0.3114{{c}}
| +2.447{{cc}}
| +2.447{{c}}
| +3.9074{{cc}}
| +3.9074{{c}}
|}
|}


Line 367: Line 367:
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.95032
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.95032


== Badtone (31 & 67) ==
== Badtone (31 & 67) ==


[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
Line 389: Line 389:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 1200.4954{{cc}}, 232.6290{{cc}}
| 1200.4954{{c}}, 232.6290{{c}}
| 1199.9347{{cc}}, 232.6275{{cc}}
| 1199.9347{{c}}, 232.6275{{c}}
| 1199.9460{{cc}}, 232.6297{{cc}}
| 1199.9460{{c}}, 232.6297{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\1, 232.6732{{cc}}
| 1\1, 232.6732{{c}}
| 1\1, 232.6392{{cc}}
| 1\1, 232.6392{{c}}
| 1\1, 232.6352{{cc}}
| 1\1, 232.6352{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\1, 232.5330{{cc}}
| 1\1, 232.5330{{c}}
| 1\1, 232.6402{{cc}}
| 1\1, 232.6402{{c}}
| 1\1, 232.6401{{cc}} <!-- x31eq rounds this to 232.6401 -->
| 1\1, 232.6401{{c}} <!-- x31eq rounds this to 232.6401 -->
|}
|}


Line 419: Line 419:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1199.9347{{cc}}
| 1199.9347{{c}}
| 1897.817{{cc}}
| 1897.817{{c}}
| 2791.5295{{cc}}
| 2791.5295{{c}}
| 3367.1765{{cc}}
| 3367.1765{{c}}
| 4150.4968{{cc}}
| 4150.4968{{c}}
| 4444.8759{{cc}}
| 4444.8759{{c}}
|-
|-
| [[Error]]
| [[Error]]
| −0.0653{{cc}}
| −0.0653{{c}}
| −4.138{{cc}}
| −4.138{{c}}
| +5.2157{{cc}}
| +5.2157{{c}}
| −1.6494{{cc}}
| −1.6494{{c}}
| −0.8211{{cc}}
| −0.8211{{c}}
| +4.3483{{cc}}
| +4.3483{{c}}
|}
|}


Line 441: Line 441:
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.52300
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.52300


== Worsetone (12f & 67) ==
== Worsetone (12f &amp; 67) ==


<p style="margin-bottom:1em">[[User:Lucius_Chiaraviglio|Lucas Chiaraviglio]] suggests the name ''[[Talk:Meantone#Unlisted_strong_7-limit_Meantone_extension_.28provisional_name_Mildtone.29.3F|mildtone]]''.</p>
<p style="margin-bottom:1em">[[User:Lucius_Chiaraviglio|Lucas Chiaraviglio]] suggests the name ''[[Talk:Meantone#Unlisted_strong_7-limit_Meantone_extension_.28provisional_name_Mildtone.29.3F|mildtone]]''.</p>
Line 465: Line 465:
| [[Optimized]]
| [[Optimized]]
| ???
| ???
| 1199.6846{{cc}}, 1897.7889{{cc}}
| 1199.6846{{c}}, 1897.7889{{c}}
| 1199.7882{{cc}}, 1898.1434{{cc}}
| 1199.7882{{c}}, 1898.1434{{c}}
| 1199.8021{{cc}}, 1898.1654{{cc}}
| 1199.8021{{c}}, 1898.1654{{c}}
|-
|-
| [[Constrained]]
| [[Constrained]]
| ???
| ???
| 1\1, 1898.2217{{cc}}
| 1\1, 1898.2217{{c}}
| 1\1, 698.4715{{cc}}
| 1\1, 698.4715{{c}}
| 1\1, 698.4553{{cc}}
| 1\1, 698.4553{{c}}
|-
|-
| [[Destretched]]
| [[Destretched]]
| ???
| ???
| 1\1, 1898.2878{{cc}}
| 1\1, 1898.2878{{c}}
| 1\1, 1898.4785{{cc}}
| 1\1, 1898.4785{{c}}
| 1\1, 1898.4785{{cc}}
| 1\1, 1898.4785{{c}}
|}
|}


Line 495: Line 495:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1199.7882{{cc}}
| 1199.7882{{c}}
| 1898.1434{{cc}}
| 1898.1434{{c}}
| 2793.4206{{cc}}
| 2793.4206{{c}}
| 3365.9316{{cc}}
| 3365.9316{{c}}
| 4153.6201{{cc}}
| 4153.6201{{c}}
| 4439.8756{{cc}}
| 4439.8756{{c}}
|-
|-
| [[Error]]
| [[Error]]
| −0.2118{{cc}}
| −0.2118{{c}}
| −3.8116{{cc}}
| −3.8116{{c}}
| +7.1069{{cc}}
| +7.1069{{c}}
| −2.8943{{cc}}
| −2.8943{{c}}
| +2.3022{{cc}}
| +2.3022{{c}}
| −0.6521{{cc}}
| −0.6521{{c}}
|}
|}


Line 537: Line 537:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 1201.1304{{cc}}, 1900.1656{{cc}}
| 1201.1304{{c}}, 1900.1656{{c}}
| 1200.9840{{cc}}, 1899.9400{{cc}}
| 1200.9840{{c}}, 1899.9400{{c}}
| 1200.9879{{cc}}, 1899.9461{{cc}}
| 1200.9879{{c}}, 1899.9461{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\1, 1898.6199{{cc}}
| 1\1, 1898.6199{{c}}
| 1\1, 1898.4197{{cc}}
| 1\1, 1898.4197{{c}}
| 1\1, 1898.4955{{cc}}
| 1\1, 1898.4955{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\1, 1898.3773{{cc}}
| 1\1, 1898.3773{{c}}
| 1\1, 1898.3833{{cc}}
| 1\1, 1898.3833{{c}}
| 1\1, 1898.3833{{cc}}
| 1\1, 1898.3833{{c}}
|}
|}


Line 570: Line 570:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1200.984{{cc}}
| 1200.984{{c}}
| 1899.940{{cc}}
| 1899.940{{c}}
| 3367.1912{{cc}}
| 3367.1912{{c}}
| 4154.903{{cc}}
| 4154.903{{c}}
| 4440.5866{{cc}}
| 4440.5866{{c}}
| 5425.2263{{cc}}
| 5425.2263{{c}}
| 5946.6708{{cc}}
| 5946.6708{{c}}
| 6251.7709{{cc}}
| 6251.7709{{c}}
| 6429.2824{{cc}}
| 6429.2824{{c}}
|-
|-
| [[Error]]
| [[Error]]
| +0.984{{cc}}
| +0.984{{c}}
| −2.015{{cc}}
| −2.015{{c}}
| −1.6347{{cc}}
| −1.6347{{c}}
| +3.585{{cc}}
| +3.585{{c}}
| +0.059{{cc}}
| +0.059{{c}}
| −3.0481{{cc}}
| −3.0481{{c}}
| +1.6352{{cc}}
| +1.6352{{c}}
| +0.4269{{cc}}
| +0.4269{{c}}
| +0.220{{cc}}
| +0.220{{c}}
|}
|}


Line 598: Line 598:
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.07545 (1.77384)
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.07545 (1.77384)


== Hemififive (68 & 94) ==
== Hemififive (68 &amp; 94) ==


[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
Line 620: Line 620:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 599.9903{{cc}}, 229.6835{{cc}}
| 599.9903{{c}}, 229.6835{{c}}
| 600.0198{{cc}}, 229.7451{{cc}}
| 600.0198{{c}}, 229.7451{{c}}
| 600.0210{{cc}}, 229.7456{{cc}}
| 600.0210{{c}}, 229.7456{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\2, 229.6914{{cc}}
| 1\2, 229.6914{{c}}
| 1\2, 229.7361{{cc}}
| 1\2, 229.7361{{c}}
| 1\2, 229.7307{{cc}}
| 1\2, 229.7307{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\2, 229.6872{{cc}}
| 1\2, 229.6872{{c}}
| 1\2, 229.7375{{cc}}
| 1\2, 229.7375{{c}}
| 1\2, 229.7376{{cc}}
| 1\2, 229.7376{{c}}
|}
|}


Line 650: Line 650:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1200.0396{{cc}}
| 1200.0396{{c}}
| 1902.6876{{cc}}
| 1902.6876{{c}}
| 2783.7665{{cc}}
| 2783.7665{{c}}
| 3370.3738{{cc}}
| 3370.3738{{c}}
| 4148.8246{{cc}}
| 4148.8246{{c}}
| 4443.2964{{cc}}
| 4443.2964{{c}}
|-
|-
| [[Error]]
| [[Error]]
| +0.0396{{cc}}
| +0.0396{{c}}
| +0.7326{{cc}}
| +0.7326{{c}}
| −2.5472{{cc}}
| −2.5472{{c}}
| +1.5479{{cc}}
| +1.5479{{c}}
| −2.4933{{cc}}
| −2.4933{{c}}
| +2.7687{{cc}}
| +2.7687{{c}}
|}
|}


Line 694: Line 694:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 599.9903{{cc}}, 229.6835{{cc}}
| 599.9903{{c}}, 229.6835{{c}}
| 600.0751{{cc}}, 229.7512{{cc}}
| 600.0751{{c}}, 229.7512{{c}}
| 600.0765{{cc}}, 229.7517{{cc}}
| 600.0765{{c}}, 229.7517{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\2, 229.6914{{cc}}
| 1\2, 229.6914{{c}}
| 1\2, 229.7168{{cc}}
| 1\2, 229.7168{{c}}
| 1\2, 229.6935{{cc}}
| 1\2, 229.6935{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\2, 229.6872{{cc}}
| 1\2, 229.6872{{c}}
| 1\2, 229.7224{{cc}}
| 1\2, 229.7224{{c}}
| 1\2, 229.7224{{cc}}
| 1\2, 229.7224{{c}}
|}
|}


Line 726: Line 726:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1200.1502{{cc}}
| 1200.1502{{c}}
| 1903.0139{{cc}}
| 1903.0139{{c}}
| 2784.2344{{cc}}
| 2784.2344{{c}}
| 3370.6993{{cc}}
| 3370.6993{{c}}
| 4149.1313{{cc}}
| 4149.1313{{c}}
| 4443.8869{{cc}}
| 4443.8869{{c}}
| 4903.3893{{cc}}
| 4903.3893{{c}}
| 5095.3563{{cc}}
| 5095.3563{{c}}
|-
|-
| [[Error]]
| [[Error]]
| +0.1502{{cc}}
| +0.1502{{c}}
| +1.0589{{cc}}
| +1.0589{{c}}
| −2.0793{{cc}}
| −2.0793{{c}}
| +1.8734{{cc}}
| +1.8734{{c}}
| −2.1867{{cc}}
| −2.1867{{c}}
| +3.3593{{cc}}
| +3.3593{{c}}
| −1.5661{{cc}}
| −1.5661{{c}}
| −2.1567{{cc}}
| −2.1567{{c}}
|}
|}


Line 774: Line 774:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 599.9903{{cc}}, 229.6835{{cc}}
| 599.9903{{c}}, 229.6835{{c}}
| 600.0744{{cc}}, 229.7474{{cc}}
| 600.0744{{c}}, 229.7474{{c}}
| 600.0758{{cc}}, 229.7480{{cc}}
| 600.0758{{c}}, 229.7480{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\2, 229.6914{{cc}}
| 1\2, 229.6914{{c}}
| 1\2, 229.7135{{cc}}
| 1\2, 229.7135{{c}}
| 1\2, 229.6908{{cc}}
| 1\2, 229.6908{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\2, 229.6872{{cc}}
| 1\2, 229.6872{{c}}
| 1\2, 229.7189{{cc}}
| 1\2, 229.7189{{c}}
| 1\2, 229.7190{{cc}}
| 1\2, 229.7190{{c}}
|}
|}


Line 807: Line 807:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1200.1488{{cc}}
| 1200.1488{{c}}
| 1903.0465{{cc}}
| 1903.0465{{c}}
| 2784.28{{cc}}
| 2784.28{{c}}
| 3370.6991{{cc}}
| 3370.6991{{c}}
| 4149.1094{{cc}}
| 4149.1094{{c}}
| 4443.9237{{cc}}
| 4443.9237{{c}}
| 4903.4186{{cc}}
| 4903.4186{{c}}
| 5095.4097{{cc}}
| 5095.4097{{c}}
| 5427.9804{{cc}}
| 5427.9804{{c}}
|-
|-
| [[Error]]
| [[Error]]
| +0.1488{{cc}}
| +0.1488{{c}}
| +1.0915{{cc}}
| +1.0915{{c}}
| −2.0337{{cc}}
| −2.0337{{c}}
| +1.8732{{cc}}
| +1.8732{{c}}
| −2.2086{{cc}}
| −2.2086{{c}}
| +3.3961{{cc}}
| +3.3961{{c}}
| −1.5368{{cc}}
| −1.5368{{c}}
| −2.1033{{cc}}
| −2.1033{{c}}
| −0.2939{{cc}}
| −0.2939{{c}}
|}
|}


Line 835: Line 835:
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.04399
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.04399


== Cyanide (140 & 441) ==
== Cyanide (140 &amp; 441) ==


[[Subgroup]]: [[2.3.5.7.13_subgroup|2.3.5.7.13]]([[2.3.5.7.13.23_subgroup|.23]])
[[Subgroup]]: [[2.3.5.7.13_subgroup|2.3.5.7.13]]([[2.3.5.7.13.23_subgroup|.23]])
Line 857: Line 857:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 171.4255{{cc}}, 231.2847{{cc}}
| 171.4255{{c}}, 231.2847{{c}}
| 171.4244{{cc}}, 231.2902{{cc}}
| 171.4244{{c}}, 231.2902{{c}}
| 171.4244{{cc}}, 231.2902{{cc}}
| 171.4244{{c}}, 231.2902{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\7, 231.2969{{cc}}
| 1\7, 231.2969{{c}}
| 1\7, 231.2963{{cc}}
| 1\7, 231.2963{{c}}
| 1\7, 231.2991{{cc}}
| 1\7, 231.2991{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\7, 231.2888{{cc}}
| 1\7, 231.2888{{c}}
| 1\7, 231.2943{{cc}}
| 1\7, 231.2943{{c}}
| 1\7, 231.2943{{cc}}
| 1\7, 231.2943{{c}}
|}
|}


Line 887: Line 887:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1199.971{{cc}}
| 1199.971{{c}}
| 1902.0144{{cc}}
| 1902.0144{{c}}
| 2786.3108{{cc}}
| 2786.3108{{c}}
| 3368.6227{{cc}}
| 3368.6227{{c}}
| 4440.6893{{cc}}
| 4440.6893{{c}}
| 5428.3714{{cc}}
| 5428.3714{{c}}
|-
|-
| [[Error]]
| [[Error]]
| −0.029{{cc}}
| −0.029{{c}}
| +0.0594{{cc}}
| +0.0594{{c}}
| −0.0029{{cc}}
| −0.0029{{c}}
| −0.2032{{cc}}
| −0.2032{{c}}
| +0.1616{{cc}}
| +0.1616{{c}}
| +0.097{{cc}}
| +0.097{{c}}
|}
|}


Line 931: Line 931:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 171.4227, 231.3059{{cc}}
| 171.4227, 231.3059{{c}}
| 171.4242, 231.3085{{cc}}
| 171.4242, 231.3085{{c}}
| 171.4242, 231.3085{{cc}}
| 171.4242, 231.3085{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\7, 231.3138{{cc}}
| 1\7, 231.3138{{c}}
| 1\7, 231.3149{{cc}}
| 1\7, 231.3149{{c}}
| 1\7, 231.3180{{cc}}
| 1\7, 231.3180{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\7, 231.3138{{cc}}
| 1\7, 231.3138{{c}}
| 1\7, 231.3144{{cc}}
| 1\7, 231.3144{{c}}
| 1\7, 231.3144{{cc}}
| 1\7, 231.3144{{c}}
|}
|}


Line 962: Line 962:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1199.9697{{cc}}
| 1199.9697{{c}}
| 1902.1239{{cc}}
| 1902.1239{{c}}
| 2786.2149{{cc}}
| 2786.2149{{c}}
| 3368.6006{{cc}}
| 3368.6006{{c}}
| 4151.6642{{cc}}
| 4151.6642{{c}}
| 4440.5731{{cc}}
| 4440.5731{{c}}
| 5427.9754{{cc}}
| 5427.9754{{c}}
|-
|-
| [[Error]]
| [[Error]]
| −0.0303{{cc}}
| −0.0303{{c}}
| +0.1689{{cc}}
| +0.1689{{c}}
| −0.0988{{cc}}
| −0.0988{{c}}
| −0.2253{{cc}}
| −0.2253{{c}}
| +0.3463{{cc}}
| +0.3463{{c}}
| +0.0454{{cc}}
| +0.0454{{c}}
| −0.2989{{cc}}
| −0.2989{{c}}
|}
|}


Line 986: Line 986:
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.21770 (1.01017)
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.21770 (1.01017)


=== 581 & 742 <small style="font-weight:400">(weak extension)</small> ===
=== 581 &amp; 742 <small style="font-weight:400">(weak extension)</small> ===


[[Subgroup]]: [[19-limit|2.3.5.7.11.13.17.19]]
[[Subgroup]]: [[19-limit|2.3.5.7.11.13.17.19]]
Line 1,008: Line 1,008:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 171.4266{{cc}}, 37.1864{{cc}}
| 171.4266{{c}}, 37.1864{{c}}
| 171.4244{{cc}}, 37.1884{{cc}}
| 171.4244{{c}}, 37.1884{{c}}
| 171.4244{{cc}}, 37.1884{{cc}}
| 171.4244{{c}}, 37.1884{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\7, 37.1867{{cc}}
| 1\7, 37.1867{{c}}
| 1\7, 37.1893{{cc}}
| 1\7, 37.1893{{c}}
| 1\7, 37.1893{{cc}}
| 1\7, 37.1893{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\7, 37.1868{{cc}}
| 1\7, 37.1868{{c}}
| 1\7, 37.1893{{cc}}
| 1\7, 37.1893{{c}}
| 1\7, 37.1893{{cc}}
| 1\7, 37.1893{{c}}
|}
|}


Line 1,040: Line 1,040:
|-
|-
| [[Tuning_map|Tuning]] ({{cent}})
| [[Tuning_map|Tuning]] ({{cent}})
| 1199.97{{cc}}
| 1199.97{{c}}
| 1901.97{{cc}}
| 1901.97{{c}}
| 2786.34{{cc}}
| 2786.34{{c}}
| 3368.63{{cc}}
| 3368.63{{c}}
| 4151.37{{cc}}
| 4151.37{{c}}
| 4440.73{{cc}}
| 4440.73{{c}}
| 4905.08{{cc}}
| 4905.08{{c}}
| 5097.39{{cc}}
| 5097.39{{c}}
|-
|-
| [[Error]]
| [[Error]]
| −0.0291{{cc}}
| −0.0291{{c}}
| +0.0195{{cc}}
| +0.0195{{c}}
| +0.0302{{cc}}
| +0.0302{{c}}
| −0.1966{{cc}}
| −0.1966{{c}}
| +0.0566{{cc}}
| +0.0566{{c}}
| +0.2014{{cc}}
| +0.2014{{c}}
| +0.1289{{cc}}
| +0.1289{{c}}
| −0.1218{{cc}}
| −0.1218{{c}}
|}
|}


Line 1,066: Line 1,066:
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.76654
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.76654


== 130 & 181 ==
== 130 &amp; 181 ==


''See also: [[User:Francium/Miscellaneous_rank-2_temperaments#311_.26_1063|7-limit]]''
''See also: [[User:Francium/Miscellaneous_rank-2_temperaments#311_.26_1063|7-limit]]''
Line 1,090: Line 1,090:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 1199.983{{cc}}, 470.7428{{cc}}
| 1199.983{{c}}, 470.7428{{c}}
| 1199.9727{{cc}}, 470.7388{{cc}}
| 1199.9727{{c}}, 470.7388{{c}}
| 1199.9727{{cc}}, 470.7388{{cc}}
| 1199.9727{{c}}, 470.7388{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\1, 470.7495{{cc}}
| 1\1, 470.7495{{c}}
| 1\1, 470.7495{{cc}}
| 1\1, 470.7495{{c}}
| 1\1, 470.7493{{cc}}
| 1\1, 470.7493{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\1, 470.7495{{cc}}
| 1\1, 470.7495{{c}}
| 1\1, 470.7495{{cc}}
| 1\1, 470.7495{{c}}
| 1\1, 470.7495{{cc}}
| 1\1, 470.7495{{c}}
|}
|}


Line 1,119: Line 1,119:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1199.9727{{cc}}
| 1199.9727{{c}}
| 1901.9687{{cc}}
| 1901.9687{{c}}
| 2786.4055{{cc}}
| 2786.4055{{c}}
| 3368.6607{{cc}}
| 3368.6607{{c}}
| 4440.6683{{cc}}
| 4440.6683{{c}}
|-
|-
| [[Error]]
| [[Error]]
| −0.0273{{cc}}
| −0.0273{{c}}
| +0.0137{{cc}}
| +0.0137{{c}}
| +0.0918{{cc}}
| +0.0918{{c}}
| −0.1652{{cc}}
| −0.1652{{c}}
| +0.1407{{cc}}
| +0.1407{{c}}
|}
|}


Line 1,161: Line 1,161:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 1199.9305{{cc}}, 470.7167{{cc}}
| 1199.9305{{c}}, 470.7167{{c}}
| 1199.9386{{cc}}, 470.7221{{cc}}
| 1199.9386{{c}}, 470.7221{{c}}
| 1199.9387{{cc}}, 470.7221{{cc}}
| 1199.9387{{c}}, 470.7221{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\1, 470.7411{{cc}}
| 1\1, 470.7411{{c}}
| 1\1, 470.746{{cc}}
| 1\1, 470.746{{c}}
| 1\1, 470.745{{cc}}
| 1\1, 470.745{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\1, 470.744{{cc}}
| 1\1, 470.744{{c}}
| 1\1, 470.7462{{cc}}
| 1\1, 470.7462{{c}}
| 1\1, 470.7461{{cc}}
| 1\1, 470.7461{{c}}
|}
|}


Line 1,191: Line 1,191:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1199.9386{{cc}}
| 1199.9386{{c}}
| 1901.9946{{cc}}
| 1901.9946{{c}}
| 2786.1202{{cc}}
| 2786.1202{{c}}
| 3368.482{{cc}}
| 3368.482{{c}}
| 4152.0197{{cc}}
| 4152.0197{{c}}
| 4440.6721{{cc}}
| 4440.6721{{c}}
|-
|-
| [[Error]]
| [[Error]]
| −0.0614{{cc}}
| −0.0614{{c}}
| +0.0396{{cc}}
| +0.0396{{c}}
| −0.1936{{cc}}
| −0.1936{{c}}
| −0.3439{{cc}}
| −0.3439{{c}}
| +0.7017{{cc}}
| +0.7017{{c}}
| +0.1445{{cc}}
| +0.1445{{c}}
|}
|}


Line 1,213: Line 1,213:
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.89734
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.89734


== Whooshlet (441 & 3072) ==
== Whooshlet (441 &amp; 3072) ==
''Main article: [[Whoosh]]''
''Main article: [[Whoosh]]''


Line 1,236: Line 1,236:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| 400.0024{{cc}}, 160.5488{{cc}}
| 400.0024{{c}}, 160.5488{{c}}
| 400.0034{{cc}}, 160.5495{{cc}}
| 400.0034{{c}}, 160.5495{{c}}
| 400.0034{{cc}}, 160.5495{{cc}}
| 400.0034{{c}}, 160.5495{{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\, 160.5478{{cc}}
| 1\, 160.5478{{c}}
| 1\, 160.5480{{cc}}
| 1\, 160.5480{{c}}
| 1\, 160.5476{{cc}}
| 1\, 160.5476{{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\3, 160.5478{{cc}}
| 1\3, 160.5478{{c}}
| 1\3, 160.5481{{cc}}
| 1\3, 160.5481{{c}}
| 1\3, 160.5481{{cc}}
| 1\3, 160.5481{{c}}
|}
|}


Line 1,265: Line 1,265:
|-
|-
| [[Tuning_map|Tuning]]
| [[Tuning_map|Tuning]]
| 1200.0102{{cc}}
| 1200.0102{{c}}
| 1901.9291{{cc}}
| 1901.9291{{c}}
| 2786.3213{{cc}}
| 2786.3213{{c}}
| 3368.7981{{cc}}
| 3368.7981{{c}}
| 4440.575{{cc}}
| 4440.575{{c}}
|-
|-
| [[Error]]
| [[Error]]
| +0.0102{{cc}}
| +0.0102{{c}}
| −0.0259{{cc}}
| −0.0259{{c}}
| +0.0076{{cc}}
| +0.0076{{c}}
| −0.0278{{cc}}
| −0.0278{{c}}
| +0.0473{{cc}}
| +0.0473{{c}}
|}
|}


Line 1,287: Line 1,287:
<!--
<!--


console.log(``.split(/\s+/).reduce((a,b)=>a+'{{cc}}\n| '+b)+'{{cc}}')
console.log(``.split(/\s+/).reduce((a,b)=>a+'{{c}}\n| '+b)+'{{c}}')
console.log(``.split(/\s+/).reduce((a,b)=>a+'{{cc}}\n| +'+b).replaceAll('+-', '−')+'{{cc}}')
console.log(``.split(/\s+/).reduce((a,b)=>a+'{{c}}\n| +'+b).replaceAll('+-', '−')+'{{c}}')


[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
Line 1,310: Line 1,310:
| [[Optimization|Optimized]]
| [[Optimization|Optimized]]
| ???
| ???
| {{cc}}, {{cc}}
| {{c}}, {{c}}
| {{cc}}, {{cc}}
| {{c}}, {{c}}
| {{cc}}, {{cc}}
| {{c}}, {{c}}
|-
|-
| [[Constrained_tuning|Constrained]]
| [[Constrained_tuning|Constrained]]
| ???
| ???
| 1\, {{cc}}
| 1\, {{c}}
| 1\, {{cc}}
| 1\, {{c}}
| 1\, {{cc}}
| 1\, {{c}}
|-
|-
| [[POTE|Destretched]]
| [[POTE|Destretched]]
| ???
| ???
| 1\, {{cc}}
| 1\, {{c}}
| 1\, {{cc}}
| 1\, {{c}}
| 1\, {{cc}}
| 1\, {{c}}
|}
|}


Revision as of 16:03, 3 March 2025

Senscape (27e & 87)

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 325/324, 441/440, 1001/1000

Mapping: [⟨3 2 2 4 1 10], ⟨0 5 9 8 17 2]]

Generator preimages: 63/50, 91/80

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 400.3122 ¢, 220.5893 ¢ 399.994 ¢, 220.6882 ¢ 399.9947 ¢, 220.6886 ¢
Constrained ??? 1\3, 220.729 ¢ 1\3, 220.6899 ¢ 1\3, 220.6873 ¢
Destretched ??? 1\3, 220.4173 ¢ 1\3, 220.6915 ¢ 1\3, 220.6915 ¢

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1
Tuning 1199.982 ¢ 1903.4292 ¢ 2786.1822 ¢ 3365.482 ¢ 4151.6942 ¢ 4441.3165 ¢
Error -0.018 ¢ +1.4742 ¢ −0.1315 ¢ −3.3439 ¢ +0.3762 ¢ +0.7889 ¢

Optimal ET sequence (TE): 27e, 60e, 87

Badness (Smith TE): 0.02560

Badness (Dirichlet): 1.05783

Tripierce (b30 & b39)

See also: BPS III

Subgroup: 3.5.7.13

Comma list: 245/243, 2197/2187

Mapping: [⟨3 1 7 7], ⟨0 2 -1 0]]

Generator preimages: 13/9, 13/7

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? ??? 634.5206 ¢, 1075.4459 ¢ 634.5215 ¢, 1075.4475 ¢
Constrained ??? 1\b3, 1074.0886 ¢ 1\b3, 1074.6634 ¢ 1\b3, 1075.1282 ¢
Destretched ??? ??? 1\b3, 1074.5381 ¢ 1\b3, 1074.5382 ¢

Prime harmonics (WE):

Prime 3/1 5/1 7/1 13/1
Tuning 1903.5617 ¢ 2785.4124 ¢ 3366.1979 ¢ 4441.6439 ¢
Error +1.6067 ¢ −0.9013 ¢ −2.628 ¢ +1.1162 ¢

Optimal ET sequence (TE): b9, b21, b30, b39, b69, b108, b177d

Badness (Smith TE): 0.00287

Badness (Dirichlet): 0.26872

Trisemipierce (b39 & b60)

See also: BPS III - weak extension

Subgroup: 3.5.7.11.13

Comma list: 245/243, 847/845, 1575/1573

Mapping: [⟨3 1 7 4 7], ⟨0 4 -2 3 0]]

Generator preimages: 13/9, 15/11

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 634.4577 ¢, 537.4447 ¢ 634.5219 ¢, 537.7266 ¢ 634.5229 ¢, 537.7274 ¢
Constrained ??? 1\b3, 537.0443 ¢ 1\b3, 537.3589 ¢ 1\b3, 537.7233 ¢
Destretched ??? 1\b3, 537.0443 ¢ 1\b3, 537.2716 ¢ 1\b3, 537.2716 ¢

Prime harmonics (WE):

Prime 3/1 5/1 7/1 11/1 13/1
Tuning 1903.5658 ¢ 2785.4282 ¢ 3366.2003 ¢ 4151.2674 ¢ 4441.6534 ¢
Error +1.6108 ¢ −0.8855 ¢ −2.6256 ¢ −0.0505 ¢ +1.1258 ¢

Optimal ET sequence (TE): b18, b21, b39, b99, b138, b177d

Badness (Smith TE): 0.00566

Badness (Dirichlet): 0.44749

Tridubhe (b69 & b78)

Subgroup: 3.5.7.13.17.19

Comma list: 245/243, 325/323, 665/663, 3213/3211

Mapping: [⟨3 3 6 7 6 7], ⟨0 4 -2 0 5 3]]

Generator preimages: 13/9, 17/15

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 634.4242 ¢, 220.2042 ¢ 634.3781 ¢, 220.1438 ¢ 634.3798 ¢, 220.1444 ¢
Constrained ??? 1\b3, 220.0518 ¢ 1\b3, 220.0811 ¢ 1\b3, 220.4087 ¢
Destretched ??? 1\b3, 220.0518 ¢ 1\b3, 220.0074 ¢ 1\b3, 220.0074 ¢

Prime harmonics (WE):

Prime 3/1 5/1 7/1 13/1 17/1 19/1
Tuning 1903.1344 ¢ 2783.7095 ¢ 3365.9811 ¢ 4440.6468 ¢ 4906.9877 ¢ 5101.0782 ¢
Error +1.1794 ¢ −2.6042 ¢ −2.8448 ¢ +0.1192 ¢ +2.0323 ¢ +3.5652 ¢

Optimal ET sequence (TE): b51, b60, b69, b147

Badness (Smith TE): 0.00523

Badness (Dirichlet): 0.48415

Trisemidubhe (b60 & b78)

Subgroup: 3.5.7.11.13.17.19

Comma list: 245/243, 325/323, 665/663, 847/845, 1575/1573

Mapping: [⟨6 2 14 8 14 7 11], ⟨0 4 -2 3 0 5 3]]

Generator preimages: 119/99, 15/11

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? ??? 317.2028 ¢, 537.3966 ¢ 317.2037 ¢, 537.3981 ¢
Constrained ??? 1\b6, 537.0443 ¢ 1\b6, 537.1261 ¢ 1\b6, 537.5291 ¢
Destretched ??? ??? 1\b6, 1074.0806 ¢ 1\b6, 1074.0806 ¢

Prime harmonics (WE):

Prime 3/1 5/1 7/1 11/1 13/1 17/1 19/1
Tuning 1903.2168 ¢ 2783.9919 ¢ 3366.0459 ¢ 4149.8121 ¢ 4440.8391 ¢ 4907.4024 ¢ 5101.4204 ¢
Error +1.2618 ¢ −2.3218 ¢ −2.780 ¢ −1.5059 ¢ +0.3114 ¢ +2.447 ¢ +3.9074 ¢

Optimal ET sequence (TE): b18, b60, b78, b138, b216

Badness (Smith TE): 0.01046

Badness (Dirichlet): 0.95032

Badtone (31 & 67)

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 144/143, 176/175, 196/195

Mapping: [⟨1 1 0 3 -1 7], ⟨0 3 12 -1 23 -17]]

Generator preimages: 2/1, 8/7

Generator tunings:

Weil Tenney WE TE
Optimized ??? 1200.4954 ¢, 232.6290 ¢ 1199.9347 ¢, 232.6275 ¢ 1199.9460 ¢, 232.6297 ¢
Constrained ??? 1\1, 232.6732 ¢ 1\1, 232.6392 ¢ 1\1, 232.6352 ¢
Destretched ??? 1\1, 232.5330 ¢ 1\1, 232.6402 ¢ 1\1, 232.6401 ¢

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1
Tuning 1199.9347 ¢ 1897.817 ¢ 2791.5295 ¢ 3367.1765 ¢ 4150.4968 ¢ 4444.8759 ¢
Error −0.0653 ¢ −4.138 ¢ +5.2157 ¢ −1.6494 ¢ −0.8211 ¢ +4.3483 ¢

Optimal ET sequence (TE): 31, 67, 98

Badness (Smith TE): 0.03686

Badness (Dirichlet): 1.52300

Worsetone (12f & 67)

Lucas Chiaraviglio suggests the name mildtone.

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 176/175, 196/195, 364/363

Mapping: [⟨1 0 -4 -32 -44 -58], ⟨0 1 4 22 30 39]]

Generator preimages: 2/1, 3/1

Generator tunings:

Weil Tenney WE TE
Optimized ??? 1199.6846 ¢, 1897.7889 ¢ 1199.7882 ¢, 1898.1434 ¢ 1199.8021 ¢, 1898.1654 ¢
Constrained ??? 1\1, 1898.2217 ¢ 1\1, 698.4715 ¢ 1\1, 698.4553 ¢
Destretched ??? 1\1, 1898.2878 ¢ 1\1, 1898.4785 ¢ 1\1, 1898.4785 ¢

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1
Tuning 1199.7882 ¢ 1898.1434 ¢ 2793.4206 ¢ 3365.9316 ¢ 4153.6201 ¢ 4439.8756 ¢
Error −0.2118 ¢ −3.8116 ¢ +7.1069 ¢ −2.8943 ¢ +2.3022 ¢ −0.6521 ¢

Optimal ET sequence: 12f, 55f, 67

Badness (Smith TE): 0.04940

Badness (Dirichlet): 2.04145

2.3.7.11.13(.17.19).23.31.37.41 subgroup

Comma list: 287/286, 297/296, 364/363, 483/481, 621/616, 3627/3626, 4797/4774, (153/152, 189/187)

Mapping: [⟨1 0 -32 -44 -58 (12 9) -73 -52 -47 -69], ⟨0 1 22 30 39 (-5 -3) 49 36 33 47]]

Generator preimages: 2/1, 3/1

Generator tunings (no-17 no-19):

Weil Tenney WE TE
Optimized ??? 1201.1304 ¢, 1900.1656 ¢ 1200.9840 ¢, 1899.9400 ¢ 1200.9879 ¢, 1899.9461 ¢
Constrained ??? 1\1, 1898.6199 ¢ 1\1, 1898.4197 ¢ 1\1, 1898.4955 ¢
Destretched ??? 1\1, 1898.3773 ¢ 1\1, 1898.3833 ¢ 1\1, 1898.3833 ¢

Prime harmonics (WE):

Prime 2/1 3/1 7/1 11/1 13/1 23/1 31/1 37/1 41/1
Tuning 1200.984 ¢ 1899.940 ¢ 3367.1912 ¢ 4154.903 ¢ 4440.5866 ¢ 5425.2263 ¢ 5946.6708 ¢ 6251.7709 ¢ 6429.2824 ¢
Error +0.984 ¢ −2.015 ¢ −1.6347 ¢ +3.585 ¢ +0.059 ¢ −3.0481 ¢ +1.6352 ¢ +0.4269 ¢ +0.220 ¢

Optimal ET sequence: 12fikm, 43deffiiikkllmm, 55film, 67, 122(h)ilm

Badness (Smith TE): 0.01056 (0.01515)

Badness (Dirichlet): 1.07545 (1.77384)

Hemififive (68 & 94)

Subgroup: 2.3.5.7.11.13

Comma list: 275/273, 325/324, 385/384, 1716/1715

Mapping: [⟨2 7 10 6 5 12], ⟨0 -10 -14 -1 5 -12]]

Generator preimages: 99/70, 8/7

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 599.9903 ¢, 229.6835 ¢ 600.0198 ¢, 229.7451 ¢ 600.0210 ¢, 229.7456 ¢
Constrained ??? 1\2, 229.6914 ¢ 1\2, 229.7361 ¢ 1\2, 229.7307 ¢
Destretched ??? 1\2, 229.6872 ¢ 1\2, 229.7375 ¢ 1\2, 229.7376 ¢

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1
Tuning 1200.0396 ¢ 1902.6876 ¢ 2783.7665 ¢ 3370.3738 ¢ 4148.8246 ¢ 4443.2964 ¢
Error +0.0396 ¢ +0.7326 ¢ −2.5472 ¢ +1.5479 ¢ −2.4933 ¢ +2.7687 ¢

Optimal ET sequence (TE): 26, 68, 94

Badness (Smith TE): 0.03418

Badness (Dirichlet): 1.41232

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 275/273, 325/324, 385/384, 561/560, 595/594

Mapping: [⟨2 7 10 6 5 12 12 15], ⟨0 -10 -14 -1 5 -12 -10 -17]]

Generator preimages: 17/12, 8/7

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 599.9903 ¢, 229.6835 ¢ 600.0751 ¢, 229.7512 ¢ 600.0765 ¢, 229.7517 ¢
Constrained ??? 1\2, 229.6914 ¢ 1\2, 229.7168 ¢ 1\2, 229.6935 ¢
Destretched ??? 1\2, 229.6872 ¢ 1\2, 229.7224 ¢ 1\2, 229.7224 ¢

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1 17/1 19/1
Tuning 1200.1502 ¢ 1903.0139 ¢ 2784.2344 ¢ 3370.6993 ¢ 4149.1313 ¢ 4443.8869 ¢ 4903.3893 ¢ 5095.3563 ¢
Error +0.1502 ¢ +1.0589 ¢ −2.0793 ¢ +1.8734 ¢ −2.1867 ¢ +3.3593 ¢ −1.5661 ¢ −2.1567 ¢

Optimal ET sequence (TE): 26, 68, 94

Badness (Smith TE): 0.01725

Badness (Dirichlet): 1.04955

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 190/189, 253/252, 275/273, 325/324, 385/384, 561/560, 595/594

Mapping: [⟨2 7 10 6 5 12 12 15 19], ⟨0 -10 -14 -1 5 -12 -10 -17 -26]]

Generator preimages: 17/12, 8/7

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 599.9903 ¢, 229.6835 ¢ 600.0744 ¢, 229.7474 ¢ 600.0758 ¢, 229.7480 ¢
Constrained ??? 1\2, 229.6914 ¢ 1\2, 229.7135 ¢ 1\2, 229.6908 ¢
Destretched ??? 1\2, 229.6872 ¢ 1\2, 229.7189 ¢ 1\2, 229.7190 ¢

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1 17/1 19/1 23/1
Tuning 1200.1488 ¢ 1903.0465 ¢ 2784.28 ¢ 3370.6991 ¢ 4149.1094 ¢ 4443.9237 ¢ 4903.4186 ¢ 5095.4097 ¢ 5427.9804 ¢
Error +0.1488 ¢ +1.0915 ¢ −2.0337 ¢ +1.8732 ¢ −2.2086 ¢ +3.3961 ¢ −1.5368 ¢ −2.1033 ¢ −0.2939 ¢

Optimal ET sequence (TE): 26i, 68, 94

Badness (Smith TE): 0.01458

Badness (Dirichlet): 1.04399

Cyanide (140 & 441)

Subgroup: 2.3.5.7.13(.23)

Comma list: 2401/2400, 4096/4095, (35000/34983), 140625/140608

Mapping: [⟨7 3 23 21 34 (60)], ⟨0 6 -5 -1 -6 (-21)]]

Generator preimages: 3675/3328, 8/7

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 171.4255 ¢, 231.2847 ¢ 171.4244 ¢, 231.2902 ¢ 171.4244 ¢, 231.2902 ¢
Constrained ??? 1\7, 231.2969 ¢ 1\7, 231.2963 ¢ 1\7, 231.2991 ¢
Destretched ??? 1\7, 231.2888 ¢ 1\7, 231.2943 ¢ 1\7, 231.2943 ¢

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 13/1 23/1
Tuning 1199.971 ¢ 1902.0144 ¢ 2786.3108 ¢ 3368.6227 ¢ 4440.6893 ¢ 5428.3714 ¢
Error −0.029 ¢ +0.0594 ¢ −0.0029 ¢ −0.2032 ¢ +0.1616 ¢ +0.097 ¢

Optimal ET sequence (TE): 21, 119, 140, 301, 441 (add-23: 21ii, 140, 301, 441, 3227f)

Badness (Smith TE): 0.01465 (0.01044)

Badness (Dirichlet): 0.50663 (0.49340)

13-limit

Subgroup: 2.3.5.7.11.13(.23)

Comma list: 1288/1287, 2080/2079, 2401/2400, 4096/4095, 35000/34983

Mapping: [⟨7 3 23 21 62 34 (60)], ⟨0 6 -5 -1 -28 -6 (-21)]]

Generator preimages: 875/792, 8/7

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 171.4227, 231.3059 ¢ 171.4242, 231.3085 ¢ 171.4242, 231.3085 ¢
Constrained ??? 1\7, 231.3138 ¢ 1\7, 231.3149 ¢ 1\7, 231.3180 ¢
Destretched ??? 1\7, 231.3138 ¢ 1\7, 231.3144 ¢ 1\7, 231.3144 ¢

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1 23/1
Tuning 1199.9697 ¢ 1902.1239 ¢ 2786.2149 ¢ 3368.6006 ¢ 4151.6642 ¢ 4440.5731 ¢ 5427.9754 ¢
Error −0.0303 ¢ +0.1689 ¢ −0.0988 ¢ −0.2253 ¢ +0.3463 ¢ +0.0454 ¢ −0.2989 ¢

Optimal ET sequence (TE): 140, 301e, 441, 581, 1022, 1603e (same sequence for add-23)

Badness (Smith TE): 0.02947 (0.01904)

Badness (Dirichlet): 1.21770 (1.01017)

581 & 742 (weak extension)

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2401/2400, 3136/3135, 4096/4095, 5985/5984, 12376/12375, 14875/14872

Mapping: [⟨7 15 13 19 24 22 31 28], ⟨0 -18 15 3 1 18 -11 8]]  (23/1: [⟨18], ⟨63]])

Generator preimages: 85/77, 143/140

Generator tunings (add-23):

Norm Weil Tenney WE TE
Optimized ??? 171.4266 ¢, 37.1864 ¢ 171.4244 ¢, 37.1884 ¢ 171.4244 ¢, 37.1884 ¢
Constrained ??? 1\7, 37.1867 ¢ 1\7, 37.1893 ¢ 1\7, 37.1893 ¢
Destretched ??? 1\7, 37.1868 ¢ 1\7, 37.1893 ¢ 1\7, 37.1893 ¢

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1 17/1 19/1
Tuning ( ¢) 1199.97 ¢ 1901.97 ¢ 2786.34 ¢ 3368.63 ¢ 4151.37 ¢ 4440.73 ¢ 4905.08 ¢ 5097.39 ¢
Error −0.0291 ¢ +0.0195 ¢ +0.0302 ¢ −0.1966 ¢ +0.0566 ¢ +0.2014 ¢ +0.1289 ¢ −0.1218 ¢

Optimal ET sequence (TE): 161, 420, 581, 742, 1323

Badness (Smith TE): 0.01260

Badness (Dirichlet): 0.76654

130 & 181

See also: 7-limit

Subgroup: 2.3.5.7.13

Comma list: 2401/2400, 4096/4095, 492128/492075

Mapping: [⟨1 11 -22 -7 19], ⟨0 -24 62 25 -39]]

Generator preimages: 2/1, 21/16

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 1199.983 ¢, 470.7428 ¢ 1199.9727 ¢, 470.7388 ¢ 1199.9727 ¢, 470.7388 ¢
Constrained ??? 1\1, 470.7495 ¢ 1\1, 470.7495 ¢ 1\1, 470.7493 ¢
Destretched ??? 1\1, 470.7495 ¢ 1\1, 470.7495 ¢ 1\1, 470.7495 ¢

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 13/1
Tuning 1199.9727 ¢ 1901.9687 ¢ 2786.4055 ¢ 3368.6607 ¢ 4440.6683 ¢
Error −0.0273 ¢ +0.0137 ¢ +0.0918 ¢ −0.1652 ¢ +0.1407 ¢

Optimal ET sequence (TE): 51, 79c, 130, 311, 441, 1453, 1894, 2335

Badness (Smith TE): 0.01381

Badness (Dirichlet): 0.47741

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 4096/4095, 4459/4455

Mapping: [⟨1 11 -22 -7 -24 19], ⟨0 -24 62 25 70 -39]]

Generator preimages: 2/1, 21/16

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 1199.9305 ¢, 470.7167 ¢ 1199.9386 ¢, 470.7221 ¢ 1199.9387 ¢, 470.7221 ¢
Constrained ??? 1\1, 470.7411 ¢ 1\1, 470.746 ¢ 1\1, 470.745 ¢
Destretched ??? 1\1, 470.744 ¢ 1\1, 470.7462 ¢ 1\1, 470.7461 ¢

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1
Tuning 1199.9386 ¢ 1901.9946 ¢ 2786.1202 ¢ 3368.482 ¢ 4152.0197 ¢ 4440.6721 ¢
Error −0.0614 ¢ +0.0396 ¢ −0.1936 ¢ −0.3439 ¢ +0.7017 ¢ +0.1445 ¢

Optimal ET sequence (TE): 51, 130, 311, 441, 752e, 1193e

Badness (Smith TE): 0.02172

Badness (Dirichlet): 0.89734

Whooshlet (441 & 3072)

Main article: Whoosh

Subgroup: 2.3.5.7.13

Comma list: 140625/140608, 250047/250000, 1990656/1990625  (9801/9800 can be added if the full 13-limit is desired)

Mapping: [⟨3 18 17 2 40], ⟨0 -33 -25 16 -72]]

Generator preimages: 63/50, 192/175

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 400.0024 ¢, 160.5488 ¢ 400.0034 ¢, 160.5495 ¢ 400.0034 ¢, 160.5495 ¢
Constrained ??? 1\, 160.5478 ¢ 1\, 160.5480 ¢ 1\, 160.5476 ¢
Destretched ??? 1\3, 160.5478 ¢ 1\3, 160.5481 ¢ 1\3, 160.5481 ¢

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 13/1
Tuning 1200.0102 ¢ 1901.9291 ¢ 2786.3213 ¢ 3368.7981 ¢ 4440.575 ¢
Error +0.0102 ¢ −0.0259 ¢ +0.0076 ¢ −0.0278 ¢ +0.0473 ¢

Optimal ET sequence (TE): 15, 411f, 426, 441, 1308, 1749, 2190, 6129, 8319, 10509  (full 13-limit: 426, 882, 1308, 2190, 5688, 7878, 17946bde)

Badness (Smith TE): 0.0142139

Badness (Dirichlet): 0.4914805


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