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

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


== 27e & 87 ==
: ''See also: [[User:Dummy_index/3rd-tritave_temperaments#BPS_III_.28b30_.26_b9.2C_b39_.26_b9.29|BPS III]]''


Subgroup: 2.3.5.7.11.13
[[Subgroup]]: [[3.5.7.13_subgroup|3.5.7.13]]


Comma list: [[196/195]], [[325/324]], [[441/440]], [[1001/1000]]
[[Comma list]]: [[245/243]], [[2197/2187]]


Mapping: [{{val|3 2 2 4 1 10}}, {{val|0 5 9 8 17 2}}]
[[Mapping]]: {{mapping|3 1 7 7|0 2 -1 0}}


TE generators: ~63/50 = 399.995, ~91/80 = 220.689
[[Transversal_generators|Generator preimages]]: [[13/9]], [[13/7]]


CTE generators: 1\3, ~91/80 = 220.687
[[Optimization|Generator tunings]]:


{{Optimal ET sequence|legend=1|27e, 60e, 87}} (idk how to calculate this)
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| ???
| 634.5206¢, 1075.4459¢
| 634.5215¢, 1075.4475¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\b3, 1074.0886¢
| 1\b3, 1074.6634¢
| 1\b3, 1075.1282¢
|-
| [[POTE|Destretched]]
| ???
| ???
| colspan="2" | 1\b3, 1074.5381¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 3/1
! 5/1
! 7/1
! 13/1
|-
| [[Tuning_map|Tuning]]
| 1903.5617¢
| 2785.4124¢
| 3366.1979¢
| 4441.6439¢
|-
| [[Error]]
| +1.6067¢
| −0.9013¢
| −2.628¢
| +1.1162¢
|}
 
[[Optimal ET sequence]] ([[TE]]): [[9edt|b9]], [[21edt|b21]], [[30edt|b30]], [[39edt|b39]], [[69edt|b69]], [[108edt|b108]], [[177edt|b177d]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.00287
 
Badness ([[Dirichlet_badness|Dirichlet]]): 0.26872
 
=== 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]]''
 
[[Subgroup]]: [[3.5.7.11.13_subgroup|3.5.7.11.13]]
 
[[Comma list]]: [[245/243]], [[847/845]], [[1575/1573]]
 
[[Mapping]]: {{mapping|3 1 7 4 7|0 4 -2 3 0}}
 
[[Transversal_generators|Generator preimages]]: [[13/9]], [[15/11]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 634.4577¢, 537.4447¢
| 634.5219¢, 537.7266¢
| 634.5229¢, 537.7274¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\b3, 537.0443¢
| 1\b3, 537.3589¢
| 1\b3, 537.7233¢
|-
| [[POTE|Destretched]]
| ???
| 1\b3, 537.0443¢
| colspan="2" | 1\b3, 537.2716¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 3/1
! 5/1
! 7/1
! 11/1
! 13/1
|-
| [[Tuning_map|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]]): [[18edt|b18]], [[21edt|b21]], [[39edt|b39]], [[99edt|b99]], [[138edt|b138]], [[177edt|b177d]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.00566
 
Badness ([[Dirichlet_badness|Dirichlet]]): 0.44749
 
=== Tri[[dubhe]] (b69 & b78) ===
 
[[Subgroup]]: [[3.5.7.13.17.19_subgroup|3.5.7.13.17.19]]


Badness: 1.058 (Sintel)
[[Comma list]]: [[245/243]], [[325/323]], [[665/663]], [[3213/3211]]


== 130 & 181 ==
[[Mapping]]: {{mapping|3 3 6 7 6 7|0 4 -2 0 5 3}}


Subgroup: 2.3.5.7.13
[[Transversal_generators|Generator preimages]]: [[13/9]], [[17/15]]


Comma list: [[2401/2400]], [[4096/4095]], 492128/492075
[[Optimization|Generator tunings]]:


Mapping: [{{val|1 11 -22 -7 19}}, {{val|0 -24 62 25 -39}}]
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 634.4242¢, 220.2042¢
| 634.3781¢, 220.1438¢
| 634.3798¢, 220.1444¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\b3, 220.0518¢
| 1\b3, 220.0811¢
| 1\b3, 220.4087¢
|-
| [[POTE|Destretched]]
| ???
| 1\b3, 220.0518¢
| colspan="2" | 1\b3, 220.0074¢
|}


TE generators: ~2/1 = 1199.973, ~21/16 = 470.739
[[Prime_interval|Prime harmonics]] ([[WE]]):


CTE generators: 2/1, ~21/16 = 470.749
: {| class="wikitable right-1"
|-
! Prime
! 3/1
! 5/1
! 7/1
! 13/1
! 17/1
! 19/1
|-
| [[Tuning_map|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|legend=1|130, 441, maybe 571, 1012, sorry idk}}
[[Optimal ET sequence]] ([[TE]]): [[51edt|b51]], [[60edt|b60]], [[69edt|b69]], [[147edt|b147]]


Badness: 0.477 (Sintel)
Badness ([[TE_logflat_badness|Smith TE]]): 0.00523


=== 13-limit ===
Badness ([[Dirichlet_badness|Dirichlet]]): 0.48415
 
=== Trisemi[[dubhe]] (b60 & b78) ===
 
: ''See also: [[Kleismic_family#13-limit_13|Clyde]]''
 
[[Subgroup]]: [[3.5.7.11.13.17.19_subgroup|3.5.7.11.13.17.19]]
 
[[Comma list]]: [[245/243]], [[325/323]], [[665/663]], [[847/845]], [[1575/1573]]
 
[[Mapping]]: {{mapping|6 2 14 8 14 7 11|0 4 -2 3 0 5 3}}
 
[[Transversal_generators|Generator preimages]]: [[119/99]], [[15/11]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| ???
| 317.2028¢, 537.3966¢
| 317.2037¢, 537.3981¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\b6, 537.0443¢
| 1\b6, 537.1261¢
| 1\b6, 537.5291¢
|-
| [[POTE|Destretched]]
| ???
| ???
| colspan="2" | 1\b6, 1074.0806¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 3/1
! 5/1
! 7/1
! 11/1
! 13/1
! 17/1
! 19/1
|-
| [[Tuning_map|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]]): [[18edt|b18]], [[60edt|b60]], [[78edt|b78]], [[138edt|b138]], [[216edt|b216]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.01046
 
Badness ([[Dirichlet_badness|Dirichlet]]): 0.95032
 
== Hemififive (68 & 94) ==
 
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
 
[[Comma list]]: [[275/273]], [[325/324]], [[385/384]], [[1716/1715]]
 
[[Mapping]]: {{mapping|2 7 10 6 5 12|0 -10 -14 -1 5 -12}}
 
[[Transversal_generators|Generator preimages]]: [[99/70]], [[8/7]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 599.9903¢, 229.6835¢
| 600.0198¢, 229.7451¢
| 600.0210¢, 229.7456¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\2, 229.6914¢
| 1\2, 229.7361¢
| 1\2, 229.7307¢
|-
| [[POTE|Destretched]]
| ???
| 1\2, 229.6872¢
| colspan="2" | 1\2, 229.7375¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 11/1
! 13/1
|-
| [[Tuning_map|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]]): [[26edo|26]], [[68edo|68]], [[94edo|94]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.03418
 
Badness ([[Dirichlet_badness|Dirichlet]]): 1.41232
 
==== 19-limit ====
 
[[Subgroup]]: [[19-limit|2.3.5.7.11.13.17.19]]
 
[[Comma list]]: [[190/189]], [[275/273]], [[325/324]], [[385/384]], [[561/560]], [[595/594]]
 
[[Mapping]]: {{mapping|2 7 10 6 5 12 12 15|0 -10 -14 -1 5 -12 -10 -17}}
 
[[Transversal_generators|Generator preimages]]: [[17/12]], [[8/7]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 599.9903¢, 229.6835¢
| 600.0751¢, 229.7512¢
| 600.0765¢, 229.7517¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\2, 229.6914¢
| 1\2, 229.7168¢
| 1\2, 229.6935¢
|-
| [[POTE|Destretched]]
| ???
| 1\2, 229.6872¢
| colspan="2" | 1\2, 229.7224¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 11/1
! 13/1
! 17/1
! 19/1
|-
| [[Tuning_map|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]]): [[26edo|26]], [[68edo|68]], [[94edo|94]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.01725
 
Badness ([[Dirichlet_badness|Dirichlet]]): 1.04955
 
==== 23-limit ====
 
[[Subgroup]]: [[23-limit|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]]: {{mapping|2 7 10 6 5 12 12 15 19|0 -10 -14 -1 5 -12 -10 -17 -26}}
 
[[Transversal_generators|Generator preimages]]: [[17/12]], [[8/7]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 599.9903¢, 229.6835¢
| 600.0744¢, 229.7474¢
| 600.0758¢, 229.7480¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\2, 229.6914¢
| 1\2, 229.7135¢
| 1\2, 229.6908¢
|-
| [[POTE|Destretched]]
| ???
| 1\2, 229.6872¢
| colspan="2" | 1\2, 229.7189¢
|}


Subgroup: 2.3.5.7.11.13
[[Prime_interval|Prime harmonics]] ([[WE]]):


Comma list: [[1575/1573]], [[2080/2079]], [[4096/4095]], [[4459/4455]]
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 11/1
! 13/1
! 17/1
! 19/1
! 23/1
|-
| [[Tuning_map|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¢
|}


Mapping: [{{val|1 11 -22 -7 -24 19}}, {{val|0 -24 62 25 70 -39}}]
[[Optimal ET sequence]] ([[TE]]): [[26edo|26i]], [[68edo|68]], [[94edo|94]]


TE generators: ~2/1 = 1199.939, ~21/16 = 470.722
Badness ([[TE_logflat_badness|Smith TE]]): 0.01458


CTE generators: 2/1, ~21/16 = 470.745
Badness ([[Dirichlet_badness|Dirichlet]]): 1.04399


{{Optimal ET sequence|legend=1|130, 311, 441, idk}}
== Worsetone (12f & 67) ==


Badness: 0.897 (Sintel)
<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>


== Worsetone (12f & 67) ==
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]


Subgroup: 2.3.5.7.11.13
[[Comma list]]: [[81/80]], [[176/175]], [[196/195]], [[364/363]]


Comma list: [[81/80]], [[176/175]], [[196/195]], [[364/363]]
[[Mapping]]: {{mapping|1 0 -4 -32 -44 -58|0 1 4 22 30 39}}


Mapping: [{{val|1 1 0 -10 -14 -19}}, {{val|0 1 4 22 30 39}}]
[[Transversal_generators|Generator preimages]]: [[2/1]], [[3/1]]


Generator tunings:
[[Optimization|Generator tunings]]:


: {| class="wikitable right-1"
: {| class="wikitable right-1"
Line 67: Line 547:
|-
|-
| [[Optimized]]
| [[Optimized]]
|
| ???
|
| 1199.6846¢, 1897.7889¢
| 1199.788, 698.355
| 1199.7882¢, 1898.1434¢
| 1199.8021, 698.3632
| 1199.8021¢, 1898.1654¢
|-
|-
| [[Constrained]]
| [[Constrained]]
|
| ???
|
| 1\1, 1898.2217¢
| 1\1, 698.471
| 1\1, 698.4715¢
| 1\1, 698.455
| 1\1, 698.4553¢
|-
|-
| [[Destretched]]
| [[Destretched]]
|
| ???
|
| 1\1, 1898.2878¢
| 1\1, 698.4784
| colspan="2" | 1\1, 1898.4785¢
| 1\1, 698.4784
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 11/1
! 13/1
|-
| [[Tuning_map|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|legend=1|12f, 55f, 67}}
{{Optimal ET sequence|legend=1|12f, 55f, 67}}


Errors (WE): −0.21{{cent}}, −3.81{{cent}}, +7.11{{cent}}, −2.89{{cent}}, +2.3{{cent}}, −0.65{{cent}}
Badness ([[TE_logflat_badness|Smith TE]]): 0.04940
 
Badness ([[Dirichlet_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]]: [{{val|1 0 -32 -44 -58 (12 9) -73 -52 -47 -69}}, {{val|0 1 22 30 39 (-5 -3) 49 36 33 47}}]
 
[[Transversal_generators|Generator preimages]]: [[2/1]], [[3/1]]
 
[[Optimization|Generator tunings]] ([[2.3.7.11.13.23.31.37.41_subgroup|no-17 no-19]]):
 
: {| class="wikitable right-1"
|-
!
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 1201.1304¢, 1900.1656¢
| 1200.9840¢, 1899.9400¢
| 1200.9879¢, 1899.9461¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\1, 1898.6199¢
| 1\1, 1898.4197¢
| 1\1, 1898.4955¢
|-
| [[POTE|Destretched]]
| ???
| 1\1, 1898.3773¢
| colspan="2" | 1\1, 1898.3833¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 7/1
! 11/1
! 13/1
! 23/1
! 31/1
! 37/1
! 41/1
|-
| [[Tuning_map|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]]: [[12edo|12fikm]], [[43edo|43deffiiikkllmm]], [[55edo|55film]], [[67edo|67]], [[122edo|122(h)ilm]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.01056 (0.01515)
 
Badness ([[Dirichlet_badness|Dirichlet]]): 1.07545 (1.77384)
 
== Senscape (27e &amp; 87) ==
 
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
 
[[Comma list]]: [[196/195]], [[325/324]], [[441/440]], [[1001/1000]]
 
[[Mapping]]: {{mapping|3 2 2 4 1 10|0 5 9 8 17 2}}
 
[[Transversal_generators|Generator preimages]]: [[63/50]], [[91/80]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 400.3122¢, 220.5893¢
| 399.994¢, 220.6882¢
| 399.9947¢, 220.6886¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\3, 220.729¢
| 1\3, 220.6899¢
| 1\3, 220.6873¢
|-
| [[POTE|Destretched]]
| ???
| 1\3, 220.4173¢
| colspan="2" | 1\3, 220.6915¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 11/1
! 13/1
|-
| [[Tuning_map|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]]): [[27edo|27e]], [[60edo|60e]], [[87edo|87]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.02560
 
Badness ([[Dirichlet_badness|Dirichlet]]): 1.05783
 
== Quadrameantone (43 &amp; 81) ==
 
Named for its [[meantone]] fifth at −4 generators, quadrameantone splits [[didacus]]'s [[14/5]] into eight [[25/22]]s. Due to the temperament's precision and complexity, the [[3/1]] from meantone is not formally included in the subgroup; if a more accurate 3/1 is desired, the temperament may be extended to the full 23-limit for a significant increase in scale size.
 
[[Subgroup]]: [[2.5.7.11 subgroup|2.5.7.11]][[2.5.7.11.17.19.23 subgroup|.17.19.23]]
 
[[Comma list]]: [[392/391]], [[476/475]], [[2300/2299]], [[3136/3125]], [[8096/8075]]
 
[[Mapping]]: {{mapping|1 12 27 21 18 23 39|0 -16 -40 -29 -23 -31 -57}}
 
[[Transversal_generators|Generator preimages]]: [[2/1]], [[25/19]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 1199.6085¢, 725.5426¢
| 1199.6487¢, 725.5704¢
| 1199.6492¢, 725.5706¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\1, 725.8157¢
| 1\1, 725.7958¢
| 1\1, 725.8170¢
|-
| [[POTE|Destretched]]
| ???
| 1\1, 725.7794¢
| colspan="2" | 1\1, 725.7828¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 5/1
! 7/1
! 11/1
! 17/1
! 19/1
! 23/1
|-
| [[Tuning_map|Tuning]]
| 1199.6487¢
| 2786.6589¢
| 3367.701¢
| 4151.0827¢
| 4905.5587¢
| 5099.2394¢
| 5428.7895¢
|-
| [[Error]]
| −0.3513¢
| +0.3452¢
| −1.1249¢
| −0.2353¢
| +0.6033¢
| +1.7264¢
| +0.5152¢
|}
 
[[Optimal ET sequence]] ([[TE]]): [[38edo|38di]], [[43edo|43]], [[81edo|81]], [[124edo|124]], [[291edo|291ghi]], [[415edo|415ghhi]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.00651
 
Badness ([[Dirichlet_badness|Dirichlet]]): 0.53307
 
==== 2.5.7.11.13.17.19.23 subgroup ====
 
[[Subgroup]]: [[2.5.7.11.13 subgroup|2.5.7.11.13]][[2.5.7.11.13.17.19.23 subgroup|.17.19.23]]
 
[[Comma list]]: [[392/391]], [[476/475]], [[1001/1000]], [[1496/1495]], [[3584/3575]], [[7744/7735]]
 
[[Mapping]]: {{mapping|1 12 27 21 -9 18 23 39|0 -16 -40 -29 21 -23 -31 -57}}
 
[[Transversal_generators|Generator preimages]]: [[2/1]], [[25/19]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 1199.6083¢, 725.5424¢
| 1199.6880¢, 725.5983¢
| 1199.6884¢, 725.5986¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\1, 725.8157¢
| 1\1, 725.7923¢
| 1\1, 725.8129¢
|-
| [[POTE|Destretched]]
| ???
| 1\1, 725.7793¢
| colspan="2" | 1\1, 725.7871¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 5/1
! 7/1
! 11/1
! 13/1
! 17/1
! 19/1
! 23/1
|-
| [[Tuning_map|Tuning]]
| 1199.6880¢
| 2786.6822¢
| 3367.6415¢
| 4151.0954¢
| 4440.3735¢
| 4905.6216¢
| 5099.2747¢
| 5428.7253¢
|-
| [[Error]]
| −0.3120¢
| +0.3684¢
| −1.1844¢
| −0.2225¢
| −0.1542¢
| +0.6662¢
| +1.7617¢
| +0.4509¢
|}
 
[[Optimal ET sequence]] ([[TE]]): [[43edo|43]], [[81edo|81]], [[124edo|124]], [[291edo|291ghi]], [[415edo|415ghhi]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.00808
 
Badness ([[Dirichlet_badness|Dirichlet]]): 0.69675
 
==== 23-limit ====
 
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]][[23-limit|.17.19.23]]
 
[[Comma list]]: [[208/207]], [[256/255]], [[364/363]], [[400/399]], [[442/441]], [[715/714]], [[1001/1000]]
 
[[Mapping]]: {{mapping|1 17 -4 -13 -8 12 -5 -8 -18|0 -39 16 40 29 -21 23 31 57}}
 
[[Transversal_generators|Generator preimages]]: [[2/1]], [[21/16]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| ???
| 1199.3238¢, 473.9951¢
| 1199.3250¢, 473.9956¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\1, 474.2641¢
| 1\1, 474.2609¢
| 1\1, 474.2484¢
|-
| [[POTE|Destretched]]
| ???
| ???
| colspan="2" | 1\1, 474.2624¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 11/1
! 13/1
! 17/1
! 19/1
! 23/1
|-
| [[Tuning_map|Tuning]]
| 1199.3238¢
| 1902.6954¢
| 2786.6265¢
| 3368.5949¢
| 4151.2677¢
| 4437.9883¢
| 4905.2684¢
| 5099.2579¢
| 5429.8927¢
|-
| [[Error]]
| −0.6762¢
| +0.7404¢
| +0.3128¢
| −0.231¢
| −0.0503¢
| −2.5393¢
| +0.313¢
| +1.7449¢
| +1.6183¢
|}
 
[[Optimal ET sequence]] ([[TE]]): [[43edo|43]], [[124edo|124]], [[167|167hi]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.02501
 
Badness ([[Dirichlet_badness|Dirichlet]]): 1.79049
 
== Cyanide (140 &amp; 441) ==
 
[[Subgroup]]: [[2.3.5.7.13_subgroup|2.3.5.7.13]]([[2.3.5.7.13.23_subgroup|.23]])
 
[[Comma list]]: [[2401/2400]], [[4096/4095]], ([[35000/34983]]), [[140625/140608]]
 
[[Mapping]]: {{mapping|7 3 23 21 34 (60)|0 6 -5 -1 -6 (-21)}}
 
[[Transversal_generators|Generator preimages]]: [[3675/3328]], [[8/7]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 171.4255¢, 231.2847¢
| 171.4244¢, 231.2902¢
| 171.4244¢, 231.2902¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\7, 231.2969¢
| 1\7, 231.2963¢
| 1\7, 231.2991¢
|-
| [[POTE|Destretched]]
| ???
| 1\7, 231.2888¢
| colspan="2" | 1\7, 231.2943¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 13/1
! 23/1
|-
| [[Tuning_map|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]]): [[21edo|21]], [[119edo|119]], [[140edo|140]], [[301edo|301]], [[441edo|441]] <small>(add-23: [[21edo|21ii]], [[140edo|140]], [[301edo|301]], [[441edo|441]], [[3227edo|3227f]])</small>
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.01465 (0.01044)
 
Badness ([[Dirichlet_badness|Dirichlet]]): 0.50663 (0.49340)
 
==== 13-limit ====
 
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]([[2.3.5.7.11.13.23_subgroup|.23]])
 
[[Comma list]]: [[1288/1287]], [[2080/2079]], [[2401/2400]], [[4096/4095]], [[35000/34983]]
 
[[Mapping]]: {{mapping|7 3 23 21 62 34 (60)|0 6 -5 -1 -28 -6 (-21)}}
 
[[Transversal_generators|Generator preimages]]: [[875/792]], [[8/7]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 171.4227, 231.3059¢
| 171.4242, 231.3085¢
| 171.4242, 231.3085¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\7, 231.3138¢
| 1\7, 231.3149¢
| 1\7, 231.3180¢
|-
| [[POTE|Destretched]]
| ???
| 1\7, 231.3138¢
| colspan="2" | 1\7, 231.3144¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 11/1
! 13/1
! 23/1
|-
| [[Tuning_map|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]]): [[140edo|140]], [[301edo|301e]], [[441edo|441]], [[581edo|581]], [[1022edo|1022]], [[1603edo|1603e]] <small>(same sequence for add-23)</small>
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.02947 (0.01904)
 
Badness ([[Dirichlet_badness|Dirichlet]]): 1.21770 (1.01017)
 
=== 581 &amp; 742 <small style="font-weight:400">(weak extension)</small> ===
 
[[Subgroup]]: [[19-limit|2.3.5.7.11.13.17.19]]
 
[[Comma list]]: [[2401/2400]], [[3136/3135]], [[4096/4095]], [[5985/5984]], [[12376/12375]], [[14875/14872]]
 
[[Mapping]]: {{mapping|7 15 13 19 24 22 31 28|0 -18 15 3 1 18 -11 8}} &nbsp;<small>(23/1: {{mapping|18|63}})</small>
 
[[Transversal_generators|Generator preimages]]: [[85/77]], [[143/140]]
 
[[Optimization|Generator tunings]] (add-23):
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 171.4266¢, 37.1864¢
| 171.4244¢, 37.1884¢
| 171.4244¢, 37.1884¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\7, 37.1867¢
| 1\7, 37.1893¢
| 1\7, 37.1893¢
|-
| [[POTE|Destretched]]
| ???
| 1\7, 37.1868¢
| colspan="2" | 1\7, 37.1893¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 11/1
! 13/1
! 17/1
! 19/1
|-
| [[Tuning_map|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]]): [[161edo|161]], [[420edo|420]], [[581edo|581]], [[742edo|742]], [[1323edo|1323]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.01260
 
Badness ([[Dirichlet_badness|Dirichlet]]): 0.76654
 
== Hemilafa (130 &amp; 181) ==
 
''See also: [[User:Francium/Miscellaneous_rank-2_temperaments#311_.26_1063|7-limit]]''
 
Hemilafa was named by [[User:FloraC|FloraC]] in 2022, after the 5-limit [[lafa]] temperament it splits.
 
[[Subgroup]]: [[2.3.5.7.13_subgroup|2.3.5.7.13]]
 
[[Comma list]]: [[2401/2400]], [[4096/4095]], [[492128/492075]]
 
[[Mapping]]: {{mapping|1 11 -22 -7 19|0 -24 62 25 -39}}
 
[[Transversal_generators|Generator preimages]]: [[2/1]], [[21/16]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 1199.983¢, 470.7428¢
| 1199.9727¢, 470.7388¢
| 1199.9727¢, 470.7388¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\1, 470.7495¢
| 1\1, 470.7495¢
| 1\1, 470.7493¢
|-
| [[POTE|Destretched]]
| ???
| 1\1, 470.7495¢
| colspan="2" | 1\1, 470.7495¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 13/1
|-
| [[Tuning_map|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]]): [[51edo|51]], [[79edo|79c]], [[130edo|130]], [[311edo|311]], [[441edo|441]], [[1453edo|1453]], [[1894edo|1894]], [[2335edo|2335]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.01381
 
Badness ([[Dirichlet_badness|Dirichlet]]): 0.47741
 
=== 13-limit ===
 
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
 
[[Comma list]]: [[1575/1573]], [[2080/2079]], [[4096/4095]], [[4459/4455]]
 
[[Mapping]]: {{mapping|1 11 -22 -7 -24 19|0 -24 62 25 70 -39}}
 
[[Transversal_generators|Generator preimages]]: [[2/1]], [[21/16]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 1199.9305¢, 470.7167¢
| 1199.9386¢, 470.7221¢
| 1199.9387¢, 470.7221¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\1, 470.7411¢
| 1\1, 470.746¢
| 1\1, 470.745¢
|-
| [[POTE|Destretched]]
| ???
| 1\1, 470.744¢
| colspan="2" | 1\1, 470.7462¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 11/1
! 13/1
|-
| [[Tuning_map|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]]): [[51edo|51]], [[130edo|130]], [[311edo|311]], [[441edo|441]], [[752edo|752e]], [[1193edo|1193e]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.02172
 
Badness ([[Dirichlet_badness|Dirichlet]]): 0.89734
 
== Whooshlet (441 &amp; 3072) ==
''Main article: [[Whoosh]]''
 
[[Subgroup]]: [[2.3.5.7.13_subgroup|2.3.5.7.13]]
 
[[Comma list]]: [[140625/140608]], [[250047/250000]], [[1990656/1990625]] &nbsp;<small>([[9801/9800]] can be added if the full 13-limit is desired)</small>
 
[[Mapping]]: {{mapping|3 18 17 2 40|0 -33 -25 16 -72}}
 
[[Transversal_generators|Generator preimages]]: [[63/50]], [[192/175]]
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 400.0024¢, 160.5488¢
| 400.0034¢, 160.5495¢
| 400.0034¢, 160.5495¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\, 160.5478¢
| 1\, 160.5480¢
| 1\, 160.5476¢
|-
| [[POTE|Destretched]]
| ???
| 1\3, 160.5478¢
| colspan="2" | 1\3, 160.5481¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 13/1
|-
| [[Tuning_map|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]]): [[15edo|15]], [[411edo|411f]], [[426edo|426]], [[441edo|441]], [[1308edo|1308]], [[1749edo|1749]], [[2190edo|2190]], [[6129edo|6129]], [[8319edo|8319]], [[10509edo|10509]] &nbsp;<small>(full 13-limit: [[426edo|426]], [[882edo|882]], [[1308edo|1308]], [[2190edo|2190]], [[5688edo|5688]], [[7878edo|7878]], [[17946edo|17946bde]])</small>
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.0142139
 
Badness ([[Dirichlet_badness|Dirichlet]]): 0.4914805
 
<!--
console.log(``.split(/\s+/).reduce((a,b) => a+'¢\n| '+b)+'¢')
console.log(``.split(/\s+/).reduce((a,b) => a+'¢\n| +'+b).replaceAll('+-', '−')+'¢')
 
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
 
[[Comma list]]:
 
[[Mapping]]: {{mapping|}}
 
[[Transversal_generators|Generator preimages]]:
 
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| ¢, ¢
| ¢, ¢
| ¢, ¢
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\, ¢
| 1\, ¢
| 1\, ¢
|-
| [[POTE|Destretched]]
| ???
| 1\, ¢
| 1\, ¢
| 1\, ¢
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):
 
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 11/1
! 13/1
|-
| [[Tuning_map|Tuning]]
|
|-
| [[Error]]
|
|}
 
[[Optimal ET sequence]] ([[TE]]):
 
Badness ([[TE_logflat_badness|Smith TE]]):
 
Badness ([[Dirichlet_badness|Dirichlet]]):


Badness: 2.041 (Sintel)
-->

Latest revision as of 15:27, 15 August 2025

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¢

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¢

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¢

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)

See also: Clyde

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¢

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

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¢

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¢

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¢

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

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¢

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¢

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)

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¢

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

Quadrameantone (43 & 81)

Named for its meantone fifth at −4 generators, quadrameantone splits didacus's 14/5 into eight 25/22s. Due to the temperament's precision and complexity, the 3/1 from meantone is not formally included in the subgroup; if a more accurate 3/1 is desired, the temperament may be extended to the full 23-limit for a significant increase in scale size.

Subgroup: 2.5.7.11.17.19.23

Comma list: 392/391, 476/475, 2300/2299, 3136/3125, 8096/8075

Mapping: [⟨1 12 27 21 18 23 39], ⟨0 -16 -40 -29 -23 -31 -57]]

Generator preimages: 2/1, 25/19

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 1199.6085¢, 725.5426¢ 1199.6487¢, 725.5704¢ 1199.6492¢, 725.5706¢
Constrained ??? 1\1, 725.8157¢ 1\1, 725.7958¢ 1\1, 725.8170¢
Destretched ??? 1\1, 725.7794¢ 1\1, 725.7828¢

Prime harmonics (WE):

Prime 2/1 5/1 7/1 11/1 17/1 19/1 23/1
Tuning 1199.6487¢ 2786.6589¢ 3367.701¢ 4151.0827¢ 4905.5587¢ 5099.2394¢ 5428.7895¢
Error −0.3513¢ +0.3452¢ −1.1249¢ −0.2353¢ +0.6033¢ +1.7264¢ +0.5152¢

Optimal ET sequence (TE): 38di, 43, 81, 124, 291ghi, 415ghhi

Badness (Smith TE): 0.00651

Badness (Dirichlet): 0.53307

2.5.7.11.13.17.19.23 subgroup

Subgroup: 2.5.7.11.13.17.19.23

Comma list: 392/391, 476/475, 1001/1000, 1496/1495, 3584/3575, 7744/7735

Mapping: [⟨1 12 27 21 -9 18 23 39], ⟨0 -16 -40 -29 21 -23 -31 -57]]

Generator preimages: 2/1, 25/19

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 1199.6083¢, 725.5424¢ 1199.6880¢, 725.5983¢ 1199.6884¢, 725.5986¢
Constrained ??? 1\1, 725.8157¢ 1\1, 725.7923¢ 1\1, 725.8129¢
Destretched ??? 1\1, 725.7793¢ 1\1, 725.7871¢

Prime harmonics (WE):

Prime 2/1 5/1 7/1 11/1 13/1 17/1 19/1 23/1
Tuning 1199.6880¢ 2786.6822¢ 3367.6415¢ 4151.0954¢ 4440.3735¢ 4905.6216¢ 5099.2747¢ 5428.7253¢
Error −0.3120¢ +0.3684¢ −1.1844¢ −0.2225¢ −0.1542¢ +0.6662¢ +1.7617¢ +0.4509¢

Optimal ET sequence (TE): 43, 81, 124, 291ghi, 415ghhi

Badness (Smith TE): 0.00808

Badness (Dirichlet): 0.69675

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 208/207, 256/255, 364/363, 400/399, 442/441, 715/714, 1001/1000

Mapping: [⟨1 17 -4 -13 -8 12 -5 -8 -18], ⟨0 -39 16 40 29 -21 23 31 57]]

Generator preimages: 2/1, 21/16

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? ??? 1199.3238¢, 473.9951¢ 1199.3250¢, 473.9956¢
Constrained ??? 1\1, 474.2641¢ 1\1, 474.2609¢ 1\1, 474.2484¢
Destretched ??? ??? 1\1, 474.2624¢

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1 17/1 19/1 23/1
Tuning 1199.3238¢ 1902.6954¢ 2786.6265¢ 3368.5949¢ 4151.2677¢ 4437.9883¢ 4905.2684¢ 5099.2579¢ 5429.8927¢
Error −0.6762¢ +0.7404¢ +0.3128¢ −0.231¢ −0.0503¢ −2.5393¢ +0.313¢ +1.7449¢ +1.6183¢

Optimal ET sequence (TE): 43, 124, 167hi

Badness (Smith TE): 0.02501

Badness (Dirichlet): 1.79049

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¢

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¢

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¢

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

Hemilafa (130 & 181)

See also: 7-limit

Hemilafa was named by FloraC in 2022, after the 5-limit lafa temperament it splits.

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¢

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¢

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¢

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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