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

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fixing names
m adding remaining tuning boxes
Line 8: Line 8:
[[Mapping]]: {{mapping|3 2 2 4 1 10|0 5 9 8 17 2}}
[[Mapping]]: {{mapping|3 2 2 4 1 10|0 5 9 8 17 2}}


[[WE]] generators: ~63/50 = 399.9940, ~91/80 = 220.6882
[[Transversal_generators|Generator preimages]]: [[63/50]], [[91/80]]


[[CWE]] generators: 1\3, ~91/80 = 220.6899
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 400.3122{{cc}}, 220.5893{{cc}}
| 399.994{{cc}}, 220.6882{{cc}}
| 399.9947{{cc}}, 220.6886{{cc}}
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\3, 220.729{{cc}}
| 1\3, 220.6899{{cc}}
| 1\3, 220.6873{{cc}}
|-
| [[POTE|Destretched]]
| ???
| 1\3, 220.4173{{cc}}
| 1\3, 220.6915{{cc}}
| 1\3, 220.6915{{cc}}
|}
 
[[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{{cc}}
| 1903.4292{{cc}}
| 2786.1822{{cc}}
| 3365.482{{cc}}
| 4151.6942{{cc}}
| 4441.3165{{cc}}
|-
| [[Error]]
| -0.018{{cc}}
| +1.4742{{cc}}
| −0.1315{{cc}}
| −3.3439{{cc}}
| +0.3762{{cc}}
| +0.7889{{cc}}
|}


[[Optimal ET sequence]] ([[TE]]): [[27edo|27e]], [[60edo|60e]], [[87edo|87]]
[[Optimal ET sequence]] ([[TE]]): [[27edo|27e]], [[60edo|60e]], [[87edo|87]]


Badness ([[TE_logflat_badness|Smith]]): 0.02560
Badness ([[TE_logflat_badness|Smith TE]]): 0.02560


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


== 130 & 181 ==
== 130 & 181 ==
''See also: [[User:Francium/Miscellaneous_rank-2_temperaments#311_.26_1063|7-limit]]''


[[Subgroup]]: [[2.3.5.7.13_subgroup|2.3.5.7.13]]
[[Subgroup]]: [[2.3.5.7.13_subgroup|2.3.5.7.13]]
Line 26: Line 84:
[[Mapping]]: {{mapping|1 11 -22 -7 19|0 -24 62 25 -39}}
[[Mapping]]: {{mapping|1 11 -22 -7 19|0 -24 62 25 -39}}


[[WE]] generators: ~2/1 = 1199.9727, ~21/16 = 470.7388
[[Transversal_generators|Generator preimages]]: [[2/1]], [[21/16]]


[[CWE]] generators: 2/1, ~21/16 = 470.7495
[[Optimization|Generator tunings]]:
 
: {| class="wikitable right-1"
|-
! Norm
! [[Minimax-lils-S|Weil]]
! [[TOP|Tenney]]
! [[WE]]
! [[TE]]
|-
| [[Optimization|Optimized]]
| ???
| 1199.983{{cc}}, 470.7428{{cc}}
| 1199.9727{{cc}}, 470.7388{{cc}}
| 1199.9727{{cc}}, 470.7388{{cc}}
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\1, 470.7495{{cc}}
| 1\1, 470.7495{{cc}}
| 1\1, 470.7493{{cc}}
|-
| [[POTE|Destretched]]
| ???
| 1\1, 470.7495{{cc}}
| 1\1, 470.7495{{cc}}
| 1\1, 470.7495{{cc}}
|}
 
[[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{{cc}}
| 1901.9687{{cc}}
| 2786.4055{{cc}}
| 3368.6607{{cc}}
| 4440.6683{{cc}}
|-
| [[Error]]
| −0.0273{{cc}}
| +0.0137{{cc}}
| +0.0918{{cc}}
| −0.1652{{cc}}
| +0.1407{{cc}}
|}


[[Optimal ET sequence]] ([[TE]]): [[51edo|51]], [[79edo|79c]], [[130edo|130]], [[311edo|311]], [[441edo|441]], [[1453edo|1453]], [[1894edo|1894]], [[2335edo|2335]]
[[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]]): 0.01381
Badness ([[TE_logflat_badness|Smith TE]]): 0.01381


Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.47741
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.47741
Line 44: Line 155:
[[Mapping]]: {{mapping|1 11 -22 -7 -24 19|0 -24 62 25 70 -39}}
[[Mapping]]: {{mapping|1 11 -22 -7 -24 19|0 -24 62 25 70 -39}}


TE generators: ~2/1 = 1199.939, ~21/16 = 470.722
[[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{{cc}}, 470.7167{{cc}}
| 1199.9386{{cc}}, 470.7221{{cc}}
| 1199.9387{{cc}}, 470.7221{{cc}}
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\1, 470.7411{{cc}}
| 1\1, 470.746{{cc}}
| 1\1, 470.745{{cc}}
|-
| [[POTE|Destretched]]
| ???
| 1\1, 470.744{{cc}}
| 1\1, 470.7462{{cc}}
| 1\1, 470.7461{{cc}}
|}
 
[[Prime_interval|Prime harmonics]] ([[WE]]):


CTE generators: 2/1, ~21/16 = 470.745
: {| class="wikitable right-1"
|-
! Prime
! 2/1
! 3/1
! 5/1
! 7/1
! 11/1
! 13/1
|-
| [[Tuning_map|Tuning]]
| 1199.9386{{cc}}
| 1901.9946{{cc}}
| 2786.1202{{cc}}
| 3368.482{{cc}}
| 4152.0197{{cc}}
| 4440.6721{{cc}}
|-
| [[Error]]
| −0.0614{{cc}}
| +0.0396{{cc}}
| −0.1936{{cc}}
| −0.3439{{cc}}
| +0.7017{{cc}}
| +0.1445{{cc}}
|}


[[Optimal ET sequence]] ([[TE]]): [[51edo|51]], [[130edo|130]], [[311edo|311]], [[441edo|441]], [[752edo|752e]], [[1193edo|1193e]]
[[Optimal ET sequence]] ([[TE]]): [[51edo|51]], [[130edo|130]], [[311edo|311]], [[441edo|441]], [[752edo|752e]], [[1193edo|1193e]]


Badness ([[TE_logflat_badness|Smith]]): 0.02172
Badness ([[TE_logflat_badness|Smith TE]]): 0.02172


Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.89734
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.89734
Line 124: Line 291:
[[Optimal ET sequence]] ([[TE]]): [[31edo|31]], [[67edo|67]], [[98edo|98]]
[[Optimal ET sequence]] ([[TE]]): [[31edo|31]], [[67edo|67]], [[98edo|98]]


Badness ([[TE_logflat_badness|Smith]]): 0.03686
Badness ([[TE_logflat_badness|Smith TE]]): 0.03686


Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.52300
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.52300
Line 198: Line 365:
{{Optimal ET sequence|legend=1|12f, 55f, 67}}
{{Optimal ET sequence|legend=1|12f, 55f, 67}}


[[Error]]s (WE): −0.21{{c}}, −3.81{{c}}, +7.11{{c}}, −2.89{{c}}, +2.3{{c}}, −0.65{{c}}
Badness ([[TE_logflat_badness|Smith TE]]): 0.04940
 
Badness ([[TE_logflat_badness|Smith]]):


Badness ([[TE_dirichlet_badness|Dirichlet]]): 2.041
Badness ([[TE_dirichlet_badness|Dirichlet]]): 2.04145


=== 2.3.7.11.13(.17.19).23.31.37.41 subgroup ===
=== 2.3.7.11.13(.17.19).23.31.37.41 subgroup ===
Line 281: Line 446:
[[Optimal ET sequence]]: [[12edo|12f(ghh)ikm]], [[43edo|43deff(ggghhh)iiikkllmm]], [[55edo|55f(gh)ilm]], [[67edo|67p]], [[122edo|122(g)ilm]]
[[Optimal ET sequence]]: [[12edo|12f(ghh)ikm]], [[43edo|43deff(ggghhh)iiikkllmm]], [[55edo|55f(gh)ilm]], [[67edo|67p]], [[122edo|122(g)ilm]]


Badness ([[TE_logflat_badness|Smith]]): (0.01156)
Badness ([[TE_logflat_badness|Smith TE]]): 0.01056 (0.01156)


Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.075 (1.35270)
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.07545 (1.35270)


== Tri[[BPS|pierce]] (b30 & b39) ==
== Tri[[BPS|pierce]] (b30 & b39) ==
Line 349: Line 514:
[[Optimal ET sequence]] ([[TE]]): [[9edt|b9]], [[21edt|b21]], [[30edt|b30]], [[39edt|b39]], [[69edt|b69]], [[108edt|b108]], [[177edt|b177d]]
[[Optimal ET sequence]] ([[TE]]): [[9edt|b9]], [[21edt|b21]], [[30edt|b30]], [[39edt|b39]], [[69edt|b69]], [[108edt|b108]], [[177edt|b177d]]


Badness ([[TE_logflat_badness|Smith]]): 0.00287
Badness ([[TE_logflat_badness|Smith TE]]): 0.00287


Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.26872
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.26872
Line 420: Line 585:
[[Optimal ET sequence]] ([[TE]]): [[18edt|b18]], [[21edt|b21]], [[39edt|b39]], [[99edt|b99]], [[138edt|b138]], [[177edt|b177d]]
[[Optimal ET sequence]] ([[TE]]): [[18edt|b18]], [[21edt|b21]], [[39edt|b39]], [[99edt|b99]], [[138edt|b138]], [[177edt|b177d]]


Badness ([[TE_logflat_badness|Smith]]): 0.00566
Badness ([[TE_logflat_badness|Smith TE]]): 0.00566


Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.44749
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.44749
Line 494: Line 659:
[[Optimal ET sequence]] ([[TE]]): [[51edt|b51]], [[60edt|b60]], [[69edt|b69]], [[147edt|b147]]
[[Optimal ET sequence]] ([[TE]]): [[51edt|b51]], [[60edt|b60]], [[69edt|b69]], [[147edt|b147]]


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


Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.48415
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.48415
Line 571: Line 736:
[[Optimal ET sequence]] ([[TE]]): [[18edt|b18]], [[60edt|b60]], [[78edt|b78]], [[138edt|b138]], [[216edt|b216]]
[[Optimal ET sequence]] ([[TE]]): [[18edt|b18]], [[60edt|b60]], [[78edt|b78]], [[138edt|b138]], [[216edt|b216]]


Badness ([[TE_logflat_badness|Smith]]): 0.01046
Badness ([[TE_logflat_badness|Smith TE]]): 0.01046


Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.95032
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.95032
Line 645: Line 810:
[[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>
[[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]]): 0.01465 (0.01044)
Badness ([[TE_logflat_badness|Smith TE]]): 0.01465 (0.01044)


Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.50663 (0.49340)
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.50663 (0.49340)
Line 722: Line 887:
[[Optimal ET sequence]] ([[TE]]): [[140edo|140]], [[301edo|301e]], [[441edo|441]], [[581edo|581]], [[1022edo|1022]], [[1603edo|1603e]] <small>(same sequence for add-23)</small>
[[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]]): 0.02947 (0.01904)
Badness ([[TE_logflat_badness|Smith TE]]): 0.02947 (0.01904)


Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.21770 (1.01017)
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.21770 (1.01017)
Line 802: Line 967:
[[Optimal ET sequence]] ([[TE]]): [[161edo|161]], [[420edo|420]], [[581edo|581]], [[742edo|742]], [[1323edo|1323]]
[[Optimal ET sequence]] ([[TE]]): [[161edo|161]], [[420edo|420]], [[581edo|581]], [[742edo|742]], [[1323edo|1323]]


Badness ([[TE_logflat_badness|Smith]]): 0.01260
Badness ([[TE_logflat_badness|Smith TE]]): 0.01260


Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.76654
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.76654
Line 868: Line 1,033:
[[Optimal ET sequence]] ([[TE]]):  
[[Optimal ET sequence]] ([[TE]]):  


Badness ([[TE_logflat_badness|Smith]]):  
Badness ([[TE_logflat_badness|Smith TE]]):  


Badness ([[TE_dirichlet_badness|Dirichlet]]):  
Badness ([[TE_dirichlet_badness|Dirichlet]]):  
-->
-->

Revision as of 15:38, 6 January 2025

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 ??? 400.3122Template:Cc, 220.5893Template:Cc 399.994Template:Cc, 220.6882Template:Cc 399.9947Template:Cc, 220.6886Template:Cc
Constrained ??? 1\3, 220.729Template:Cc 1\3, 220.6899Template:Cc 1\3, 220.6873Template:Cc
Destretched ??? 1\3, 220.4173Template:Cc 1\3, 220.6915Template:Cc 1\3, 220.6915Template:Cc

Prime harmonics (WE):

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

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

Badness (Smith TE): 0.02560

Badness (Dirichlet): 1.05783

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 ??? 1199.983Template:Cc, 470.7428Template:Cc 1199.9727Template:Cc, 470.7388Template:Cc 1199.9727Template:Cc, 470.7388Template:Cc
Constrained ??? 1\1, 470.7495Template:Cc 1\1, 470.7495Template:Cc 1\1, 470.7493Template:Cc
Destretched ??? 1\1, 470.7495Template:Cc 1\1, 470.7495Template:Cc 1\1, 470.7495Template:Cc

Prime harmonics (WE):

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

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 ??? 1199.9305Template:Cc, 470.7167Template:Cc 1199.9386Template:Cc, 470.7221Template:Cc 1199.9387Template:Cc, 470.7221Template:Cc
Constrained ??? 1\1, 470.7411Template:Cc 1\1, 470.746Template:Cc 1\1, 470.745Template:Cc
Destretched ??? 1\1, 470.744Template:Cc 1\1, 470.7462Template:Cc 1\1, 470.7461Template:Cc

Prime harmonics (WE):

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

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

Badness (Smith TE): 0.02172

Badness (Dirichlet): 0.89734

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 ??? 1200.4954Template:Cc, 232.6290Template:Cc 1199.9347Template:Cc, 232.6275Template:Cc 1199.9460Template:Cc, 232.6297Template:Cc
Constrained ??? 1\1, 232.6732Template:Cc 1\1, 232.6392Template:Cc 1\1, 232.6352Template:Cc
Destretched ??? 1\1, 232.5330Template:Cc 1\1, 232.6402Template:Cc 1\1, 232.6401Template:Cc

Prime harmonics (WE):

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

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

Badness (Smith TE): 0.03686

Badness (Dirichlet): 1.52300

Worsetone (12f & 67)

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 ??? 1199.6846Template:Cc, 1897.7889Template:Cc 1199.7882Template:Cc, 1898.1434Template:Cc 1199.8021Template:Cc, 1898.1654Template:Cc
Constrained ??? 1\1, 1898.2217Template:Cc 1\1, 698.4715Template:Cc 1\1, 698.4553Template:Cc
Destretched ??? 1\1, 1898.2878Template:Cc 1\1, 1898.4785Template:Cc 1\1, 1898.4785Template:Cc

Prime harmonics (WE):

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

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, 273/272)

Mapping: [⟨1 0 -32 -44 -58 (-94 -97) -73 -52 -47 -69], ⟨0 1 22 30 39 (62 64) 49 36 33 47]]

Generator preimages: 2/1, 3/1

Generator tunings (no-17 no-19):

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

Prime harmonics (WE):

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

Optimal ET sequence: 12f(ghh)ikm, 43deff(ggghhh)iiikkllmm, 55f(gh)ilm, 67p, 122(g)ilm

Badness (Smith TE): 0.01056 (0.01156)

Badness (Dirichlet): 1.07545 (1.35270)

Tripierce (b30 & b39)

Subgroup: 3.5.7.13

Comma list: 245/243, 2197/2187

Mapping: [⟨3 1 7 7], ⟨0 4 -2 0]]

Generator preimages: 13/9, 13/7

Generator tunings:

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

Prime harmonics (WE):

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

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

Badness (Smith TE): 0.00287

Badness (Dirichlet): 0.26872

Trisemipierce (b39 & b60)

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 ??? 634.4577Template:Cc, 537.4447Template:Cc 634.5219Template:Cc, 537.7266Template:Cc 634.5229Template:Cc, 537.7274Template:Cc
Constrained ??? 1\b3, 537.0443Template:Cc 1\b3, 537.3589Template:Cc 1\b3, 537.7233Template:Cc
Destretched ??? 1\b3, 537.0443Template:Cc 1\b3, 537.2716Template:Cc 1\b3, 537.2716Template:Cc

Prime harmonics (WE):

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

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 ??? 634.4242Template:Cc, 220.2042Template:Cc 634.3781Template:Cc, 220.1438Template:Cc 634.3798Template:Cc, 220.1444Template:Cc
Constrained ??? 1\b3, 220.0518Template:Cc 1\b3, 220.0811Template:Cc 1\b3, 220.4087Template:Cc
Destretched ??? 1\b3, 220.0518Template:Cc 1\b3, 220.0074Template:Cc 1\b3, 220.0074Template:Cc

Prime harmonics (WE):

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

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 ??? ??? 317.2028Template:Cc, 537.3966Template:Cc 317.2037Template:Cc, 537.3981Template:Cc
Constrained ??? 1\b6, 537.0443Template:Cc 1\b6, 537.1261Template:Cc 1\b6, 537.5291Template:Cc
Destretched ??? ??? 1\b6, 1074.0806Template:Cc 1\b6, 1074.0806Template:Cc

Prime harmonics (WE):

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

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

Badness (Smith TE): 0.01046

Badness (Dirichlet): 0.95032

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 ??? 171.4255Template:Cc, 231.2847Template:Cc 171.4244Template:Cc, 231.2902Template:Cc 171.4244Template:Cc, 231.2902Template:Cc
Constrained ??? 1\7, 231.2969Template:Cc 1\7, 231.2963Template:Cc 1\7, 231.2991Template:Cc
Destretched ??? 1\7, 231.2888Template:Cc 1\7, 231.2943Template:Cc 1\7, 231.2943Template:Cc

Prime harmonics (WE):

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

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, 231.3059Template:Cc 171.4242, 231.3085Template:Cc 171.4242, 231.3085Template:Cc
Constrained ??? 1\7, 231.3138Template:Cc 1\7, 231.3149Template:Cc 1\7, 231.3180Template:Cc
Destretched ??? 1\7, 231.3138Template:Cc 1\7, 231.3144Template:Cc 1\7, 231.3144Template:Cc

Prime harmonics (WE):

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

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 ??? 171.4266Template:Cc, 37.1864Template:Cc 171.4244Template:Cc, 37.1884Template:Cc 171.4244Template:Cc, 37.1884Template:Cc
Constrained ??? 1\7, 37.1867Template:Cc 1\7, 37.1893Template:Cc 1\7, 37.1893Template:Cc
Destretched ??? 1\7, 37.1868Template:Cc 1\7, 37.1893Template:Cc 1\7, 37.1893Template:Cc

Prime harmonics (WE):

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

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

Badness (Smith TE): 0.01260

Badness (Dirichlet): 0.76654


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