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

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== 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]]''


[[Subgroup]]: [[3.5.7.13_subgroup|3.5.7.13]]
[[Subgroup]]: [[3.5.7.13_subgroup|3.5.7.13]]
Line 80: Line 82:
[[Comma list]]: [[245/243]], [[2197/2187]]
[[Comma list]]: [[245/243]], [[2197/2187]]


[[Mapping]]: {{mapping|3 1 7 7|0 4 -2 0}}
[[Mapping]]: {{mapping|3 1 7 7|0 2 -1 0}}


[[Transversal_generators|Generator preimages]]: [[13/9]], [[13/7]]
[[Transversal_generators|Generator preimages]]: [[13/9]], [[13/7]]
Line 143: Line 145:


=== 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]]''


[[Subgroup]]: [[3.5.7.11.13_subgroup|3.5.7.11.13]]
[[Subgroup]]: [[3.5.7.11.13_subgroup|3.5.7.11.13]]
Line 514: Line 517:
Badness ([[TE_dirichlet_badness|Dirichlet]]): 2.04145
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 ====


[[Comma list]]: [[287/286]], [[297/296]], [[364/363]], [[483/481]], [[621/616]], [[3627/3626]], [[4797/4774]], ([[153/152]], [[273/272]])
[[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 (-94 -97) -73 -52 -47 -69}}, {{val|0 1 22 30 39 (62 64) 49 36 33 47}}]
[[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]]
[[Transversal_generators|Generator preimages]]: [[2/1]], [[3/1]]
Line 589: Line 592:
|}
|}


[[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|12fikm]], [[43edo|43deffiiikkllmm]], [[55edo|55film]], [[67edo|67]], [[122edo|122(h)ilm]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.01056 (0.01515)
 
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.07545 (1.77384)
 
== 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{{cc}}, 229.6835{{cc}}
| 600.0198{{cc}}, 229.7451{{cc}}
| 600.0210{{cc}}, 229.7456{{cc}}
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\2, 229.6914{{cc}}
| 1\2, 229.7361{{cc}}
| 1\2, 229.7307{{cc}}
|-
| [[POTE|Destretched]]
| ???
| 1\2, 229.6872{{cc}}
| 1\2, 229.7375{{cc}}
| 1\2, 229.7376{{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]]
| 1200.0396{{cc}}
| 1902.6876{{cc}}
| 2783.7665{{cc}}
| 3370.3738{{cc}}
| 4148.8246{{cc}}
| 4443.2964{{cc}}
|-
| [[Error]]
| +0.0396{{cc}}
| +0.7326{{cc}}
| −2.5472{{cc}}
| +1.5479{{cc}}
| −2.4933{{cc}}
| +2.7687{{cc}}
|}
 
[[Optimal ET sequence]] ([[TE]]): [[26edo|26]], [[68edo|68]], [[94edo|94]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.03418
 
Badness ([[TE_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{{cc}}, 229.6835{{cc}}
| 600.0751{{cc}}, 229.7512{{cc}}
| 600.0765{{cc}}, 229.7517{{cc}}
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\2, 229.6914{{cc}}
| 1\2, 229.7168{{cc}}
| 1\2, 229.6935{{cc}}
|-
| [[POTE|Destretched]]
| ???
| 1\2, 229.6872{{cc}}
| 1\2, 229.7224{{cc}}
| 1\2, 229.7224{{cc}}
|}
 
[[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{{cc}}
| 1903.0139{{cc}}
| 2784.2344{{cc}}
| 3370.6993{{cc}}
| 4149.1313{{cc}}
| 4443.8869{{cc}}
| 4903.3893{{cc}}
| 5095.3563{{cc}}
|-
| [[Error]]
| +0.1502{{cc}}
| +1.0589{{cc}}
| −2.0793{{cc}}
| +1.8734{{cc}}
| −2.1867{{cc}}
| +3.3593{{cc}}
| −1.5661{{cc}}
| −2.1567{{cc}}
|}
 
[[Optimal ET sequence]] ([[TE]]): [[26edo|26]], [[68edo|68]], [[94edo|94]]
 
Badness ([[TE_logflat_badness|Smith TE]]): 0.01725
 
Badness ([[TE_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{{cc}}, 229.6835{{cc}}
| 600.0744{{cc}}, 229.7474{{cc}}
| 600.0758{{cc}}, 229.7480{{cc}}
|-
| [[Constrained_tuning|Constrained]]
| ???
| 1\2, 229.6914{{cc}}
| 1\2, 229.7135{{cc}}
| 1\2, 229.6908{{cc}}
|-
| [[POTE|Destretched]]
| ???
| 1\2, 229.6872{{cc}}
| 1\2, 229.7189{{cc}}
| 1\2, 229.7190{{cc}}
|}
 
[[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]]
| 1200.1488{{cc}}
| 1903.0465{{cc}}
| 2784.28{{cc}}
| 3370.6991{{cc}}
| 4149.1094{{cc}}
| 4443.9237{{cc}}
| 4903.4186{{cc}}
| 5095.4097{{cc}}
| 5427.9804{{cc}}
|-
| [[Error]]
| +0.1488{{cc}}
| +1.0915{{cc}}
| −2.0337{{cc}}
| +1.8732{{cc}}
| −2.2086{{cc}}
| +3.3961{{cc}}
| −1.5368{{cc}}
| −2.1033{{cc}}
| −0.2939{{cc}}
|}
 
[[Optimal ET sequence]] ([[TE]]): [[26edo|26i]], [[68edo|68]], [[94edo|94]]


Badness ([[TE_logflat_badness|Smith TE]]): 0.01056 (0.01156)
Badness ([[TE_logflat_badness|Smith TE]]): 0.01458


Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.07545 (1.35270)
Badness ([[TE_dirichlet_badness|Dirichlet]]): 1.04399


== Cyanide (140 & 441) ==
== Cyanide (140 & 441) ==
Line 669: Line 909:
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.50663 (0.49340)
Badness ([[TE_dirichlet_badness|Dirichlet]]): 0.50663 (0.49340)


=== 13-limit ===
==== 13-limit ====


[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]([[2.3.5.7.11.13.23_subgroup|.23]])
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]([[2.3.5.7.11.13.23_subgroup|.23]])

Revision as of 17:31, 1 February 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 ??? 400.3122Template: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

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 ??? ??? 634.5206Template: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)

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 ??? 634.4577Template: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

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)

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 ??? 1199.6846Template: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, 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 ??? 1201.1304Template: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: 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 ??? 599.9903Template:Cc, 229.6835Template:Cc 600.0198Template:Cc, 229.7451Template:Cc 600.0210Template:Cc, 229.7456Template:Cc
Constrained ??? 1\2, 229.6914Template:Cc 1\2, 229.7361Template:Cc 1\2, 229.7307Template:Cc
Destretched ??? 1\2, 229.6872Template:Cc 1\2, 229.7375Template:Cc 1\2, 229.7376Template:Cc

Prime harmonics (WE):

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

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 ??? 599.9903Template:Cc, 229.6835Template:Cc 600.0751Template:Cc, 229.7512Template:Cc 600.0765Template:Cc, 229.7517Template:Cc
Constrained ??? 1\2, 229.6914Template:Cc 1\2, 229.7168Template:Cc 1\2, 229.6935Template:Cc
Destretched ??? 1\2, 229.6872Template:Cc 1\2, 229.7224Template:Cc 1\2, 229.7224Template:Cc

Prime harmonics (WE):

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

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 ??? 599.9903Template:Cc, 229.6835Template:Cc 600.0744Template:Cc, 229.7474Template:Cc 600.0758Template:Cc, 229.7480Template:Cc
Constrained ??? 1\2, 229.6914Template:Cc 1\2, 229.7135Template:Cc 1\2, 229.6908Template:Cc
Destretched ??? 1\2, 229.6872Template:Cc 1\2, 229.7189Template:Cc 1\2, 229.7190Template:Cc

Prime harmonics (WE):

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

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 ??? 171.4255Template: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

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

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 ??? 400.0024Template:Cc, 160.5488Template:Cc 400.0034Template:Cc, 160.5495Template:Cc 400.0034Template:Cc, 160.5495Template:Cc
Constrained ??? 1\, 160.5478Template:Cc 1\, 160.5480Template:Cc 1\, 160.5476Template:Cc
Destretched ??? 1\3, 160.5478Template:Cc 1\3, 160.5481Template:Cc 1\3, 160.5481Template:Cc

Prime harmonics (WE):

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

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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