Hemifamity family: Difference between revisions

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== Hemifamity ==
== Hemifamity ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[5120/5103]]
[[Comma list]]: [[5120/5103]]


[[Mapping]]: [{{val| 1 0 0 10 }}, {{val| 0 1 0 -6 }}, {{val| 0 0 1 1 }}]
{{Mapping|legend=1| 1 0 0 10 | 0 1 0 -6 | 0 0 1 1 }}


Mapping generators: ~2, ~3, ~5
: mapping generators: ~2, ~3, ~5


[[Mapping to lattice]]: [{{val| 0 1 2 -4 }}, {{val| 0 0 1 1 }}]
[[Mapping to lattice]]: [{{val| 0 1 2 -4 }}, {{val| 0 0 1 1 }}]
Line 18: Line 18:
[[Minimax tuning]]: c = 5120/5103
[[Minimax tuning]]: c = 5120/5103
* [[7-odd-limit]]: 3 and 7 1/7c sharp, 5 just
* [[7-odd-limit]]: 3 and 7 1/7c sharp, 5 just
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 1/7 1/7 -1/7 }}, {{monzo| 0 0 1 0 }}, {{monzo| 10/7 -6/7 1/7 6/7 }}]
: {{monzo list| 1 0 0 0 | 10/7 1/7 1/7 -1/7 | 0 0 1 0 | 10/7 -6/7 1/7 6/7 }}
: [[Eigenmonzo]]s (unchanged-intervals): 2, 5/4, 7/6
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7/3
* [[9-odd-limit]]: 3 1/8c sharp, 5 just, 7 1/4c sharp
* [[9-odd-limit]]: 3 1/8c sharp, 5 just, 7 1/4c sharp
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/4 1/4 1/8 -1/8 }}, {{monzo| 0 0 1 0 }}, {{monzo| 5/2 -3/2 1/4 3/4 }}]
: {{monzo list| 1 0 0 0 | 5/4 1/4 1/8 -1/8 | 0 0 1 0 | 5/2 -3/2 1/4 3/4 }}
: [[Eigenmonzo]]s (unchanged-intervals): 2, 5/4, 9/7
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.9/7


{{Optimal ET sequence|legend=1| 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd }}
{{Optimal ET sequence|legend=1| 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd }}
Line 31: Line 31:


; Music
; Music
* [http://www.archive.org/details/Choraled Choraled] [http://www.archive.org/download/Choraled/Genewardsmith-Choraled.mp3 play] by [[Gene Ward Smith]]
* [http://www.archive.org/details/Choraled ''Choraled''] [http://www.archive.org/download/Choraled/Genewardsmith-Choraled.mp3 play] by [[Gene Ward Smith]]
* [http://clones.soonlabel.com/public/micro/hemifamity27/hemifamity27-IF-20100917.mp3 Hemifamity27] by [[Chris Vaisvil]]
* [http://clones.soonlabel.com/public/micro/hemifamity27/hemifamity27-IF-20100917.mp3 ''Hemifamity27''] by [[Chris Vaisvil]]


== Pele ==
== Pele ==
{{See also| Pentacircle clan }}
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 441/440, 896/891
[[Comma list]]: 441/440, 896/891


[[Mapping]]: [{{val| 1 0 0 10 17 }}, {{val| 0 1 0 -6 -10 }}, {{val| 0 0 1 1 1 }}]
{{Mapping|legend=1| 1 0 0 10 17 | 0 1 0 -6 -10 | 0 0 1 1 1 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 17/10 0 1/10 0 -1/10 }}, {{monzo| 17/5 -2 6/5 0 -1/5 }}, {{monzo| 16/5 -2 3/5 0 2/5 }}, {{monzo| 17/5 -2 1/5 0 4/5 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 17/10 0 1/10 0 -1/10 }}, {{monzo| 17/5 -2 6/5 0 -1/5 }}, {{monzo| 16/5 -2 3/5 0 2/5 }}, {{monzo| 17/5 -2 1/5 0 4/5 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 10/9, 11/9
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5.11/9


[[Mapping to lattice]]: [{{val| 0 1 4 -2 -6 }}, {{val| 0 0 -1 -1 -1 }}]
[[Mapping to lattice]]: [{{val| 0 1 4 -2 -6 }}, {{val| 0 0 -1 -1 -1 }}]
Line 63: Line 65:
Comma list: 196/195, 352/351, 364/363
Comma list: 196/195, 352/351, 364/363


Mapping: [{{val| 1 0 0 10 17 22 }}, {{val| 0 1 0 -6 -10 -13 }}, {{val| 0 0 1 1 1 1 }}]
Mapping: {{mapping| 1 0 0 10 17 22 | 0 1 0 -6 -10 -13 | 0 0 1 1 1 1 }}


Minimax tuning:  
Minimax tuning:  
* 13-odd-limit eigenmonzos (unchanged-intervals): 2, 10/9, 13/10
* 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/5.13/9
* 15-odd-limit eigenmonzos (unchanged-intervals): 2, 15/13, 6/5
* 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.5/3.13/9


{{Optimal ET sequence|legend=1| 29, 41, 46, 58, 87, 145, 232 }}
{{Optimal ET sequence|legend=1| 29, 41, 46, 58, 87, 145, 232 }}
Line 74: Line 76:


== Laka ==
== Laka ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 540/539, 5120/5103
[[Comma list]]: 540/539, 5120/5103


[[Mapping]]: [{{val| 1 0 0 10 -18 }}, {{val| 0 1 0 -6 15 }}, {{val| 0 0 1 1 -1 }}]
{{Mapping|legend=1| 1 0 0 10 -18 | 0 1 0 -6 15 | 0 0 1 1 -1 }}


[[Minimax tuning]]
[[Minimax tuning]]
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 0 2/21 -1/21 1/21 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 2 0 3/7 2/7 -2/7 }}, {{monzo| 2 0 3/7 -5/7 5/7 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 4/3 0 2/21 -1/21 1/21 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 2 0 3/7 2/7 -2/7 }}, {{monzo| 2 0 3/7 -5/7 5/7 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 5/4, 14/11
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.11/7


{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee }}
{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee }}
Line 93: Line 95:


Comma list: 352/351, 540/539, 847/845
Comma list: 352/351, 540/539, 847/845
Mapping: {{mapping| 1 0 0 10 -18 -13 | 0 1 0 -6 15 12 | 0 0 1 1 -1 -1 }}


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit
* 13- and 15-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 13/8 -1/2 1/8 0 0 1/8 }}, {{monzo| 13/4 -3 5/4 0 0 1/4 }}, {{monzo| 7/2 0 1/2 0 0 -1/2 }}, {{monzo| 25/8 -9/2 5/8 0 0 13/8 }}, {{monzo| 13/4 -3 1/4 0 0 5/4 }}]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 13/8 -1/2 1/8 0 0 1/8 }}, {{monzo| 13/4 -3 5/4 0 0 1/4 }}, {{monzo| 7/2 0 1/2 0 0 -1/2 }}, {{monzo| 25/8 -9/2 5/8 0 0 13/8 }}, {{monzo| 13/4 -3 1/4 0 0 5/4 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/8, 14/13
: eigenmonzo (unchanged-interval) basis: 2.11.13/7


{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 111, 152f, 415de }}*
{{Optimal ET sequence|legend=1| 41, 53, 58, 94, 111, 152f, 415de }}*
Line 108: Line 112:
Comma list: 352/351, 442/441, 540/539, 847/845
Comma list: 352/351, 442/441, 540/539, 847/845


Mapping: [{{val| 1 0 0 10 -18 -13 32 }}, {{val| 0 1 0 -6 15 12 -22 }}, {{val| 0 0 1 1 -1 -1 3 }}]
Mapping: {{mapping| 1 0 0 10 -18 -13 32 | 0 1 0 -6 15 12 -22 | 0 0 1 1 -1 -1 3 }}


Minimax tuning:  
Minimax tuning:  
* 17-odd-limit
* 17-odd-limit
: [{{monzo| 1 0 0 0 0 0 0 }}, {{monzo| 13/12 0 0 1/12 1/6 -1/12 0 }}, {{monzo| -7/4 0 0 5/4 3/2 -5/4 0 }}, {{monzo| 7/4 0 0 3/4 1/2 -3/4 0 }}, {{monzo| 0 0 0 0 1 0 0 }}, {{monzo| 7/4 0 0 -1/4 1/2 1/4 0 }}, {{monzo| 35/12 0 0 23/12 5/6 -23/12 0 }}]
: [{{monzo| 1 0 0 0 0 0 0 }}, {{monzo| 13/12 0 0 1/12 1/6 -1/12 0 }}, {{monzo| -7/4 0 0 5/4 3/2 -5/4 0 }}, {{monzo| 7/4 0 0 3/4 1/2 -3/4 0 }}, {{monzo| 0 0 0 0 1 0 0 }}, {{monzo| 7/4 0 0 -1/4 1/2 1/4 0 }}, {{monzo| 35/12 0 0 23/12 5/6 -23/12 0 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/8, 14/13
: eigenmonzo (unchanged-interval) basis: 2.11.13/7


{{Optimal ET sequence|legend=1| 58, 94, 111, 152f, 205, 263df }}
{{Optimal ET sequence|legend=1| 58, 94, 111, 152f, 205, 263df }}


== Akea ==
== Akea ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 385/384, 2200/2187
[[Comma list]]: 385/384, 2200/2187


[[Mapping]]: [{{val| 1 0 0 10 -3 }}, {{val| 0 1 0 -6 7 }}, {{val| 0 0 1 1 -2 }}]
{{Mapping|legend=1| 1 0 0 10 -3 | 0 1 0 -6 7 | 0 0 1 1 -2 }}


Mapping generators: ~2, ~3, ~5
: mapping generators: ~2, ~3, ~5


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3>, {{monzo| 26/9 0 -5/18 11/18 -1/3 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 }}]
: [[Eigenmonzo]]s (unchanged-intervals): ~2, ~14/11, ~7/5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5.11/5


{{Optimal ET sequence|legend=1| 34, 41, 53, 87, 140, 181, 321 }}
{{Optimal ET sequence|legend=1| 34, 41, 53, 87, 140, 181, 321 }}


Badness: 0.998 × 10<sup>-3</sup>
[[Badness]]: 0.998 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Line 140: Line 144:
Comma list: 325/324, 352/351, 385/384
Comma list: 325/324, 352/351, 385/384


Mapping: [{{val| 1 0 0 10 -3 2 }}, {{val| 0 1 0 -6 7 4 }}, {{val| 0 0 1 1 -2 -2 }}]
Mapping: {{mapping| 1 0 0 10 -3 2 | 0 1 0 -6 7 4 | 0 0 1 1 -2 -2 }}


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit
* 13- and 15-odd-limit
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 0 }}, {{monzo| 26/9 0 -7/9 1/9 2/3 0 }}]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 5/3 0 1/6 -1/6 0 0 }}, {{monzo| 26/9 0 13/18 -7/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 11/18 -1/3 0 }}, {{monzo| 26/9 0 -5/18 -7/18 2/3 0 }}, {{monzo| 26/9 0 -7/9 1/9 2/3 0 }}]
: Eigenmonzos (unchanged-intervals): 2, 14/11, 7/5
: eigenmonzo (unchanged-interval) basis: 2.7/5.11/5


Lattice basis:  
Lattice basis:  
Line 160: Line 164:


== Lono ==
== Lono ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 176/175, 5120/5103
[[Comma list]]: 176/175, 5120/5103


[[Mapping]]: [{{val| 1 0 0 10 6 }}, {{val| 0 1 0 -6 -6 }}, {{val| 0 0 1 1 3 }}]
{{Mapping|legend=1| 1 0 0 10 6 | 0 1 0 -6 -6 | 0 0 1 1 3 }}


{{Optimal ET sequence|legend=1| 46, 53, 58, 99, 111, 268cd }}
{{Optimal ET sequence|legend=1| 46, 53, 58, 99, 111, 268cd }}
Line 175: Line 179:
Comma list: 176/175, 351/350, 847/845
Comma list: 176/175, 351/350, 847/845


Mapping: [{{val| 1 0 0 10 6 11 }}, {{val| 0 1 0 -6 -6 -9 }}, {{val| 0 0 1 1 3 3 }}]
Mapping: {{mapping| 1 0 0 10 6 11 | 0 1 0 -6 -6 -9 | 0 0 1 1 3 3 }}


{{Optimal ET sequence|legend=1| 46, 53, 58, 99, 104c, 111, 268cd }}
{{Optimal ET sequence|legend=1| 46, 53, 58, 99, 104c, 111, 268cd }}
Line 182: Line 186:


== Kapo ==
== Kapo ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3025/3024, 5120/5103
[[Comma list]]: 3025/3024, 5120/5103


[[Mapping]]: [{{val| 1 0 0 10 7 }}, {{val| 0 1 1 -5 -2 }}, {{val| 0 0 2 2 -1 }}]
{{Mapping|legend=1| 1 0 0 10 7 | 0 1 1 -5 -2 | 0 0 2 2 -1 }}


Mapping generators: ~2, ~3, ~128/99
: mapping generators: ~2, ~3, ~128/99


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]:  
* [[11-odd-limit]]:  
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/5 2/5 0 -1/15 -2/15 }}, {{monzo| 14/5 6/5 0 7/15 -16/15 }}, {{monzo| 16/5 -6/5 0 13/15 -4/15 }}, {{monzo| 16/5 -6/5 0 -2/15 11/15 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/5 2/5 0 -1/15 -2/15 }}, {{monzo| 14/5 6/5 0 7/15 -16/15 }}, {{monzo| 16/5 -6/5 0 13/15 -4/15 }}, {{monzo| 16/5 -6/5 0 -2/15 11/15 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/9, 9/7
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7.11/9


{{Optimal ET sequence|legend=1| 41, 87, 111, 152, 239, 391 }}
{{Optimal ET sequence|legend=1| 41, 87, 111, 152, 239, 391 }}
Line 200: Line 204:


== Namaka ==
== Namaka ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 3388/3375, 5120/5103
[[Comma list]]: 3388/3375, 5120/5103


[[Mapping]]: [{{val| 1 0 0 10 -6 }}, {{val| 0 2 0 -12 9 }}, {{val| 0 0 1 1 1 }}]
{{Mapping|legend=1| 1 0 0 10 -6 | 0 2 0 -12 9 | 0 0 1 1 1 }}
 
: mapping generators: ~2, ~400/231, ~5


{{Optimal ET sequence|legend=1| 29, 53, 58, 87, 111, 140, 198 }}
{{Optimal ET sequence|legend=1| 29, 53, 58, 87, 111, 140, 198 }}
Line 215: Line 221:
Comma list: 352/351, 676/675, 847/845
Comma list: 352/351, 676/675, 847/845


Mapping: [{{val| 1 0 0 10 -6 -1 }}, {{val| 0 2 0 -12 9 3 }}, {{val| 0 0 1 1 1 1 }}]
Mapping: {{mapping| 1 0 0 10 -6 -1 | 0 2 0 -12 9 3 | 0 0 1 1 1 1 }}


{{Optimal ET sequence|legend=1| 29, 53, 58, 87, 111, 140, 198 }}
{{Optimal ET sequence|legend=1| 29, 53, 58, 87, 111, 140, 198 }}
Line 223: Line 229:
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Hemifamity family| ]] <!-- main article -->
[[Category:Hemifamity family| ]] <!-- main article -->
[[Category:Hemifamity| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
[[Category:Listen]]
[[Category:Listen]]

Revision as of 07:48, 11 September 2023

The hemifamity family of temperaments are rank-3 temperaments tempering out 5120/5103.

Hemifamity

Subgroup: 2.3.5.7

Comma list: 5120/5103

Mapping: [⟨1 0 0 10], ⟨0 1 0 -6], ⟨0 0 1 1]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 1 2 -4], ⟨0 0 1 1]]

Lattice basis:

3/2 length = 0.5670, 10/9 length = 1.8063
Angle (3/2, 10/9) = 82.112 degrees

Minimax tuning: c = 5120/5103

[[1 0 0 0⟩, [10/7 1/7 1/7 -1/7⟩, [0 0 1 0⟩, [10/7 -6/7 1/7 6/7⟩]
eigenmonzo (unchanged-interval) basis: 2.5.7/3
[[1 0 0 0⟩, [5/4 1/4 1/8 -1/8⟩, [0 0 1 0⟩, [5/2 -3/2 1/4 3/4⟩]
eigenmonzo (unchanged-interval) basis: 2.5.9/7

Optimal ET sequence: 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd

Badness: 0.153 × 10-3

Projection pairs: 7 5120/729

Music

Pele

See also: Pentacircle clan

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891

Mapping: [⟨1 0 0 10 17], ⟨0 1 0 -6 -10], ⟨0 0 1 1 1]]

Minimax tuning:

[[1 0 0 0 0⟩, [17/10 0 1/10 0 -1/10⟩, [17/5 -2 6/5 0 -1/5⟩, [16/5 -2 3/5 0 2/5⟩, [17/5 -2 1/5 0 4/5⟩]
eigenmonzo (unchanged-interval) basis: 2.9/5.11/9

Mapping to lattice: [⟨0 1 4 -2 -6], ⟨0 0 -1 -1 -1]]

Lattice basis:

3/2 length = 0.3812, 56/55 length = 1.5893
Angle(3/2, 56/55) = 90.4578 degrees

Optimal ET sequence: 29, 41, 58, 87, 99e, 145, 186e

Badness: 0.648 × 10-3

Projection pairs: 7 5120/729 11 655360/59049

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363

Mapping: [⟨1 0 0 10 17 22], ⟨0 1 0 -6 -10 -13], ⟨0 0 1 1 1 1]]

Minimax tuning:

  • 13-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/5.13/9
  • 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.5/3.13/9

Optimal ET sequence: 29, 41, 46, 58, 87, 145, 232

Badness: 0.703 × 10-3

Laka

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5120/5103

Mapping: [⟨1 0 0 10 -18], ⟨0 1 0 -6 15], ⟨0 0 1 1 -1]]

Minimax tuning

[[1 0 0 0 0⟩, [4/3 0 2/21 -1/21 1/21⟩, [0 0 1 0 0⟩, [2 0 3/7 2/7 -2/7⟩, [2 0 3/7 -5/7 5/7⟩]
eigenmonzo (unchanged-interval) basis: 2.5.11/7

Optimal ET sequence: 41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee

Projection pairs: 5120/729 11 14348907/1310720

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 847/845

Mapping: [⟨1 0 0 10 -18 -13], ⟨0 1 0 -6 15 12], ⟨0 0 1 1 -1 -1]]

Minimax tuning:

  • 13- and 15-odd-limit
[[1 0 0 0 0 0⟩, [13/8 -1/2 1/8 0 0 1/8⟩, [13/4 -3 5/4 0 0 1/4⟩, [7/2 0 1/2 0 0 -1/2⟩, [25/8 -9/2 5/8 0 0 13/8⟩, [13/4 -3 1/4 0 0 5/4⟩]
eigenmonzo (unchanged-interval) basis: 2.11.13/7

Optimal ET sequence: 41, 53, 58, 94, 111, 152f, 415de*

* optimal patent val: 205

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 442/441, 540/539, 847/845

Mapping: [⟨1 0 0 10 -18 -13 32], ⟨0 1 0 -6 15 12 -22], ⟨0 0 1 1 -1 -1 3]]

Minimax tuning:

  • 17-odd-limit
[[1 0 0 0 0 0 0⟩, [13/12 0 0 1/12 1/6 -1/12 0⟩, [-7/4 0 0 5/4 3/2 -5/4 0⟩, [7/4 0 0 3/4 1/2 -3/4 0⟩, [0 0 0 0 1 0 0⟩, [7/4 0 0 -1/4 1/2 1/4 0⟩, [35/12 0 0 23/12 5/6 -23/12 0⟩]
eigenmonzo (unchanged-interval) basis: 2.11.13/7

Optimal ET sequence: 58, 94, 111, 152f, 205, 263df

Akea

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187

Mapping: [⟨1 0 0 10 -3], ⟨0 1 0 -6 7], ⟨0 0 1 1 -2]]

mapping generators: ~2, ~3, ~5

Minimax tuning:

[[1 0 0 0 0⟩, [5/3 0 1/6 -1/6 0⟩, [26/9 0 13/18 -7/18 -1/3⟩, [26/9 0 -5/18 11/18 -1/3⟩, [26/9 0 -5/18 -7/18 2/3⟩]
eigenmonzo (unchanged-interval) basis: 2.7/5.11/5

Optimal ET sequence: 34, 41, 53, 87, 140, 181, 321

Badness: 0.998 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384

Mapping: [⟨1 0 0 10 -3 2], ⟨0 1 0 -6 7 4], ⟨0 0 1 1 -2 -2]]

Minimax tuning:

  • 13- and 15-odd-limit
[[1 0 0 0 0 0⟩, [5/3 0 1/6 -1/6 0 0⟩, [26/9 0 13/18 -7/18 -1/3 0⟩, [26/9 0 -5/18 11/18 -1/3 0⟩, [26/9 0 -5/18 -7/18 2/3 0⟩, [26/9 0 -7/9 1/9 2/3 0⟩]
eigenmonzo (unchanged-interval) basis: 2.7/5.11/5

Lattice basis:

3/2 length = 0.5354, 27/20 length = 1.0463
Angle (3/2, 27/20) = 80.5628 degrees

Mapping to lattice: [⟨0 1 3 -3 1 -2], ⟨0 0 -1 -1 2 2]]

Optimal ET sequence: 34, 41, 46, 53, 87, 140, 321, 461e

Badness: 0.822 × 10-3

Scales: akea46_13

Lono

Subgroup: 2.3.5.7.11

Comma list: 176/175, 5120/5103

Mapping: [⟨1 0 0 10 6], ⟨0 1 0 -6 -6], ⟨0 0 1 1 3]]

Optimal ET sequence: 46, 53, 58, 99, 111, 268cd

Badness: 1.18 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845

Mapping: [⟨1 0 0 10 6 11], ⟨0 1 0 -6 -6 -9], ⟨0 0 1 1 3 3]]

Optimal ET sequence: 46, 53, 58, 99, 104c, 111, 268cd

Badness: 0.908 × 10-3

Kapo

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5120/5103

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

mapping generators: ~2, ~3, ~128/99

Minimax tuning:

[[1 0 0 0 0⟩, [8/5 2/5 0 -1/15 -2/15⟩, [14/5 6/5 0 7/15 -16/15⟩, [16/5 -6/5 0 13/15 -4/15⟩, [16/5 -6/5 0 -2/15 11/15⟩]
eigenmonzo (unchanged-interval) basis: 2.9/7.11/9

Optimal ET sequence: 41, 87, 111, 152, 239, 391

Badness: 0.994 × 10-3

Namaka

Subgroup: 2.3.5.7.11

Comma list: 3388/3375, 5120/5103

Mapping: [⟨1 0 0 10 -6], ⟨0 2 0 -12 9], ⟨0 0 1 1 1]]

mapping generators: ~2, ~400/231, ~5

Optimal ET sequence: 29, 53, 58, 87, 111, 140, 198

Badness: 1.74 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845

Mapping: [⟨1 0 0 10 -6 -1], ⟨0 2 0 -12 9 3], ⟨0 0 1 1 1 1]]

Optimal ET sequence: 29, 53, 58, 87, 111, 140, 198

Badness: 0.781 × 10-3

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