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Archytas clan: Difference between revisions

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+CTE tunings. Review optimal ET sequences.
Continue to work out the tunings for the rest
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{{Multival|legend=1| 2 -9 -4 -19 -12 16 }}
{{Multival|legend=1| 2 -9 -4 -19 -12 16 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 355.904
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~49/40 = 356.6336
* [[POTE]]: ~2 = 1\1, ~49/40 = 355.904


{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bcd }}
{{Optimal ET sequence|legend=1| 10, 17c, 27, 64b, 91bcd, 118bccd }}


[[Badness]]: 0.045872
[[Badness]]: 0.045872
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Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}
Mapping: {{mapping| 1 1 5 4 10 | 0 2 -9 -4 -22 }}


Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 356.140
Optimal tunings:
* CTE: ~2 = 1\1, ~49/40 = 356.7189
* POTE: ~2 = 1\1, ~49/40 = 356.140


{{Optimal ET sequence|legend=1| 27e, 37, 64be, 91bcde }}
Optimal ET sequence: {{Optimal ET sequence| 10e, 17cee, 27e, 64be, 91bcdee }}


Badness: 0.045639
Badness: 0.045639
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Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}
Mapping: {{mapping| 1 1 5 4 10 4 | 0 2 -9 -4 -22 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 356.229
Optimal tunings:
* CTE: ~2 = 1\1, ~16/13 = 356.7223
* POTE: ~2 = 1\1, ~16/13 = 356.229


{{Optimal ET sequence|legend=1| 27e, 37, 64be }}
Optimal ET sequence: {{Optimal ET sequence| 10e, 27e, 37, 64be }}


Badness: 0.030161
Badness: 0.030161
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Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}
Mapping: {{mapping| 1 1 5 4 2 | 0 2 -9 -4 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 355.419
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 355.9918
* POTE: ~2 = 1\1, ~11/9 = 355.419


{{Optimal ET sequence|legend=1| 10, 17c, 27e }}
Optimal ET sequence: {{Optimal ET sequence| 10, 17c, 27e }}


Badness: 0.032863
Badness: 0.032863
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Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}
Mapping: {{mapping| 1 1 5 4 2 4 | 0 2 -9 -4 5 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 355.456
Optimal tunings:
* CTE: ~2 = 1\1, ~11/9 = 356.0040
* POTE: ~2 = 1\1, ~11/9 = 355.456


{{Optimal ET sequence|legend=1| 10, 17c, 27e }}
Optimal ET sequence: {{Optimal ET sequence| 10, 17c, 27e }}


Badness: 0.022619
Badness: 0.022619
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Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15}}
Mapping: {{mapping| 1 1 5 4 -1 | 0 2 -9 -4 15}}


Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 356.710
Optimal tunings:
* CTE: ~2 = 1\1, ~49/40 = 356.6943
* POTE: ~2 = 1\1, ~49/40 = 356.710


{{Optimal ET sequence|legend=1| 10, 27, 37 }}
Optimal ET sequence: {{Optimal ET sequence| 10, 27, 37 }}


Badness: 0.058084
Badness: 0.058084
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Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}
Mapping: {{mapping| 1 1 5 4 -1 4 | 0 2 -9 -4 15 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 356.701
Optimal tunings:
* CTE: ~2 = 1\1, ~16/13 = 356.6997
* POTE: ~2 = 1\1, ~16/13 = 356.701


{{Optimal ET sequence|legend=1| 10, 27, 37 }}
Optimal ET sequence: {{Optimal ET sequence| 10, 27, 37 }}


Badness: 0.033971
Badness: 0.033971
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{{Mapping|legend=1| 1 4 -2 -2 | 0 -5 9 10 }}
{{Mapping|legend=1| 1 4 -2 -2 | 0 -5 9 10 }}
: mapping generators: ~2, ~7/5


{{Multival|legend=1| 5 -9 -10 -26 -30 2 }}
{{Multival|legend=1| 5 -9 -10 -26 -30 2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 577.776
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~7/5 = 577.3535
* [[POTE]]: ~2 = 1\1, ~7/5 = 577.776


{{Optimal ET sequence|legend=1| 2, 25, 27 }}
{{Optimal ET sequence|legend=1| 2, 25, 27 }}
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Mapping: {{mapping| 1 4 -2 -2 3 | 0 -5 9 10 1 }}
Mapping: {{mapping| 1 4 -2 -2 3 | 0 -5 9 10 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 577.850
Optimal tunings:
* CTE: ~2 = 1\1, ~7/5 = 577.2958
* POTE: ~2 = 1\1, ~7/5 = 577.850


{{Optimal ET sequence|legend=1| 2, 25e, 27e }}
Optimal ET sequence: {{Optimal ET sequence| 2, 25e, 27e }}


Badness: 0.052054
Badness: 0.052054
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Mapping: {{mapping| 1 4 -2 -2 3 -4 | 0 -5 9 10 1 16 }}
Mapping: {{mapping| 1 4 -2 -2 3 -4 | 0 -5 9 10 1 16 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 578.060
Optimal tunings:
* CTE: ~2 = 1\1, ~7/5 = 577.3744
* POTE: ~2 = 1\1, ~7/5 = 578.060


{{Optimal ET sequence|legend=1| 2f, 25ef, 27e }}
Optimal ET sequence: {{Optimal ET sequence| 2f, 27e }}


Badness: 0.039705
Badness: 0.039705
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{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
{{Mapping|legend=1| 1 0 5 6 | 0 3 -5 -6 }}
: mapping generators: ~2, ~10/7


{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}
{{Multival|legend=1| 3 -5 -6 -15 -18 0 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 562.122
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~10/7 = 638.7819
* [[POTE]]: ~2 = 1\1, ~10/7 = 637.878


{{Optimal ET sequence|legend=1| 2, 13, 15, 32c, 79bcc, 111bcc }}
{{Optimal ET sequence|legend=1| 2, 13, 15, 32c }}


[[Badness]]: 0.066400
[[Badness]]: 0.066400
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Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}
Mapping: {{mapping| 1 0 5 6 4 | 0 3 -5 -6 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 562.085
Optimal tunings:
* CTE: ~2 = 1\1, ~10/7 = 638.8462
* POTE: ~2 = 1\1, ~10/7 = 637.915


{{Optimal ET sequence|legend=1| 2, 13, 15, 32c, 47bc, 79bcce }}
Optimal ET sequence: {{Optimal ET sequence| 2, 13, 15, 32c, 47bc }}


Badness: 0.031036
Badness: 0.031036
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Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}
Mapping: {{mapping| 1 0 5 6 4 0 | 0 3 -5 -6 -1 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 562.365
Optimal tunings:
* CTE: ~2 = 1\1, ~10/7 = 637.8705
* POTE: ~2 = 1\1, ~10/7 = 637.635


{{Optimal ET sequence|legend=1| 15, 17c, 32cf }}
Optimal ET sequence: {{Optimal ET sequence| 15, 17c, 32cf }}


Badness: 0.026214
Badness: 0.026214
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Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}
Mapping: {{mapping| 1 0 5 6 4 9 | 0 3 -5 -6 -1 -10 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 563.239
Optimal tunings:
* CTE: ~2 = 1\1, ~10/7 = 637.7968
* POTE: ~2 = 1\1, ~10/7 = 636.761


{{Optimal ET sequence|legend=1| 15f, 17c, 32c, 49c }}
Optimal ET sequence: {{Optimal ET sequence| 2f, 15f, 17c }}


Badness: 0.032721
Badness: 0.032721
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{{See also| Dual-fifth temperaments #Dual-3 Sixix }}
{{See also| Dual-fifth temperaments #Dual-3 Sixix }}


Sixix is related to the [[Kleismic]] family in a way similar to the one between [[Meantone]] and [[Mavila]]. In both cases the generator is nominally a 6/5 and the complexity to generate major and minor chords is the same, but in sixix it is tuned extremely sharply, to the point where the 3rd and 5th harmonics are reached by going down instead of up, inverting the logic of chord construction.
Sixix is related to the [[kleismic]] family in a way similar to the one between [[meantone]] and [[mavila]]. In both cases the generator is nominally a 6/5 and the complexity to generate major and minor chords is the same, but in sixix it is tuned extremely sharply, to the point where the 3rd and 5th harmonics are reached by going down instead of up, inverting the logic of chord construction.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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{{Mapping|legend=1| 1 3 4 | 0 -5 -6 }}
{{Mapping|legend=1| 1 3 4 | 0 -5 -6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 338.365
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~6/5 = 338.005
* [[POTE]]: ~2 = 1\1, ~6/5 = 338.365


{{Optimal ET sequence|legend=1| 7, 25, 32 }}
{{Optimal ET sequence|legend=1| 7, 25, 32, 39c }}


[[Badness]]: 0.153088
[[Badness]]: 0.153088
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{{Multival|legend=1| 5 6 -10 -2 -30 -40 }}
{{Multival|legend=1| 5 6 -10 -2 -30 -40 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 337.442
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~6/5 = 337.5192
* [[POTE]]: ~2 = 1\1, ~6/5 = 337.442


{{Optimal ET sequence|legend=1| 7, 25, 32 }}
{{Optimal ET sequence|legend=1| 7, 18d, 25, 32 }}


[[Badness]]: 0.158903
[[Badness]]: 0.158903
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Mapping: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}
Mapping: {{mapping| 1 3 4 0 6 | 0 -5 -6 10 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 337.564
Optimal tunings:
* CTE: ~2 = 1\1, ~6/5 = 337.7486
* POTE: ~2 = 1\1, ~6/5 = 337.564


Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32 }}
Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32 }}
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Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}
Mapping: {{mapping| 1 3 4 0 6 4 | 0 -5 -6 10 -9 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 337.483
Optimal tunings:
* CTE: ~2 = 1\1, ~6/5 = 337.7925
* POTE: ~2 = 1\1, ~6/5 = 337.483


Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32f }}
Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32f }}
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Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}
Mapping: {{mapping| 1 3 4 0 6 4 1 | 0 -5 -6 10 -9 -1 11 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 337.513
Optimal tunings:
* CTE: ~2 = 1\1, ~6/5 = 337.6293
* POTE: ~2 = 1\1, ~6/5 = 337.513


Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32f }}
Optimal ET sequence: {{Optimal ET sequence| 7, 25e, 32f }}
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