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* Godzilla, with normal list [{{Monzo| -4 4 -1 }}, [[49/48|{{Monzo| -4 -1 0 2 }}]]],
* Godzilla, with normal list [{{Monzo| -4 4 -1 }}, [[49/48|{{Monzo| -4 -1 0 2 }}]]],
* Mothra, with normal list [{{Monzo| -4 4 -1 }}, [[1029/1024|{{Monzo| -10 1 0 3 }}]]],
* Mothra, with normal list [{{Monzo| -4 4 -1 }}, [[1029/1024|{{Monzo| -10 1 0 3 }}]]],
* Liese, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -9 11 0 -3 }}],
* Squares, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -3 9 0 -4 }}],  
* Squares, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -3 9 0 -4 }}],  
* Liese, with normal list [{{Monzo| -4 4 -1 }}, {{Monzo| -9 11 0 -3 }}],


all considered below.  
all considered below.  
Line 91: Line 91:
POTE generator: ~11/9 = 348.9155
POTE generator: ~11/9 = 348.9155


Vals: {{Val list| 7, 17c, 24, 31, 55, 86ef, 141ceff }}
Optimal GPV sequence: {{Val list| 7, 17c, 24, 31, 55, 86ef, 141ceff }}


Scales: [[mohaha7]], [[mohaha10]]
Scales: [[mohaha7]], [[mohaha10]]
Line 154: Line 154:
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.


Vals: {{Val list| 12, 19e, 31, 105, 136b, 167be, 198be }}
Optimal GPV sequence: {{Val list| 12, 19e, 31, 105, 136b, 167be, 198be }}


Badness: 0.017027
Badness: 0.017027
Line 174: Line 174:
: Eigenmonzos (unchanged intervals): 2, 11/9
: Eigenmonzos (unchanged intervals): 2, 11/9


Vals: {{Val list| 12f, 19e, 31 }}
Optimal GPV sequence: {{Val list| 12f, 19e, 31 }}


Badness: 0.018048
Badness: 0.018048
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* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 697.674]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 697.674]


Vals: {{Val list| 12, 19ef, 31, 43, 74 }}
Optimal GPV sequence: {{Val list| 12, 19ef, 31, 43, 74 }}


Badness: 0.025899
Badness: 0.025899
Line 213: Line 213:
: Eigenmonzos (unchanged intervals): 2, 18/13
: Eigenmonzos (unchanged intervals): 2, 18/13


Vals: {{Val list| 12f, 31f, 43 }}
Optimal GPV sequence: {{Val list| 12f, 31f, 43 }}


Badness: 0.026421
Badness: 0.026421
Line 228: Line 228:
POTE generator: ~15/13 = 251.535
POTE generator: ~15/13 = 251.535


Vals: {{Val list| 19e, 43, 62, 167bef }}
Optimal GPV sequence: {{Val list| 19e, 43, 62, 167bef }}


Badness: 0.031433
Badness: 0.031433
Line 257: Line 257:
Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.


Vals: {{Val list| 12e, 19, 31, 81 }}
Optimal GPV sequence: {{Val list| 12e, 19, 31, 81 }}


Badness: 0.021543
Badness: 0.021543
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* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 696.774]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 696.774]


Vals: {{Val list| 12ef, 19, 31, 50, 81, 131bd, 212bbddf }}
Optimal GPV sequence: {{Val list| 12ef, 19, 31, 50, 81, 131bd, 212bbddf }}


Badness: 0.020883
Badness: 0.020883
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: Eigenmonzos (unchanged intervals): 2, 11
: Eigenmonzos (unchanged intervals): 2, 11


Vals: {{Val list| 12e, 19, 31f, 50ff, 81fff }}
Optimal GPV sequence: {{Val list| 12e, 19, 31f, 50ff, 81fff }}


Badness: 0.027666
Badness: 0.027666
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* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]


Vals: {{Val list| 7d, 12, 19, 31e, 50ee }}
Optimal GPV sequence: {{Val list| 7d, 12, 19, 31e, 50ee }}


Badness: 0.021423
Badness: 0.021423
Line 333: Line 333:
POTE generator: ~3/2 = 696.146
POTE generator: ~3/2 = 696.146


Vals: {{Val list| 12f, 19, 31e, 50ee }}
Optimal GPV sequence: {{Val list| 12f, 19, 31e, 50ee }}


Badness: 0.021182
Badness: 0.021182
Line 346: Line 346:
POTE generator: ~3/2 = 695.060
POTE generator: ~3/2 = 695.060


Vals: {{Val list| 7d, 12, 19 }}
Optimal GPV sequence: {{Val list| 7d, 12, 19 }}


Badness: 0.024763
Badness: 0.024763
Line 359: Line 359:
POTE generator: ~3/2 = 695.858
POTE generator: ~3/2 = 695.858


Vals: {{Val list| 7d, 12, 19 }}
Optimal GPV sequence: {{Val list| 7d, 12, 19 }}


Badness: 0.025535
Badness: 0.025535
Line 372: Line 372:
POTE generator: ~3/2 = 696.131
POTE generator: ~3/2 = 696.131


Vals: {{Val list| 7d, 12, 19 }}
Optimal GPV sequence: {{Val list| 7d, 12, 19 }}


Badness: 0.022302
Badness: 0.022302
Line 385: Line 385:
POTE generator: ~3/2 = 696.044
POTE generator: ~3/2 = 696.044


Vals: {{Val list| 7d, 12, 19 }}
Optimal GPV sequence: {{Val list| 7d, 12, 19 }}


Badness: 0.020139
Badness: 0.020139
Line 398: Line 398:
POTE generator: ~3/2 = 695.913
POTE generator: ~3/2 = 695.913


Vals: {{Val list| 7d, 12, 19 }}
Optimal GPV sequence: {{Val list| 7d, 12, 19 }}


Badness: 0.018168
Badness: 0.018168
Line 411: Line 411:
POTE generator: ~3/2 = 695.750
POTE generator: ~3/2 = 695.750


Vals: {{Val list| 7d, 12, 19 }}
Optimal GPV sequence: {{Val list| 7d, 12, 19 }}


Badness: 0.017069
Badness: 0.017069
Line 424: Line 424:
POTE generator: ~3/2 = 695.603
POTE generator: ~3/2 = 695.603


Vals: {{Val list| 7d, 12, 19 }}
Optimal GPV sequence: {{Val list| 7d, 12, 19 }}


Badness: 0.016129
Badness: 0.016129
Line 437: Line 437:
POTE generator: ~3/2 = 695.696
POTE generator: ~3/2 = 695.696


Vals: {{Val list| 7d, 12, 19 }}
Optimal GPV sequence: {{Val list| 7d, 12, 19 }}


Badness: 0.015356
Badness: 0.015356
Line 450: Line 450:
POTE generator: ~3/2 = 695.688
POTE generator: ~3/2 = 695.688


Vals: {{Val list| 7d, 12, 19 }}
Optimal GPV sequence: {{Val list| 7d, 12, 19 }}


Badness: 0.013906
Badness: 0.013906
Line 463: Line 463:
POTE generator: ~3/2 = 695.676
POTE generator: ~3/2 = 695.676


Vals: {{Val list| 7d, 12, 19 }}
Optimal GPV sequence: {{Val list| 7d, 12, 19 }}


Badness: 0.013818
Badness: 0.013818
Line 476: Line 476:
POTE generator: ~3/2 = 697.254
POTE generator: ~3/2 = 697.254


Vals: {{Val list| 7d, 12f, 19f, 31eff }}
Optimal GPV sequence: {{Val list| 7d, 12f, 19f, 31eff }}


Badness: 0.024243
Badness: 0.024243
Line 489: Line 489:
POTE generator: ~3/2 = 694.689
POTE generator: ~3/2 = 694.689


Vals: {{Val list| 7d, 12e, 19e }}
Optimal GPV sequence: {{Val list| 7d, 12e, 19e }}


Badness: 0.031539
Badness: 0.031539
Line 502: Line 502:
POTE generator: ~3/2 = 694.764
POTE generator: ~3/2 = 694.764


Vals: {{Val list| 7d, 12e, 19e }}
Optimal GPV sequence: {{Val list| 7d, 12e, 19e }}


Badness: 0.026288
Badness: 0.026288
Line 517: Line 517:
POTE generator: ~11/9 = 348.182
POTE generator: ~11/9 = 348.182


Vals: {{Val list| 7d, 24d, 31, 100de, 131bdee, 162bdee }}
Optimal GPV sequence: {{Val list| 7d, 24d, 31, 100de, 131bdee, 162bdee }}


Badness: 0.025516
Badness: 0.025516
Line 532: Line 532:
POTE generator: ~11/9 = 348.490
POTE generator: ~11/9 = 348.490


Vals: {{Val list| 7d, 24d, 31, 55d }}
Optimal GPV sequence: {{Val list| 7d, 24d, 31, 55d }}


Badness: 0.028071
Badness: 0.028071
Line 549: Line 549:
POTE generator: ~3/2 = 696.016
POTE generator: ~3/2 = 696.016


Vals: {{Val list| 12, 26de, 38d, 50 }}
Optimal GPV sequence: {{Val list| 12, 26de, 38d, 50 }}


Badness: 0.038122
Badness: 0.038122
Line 564: Line 564:
POTE generator: ~3/2 = 695.836
POTE generator: ~3/2 = 695.836


Vals: {{Val list| 12f, 26deff, 38df, 50 }}
Optimal GPV sequence: {{Val list| 12f, 26deff, 38df, 50 }}


Badness: 0.028817
Badness: 0.028817
Line 579: Line 579:
POTE generator: ~3/2 = 695.783
POTE generator: ~3/2 = 695.783


Vals: {{Val list| 12f, 26deff, 38df, 50 }}
Optimal GPV sequence: {{Val list| 12f, 26deff, 38df, 50 }}


Badness: 0.022666
Badness: 0.022666
Line 633: Line 633:
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]


Vals: {{Val list| 7, 19, 26, 45, 71bc, 116bcde }}
Optimal GPV sequence: {{Val list| 7, 19, 26, 45, 71bc, 116bcde }}


Badness: 0.033839
Badness: 0.033839
Line 653: Line 653:
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]


Vals: {{Val list| 7, 19, 26, 45f, 71bcf, 116bcdef }}
Optimal GPV sequence: {{Val list| 7, 19, 26, 45f, 71bcf, 116bcdef }}


Badness: 0.022260
Badness: 0.022260
Line 668: Line 668:
POTE generator: ~11/9 = 346.922
POTE generator: ~11/9 = 346.922


Vals: {{Val list| 7, 31dd, 38d, 45e, 83bcddee }}
Optimal GPV sequence: {{Val list| 7, 31dd, 38d, 45e, 83bcddee }}


Badness: 0.058785
Badness: 0.058785
Line 681: Line 681:
POTE generator: ~11/9 = 346.910
POTE generator: ~11/9 = 346.910


Vals: {{Val list| 7, 31ddf, 38df, 45ef, 83bcddeeff }}
Optimal GPV sequence: {{Val list| 7, 31ddf, 38df, 45ef, 83bcddeeff }}


Badness: 0.034316
Badness: 0.034316
Line 722: Line 722:
POTE generator: ~3/2 = 703.254
POTE generator: ~3/2 = 703.254


Vals: {{Val list| 5, 12, 17c, 29cde }}
Optimal GPV sequence: {{Val list| 5, 12, 17c, 29cde }}


Badness: 0.024180
Badness: 0.024180
Line 740: Line 740:
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = 705.882
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = 705.882


Vals: {{Val list| 12f, 17c, 29cdef }}
Optimal GPV sequence: {{Val list| 12f, 17c, 29cdef }}


Badness: 0.024108
Badness: 0.024108
Line 751: Line 751:
Mapping: [{{val| 1 0 -4 6 13 -9 }}, {{val| 0 1 4 -2 -6 8 }}]
Mapping: [{{val| 1 0 -4 6 13 -9 }}, {{val| 0 1 4 -2 -6 8 }}]


Vals: {{Val list| 5, 12, 17c, 46cde }}
POTE generator: ~3/2 = 704.905


POTE generator: ~3/2 = 704.905
Optimal GPV sequence: {{Val list| 5, 12, 17c, 46cde }}


Badness: 0.027295
Badness: 0.027295
Line 766: Line 766:
POTE generator: ~3/2 = 698.776
POTE generator: ~3/2 = 698.776


Vals: {{Val list| 5e, 7, 12, 19d, 43de }}
Optimal GPV sequence: {{Val list| 5e, 7, 12, 19d, 43de }}


Badness: 0.021978
Badness: 0.021978
Line 779: Line 779:
POTE generator: ~3/2 = 695.762
POTE generator: ~3/2 = 695.762


Vals: {{Val list| 5ef, 7, 12, 19d, 31def }}
Optimal GPV sequence: {{Val list| 5ef, 7, 12, 19d, 31def }}


Badness: 0.027039
Badness: 0.027039
Line 792: Line 792:
POTE generator: ~3/2 = 696.115
POTE generator: ~3/2 = 696.115


Vals: {{Val list| 5ef, 7, 12, 19d, 31def }}
Optimal GPV sequence: {{Val list| 5ef, 7, 12, 19d, 31def }}


Badness: 0.024539
Badness: 0.024539
Line 805: Line 805:
POTE generator: ~3/2 = 696.217
POTE generator: ~3/2 = 696.217


Vals: {{Val list| 5ef, 7, 12, 19d, 31def }}
Optimal GPV sequence: {{Val list| 5ef, 7, 12, 19d, 31def }}


Badness: 0.020398
Badness: 0.020398
Line 818: Line 818:
POTE generator: ~3/2 = 698.544
POTE generator: ~3/2 = 698.544


Vals: {{Val list| 5e, 7, 12f, 19df }}
Optimal GPV sequence: {{Val list| 5e, 7, 12f, 19df }}


Badness: 0.018289
Badness: 0.018289
Line 831: Line 831:
POTE generator: ~3/2 = 705.004
POTE generator: ~3/2 = 705.004


Vals: {{Val list| 5e, 12e, 17c, 46cd }}
Optimal GPV sequence: {{Val list| 5e, 12e, 17c, 46cd }}


Badness: 0.036562
Badness: 0.036562
Line 844: Line 844:
POTE generator: ~3/2 = 705.496
POTE generator: ~3/2 = 705.496


Vals: {{Val list| 5e, 12e, 17c }}
Optimal GPV sequence: {{Val list| 5e, 12e, 17c }}


Badness: 0.027435
Badness: 0.027435
Line 857: Line 857:
POTE generator: ~3/2 = 698.491
POTE generator: ~3/2 = 698.491


Vals: {{Val list| 5, 7, 12e }}
Optimal GPV sequence: {{Val list| 5, 7, 12e }}


Badness: 0.026141
Badness: 0.026141
Line 870: Line 870:
POTE generator: ~3/2 = 696.743
POTE generator: ~3/2 = 696.743


Vals: {{Val list| 5, 7, 12ef, 19def }}
Optimal GPV sequence: {{Val list| 5, 7, 12ef, 19def }}


Badness: 0.023300
Badness: 0.023300
Line 883: Line 883:
POTE generator: ~3/2 = 696.978
POTE generator: ~3/2 = 696.978


Vals: {{Val list| 5, 7, 12ef, 19def }}
Optimal GPV sequence: {{Val list| 5, 7, 12ef, 19def }}


Badness: 0.024535
Badness: 0.024535
Line 896: Line 896:
POTE generator: ~3/2 = 697.068
POTE generator: ~3/2 = 697.068


Vals: {{Val list| 5, 7, 12ef, 19def }}
Optimal GPV sequence: {{Val list| 5, 7, 12ef, 19def }}


Badness: 0.021098
Badness: 0.021098
Line 916: Line 916:
POTE generator: ~11/9 = 350.934
POTE generator: ~11/9 = 350.934


Vals: {{Val list| 7, 17c, 24d, 41cd }}
Optimal GPV sequence: {{Val list| 7, 17c, 24d, 41cd }}


Badness: 0.040240
Badness: 0.040240
Line 931: Line 931:
POTE generator: ~11/9 = 350.816
POTE generator: ~11/9 = 350.816


Vals: {{Val list| 7, 17c, 24d, 41cd }}
Optimal GPV sequence: {{Val list| 7, 17c, 24d, 41cd }}


Badness: 0.027214
Badness: 0.027214
Line 961: Line 961:
POTE generator: ~3/2 = 696.615
POTE generator: ~3/2 = 696.615


Vals: {{Val list| 5, 7d, 12de }}
Optimal GPV sequence: {{Val list| 5, 7d, 12de }}


Badness: 0.025167
Badness: 0.025167
Line 989: Line 989:
POTE generator: ~3/2 = 685.234
POTE generator: ~3/2 = 685.234


Vals: {{Val list| 2cde, 5de, 7 }}
Optimal GPV sequence: {{Val list| 2cde, 5de, 7 }}


Badness: 0.032521
Badness: 0.032521
Line 1,015: Line 1,015:
POTE generator: ~3/2 = 705.096
POTE generator: ~3/2 = 705.096


Vals: {{Val list| 5de, 12de, 17c, 29c }}
Optimal GPV sequence: {{Val list| 5de, 12de, 17c, 29c }}


Badness: 0.063262
Badness: 0.063262
Line 1,028: Line 1,028:
POTE generator: ~3/2 = 705.094
POTE generator: ~3/2 = 705.094


Vals: {{Val list| 5de, 12de, 17c, 29c }}
Optimal GPV sequence: {{Val list| 5de, 12de, 17c, 29c }}


Badness: 0.040324
Badness: 0.040324
Line 1,075: Line 1,075:
* 11-odd-limit diamond monotone and tradeoff: ~7/6 = [252.632, 257.143]
* 11-odd-limit diamond monotone and tradeoff: ~7/6 = [252.632, 257.143]


Vals: {{Val list| 14c, 19, 33cd, 52cd }}
Optimal GPV sequence: {{Val list| 14c, 19, 33cd, 52cd }}


Badness: 0.028947
Badness: 0.028947
Line 1,095: Line 1,095:
* 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = 252.632
* 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = 252.632


Vals: {{Val list| 14cf, 19, 33cdff, 52cdff }}
Optimal GPV sequence: {{Val list| 14cf, 19, 33cdff, 52cdff }}


Badness: 0.022503
Badness: 0.022503
Line 1,110: Line 1,110:
POTE generator: ~8/7 = 254.042
POTE generator: ~8/7 = 254.042


Vals: {{Val list| 14c, 19e, 33cdee }}
Optimal GPV sequence: {{Val list| 14c, 19e, 33cdee }}


Badness: 0.028510
Badness: 0.028510
Line 1,125: Line 1,125:
POTE generator: ~8/7 = 251.079
POTE generator: ~8/7 = 251.079


Vals: {{Val list| 19e, 24, 43de }}
Optimal GPV sequence: {{Val list| 19e, 24, 43de }}


Badness: 0.039647
Badness: 0.039647
Line 1,140: Line 1,140:
POTE generator: ~8/7 = 251.165
POTE generator: ~8/7 = 251.165


Vals: {{Val list| 19e, 24, 43de }}
Optimal GPV sequence: {{Val list| 19e, 24, 43de }}


Badness: 0.025676
Badness: 0.025676
Line 1,155: Line 1,155:
POTE generator: ~8/7 = 251.173
POTE generator: ~8/7 = 251.173


Vals: {{Val list| 19, 24, 43d }}
Optimal GPV sequence: {{Val list| 5, 14ce, 19, 24, 43d }}


Badness: 0.035673
Badness: 0.035673


== Injera ==
==== 13-limit ====
Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo|38EDO]], which is two parallel [[19edo|19EDOs]], is an excellent tuning for injera.
Subgroup: 2.3.5.7.11.13


[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]
Comma list: 49/48, 56/55, 81/80, 91/90


Subgroup: 2.3.5.7
Mapping: [{{val| 1 0 -4 2 9 -5 }}, {{val| 0 2 8 1 -7 11 }}]


[[Comma list]]: 50/49, 81/80
Mapping generators: ~2, ~7/4


[[Mapping]]: [{{val| 2 0 -8 -7 }}, {{val| 0 1 4 4 }}]
POTE generator: ~8/7 = 251.198


Mapping generators: ~7/5, ~3
Optimal GPV sequence: {{Val list| 5, 14cef, 19, 24, 43d }}


{{Multival|legend=1| 2 8 8 8 7 -4 }}
Badness: 0.026703


[[POTE generator]]: ~3/2 = 694.375
== Mohajira ==
{{Main| Mohajira }}


[[Tuning ranges]]:
Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. [[31edo|31EDO]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31.  
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [688.957, 700.000]
* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [685.714, 700.000]


{{Val list|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}
Subgroup: 2.3.5.7


[[Badness]]: 0.031130
[[Comma list]]: 81/80, 6144/6125


; Music
[[Mapping]]: [{{val| 1 1 0 6 }}, {{val| 0 2 8 -11 }}]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo|26EDO]]) by [[Igliashon Jones]]


=== 11-limit ===
Mapping generators: ~2, ~128/105
Subgroup: 2.3.5.7.11


Comma list: 45/44, 50/49, 81/80
{{Multival|legend=1| 2 8 -11 8 -23 -48 }}


Mapping: [{{val| 2 0 -8 -7 -12 }}, {{val| 0 1 4 4 6 }}]
[[POTE generator]]: ~128/105 = 348.415


Mapping generators: ~7/5, ~3
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{Monzo| 0 0 1/8 }}
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 6 0 -11/8 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5


POTE generator: ~3/2 = 692.840
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978]
* 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]
* 7-odd-limit diamond monotone and tradeoff: ~128/105 = [347.393, 350.000]
* 9-odd-limit diamond monotone and tradeoff: ~128/105 = [347.368, 350.000]


Tuning ranges:  
[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.
* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [685.714, 700.000]


Vals: {{Val list| 12, 14c, 26, 90bce, 116bcce }}
{{Val list|legend=1| 7, 24, 31 }}


Badness: 0.023124
[[Badness]]: 0.055714


==== 13-limit ====
Scales: [[mohaha7]], [[mohaha10]]
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 50/49, 78/77, 81/80
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 0 -8 -7 -12 -21 }}, {{val| 0 1 4 4 6 9 }}]
Comma list: 81/80, 121/120, 176/175


Mapping generators: ~7/5, ~3
Mapping: [{{val| 1 1 0 6 2 }}, {{val| 0 2 8 -11 5 }}]


POTE generator: ~3/2 = 692.673
Mapping generators: ~2, ~11/9


Tuning ranges:
POTE generator: ~11/9 = 348.477
* 13- and 15-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = 692.308


Vals: {{Val list| 12f, 14cf, 26, 38e }}
Minimax tuning:  
* [[11-odd-limit]]: ~11/9 = {{Monzo| 0 0 1/8 }}
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| 6 0 -11/8 0 0 }}, {{Monzo| 2 0 5/8 0 0 }}]
: Eigenmonzos (unchanged intervals): 2, 5


Badness: 0.021565
[[Tuning ranges]]:
* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]
* 11-odd-limit diamond monotone and tradeoff: ~11/9 = [348.387, 350.000]


===== 17-limit =====
Optimal GPV sequence: {{Val list| 7, 24, 31 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 45/44, 50/49, 78/77, 81/80, 85/84
Badness: 0.026064


Mapping: [{{val| 2 0 -8 -7 -12 -21 5 }}, {{val| 0 1 4 4 6 9 1 }}]
Scales: [[mohaha7]], [[mohaha10]]


POTE generator: ~3/2 = 692.487
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 12f, 14cf, 26 }}
Comma list: 66/65, 81/80, 105/104, 121/120


Badness: 0.018358
Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 8 -11 5 -1 }}]


===== 19-limit =====
Mapping generators: ~2, ~11/9
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84
POTE generator: ~11/9 = 348.558


Mapping: [{{val| 2 0 -8 -7 -12 -21 5 -1 }}, {{val| 0 1 4 4 6 9 1 3 }}]
Optimal GPV sequence: {{Val list| 7, 24, 31 }}


POTE generator: ~3/2 = 692.299
Badness: 0.023388


Vals: {{Val list| 12f, 14cf, 26 }}
Scales: [[mohaha7]], [[mohaha10]]


Badness: 0.015118
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


==== Enjera ====
Comma list: 66/65, 81/80, 105/104, 121/120, 154/153
Subgroup: 2.3.5.7.11.13


Comma list: 27/26, 40/39, 45/44, 50/49
Mapping: [{{val| 1 1 0 6 2 4 7 }}, {{val| 0 2 8 -11 5 -1 -10 }}]


Mapping: [{{val| 2 0 -8 -7 -12 -2 }}, {{val| 0 1 4 4 6 3 }}]
Mapping generators: ~2, ~11/9


Mapping generators: ~7/5, ~3
POTE generator: ~11/9 = 348.736


POTE generator: ~3/2 = 694.121
Optimal GPV sequence: {{Val list| 7, 24, 31, 86ef }}


Vals: {{Val list| 12f, 14c, 26f, 38eff }}
Badness: 0.020576


Badness: 0.026542
Scales: [[mohaha7]], [[mohaha10]]


=== Injerous ===
=== 19-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 33/32, 50/49, 55/54
Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152


Mapping: [{{val| 2 0 -8 -7 10 }}, {{val| 0 1 4 4 -1 }}]
Mapping: [{{val| 1 1 0 6 2 4 7 6 }}, {{val| 0 2 8 -11 5 -1 -10 -6 }}]


Mapping generators: ~7/5, ~3
Mapping generators: ~2, ~11/9


POTE generator: ~3/2 = 690.548
POTE generator: ~11/9 = 348.810


Vals: {{Val list| 12e, 14c, 26e, 40cee }}
Optimal GPV sequence: {{Val list| 7, 24, 31, 55, 86efh }}


Badness: 0.038577
Badness: 0.017302


=== Lahoh ===
Scales: [[mohaha7]], [[mohaha10]]
Subgroup: 2.3.5.7.11


Comma list: 50/49, 56/55, 81/77
== Mohamaq ==
Subgroup: 2.3.5.7


Mapping: [{{val| 2 0 -8 -7 7 }}, {{val| 0 1 4 4 0 }}]
[[Comma list]]: 81/80, 392/375


Mapping generators: ~7/5, ~3
[[Mapping]]: [{{val| 1 1 0 -1 }}, {{val| 0 2 8 13 }}]


POTE generator: ~3/2 = 699.001
Mapping generators: ~2, ~25/21


Vals: {{Val list| 2cd, 10cd, 12 }}
[[POTE generator]]: ~25/21 = 350.586


Badness: 0.043062
{{Val list|legend=1| 7d, 17c, 24, 65cc, 89ccd }}


=== Teff ===
[[Badness]]: 0.077734
{{Main| Teff }}


Teff (found by Mason Green) is to injera what mohajira is to meantone; it splits the generator in half in order to accommodate higher limit intervals, creating a half-octave quarter-tone temperament.
Scales: [[mohaha7]], [[mohaha10]]


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 50/49, 81/80, 864/847
Comma list: 56/55, 77/75, 243/242
 
Mapping: [{{val| 1 1 0 -1 2 }}, {{val| 0 2 8 13 5 }}]


Mapping: [{{val| 2 1 -4 -3 8 }}, {{val| 0 2 8 8 -1 }}]
Mapping generators: ~2, ~11/9


Mapping generators: ~7/5, ~16/11
POTE generator: ~11/9 = 350.565


POTE generator: ~11/8 = 552.5303
Optimal GPV sequence: {{Val list| 7d, 17c, 24, 65cc, 89ccd }}


Vals: {{Val list| 24d, 26, 50d }}
Badness: 0.036207


Badness: 0.070689
Scales: [[mohaha7]], [[mohaha10]]


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 50/49, 78/77, 81/80, 144/143
Comma list: 56/55, 66/65, 77/75, 243/242


Mapping: [{{val| 2 1 -4 -3 8 2 }}, {{val| 0 2 8 8 -1 5 }}]
Mapping: [{{val| 1 1 0 -1 2 4 }}, {{val| 0 2 8 13 5 -1 }}]


POTE generator: ~11/8 = 552.5324
Mapping generators: ~2, ~11/9


Vals: {{Val list| 24d, 26, 50d }}
POTE generator: ~11/9 = 350.745


Badness: 0.040047
Optimal GPV sequence: {{Val list| 7d, 17c, 24, 41c, 65cc }}


==== 17-limit ====
Badness: 0.028738
Subgroup: 2.3.5.7.11.13.17


Comma list: 50/49, 78/77, 81/80, 85/84, 144/143
Scales: [[mohaha7]], [[mohaha10]]


Mapping: [{{val| 2 1 -4 -3 8 2 6 }}, {{val| 0 2 8 8 -1 5 2 }}]
== Mothra ==
Mothra splits the fifth into three 8/7 generators. It uses [[1029/1024]], the gamelisma, to accomplish this deed and also tempers out [[1728/1715]], the orwell comma. Using [[31edo|31EDO]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7 subgroup, mothra is identical to [[slendric]].


POTE generator: ~11/8 = 552.6558
Note that mothra can also be called '''cynder''' in the 7-limit, which can be a little confusing sometimes.  


Vals: {{Val list| 24d, 26 }}
Subgroup: 2.3.5.7


Badness: 0.029499
[[Comma list]]: 81/80, 1029/1024


==== 19-limit ====
[[Mapping]]: [{{val| 1 1 0 3 }}, {{val| 0 3 12 -1 }}]
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143
Mapping generators: ~2, ~8/7


Mapping: [{{val| 2 1 -4 -3 8 2 6 2 }}, {{val| 0 2 8 8 -1 5 2 6 }}]
{{Multival|legend=1| 3 12 -1 12 -10 -36 }}


POTE generator: ~11/8 = 552.6382
[[POTE generator]]: ~8/7 = 232.193


Vals: {{Val list| 24d, 26 }}
[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.


Badness: 0.023133
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{Monzo| 0 0 1/12 }}
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 3 0 -1/12 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5


== Pombe ==
{{Val list|legend=1| 5, 26, 31 }}
Pombe (named after the African millet beer) is a variant of [[#Teff]] by Kaiveran Lugheidh that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.


Subgroup: 2.3.5.7
[[Badness]]: 0.037146


[[Comma list]]: 81/80, 300125/294912
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val| 2 1 -4 11 }}, {{val| 0 2 8 -5 }}]
Comma list: 81/80, 99/98, 385/384


Mapping generators: ~735/512, ~35/24
Mapping: [{{val| 1 1 0 3 5 }}, {{val| 0 3 12 -1 -8 }}]


{{Multival|legend=1| 4 16 -10 16 -27 -68 }}
Mapping generators: ~2, ~8/7


[[POTE generator]]: ~48/35 = 552.2206
POTE generator: ~8/7 = 232.031


{{Val list|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}
Optimal GPV sequence: {{Val list| 5, 26, 31, 88, 150be, 181bee }}


[[Badness]]: 0.116104
Badness: 0.025642


=== 11-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 245/242, 385/384
Comma list: 81/80, 99/98, 105/104, 144/143


Mapping: [{{val| 2 1 -4 11 8 }}, {{val| 0 2 8 -5 -1 }}]
Mapping: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 12 -1 -8 14 }}]


POTE generator: ~11/8 = 552.0929
Mapping generators: ~2, ~8/7


Vals: {{Val list| 24, 26, 50 }}
POTE generator: ~8/7 = 231.811


Badness: 0.052099
Optimal GPV sequence: {{Val list| 5, 26, 31, 57, 88 }}


=== 13-limit ===
Badness: 0.023954
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 105/104, 144/143, 245/242
; Music:
* [http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 Prelude for solo piano in mothra16, brat 4 tuning] by [http://chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ Chris Vaisvil]


Mapping: [{{val| 2 1 -4 11 8 2 }}, {{val| 0 2 8 -5 -1 5 }}]
=== Cynder ===
Subgroup: 2.3.5.7.11


POTE generator: ~11/8 = 552.1498
Comma list: 45/44, 81/80, 1029/1024


Vals: {{Val list| 24, 26, 50 }}
Mapping: [{{val| 1 1 0 3 0 }}, {{val| 0 3 12 -1 18 }}]


Badness: 0.031039
Mapping generators: ~2, ~8/7


=== 17-limit ===
POTE generator: ~8/7 = 231.317
Subgroup: 2.3.5.7.11.13.17


Comma list: 81/80, 105/104, 144/143, 245/242, 273/272
Optimal GPV sequence: {{Val list| 5e, 26, 57e, 83bce }}


Mapping: [{{val| 2 1 -4 11 8 2 6 }}, {{val| 0 2 8 -5 -1 5 2 }}]
Badness: 0.055706


POTE generator: ~11/8 = 552.1579
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 24, 26, 50 }}
Comma list: 45/44, 78/77, 81/80, 640/637


Badness: 0.021260
Mapping: [{{val| 1 1 0 3 0 1 }}, {{val| 0 3 12 -1 18 14 }}]


=== 19-limit ===
Mapping generators: ~2, ~8/7
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209
POTE generator: ~8/7 = 231.293


Mapping: [{{val| 2 1 -4 11 8 2 6 2 }}, {{val| 0 2 8 -5 -1 5 2 6 }}]
Optimal GPV sequence: {{Val list| 5e, 26, 57e, 83bce }}


POTE generator: ~11/8 = 552.1196
Badness: 0.034124


Vals: {{Val list| 24, 26, 50 }}
=== Mosura ===
Subgroup: 2.3.5.7.11


Badness: 0.016548
Comma list: 81/80, 176/175, 540/539


== Mohajira ==
Mapping: [{{val| 1 1 0 3 -1 }}, {{val| 0 3 12 -1 23 }}]
{{Main| Mohajira }}


Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. [[31edo|31EDO]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31.
Mapping generators: ~2, ~8/7


Subgroup: 2.3.5.7
POTE generator: ~8/7 = 232.419


[[Comma list]]: 81/80, 6144/6125
Optimal GPV sequence: {{Val list| 31, 129, 160be, 191bce, 222bce, 253bcee }}


[[Mapping]]: [{{val| 1 1 0 6 }}, {{val| 0 2 8 -11 }}]
Badness: 0.031334


Mapping generators: ~2, ~128/105
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Multival|legend=1| 2 8 -11 8 -23 -48 }}
Comma list: 81/80, 144/143, 176/175, 196/195


[[POTE generator]]: ~128/105 = 348.415
Mapping: [{{val| 1 1 0 3 -1 7 }}, {{val| 0 3 12 -1 23 -17 }}]


[[Minimax tuning]]:
Mapping generators: ~2, ~8/7
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{Monzo| 0 0 1/8 }}
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 6 0 -11/8 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5


[[Tuning ranges]]:
POTE generator: ~8/7 = 232.640
* 7- and 9-odd-limit [[diamond monotone]]: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978]
* 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]
* 7-odd-limit diamond monotone and tradeoff: ~128/105 = [347.393, 350.000]
* 9-odd-limit diamond monotone and tradeoff: ~128/105 = [347.368, 350.000]


[[Algebraic generator]]: Mohabis, real root of 3''x''<sup>3</sup> - 3''x''<sup>2</sup> - 1, 348.6067 cents. Corresponding recurrence converges quickly.
Optimal GPV sequence: {{Val list| 31, 36, 67, 98 }}


{{Val list|legend=1| 7, 24, 31 }}
Badness: 0.036857


[[Badness]]: 0.055714
== Liese ==
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>


Scales: [[mohaha7]], [[mohaha10]]
Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo|74EDO]] makes for a good liese tuning, though [[19edo|19EDO]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.
 
Subgroup: 2.3.5.7


=== 11-limit ===
[[Comma list]]: 81/80, 686/675
Subgroup: 2.3.5.7.11


Comma list: 81/80, 121/120, 176/175
[[Mapping]]: [{{val| 1 0 -4 -3 }}, {{val| 0 3 12 11 }}]


Mapping: [{{val| 1 1 0 6 2 }}, {{val| 0 2 8 -11 5 }}]
Mapping generators: ~2, ~10/7


Mapping generators: ~2, ~11/9
{{Multival|legend=1| 3 12 11 12 9 -8 }}


POTE generator: ~11/9 = 348.477
[[POTE generator]]: ~10/7 = 632.406


Minimax tuning:  
Minimax tuning:  
* [[11-odd-limit]]: ~11/9 = {{Monzo| 0 0 1/8 }}
* 7- and 9-odd-limit: ~10/7 = {{Monzo| 1/3 0 1/12 }}
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| 6 0 -11/8 0 0 }}, {{Monzo| 2 0 5/8 0 0 }}]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 2/3 0 11/12 0 }}]
: Eigenmonzos (unchanged intervals): 2, 5
: [[Eigenmonzo]]s (unchanged intervals): 2, 5


[[Tuning ranges]]:
[[Algebraic generator]]: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges.
* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]
* 11-odd-limit diamond monotone and tradeoff: ~11/9 = [348.387, 350.000]


Vals: {{Val list| 7, 24, 31 }}
{{Val list|legend=1| 17c, 19, 55, 74d }}


Badness: 0.026064
[[Badness]]: 0.046706


Scales: [[mohaha7]], [[mohaha10]]
=== Liesel ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 56/55, 81/80, 540/539
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 81/80, 105/104, 121/120
Mapping: [{{val| 1 0 -4 -3 4 }}, {{val| 0 3 12 11 -1 }}]


Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 8 -11 5 -1 }}]
POTE generator: ~10/7 = 633.073


Mapping generators: ~2, ~11/9
Optimal GPV sequence: {{Val list| 17c, 19, 36, 91cee }}


POTE generator: ~11/9 = 348.558
Badness: 0.040721


Vals: {{Val list| 7, 24, 31 }}
==== 13-limit ====
Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.


Badness: 0.023388
Subgroup: 2.3.5.7.11.13


Scales: [[mohaha7]], [[mohaha10]]
Comma list: 56/55, 78/77, 81/80, 91/90


=== 17-limit ===
Mapping: [{{val| 1 0 -4 -3 4 0 }}, {{val| 0 3 12 11 -1 7 }}]
Subgroup: 2.3.5.7.11.13.17


Comma list: 66/65, 81/80, 105/104, 121/120, 154/153
POTE generator: ~10/7 = 633.042


Mapping: [{{val| 1 1 0 6 2 4 7 }}, {{val| 0 2 8 -11 5 -1 -10 }}]
Optimal GPV sequence: {{Val list| 17c, 19, 36, 91ceef }}


Mapping generators: ~2, ~11/9
Badness: 0.027304


POTE generator: ~11/9 = 348.736
=== Elisa ===
Subgroup: 2.3.5.7.11


Vals: {{Val list| 7, 24, 31, 86ef }}
Comma list: 77/75, 81/80, 99/98


Badness: 0.020576
Mapping: [{{val| 1 0 -4 -3 -5 }}, {{val| 0 3 12 11 16 }}]


Scales: [[mohaha7]], [[mohaha10]]
POTE generator: ~10/7 = 633.061


=== 19-limit ===
Optimal GPV sequence: {{Val list| 17c, 19e, 36e }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
Badness: 0.041592


Mapping: [{{val| 1 1 0 6 2 4 7 6 }}, {{val| 0 2 8 -11 5 -1 -10 -6 }}]
==== 13-limit ====
Subgroup: 2.3.5.7.11


Mapping generators: ~2, ~11/9
Comma list: 66/65, 77/75, 81/80, 99/98


POTE generator: ~11/9 = 348.810
Mapping: [{{val| 1 0 -4 -3 -5 0 }}, {{val| 0 3 12 11 16 7 }}]


Vals: {{Val list| 7, 24, 31, 55, 86efh }}
POTE generator: ~10/7 = 632.991


Badness: 0.017302
Optimal GPV sequence: {{Val list| 17c, 19e, 36e }}


Scales: [[mohaha7]], [[mohaha10]]
Badness: 0.026922


== Mohamaq ==
=== Lisa ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 81/80, 392/375
Comma list: 45/44, 81/80, 343/330


[[Mapping]]: [{{val| 1 1 0 -1 }}, {{val| 0 2 8 13 }}]
Mapping: [{{val| 1 0 -4 -3 -6 }}, {{val| 0 3 12 11 18 }}]


Mapping generators: ~2, ~25/21
POTE generator: ~10/7 = 631.370


[[POTE generator]]: ~25/21 = 350.586
Optimal GPV sequence: {{Val list| 17cee, 19 }}


{{Val list|legend=1| 7d, 17c, 24, 65cc, 89ccd }}
Badness: 0.054829


[[Badness]]: 0.077734
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Scales: [[mohaha7]], [[mohaha10]]
Comma list: 45/44, 81/80, 91/88, 147/143


=== 11-limit ===
Mapping: [{{val| 1 0 -4 -3 -6 0 }}, {{val| 0 3 12 11 18 7 }}]
Subgroup: 2.3.5.7.11


Comma list: 56/55, 77/75, 243/242
POTE generator: ~10/7 = 631.221


Mapping: [{{val| 1 1 0 -1 2 }}, {{val| 0 2 8 13 5 }}]
Optimal GPV sequence: {{Val list| 17cee, 19 }}


Mapping generators: ~2, ~11/9
Badness: 0.036144


POTE generator: ~11/9 = 350.565
== Squares ==
{{Main| Squares }}


Vals: {{Val list| 7d, 17c, 24, 65cc, 89ccd }}
Squares splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[31edo|31EDO]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].


Badness: 0.036207
Subgroup: 2.3.5.7


Scales: [[mohaha7]], [[mohaha10]]
[[Comma list]]: 81/80, 2401/2400


=== 13-limit ===
[[Mapping]]: [{{val| 1 3 8 6 }}, {{val| 0 -4 -16 -9 }}]
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 66/65, 77/75, 243/242
Mapping generators: ~2, ~9/7


Mapping: [{{val| 1 1 0 -1 2 4 }}, {{val| 0 2 8 13 5 -1 }}]
{{Multival|legend=1| 4 16 9 16 3 -24 }}


Mapping generators: ~2, ~11/9
[[POTE generator]]: ~9/7 = 425.942


POTE generator: ~11/9 = 350.745
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3/2 0 9/16 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5


Vals: {{Val list| 7d, 17c, 24, 41c, 65cc }}
[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.


Badness: 0.028738
{{Val list|legend=1| 14c, 17c, 31 }}


Scales: [[mohaha7]], [[mohaha10]]
[[Badness]]: 0.045993


== Orphic ==
Scales: [[skwares8]], [[skwares11]], [[skwares14]]
Subgroup: 2.3.5.7


[[Comma list]]: 81/80, 5898240/5764801
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val| 2 5 12 7 }}, {{val| 0 -4 -16 -3 }}]
Comma list: 81/80, 99/98, 121/120


Mapping generators: ~2401/1728, ~7/6
Mapping: [{{val| 1 3 8 6 7 }}, {{val| 0 -4 -16 -9 -10 }}]


{{Multival|legend=1| 8 32 6 32 -13 -76 }}
POTE generator: ~9/7 = 425.957


[[POTE generator]]: ~7/6 = 275.794
Optimal GPV sequence: {{Val list| 14c, 17c, 31 }}


{{Val list|legend=1| 26, 48c, 74, 174bd, 248bbd }}
Badness: 0.021636


[[Badness]]: 0.258825
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 11-limit ===
Comma list: 66/65, 81/80, 99/98, 121/120
Subgroup: 2.3.5.7.11


Comma list: 81/80, 99/98, 73728/73205
Mapping: [{{val| 1 3 8 6 7 3 }}, {{val| 0 -4 -16 -9 -10 2 }}]


Mapping: [{{val| 2 5 12 7 6 }}, {{val| 0 -4 -16 -3 2 }}]
POTE generator: ~9/7 = 425.550


Mapping generators: ~363/256, ~7/6
Optimal GPV sequence: {{Val list| 14c, 17c, 31, 79cf }}


POTE generator: ~7/6 = 275.762
Badness: 0.025514


Vals: {{Val list| 26, 48c, 74, 248bbd, 322bbdd }}
==== Squad ====
 
Badness: 0.101499
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 99/98, 144/143, 2200/2197
Comma list: 78/77, 81/80, 91/90, 99/98


Mapping: [{{val| 2 5 12 7 6 12 }}, {{val| 0 -4 -16 -3 2 -10 }}]
Mapping: [{{val| 1 3 8 6 7 9 }}, {{val| 0 -4 -16 -9 -10 -15 }}]


Mapping generators: ~55/39, ~7/6
POTE generator: ~9/7 = 425.7516


POTE generator: ~7/6 = 275.774
Optimal GPV sequence: {{Val list| 14cf, 17c, 31f }}


Vals: {{Val list| 26, 48c, 74, 174bd, 248bbd, 322bbdd }}
Badness: 0.026877


Badness: 0.053482
==== Agora ====
Subgroup: 2.3.5.7.11.13


== Mothra ==
Comma list: 81/80, 99/98, 105/104, 121/120
Mothra splits the fifth into three 8/7 generators. It uses [[1029/1024]], the gamelisma, to accomplish this deed and also tempers out [[1728/1715]], the orwell comma. Using [[31edo|31EDO]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7 subgroup, mothra is identical to [[slendric]].


Note that mothra can also be called '''cynder''' in the 7-limit, which can be a little confusing sometimes.
Mapping: [{{val| 1 3 8 6 7 14 }}, {{val| 0 -4 -16 -9 -10 -29 }}]


Subgroup: 2.3.5.7
POTE generator: ~9/7 = 426.276


[[Comma list]]: 81/80, 1029/1024
Optimal GPV sequence: {{Val list| 14cf, 31, 45ef, 76e }}


[[Mapping]]: [{{val| 1 1 0 3 }}, {{val| 0 3 12 -1 }}]
Badness: 0.024522


Mapping generators: ~2, ~8/7
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


{{Multival|legend=1| 3 12 -1 12 -10 -36 }}
Comma list: 81/80, 99/98, 105/104, 120/119, 121/119


[[POTE generator]]: ~8/7 = 232.193
Mapping: [{{val| 1 3 8 6 7 14 8 }}, {{val| 0 -4 -16 -9 -10 -29 -11 }}]


[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
POTE generator: ~9/7 = 426.187


[[Minimax tuning]]:  
Optimal GPV sequence: {{Val list| 14cf, 31, 76e }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{Monzo| 0 0 1/12 }}
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 3 0 -1/12 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5


{{Val list|legend=1| 5, 26, 31 }}
Badness: 0.022573


[[Badness]]: 0.037146
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


=== 11-limit ===
Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
Subgroup: 2.3.5.7.11


Comma list: 81/80, 99/98, 385/384
Mapping: [{{val| 1 3 8 6 7 14 8 11 }}, {{val| 0 -4 -16 -9 -10 -29 -11 -19 }}]


Mapping: [{{val| 1 1 0 3 5 }}, {{val| 0 3 12 -1 -8 }}]
POTE generator: ~9/7 = 426.225


Mapping generators: ~2, ~8/7
Optimal GPV sequence: {{Val list| 14cf, 31, 76e }}


POTE generator: ~8/7 = 232.031
Badness: 0.018839


Vals: {{Val list| 5, 26, 31, 88, 150be, 181bee }}
=== Cuboctahedra ===
Subgroup: 2.3.5.7.11


Badness: 0.025642
Comma list: 81/80, 385/384, 1375/1372


==== 13-limit ====
Mapping: [{{val| 1 3 8 6 -4 }}, {{val| 0 -4 -16 -9 21 }}]
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 99/98, 105/104, 144/143
POTE generator: ~9/7 = 425.993


Mapping: [{{val| 1 1 0 3 5 1 }}, {{val| 0 3 12 -1 -8 14 }}]
Optimal GPV sequence: {{Val list| 14ce, 17ce, 31, 107b, 138b, 169be, 200be }}


Mapping generators: ~2, ~8/7
Badness: 0.056826


POTE generator: ~8/7 = 231.811
== Jerome ==
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.


Vals: {{Val list| 5, 26, 31, 57, 88 }}
Subgroup: 2.3.5.7


Badness: 0.023954
[[Comma list]]: 81/80, 17280/16807


; Music:  
[[Mapping]]: [{{val| 1 1 0 2 }}, {{val| 0 5 20 7 }}]
* [http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 Prelude for solo piano in mothra16, brat 4 tuning] by [http://chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ Chris Vaisvil]


=== Cynder ===
Mapping generators: ~2, ~54/49
Subgroup: 2.3.5.7.11


Comma list: 45/44, 81/80, 1029/1024
{{Multival|legend=1| 5 20 7 20 -3 -40 }}


Mapping: [{{val| 1 1 0 3 0 }}, {{val| 0 3 12 -1 18 }}]
[[POTE generator]]: ~54/49 = 139.343


Mapping generators: ~2, ~8/7
{{Val list|legend=1| 17c, 26, 43, 69, 112bd }}


POTE generator: ~8/7 = 231.317
[[Badness]]: 0.108656


Vals: {{Val list| 5e, 26, 57e, 83bce }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.055706
Comma list: 81/80, 99/98, 864/847


==== 13-limit ====
Mapping: [{{val| 1 1 0 2 3 }}, {{val| 0 5 20 7 4 }}]
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 78/77, 81/80, 640/637
Mapping generators: ~2, ~12/11


Mapping: [{{val| 1 1 0 3 0 1 }}, {{val| 0 3 12 -1 18 14 }}]
POTE generator: ~12/11 = 139.428


Mapping generators: ~2, ~8/7
Optimal GPV sequence: {{Val list| 17c, 26, 43, 69 }}


POTE generator: ~8/7 = 231.293
Badness: 0.047914


Vals: {{Val list| 5e, 26, 57e, 83bce }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.034124
Comma list: 78/77, 81/80, 99/98, 144/143


=== Mosura ===
Mapping: [{{val| 1 1 0 2 3 3 }}, {{val| 0 5 20 7 4 6 }}]
Subgroup: 2.3.5.7.11


Comma list: 81/80, 176/175, 540/539
Mapping generators: ~2, ~12/11


Mapping: [{{val| 1 1 0 3 -1 }}, {{val| 0 3 12 -1 23 }}]
POTE generator: ~12/11 = 139.387


Mapping generators: ~2, ~8/7
Optimal GPV sequence: {{Val list| 17c, 26, 43, 69 }}


POTE generator: ~8/7 = 232.419
Badness: 0.029285


Vals: {{Val list| 31, 129, 160be, 191bce, 222bce, 253bcee }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0313
Comma list: 78/77, 81/80, 99/98, 144/143, 189/187


==== 13-limit ====
Mapping: [{{val| 1 1 0 2 3 3 2 }}, {{val| 0 5 20 7 4 6 18 }}]
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 144/143, 176/175, 196/195
Mapping generators: ~2, ~12/11


Mapping: [{{val| 1 1 0 3 -1 7 }}, {{val| 0 3 12 -1 23 -17 }}]
POTE generator: ~12/11 = 139.362


Mapping generators: ~2, ~8/7
Optimal GPV sequence: {{Val list| 17cg, 26, 43, 69 }}


POTE generator: ~8/7 = 232.640
Badness: 0.020878


Vals: {{Val list| 31, 36, 67, 98 }}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.036857
Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143


== Squares ==
Mapping: [{{val| 1 1 0 2 3 3 2 1 }}, {{val| 0 5 20 7 4 6 18 28 }}]
{{Main| Squares }}


Squares splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[31edo|31EDO]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].
Mapping generators: ~2, ~12/11


Subgroup: 2.3.5.7
POTE generator: ~12/11 = 139.313


[[Comma list]]: 81/80, 2401/2400
Optimal GPV sequence: {{Val list| 17cgh, 26, 43, 69 }}


[[Mapping]]: [{{val| 1 3 8 6 }}, {{val| 0 -4 -16 -9 }}]
Badness: 0.018229


Mapping generators: ~2, ~9/7
== Injera ==
Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo|38EDO]], which is two parallel [[19edo|19EDOs]], is an excellent tuning for injera.


{{Multival|legend=1| 4 16 9 16 3 -24 }}
[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]


[[POTE generator]]: ~9/7 = 425.942
Subgroup: 2.3.5.7


[[Minimax tuning]]:  
[[Comma list]]: 50/49, 81/80
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3/2 0 9/16 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5


[[Algebraic generator]]: Sceptre2, the positive root of 9''x''<sup>2</sup> + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
[[Mapping]]: [{{val| 2 0 -8 -7 }}, {{val| 0 1 4 4 }}]


{{Val list|legend=1| 14c, 17c, 31 }}
Mapping generators: ~7/5, ~3


[[Badness]]: 0.045993
{{Multival|legend=1| 2 8 8 8 7 -4 }}


Scales: [[skwares8]], [[skwares11]], [[skwares14]]
[[POTE generator]]: ~3/2 = 694.375


=== 11-limit ===
[[Tuning ranges]]:
Subgroup: 2.3.5.7.11
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [688.957, 700.000]
* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [685.714, 700.000]


Comma list: 81/80, 99/98, 121/120
{{Val list|legend=1| 12, 26, 38, 102bcd, 140bccd, 178bbccdd }}


Mapping: [{{val| 1 3 8 6 7 }}, {{val| 0 -4 -16 -9 -10 }}]
[[Badness]]: 0.031130


POTE generator: ~9/7 = 425.957
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 Two Pairs of Socks] (in [[26edo|26EDO]]) by [[Igliashon Jones]]


Vals: {{Val list| 14c, 17c, 31 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.021636
Comma list: 45/44, 50/49, 81/80


==== 13-limit ====
Mapping: [{{val| 2 0 -8 -7 -12 }}, {{val| 0 1 4 4 6 }}]
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 81/80, 99/98, 121/120
Mapping generators: ~7/5, ~3


Mapping: [{{val| 1 3 8 6 7 3 }}, {{val| 0 -4 -16 -9 -10 2 }}]
POTE generator: ~3/2 = 692.840


POTE generator: ~9/7 = 425.550
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [685.714, 700.000]


Vals: {{Val list| 14c, 17c, 31, 79cf }}
Optimal GPV sequence: {{Val list| 12, 14c, 26, 90bce, 116bcce }}


Badness: 0.025514
Badness: 0.023124


==== Squad ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 81/80, 91/90, 99/98
Comma list: 45/44, 50/49, 78/77, 81/80


Mapping: [{{val| 1 3 8 6 7 9 }}, {{val| 0 -4 -16 -9 -10 -15 }}]
Mapping: [{{val| 2 0 -8 -7 -12 -21 }}, {{val| 0 1 4 4 6 9 }}]


POTE generator: ~9/7 = 425.7516
Mapping generators: ~7/5, ~3


Vals: {{Val list| 14cf, 17c, 31f }}
POTE generator: ~3/2 = 692.673


Badness: 0.026877
Tuning ranges:  
* 13- and 15-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = 692.308


==== Agora ====
Optimal GPV sequence: {{Val list| 12f, 14cf, 26, 38e }}
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 99/98, 105/104, 121/120
Badness: 0.021565
 
Mapping: [{{val| 1 3 8 6 7 14 }}, {{val| 0 -4 -16 -9 -10 -29 }}]
 
POTE generator: ~9/7 = 426.276
 
Vals: {{Val list| 14cf, 31, 45ef, 76e }}
 
Badness: 0.024522


===== 17-limit =====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 81/80, 99/98, 105/104, 120/119, 121/119
Comma list: 45/44, 50/49, 78/77, 81/80, 85/84


Mapping: [{{val| 1 3 8 6 7 14 8 }}, {{val| 0 -4 -16 -9 -10 -29 -11 }}]
Mapping: [{{val| 2 0 -8 -7 -12 -21 5 }}, {{val| 0 1 4 4 6 9 1 }}]


POTE generator: ~9/7 = 426.187
POTE generator: ~3/2 = 692.487


Vals: {{Val list| 14cf, 31, 76e }}
Optimal GPV sequence: {{Val list| 12f, 14cf, 26 }}


Badness: 0.022573
Badness: 0.018358


===== 19-limit =====
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84


Mapping: [{{val| 1 3 8 6 7 14 8 11 }}, {{val| 0 -4 -16 -9 -10 -29 -11 -19 }}]
Mapping: [{{val| 2 0 -8 -7 -12 -21 5 -1 }}, {{val| 0 1 4 4 6 9 1 3 }}]


POTE generator: ~9/7 = 426.225
POTE generator: ~3/2 = 692.299


Vals: {{Val list| 14cf, 31, 76e }}
Optimal GPV sequence: {{Val list| 12f, 14cf, 26 }}


Badness: 0.018839
Badness: 0.015118


=== Cuboctahedra ===
==== Enjera ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 385/384, 1375/1372
Comma list: 27/26, 40/39, 45/44, 50/49


Mapping: [{{val| 1 3 8 6 -4 }}, {{val| 0 -4 -16 -9 21 }}]
Mapping: [{{val| 2 0 -8 -7 -12 -2 }}, {{val| 0 1 4 4 6 3 }}]


POTE generator: ~9/7 = 425.993
Mapping generators: ~7/5, ~3


Vals: {{Val list| 14ce, 17ce, 31, 107b, 138b, 169be, 200be }}
POTE generator: ~3/2 = 694.121


Badness: 0.056826
Optimal GPV sequence: {{Val list| 12f, 14c, 26f, 38eff }}


== Liese ==
Badness: 0.026542
<span style="display: block; text-align: right;">[[:de:Liese|Deutsch]]</span>


Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo|74EDO]] makes for a good liese tuning, though [[19edo|19EDO]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.
=== Injerous ===
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5.7
Comma list: 33/32, 50/49, 55/54


[[Comma list]]: 81/80, 686/675
Mapping: [{{val| 2 0 -8 -7 10 }}, {{val| 0 1 4 4 -1 }}]


[[Mapping]]: [{{val| 1 0 -4 -3 }}, {{val| 0 3 12 11 }}]
Mapping generators: ~7/5, ~3


Mapping generators: ~2, ~10/7
POTE generator: ~3/2 = 690.548


{{Multival|legend=1| 3 12 11 12 9 -8 }}
Optimal GPV sequence: {{Val list| 12e, 14c, 26e, 40cee }}


[[POTE generator]]: ~10/7 = 632.406
Badness: 0.038577


Minimax tuning:
=== Lahoh ===
* 7- and 9-odd-limit: ~10/7 = {{Monzo| 1/3 0 1/12 }}
Subgroup: 2.3.5.7.11
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| 2/3 0 11/12 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 5


[[Algebraic generator]]: Radix, the real root of ''x''<sup>5</sup> - 2''x''<sup>4</sup> + 2''x''<sup>3</sup> - 2''x''<sup>2</sup> + 2''x'' - 2, also a root of ''x''<sup>6</sup> - ''x''<sup>5</sup> - 2. The recurrence converges.
Comma list: 50/49, 56/55, 81/77


{{Val list|legend=1| 17c, 19, 55, 74d }}
Mapping: [{{val| 2 0 -8 -7 7 }}, {{val| 0 1 4 4 0 }}]


[[Badness]]: 0.046706
Mapping generators: ~7/5, ~3


=== Liesel ===
POTE generator: ~3/2 = 699.001
Subgroup: 2.3.5.7.11


Comma list: 56/55, 81/80, 540/539
Optimal GPV sequence: {{Val list| 2cd, 10cd, 12 }}


Mapping: [{{val| 1 0 -4 -3 4 }}, {{val| 0 3 12 11 -1 }}]
Badness: 0.043062


Mapping generators: ~2, ~10/7
=== Teff ===
{{Main| Teff }}


POTE generator: ~10/7 = 633.073
Teff (found by Mason Green) is to injera what mohajira is to meantone; it splits the generator in half in order to accommodate higher limit intervals, creating a half-octave quarter-tone temperament.


Vals: {{Val list| 17c, 19, 36, 91cee }}
Subgroup: 2.3.5.7.11


Badness: 0.040721
Comma list: 50/49, 81/80, 864/847


==== 13-limit ====
Mapping: [{{val| 2 1 -4 -3 8 }}, {{val| 0 2 8 8 -1 }}]
Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.


Subgroup: 2.3.5.7.11.13
Mapping generators: ~7/5, ~16/11


Comma list: 56/55, 78/77, 81/80, 91/90
POTE generator: ~11/8 = 552.5303


Mapping: [{{val| 1 0 -4 -3 4 0 }}, {{val| 0 3 12 11 -1 7 }}]
Optimal GPV sequence: {{Val list| 24d, 26, 50d }}


Mapping generators: ~2, ~10/7
Badness: 0.070689


POTE generator: ~10/7 = 633.042
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 17c, 19, 36, 91ceef }}
Comma list: 50/49, 78/77, 81/80, 144/143


Badness: 0.027304
Mapping: [{{val| 2 1 -4 -3 8 2 }}, {{val| 0 2 8 8 -1 5 }}]


=== Elisa ===
POTE generator: ~11/8 = 552.5324
Subgroup: 2.3.5.7.11


Comma list: 77/75, 81/80, 99/98
Optimal GPV sequence: {{Val list| 24d, 26, 50d }}


Mapping: [{{val| 1 0 -4 -3 -5 }}, {{val| 0 3 12 11 16 }}]
Badness: 0.040047


Mapping generators: ~2, ~10/7
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~10/7 = 633.061
Comma list: 50/49, 78/77, 81/80, 85/84, 144/143


Vals: {{Val list| 17c, 19e, 36e }}
Mapping: [{{val| 2 1 -4 -3 8 2 6 }}, {{val| 0 2 8 8 -1 5 2 }}]


Badness: 0.041592
POTE generator: ~11/8 = 552.6558


==== 13-limit ====
Optimal GPV sequence: {{Val list| 24d, 26 }}
Subgroup: 2.3.5.7.11


Comma list: 66/65, 77/75, 81/80, 99/98
Badness: 0.029499


Mapping: [{{val| 1 0 -4 -3 -5 0 }}, {{val| 0 3 12 11 16 7 }}]
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Mapping generators: ~2, ~10/7
Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143


POTE generator: ~10/7 = 632.991
Mapping: [{{val| 2 1 -4 -3 8 2 6 2 }}, {{val| 0 2 8 8 -1 5 2 6 }}]


Vals: {{Val list| 17c, 19e, 36e }}
POTE generator: ~11/8 = 552.6382


Badness: 0.026922
Optimal GPV sequence: {{Val list| 24d, 26 }}


=== Lisa ===
Badness: 0.023133
Subgroup: 2.3.5.7.11


Comma list: 45/44, 81/80, 343/330
== Pombe ==
Pombe (named after the African millet beer) is a variant of [[#Teff]] by Kaiveran Lugheidh that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.


Mapping: [{{val| 1 0 -4 -3 -6 }}, {{val| 0 3 12 11 18 }}]
Subgroup: 2.3.5.7


Mapping generators: ~2, ~10/7
[[Comma list]]: 81/80, 300125/294912


POTE generator: ~10/7 = 631.370
[[Mapping]]: [{{val| 2 1 -4 11 }}, {{val| 0 2 8 -5 }}]


Vals: {{Val list| 17cee, 19 }}
Mapping generators: ~735/512, ~35/24


Badness: 0.054829
{{Multival|legend=1| 4 16 -10 16 -27 -68 }}


==== 13-limit ====
[[POTE generator]]: ~48/35 = 552.2206
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 81/80, 91/88, 147/143
{{Val list|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}


Mapping: [{{val| 1 0 -4 -3 -6 0 }}, {{val| 0 3 12 11 18 7 }}]
[[Badness]]: 0.116104


Mapping generators: ~2, ~10/7
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~10/7 = 631.221
Comma list: 81/80, 245/242, 385/384


Vals: {{Val list| 17cee, 19 }}
Mapping: [{{val| 2 1 -4 11 8 }}, {{val| 0 2 8 -5 -1 }}]


Badness: 0.036144
POTE generator: ~11/8 = 552.0929


== Jerome ==
Optimal GPV sequence: {{Val list| 24, 26, 50 }}
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.


Subgroup: 2.3.5.7
Badness: 0.052099


[[Comma list]]: 81/80, 17280/16807
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val| 1 1 0 2 }}, {{val| 0 5 20 7 }}]
Comma list: 81/80, 105/104, 144/143, 245/242


Mapping generators: ~2, ~54/49
Mapping: [{{val| 2 1 -4 11 8 2 }}, {{val| 0 2 8 -5 -1 5 }}]


{{Multival|legend=1| 5 20 7 20 -3 -40 }}
POTE generator: ~11/8 = 552.1498


[[POTE generator]]: ~54/49 = 139.343
Optimal GPV sequence: {{Val list| 24, 26, 50 }}


{{Val list|legend=1| 17c, 26, 43, 69, 112bd }}
Badness: 0.031039


[[Badness]]: 0.108656
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


=== 11-limit ===
Comma list: 81/80, 105/104, 144/143, 245/242, 273/272
Subgroup: 2.3.5.7.11


Comma list: 81/80, 99/98, 864/847
Mapping: [{{val| 2 1 -4 11 8 2 6 }}, {{val| 0 2 8 -5 -1 5 2 }}]


Mapping: [{{val| 1 1 0 2 3 }}, {{val| 0 5 20 7 4 }}]
POTE generator: ~11/8 = 552.1579


Mapping generators: ~2, ~12/11
Optimal GPV sequence: {{Val list| 24, 26, 50 }}


POTE generator: ~12/11 = 139.428
Badness: 0.021260


Vals: {{Val list| 17c, 26, 43, 69 }}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.047914
Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209


=== 13-limit ===
Mapping: [{{val| 2 1 -4 11 8 2 6 2 }}, {{val| 0 2 8 -5 -1 5 2 6 }}]
Subgroup: 2.3.5.7.11.13
 
POTE generator: ~11/8 = 552.1196
 
Optimal GPV sequence: {{Val list| 24, 26, 50 }}
 
Badness: 0.016548
 
== Orphic ==
Subgroup: 2.3.5.7
 
[[Comma list]]: 81/80, 5898240/5764801


Comma list: 78/77, 81/80, 99/98, 144/143
[[Mapping]]: [{{val| 2 5 12 7 }}, {{val| 0 -4 -16 -3 }}]


Mapping: [{{val| 1 1 0 2 3 3 }}, {{val| 0 5 20 7 4 6 }}]
Mapping generators: ~2401/1728, ~7/6


Mapping generators: ~2, ~12/11
{{Multival|legend=1| 8 32 6 32 -13 -76 }}


POTE generator: ~12/11 = 139.387
[[POTE generator]]: ~7/6 = 275.794


Vals: {{Val list| 17c, 26, 43, 69 }}
{{Val list|legend=1| 26, 48c, 74, 174bd, 248bbd }}


Badness: 0.029285
[[Badness]]: 0.258825


=== 17-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11


Comma list: 78/77, 81/80, 99/98, 144/143, 189/187
Comma list: 81/80, 99/98, 73728/73205


Mapping: [{{val| 1 1 0 2 3 3 2 }}, {{val| 0 5 20 7 4 6 18 }}]
Mapping: [{{val| 2 5 12 7 6 }}, {{val| 0 -4 -16 -3 2 }}]


Mapping generators: ~2, ~12/11
Mapping generators: ~363/256, ~7/6


POTE generator: ~12/11 = 139.362
POTE generator: ~7/6 = 275.762


Vals: {{Val list| 17cg, 26, 43, 69 }}
Optimal GPV sequence: {{Val list| 26, 48c, 74, 248bbd, 322bbdd }}


Badness: 0.020878
Badness: 0.101499


=== 19-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143
Comma list: 81/80, 99/98, 144/143, 2200/2197


Mapping: [{{val| 1 1 0 2 3 3 2 1 }}, {{val| 0 5 20 7 4 6 18 28 }}]
Mapping: [{{val| 2 5 12 7 6 12 }}, {{val| 0 -4 -16 -3 2 -10 }}]


Mapping generators: ~2, ~12/11
Mapping generators: ~55/39, ~7/6


POTE generator: ~12/11 = 139.313
POTE generator: ~7/6 = 275.774


Vals: {{Val list| 17cgh, 26, 43, 69 }}
Optimal GPV sequence: {{Val list| 26, 48c, 74, 174bd, 248bbd, 322bbdd }}


Badness: 0.018229
Badness: 0.053482


== Cloudtone ==
== Cloudtone ==
Line 2,114: Line 2,117:
POTE generator: ~3/2 = 696.536
POTE generator: ~3/2 = 696.536


Vals: {{Val list| 5, 45, 50, 155bdd, 205bddd }}
Optimal GPV sequence: {{Val list| 5, 45, 50, 155bdd, 205bddd }}


Badness: 0.070378
Badness: 0.070378
Line 2,129: Line 2,132:
POTE generator: ~3/2 = 696.162
POTE generator: ~3/2 = 696.162


Vals: {{Val list| 5, 45f, 50 }}
Optimal GPV sequence: {{Val list| 5, 45f, 50 }}


Badness: 0.048829
Badness: 0.048829
Line 2,161: Line 2,164:
POTE generator: ~8/7 = 233.486
POTE generator: ~8/7 = 233.486


Vals: {{Val list| 19, 38, 57, 76 }}
Optimal GPV sequence: {{Val list| 19, 38, 57, 76 }}


Badness: 0.066829
Badness: 0.066829
Line 2,176: Line 2,179:
POTE generator: ~8/7 = 234.890
POTE generator: ~8/7 = 234.890


Vals: {{Val list| 19, 38, 57, 76 }}
Optimal GPV sequence: {{Val list| 19, 38, 57, 76 }}


Badness: 0.045844
Badness: 0.045844
Line 2,206: Line 2,209:
POTE generator: ~11/8 = 537.061
POTE generator: ~11/8 = 537.061


Vals: {{Val list| 19, 38df }}
Optimal GPV sequence: {{Val list| 19, 38df }}


Badness: 0.022933
Badness: 0.022933
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