Kleismic family: Difference between revisions

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[[de:Hanson_Kleismisch]]</span>
[[de:Hanson_Kleismisch]]</span>


The [[5-limit]] parent comma for the '''kleismic family''' is 15625/15552, the kleisma. Its monzo is |-6 -5 6&gt;, and flipping that yields &lt;&lt;6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)^5 = 5/2 * 15625/15552. This 5-limit temperament (virtually a microtemperament) is commonly called '''Hanson''', and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].
The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma. Its monzo is {{monzo| -6 -5 6 }}, and flipping that yields &lt;&lt;6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)<sup>5</sup> = 5/2 × 15625/15552. This 5-limit temperament (virtually a [[microtemperament]]) is commonly called '''Hanson''', and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].


[[Tuning Ranges of Regular Temperaments|valid range]]: [300.000, 327.273] (4 to 11b)
[[Tuning Ranges of Regular Temperaments|valid range]]: [300.000, 327.273] (4 to 11b)
Line 14: Line 14:
Map: [&lt;1 0 1|, &lt;0 6 5|]
Map: [&lt;1 0 1|, &lt;0 6 5|]


EDOs: {{EDOs|15, 19, 34, 53, 458, 882c}}
EDOs: {{EDOs| 15, 19, 34, 53, 458, 882c }}


Music:
Music:
Line 44: Line 44:
[[Wedgie]]: &lt;&lt;6 5 3 -6 -12 -7||
[[Wedgie]]: &lt;&lt;6 5 3 -6 -12 -7||


EDOs: {{EDOs|15, 19, 53d, 72dd, 91dd}}
EDOs: {{EDOs| 15, 19, 53d, 72dd, 91dd }}


[[Badness]]: 0.0274
[[Badness]]: 0.0274
Line 61: Line 61:
Map: [&lt;1 0 1 2 4|, &lt;0 6 5 3 -2|]
Map: [&lt;1 0 1 2 4|, &lt;0 6 5 3 -2|]


EDOs: {{EDOs|4, 15, 19, 34}}
EDOs: {{EDOs| 4, 15, 19, 34 }}


Badness: 0.0274
Badness: 0.0274
Line 78: Line 78:
Map: [&lt;1 0 1 2 4 5|, &lt;0 6 5 3 -2 -5|]
Map: [&lt;1 0 1 2 4 5|, &lt;0 6 5 3 -2 -5|]


EDOs: {{EDOs|4, 15f, 19, 53def, 72def}}
EDOs: {{EDOs| 4, 15f, 19, 53def, 72def }}


Badness: 0.0297
Badness: 0.0297
Line 95: Line 95:
Map: [&lt;1 0 1 2 4 0|, &lt;0 6 5 3 -2 14|]
Map: [&lt;1 0 1 2 4 0|, &lt;0 6 5 3 -2 14|]


EDOs: {{EDOs|15, 19, 34, 87ddee}}
EDOs: {{EDOs| 15, 19, 34, 87ddee }}


Badness: 0.0227
Badness: 0.0227
Line 106: Line 106:
Map: [&lt;1 0 1 2 4 4|, &lt;0 6 5 3 -2 -1|]
Map: [&lt;1 0 1 2 4 4|, &lt;0 6 5 3 -2 -1|]


EDOs: {{EDOs|4, 15, 19f, 34ff}}
EDOs: {{EDOs| 4, 15, 19f, 34ff }}


Badness: 0.0316
Badness: 0.0316
Line 117: Line 117:
Map: [&lt;1 0 1 2 -1|, &lt;0 6 5 3 17|]
Map: [&lt;1 0 1 2 -1|, &lt;0 6 5 3 17|]


EDOs: {{EDOs|4e, 19, 42bcd, 61bcdd}}
EDOs: {{EDOs| 4e, 19, 42bcd, 61bcdd }}


Badness: 0.0401
Badness: 0.0401
Line 128: Line 128:
Map: [&lt;1 0 1 2 -1 0|, &lt;0 6 5 3 17 14|]
Map: [&lt;1 0 1 2 -1 0|, &lt;0 6 5 3 17 14|]


EDOs: {{EDOs|4ef, 19}}
EDOs: {{EDOs| 4ef, 19 }}


Badness: 0.0294
Badness: 0.0294
Line 139: Line 139:
Map: [&lt;1 0 1 2 0|, &lt;0 6 5 3 13|]
Map: [&lt;1 0 1 2 0|, &lt;0 6 5 3 13|]


EDOs: {{EDOs|15, 19e, 34e}}
EDOs: {{EDOs| 15, 19e, 34e }}


Badness: 0.0276
Badness: 0.0276
Line 150: Line 150:
Map: [&lt;1 0 1 2 0 0|, &lt;0 6 5 3 13 14|]
Map: [&lt;1 0 1 2 0 0|, &lt;0 6 5 3 13 14|]


EDOs: {{EDOs|15, 19e, 34e, 53dee}}
EDOs: {{EDOs| 15, 19e, 34e, 53dee }}


Badness: 0.02144
Badness: 0.02144
Line 163: Line 163:
Wedgie: &lt;&lt;6 5 -12 -6 -36 -42||
Wedgie: &lt;&lt;6 5 -12 -6 -36 -42||


EDOs: {{EDOs|15, 34d, 49, 132bcd}}
EDOs: {{EDOs| 15, 34d, 49, 132bcd }}


Badness: 0.0949
Badness: 0.0949
Line 174: Line 174:
Map: [&lt;1 0 1 6 4|, &lt;0 6 5 -12 -2|]
Map: [&lt;1 0 1 6 4|, &lt;0 6 5 -12 -2|]


EDOs: {{EDOs|15, 34d, 49}}
EDOs: {{EDOs| 15, 34d, 49 }}


Badness: 0.0369
Badness: 0.0369
Line 193: Line 193:
[[Wedgie]]: &lt;&lt;6 5 22 -6 18 37||
[[Wedgie]]: &lt;&lt;6 5 22 -6 18 37||


EDOs: {{EDOs|19, 53, 72, 197, 269c}}
EDOs: {{EDOs| 19, 53, 72, 197, 269c }}


[[Badness]]: 0.0215
[[Badness]]: 0.0215
Line 210: Line 210:
Map: [&lt;1 0 1 -3 9|, &lt;0 6 5 22 -21|]
Map: [&lt;1 0 1 -3 9|, &lt;0 6 5 22 -21|]


EDOs: {{EDOs|19, 53, 72, 197e, 269ce, 341ce, 610bce}}
EDOs: {{EDOs| 19, 53, 72, 197e, 269ce, 341ce, 610bce }}


[[Badness]]: 0.0218
[[Badness]]: 0.0218
Line 227: Line 227:
Map: [&lt;1 0 1 -3 9 0|, &lt;0 6 5 22 -21 14|]
Map: [&lt;1 0 1 -3 9 0|, &lt;0 6 5 22 -21 14|]


EDOs: {{EDOs|19, 53, 72, 125f, 197ef, 269cef}}
EDOs: {{EDOs| 19, 53, 72, 125f, 197ef, 269cef }}


[[Badness]]: 0.0169
[[Badness]]: 0.0169
Line 238: Line 238:
Map: [&lt;1 0 1 -3 -5|, &lt;0 6 5 22 32|]
Map: [&lt;1 0 1 -3 -5|, &lt;0 6 5 22 32|]


EDOs: {{EDOs|53, 87d, 140d, 171de, 181de, 193de, 224de, 246de, 277de}}
EDOs: {{EDOs| 53, 87d, 140d, 171de, 181de, 193de, 224de, 246de, 277de }}


Badness: 0.0400
Badness: 0.0400
Line 249: Line 249:
Map: [&lt;1 0 1 -3 -5 0|, &lt;0 6 5 22 32 14|]
Map: [&lt;1 0 1 -3 -5 0|, &lt;0 6 5 22 32 14|]


EDOs: {{EDOs|53, 87d, 140d, 193de, 246de}}
EDOs: {{EDOs| 53, 87d, 140d, 193de, 246de }}


Badness: 0.0226
Badness: 0.0226
Line 260: Line 260:
Map: [&lt;1 0 1 -3 -10|, &lt;0 6 5 22 51|]
Map: [&lt;1 0 1 -3 -10|, &lt;0 6 5 22 51|]


EDOs: {{EDOs|53e, 72}}
EDOs: {{EDOs| 53e, 72 }}


Badness: 0.0304
Badness: 0.0304
Line 271: Line 271:
Map: [&lt;1 0 1 -3 -10 0|, &lt;0 6 5 22 51 14|]
Map: [&lt;1 0 1 -3 -10 0|, &lt;0 6 5 22 51 14|]


EDOs: {{EDOs|19e, 53e, 72}}
EDOs: {{EDOs| 19e, 53e, 72 }}


Badness: 0.0223
Badness: 0.0223
Line 282: Line 282:
Map: [&lt;1 0 1 -3 4|, &lt;0 6 5 22 -2|]
Map: [&lt;1 0 1 -3 4|, &lt;0 6 5 22 -2|]


EDOs: {{EDOs|19, 34d, 53e}}
EDOs: {{EDOs| 19, 34d, 53e }}


Badness: 0.0443
Badness: 0.0443
Line 293: Line 293:
Map: [&lt;1 0 1 -3 4 0|, &lt;0 6 5 22 -2 14|]
Map: [&lt;1 0 1 -3 4 0|, &lt;0 6 5 22 -2 14|]


EDOs: {{EDOs|19, 34d, 53e, 87de}}
EDOs: {{EDOs| 19, 34d, 53e, 87de }}


Badness: 0.0273
Badness: 0.0273
Line 306: Line 306:
[[Wedgie]]: &lt;&lt;6 5 -31 -6 -66 -86||
[[Wedgie]]: &lt;&lt;6 5 -31 -6 -66 -86||


EDOs: {{EDOs|19d, 34, 53, 87, 140, 333, 473, 806b}}
EDOs: {{EDOs| 19d, 34, 53, 87, 140, 333, 473, 806b }}


[[Badness]]: 0.0521
[[Badness]]: 0.0521
Line 317: Line 317:
Map: [&lt;1 0 1 11 -5|, &lt;0 6 5 -31 32|]
Map: [&lt;1 0 1 11 -5|, &lt;0 6 5 -31 32|]


EDOs: {{EDOs|34, 53, 87, 140, 227}}
EDOs: {{EDOs| 34, 53, 87, 140, 227 }}


[[Badness]]: 0.0398
[[Badness]]: 0.0398
Line 328: Line 328:
Map: [&lt;1 0 1 11 -5 0|, &lt;0 6 5 -31 32 14|]
Map: [&lt;1 0 1 11 -5 0|, &lt;0 6 5 -31 32 14|]


EDOs: {{EDOs|34, 53, 87, 140, 227, 367e, 507e}}
EDOs: {{EDOs| 34, 53, 87, 140, 227, 367e, 507e }}


[[Badness]]: 0.0202
[[Badness]]: 0.0202
Line 341: Line 341:
Wedgie: &lt;&lt;6 5 56 -6 72 116||
Wedgie: &lt;&lt;6 5 56 -6 72 116||


EDOs: {{EDOs|34d, 53d, 87, 121, 208}}
EDOs: {{EDOs| 34d, 53d, 87, 121, 208 }}


[[Badness]]: 0.1635
[[Badness]]: 0.1635
Line 352: Line 352:
Map: [&lt;1 0 1 -12 -5|, &lt;0 6 5 56 32|]
Map: [&lt;1 0 1 -12 -5|, &lt;0 6 5 56 32|]


EDOs: {{EDOs|34d, 53d, 87, 121, 208}}
EDOs: {{EDOs| 34d, 53d, 87, 121, 208 }}


[[Badness]]: 0.0486
[[Badness]]: 0.0486
Line 363: Line 363:
Map: [&lt;1 0 1 -12 -5 0|, &lt;0 6 5 56 32 14|]
Map: [&lt;1 0 1 -12 -5 0|, &lt;0 6 5 56 32 14|]


EDOs: {{EDOs|34d, 53d, 87, 121, 208}}
EDOs: {{EDOs| 34d, 53d, 87, 121, 208 }}


[[Badness]]: 0.0244
[[Badness]]: 0.0244
Line 374: Line 374:
Map: [&lt;1 0 1 4|, &lt;0 12 10 -9|]
Map: [&lt;1 0 1 4|, &lt;0 12 10 -9|]


EDOs: {{EDOs|15, 53, 121}}
EDOs: {{EDOs| 15, 53, 121 }}


Badness: 0.0521
Badness: 0.0521
Line 385: Line 385:
Map: [&lt;1 0 1 4 2|, &lt;0 12 10 -9 11|]
Map: [&lt;1 0 1 4 2|, &lt;0 12 10 -9 11|]


EDOs: {{EDOs|15, 38, 53, 68, 121e}}
EDOs: {{EDOs| 15, 38, 53, 68, 121e }}


Badness: 0.0380
Badness: 0.0380
Line 396: Line 396:
Map: [&lt;1 0 1 4 2 0|, &lt;0 12 10 -9 11 28|]
Map: [&lt;1 0 1 4 2 0|, &lt;0 12 10 -9 11 28|]


EDOs: {{EDOs|15, 53, 121e}}
EDOs: {{EDOs| 15, 53, 121e }}


Badness: 0.0260
Badness: 0.0260
Line 415: Line 415:
Map: [&lt;1 6 6 12|, &lt;0 -12 -10 -25|]
Map: [&lt;1 6 6 12|, &lt;0 -12 -10 -25|]


[[generator|Generators]]: 2, 9/7
[[Generator|Generators]]: 2, 9/7


[[EDO|Edos]]: {{EDOs|19, 49, 68, 87, 155}}
[[EDO|EDOs]]: {{EDOs| 19, 49, 68, 87, 155 }}


Badness: 0.0473
Badness: 0.0473
Line 428: Line 428:
Map: [&lt;1 6 6 12 -5|, &lt;0 -12 -10 -25 23|]
Map: [&lt;1 6 6 12 -5|, &lt;0 -12 -10 -25 23|]


EDOs: {{EDOs|19, 68, 87, 329bd, 419bd, 503bd, 590bd}}
EDOs: {{EDOs| 19, 68, 87, 329bd, 419bd, 503bd, 590bd }}


Badness: 0.0474
Badness: 0.0474
Line 439: Line 439:
Map: [&lt;1 6 6 12 -5 14|, &lt;0 -12 -10 -25 23 -28|]
Map: [&lt;1 6 6 12 -5 14|, &lt;0 -12 -10 -25 23 -28|]


EDOs: {{EDOs|19, 68, 87, 503bdf, 590bdf}}
EDOs: {{EDOs| 19, 68, 87, 503bdf, 590bdf }}


Badness: 0.0268
Badness: 0.0268
Line 450: Line 450:
Map: [&lt;2 0 2 -6 -1|, &lt;0 6 5 22 15|]
Map: [&lt;2 0 2 -6 -1|, &lt;0 6 5 22 15|]


EDOs: {{EDOs|72, 106, 178, 250, 322c, 394c, 466bc, 538bc, 610bc}}
EDOs: {{EDOs| 72, 106, 178, 250, 322c, 394c, 466bc, 538bc, 610bc }}


Badness: 0.0293
Badness: 0.0293
Line 461: Line 461:
Map: [&lt;2 0 2 -6 -1 0|, &lt;0 6 5 22 15 14|]
Map: [&lt;2 0 2 -6 -1 0|, &lt;0 6 5 22 15 14|]


EDOs: {{EDOs|72, 106, 322cff, 394cff, 466bcff, 538bcfff}}
EDOs: {{EDOs| 72, 106, 322cff, 394cff, 466bcff, 538bcfff }}


Badness: 0.0218
Badness: 0.0218
Line 476: Line 476:
[[Wedgie]]: &lt;&lt;18 15 -6 -18 -60 -56||
[[Wedgie]]: &lt;&lt;18 15 -6 -18 -60 -56||


EDOs: {{EDOs|15, 57, 72, 159, 231}}
EDOs: {{EDOs| 15, 57, 72, 159, 231 }}


[[Badness]]: 0.0563
[[Badness]]: 0.0563
Line 487: Line 487:
Map: [&lt;3 0 3 10 8|, &lt;0 6 5 -2 3|]
Map: [&lt;3 0 3 10 8|, &lt;0 6 5 -2 3|]


EDOs: {{EDOs|15, 57, 72, 159, 231}}
EDOs: {{EDOs| 15, 57, 72, 159, 231 }}


[[Badness]]: 0.0193
[[Badness]]: 0.0193
Line 498: Line 498:
Map: [&lt;3 0 3 10 8 0|, &lt;0 6 5 -2 3 14|]
Map: [&lt;3 0 3 10 8 0|, &lt;0 6 5 -2 3 14|]


EDOs: {{EDOs|15, 72, 87, 159, 867, 1026}}
EDOs: {{EDOs| 15, 72, 87, 159, 867, 1026 }}


[[Badness]]: 0.0157
[[Badness]]: 0.0157
Line 511: Line 511:
Wedgie: &lt;&lt;24 20 16 -24 -42 -19||
Wedgie: &lt;&lt;24 20 16 -24 -42 -19||


EDOs: {{EDOs|68, 72, 140, 212, 1200}}
EDOs: {{EDOs| 68, 72, 140, 212, 1200 }}


[[Badness]]: 0.0392
[[Badness]]: 0.0392
Line 522: Line 522:
Map: [&lt;4 0 4 7 17|, &lt;0 6 5 4 -3|]
Map: [&lt;4 0 4 7 17|, &lt;0 6 5 4 -3|]


EDOs: {{EDOs|68, 72, 140, 212, 284, 496, 780}}
EDOs: {{EDOs| 68, 72, 140, 212, 284, 496, 780 }}


[[Badness]]: 0.0234
[[Badness]]: 0.0234
Line 533: Line 533:
Map: [&lt;4 0 4 7 17 0|, &lt;0 6 5 4 -3 14|]
Map: [&lt;4 0 4 7 17 0|, &lt;0 6 5 4 -3 14|]


EDOs: {{EDOs|68, 72, 140, 212}}
EDOs: {{EDOs| 68, 72, 140, 212 }}


[[Badness]]: 0.0187
[[Badness]]: 0.0187
Line 546: Line 546:
Wedgie: &lt;&lt;18 15 13 -18 -30 -12||
Wedgie: &lt;&lt;18 15 13 -18 -30 -12||


EDOs: {{EDOs|49, 53, 314d}}
EDOs: {{EDOs| 49, 53, 314d }}


Badness: 0.0765
Badness: 0.0765
Line 557: Line 557:
Map: [&lt;1 6 6 6 14|, &lt;0 -18 -15 -13 -43|]
Map: [&lt;1 6 6 6 14|, &lt;0 -18 -15 -13 -43|]


EDOs: {{EDOs|49, 53, 102d, 155d}}
EDOs: {{EDOs| 49, 53, 102d, 155d }}


Badness: 0.0528
Badness: 0.0528
Line 568: Line 568:
Map: [&lt;1 6 6 6 14 14|, &lt;0 -18 -15 -13 -43 -42|]
Map: [&lt;1 6 6 6 14 14|, &lt;0 -18 -15 -13 -43 -42|]


EDOs: {{EDOs|53, 102df, 155d}}
EDOs: {{EDOs| 53, 102df, 155d }}


Badness: 0.0311
Badness: 0.0311
Line 581: Line 581:
Wedgie: &lt;&lt;54 45 54 -54 -66 -1||
Wedgie: &lt;&lt;54 45 54 -54 -66 -1||


EDOs: {{EDOs|72, 261, 333, 405, 477c, 882c}}
EDOs: {{EDOs| 72, 261, 333, 405, 477c, 882c }}


Badness: 0.1934
Badness: 0.1934
Line 592: Line 592:
Map: [&lt;9 0 9 11 24|, &lt;0 6 5 6 3|]
Map: [&lt;9 0 9 11 24|, &lt;0 6 5 6 3|]


EDOs: {{EDOs|72, 261, 333, 405, 882c}}
EDOs: {{EDOs| 72, 261, 333, 405, 882c }}


Badness: 0.05172
Badness: 0.05172
Line 603: Line 603:
Map: [&lt;9 0 9 11 24 0|, &lt;0 6 5 6 3 14|]
Map: [&lt;9 0 9 11 24 0|, &lt;0 6 5 6 3 14|]


EDOs: {{EDOs|72, 261, 333, 738cf, 1071bcf}}
EDOs: {{EDOs| 72, 261, 333, 738cf, 1071bcf }}


Badness: 0.0391
Badness: 0.0391
Line 616: Line 616:
Map: [&lt;1 12 11 16|, &lt;0 -30 -25 -38|]
Map: [&lt;1 12 11 16|, &lt;0 -30 -25 -38|]


EDOs: {{EDOs|49, 72, 193, 265}}
EDOs: {{EDOs| 49, 72, 193, 265 }}


Badness: 0.0704
Badness: 0.0704
Line 627: Line 627:
Map: [&lt;1 12 11 16 17|, &lt;0 -30 -25 -38 -39|]
Map: [&lt;1 12 11 16 17|, &lt;0 -30 -25 -38 -39|]


EDOs: {{EDOs|49, 72, 193, 265}}
EDOs: {{EDOs| 49, 72, 193, 265 }}


Badness: 0.0255
Badness: 0.0255
Line 638: Line 638:
Map: [&lt;1 12 11 16 17 28|, &lt;0 -30 -25 -38 -39 -70|]
Map: [&lt;1 12 11 16 17 28|, &lt;0 -30 -25 -38 -39 -70|]


EDOs: {{EDOs|72, 121, 193}}
EDOs: {{EDOs| 72, 121, 193 }}


Badness: 0.0200
Badness: 0.0200
Line 649: Line 649:
Map: [&lt;1 12 11 16 17 28 27|, &lt;0 -30 -25 -38 -39 -70 -66|]
Map: [&lt;1 12 11 16 17 28 27|, &lt;0 -30 -25 -38 -39 -70 -66|]


EDOs: {{EDOs|72, 121, 193}}
EDOs: {{EDOs| 72, 121, 193 }}


Badness: 0.0130
Badness: 0.0130
Line 660: Line 660:
Map: [&lt;1 12 11 16 17 28 27 -2|, &lt;0 -30 -25 -38 -39 -70 -66 18|]
Map: [&lt;1 12 11 16 17 28 27 -2|, &lt;0 -30 -25 -38 -39 -70 -66 18|]


EDOs: {{EDOs|72, 121, 193}}
EDOs: {{EDOs| 72, 121, 193 }}


Badness: 0.0147
Badness: 0.0147

Revision as of 12:17, 11 November 2020

The 5-limit parent comma for the kleismic family is 15625/15552, the kleisma. Its monzo is [-6 -5 6⟩, and flipping that yields <<6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)5 = 5/2 × 15625/15552. This 5-limit temperament (virtually a microtemperament) is commonly called Hanson, and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include 72edo, 87edo and 140edo.

valid range: [300.000, 327.273] (4 to 11b)

nice range: [315.641, 317.263]

strict range: [315.641, 317.263]

POTE generator: 317.007

Map: [<1 0 1|, <0 6 5|]

EDOs: 15, 19, 34, 53, 458, 882c

Music:

Hanson Improv by Petr Parizek

In Hanson11 by Chris Vaisvil

Analysis and diagrams:

11 note chain-of-minor-thirds scale, by David Keenan

Seven limit children

The second comma of the normal comma list defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives keemun, 4375/4374, the ragisma, gives catakleismic, 5120/5103, hemifamity, gives countercata, 6144/6125, the porwell comma, gives hemikleismic, 245/243, sensamagic, gives clyde, 1029/1024, the gamelisma, gives tritikleismic, and 2401/2400, the breedsma, gives quadritikleismic. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3 octave period with minor third generator, and quadritikleismic a 1/4 octave period with the minor third generator.

Keemun

Commas: 49/48, 126/125

valid range: [300.000, 327.273] (4 to 11b)

nice range: [308.744, 322.942]

strict range: [308.744, 322.942]

POTE generator: ~6/5 = 316.473

Map: [<1 0 1 2|, <0 6 5 3|]

Wedgie: <<6 5 3 -6 -12 -7||

EDOs: 15, 19, 53d, 72dd, 91dd

Badness: 0.0274

11-limit

Commas: 49/48, 56/55, 100/99

valid range: [315.789, 320.000] (19 to 15)

nice range: [308.744, 324.341]

strict range: [315.789, 320.000]

POTE generator: ~6/5 = 317.576

Map: [<1 0 1 2 4|, <0 6 5 3 -2|]

EDOs: 4, 15, 19, 34

Badness: 0.0274

13-limit

Commas: 49/48, 56/55, 78/77, 100/99

valid range: 315.789 (19)

nice range: [303.597, 324.341]

strict range: 315.789

POTE generator: ~6/5 = 316.611

Map: [<1 0 1 2 4 5|, <0 6 5 3 -2 -5|]

EDOs: 4, 15f, 19, 53def, 72def

Badness: 0.0297

Kema

Commas: 49/48, 56/55, 91/90, 100/99

valid range: [315.789, 320.000] (19 to 15)

nice range: [308.744, 324.341]

strict range: [315.789, 320.000]

POTE generator: ~6/5 = 317.423

Map: [<1 0 1 2 4 0|, <0 6 5 3 -2 14|]

EDOs: 15, 19, 34, 87ddee

Badness: 0.0227

Kumbaya

Commas: 40/39, 49/48, 56/55, 66/65

POTE generator: ~6/5 = 318.595

Map: [<1 0 1 2 4 4|, <0 6 5 3 -2 -1|]

EDOs: 4, 15, 19f, 34ff

Badness: 0.0316

Qeema

Commas: 45/44, 49/48, 126/125

POTE generator: ~6/5 = 314.730

Map: [<1 0 1 2 -1|, <0 6 5 3 17|]

EDOs: 4e, 19, 42bcd, 61bcdd

Badness: 0.0401

13-limit

Commas: 45/44, 49/48, 78/77, 126/125

POTE generator: ~6/5 = 315.044

Map: [<1 0 1 2 -1 0|, <0 6 5 3 17 14|]

EDOs: 4ef, 19

Badness: 0.0294

Darjeeling

Commas: 49/48, 55/54, 77/75

POTE generator: ~6/5 = 317.656

Map: [<1 0 1 2 0|, <0 6 5 3 13|]

EDOs: 15, 19e, 34e

Badness: 0.0276

13-limit

Commas: 49/48, 55/54, 66/65, 77/75

POTE generator: ~6/5 = 317.298

Map: [<1 0 1 2 0 0|, <0 6 5 3 13 14|]

EDOs: 15, 19e, 34e, 53dee

Badness: 0.02144

Catalan

Commas: 64/63, 15625/15552

POTE generator: ~6/5 = 318.267

Map: [<1 0 1 6|, <0 6 5 -12|]

Wedgie: <<6 5 -12 -6 -36 -42||

EDOs: 15, 34d, 49, 132bcd

Badness: 0.0949

11-limit

Commas: 64/63, 100/99, 1331/1323

POTE generator: ~6/5 = 318.282

Map: [<1 0 1 6 4|, <0 6 5 -12 -2|]

EDOs: 15, 34d, 49

Badness: 0.0369

Catakleismic

Commas: 225/224, 4375/4374

valid range: [315.789, 317.647] (19 to 34)

nice range: [315.641, 317.263]

strict range: [315.789, 317.263]

POTE generator: 316.732

Map: [<1 0 1 -3|, <0 6 5 22|]

Wedgie: <<6 5 22 -6 18 37||

EDOs: 19, 53, 72, 197, 269c

Badness: 0.0215

11-limit

Commas: 225/224, 385/384, 4375/4374

valid range: [315.789, 316.981] (19 to 53)

nice range: [315.641, 317.263]

strict range: [315.789, 316.981]

POTE generator: 316.719

Map: [<1 0 1 -3 9|, <0 6 5 22 -21|]

EDOs: 19, 53, 72, 197e, 269ce, 341ce, 610bce

Badness: 0.0218

13-limit

Commas: 169/168, 225/224, 325/324, 540/539

valid range: [315.789, 316.981] (19 to 53)

nice range: [315.641, 318.309]

strict range: [315.789, 316.981]

POTE generator: 316.738

Map: [<1 0 1 -3 9 0|, <0 6 5 22 -21 14|]

EDOs: 19, 53, 72, 125f, 197ef, 269cef

Badness: 0.0169

Cataclysmic

Commas: 99/98, 176/175, 2200/2187

POTE generator: ~6/5 = 317.042

Map: [<1 0 1 -3 -5|, <0 6 5 22 32|]

EDOs: 53, 87d, 140d, 171de, 181de, 193de, 224de, 246de, 277de

Badness: 0.0400

13-limit

Commas: 99/98, 169/168, 176/175, 275/273

POTE generator: ~6/5 = 317.036

Map: [<1 0 1 -3 -5 0|, <0 6 5 22 32 14|]

EDOs: 53, 87d, 140d, 193de, 246de

Badness: 0.0226

Catalytic

Commas: 225/224, 441/440, 4375/4374

POTE generator: ~6/5 = 316.653

Map: [<1 0 1 -3 -10|, <0 6 5 22 51|]

EDOs: 53e, 72

Badness: 0.0304

13-limit

Commas: 169/168 225/224 325/324 1716/1715

POTE generator: ~6/5 = 316.639

Map: [<1 0 1 -3 -10 0|, <0 6 5 22 51 14|]

EDOs: 19e, 53e, 72

Badness: 0.0223

Cataleptic

Commas: 100/99, 225/224, 864/847

POTE generator: ~6/5 = 317.083

Map: [<1 0 1 -3 4|, <0 6 5 22 -2|]

EDOs: 19, 34d, 53e

Badness: 0.0443

13-limit

Commas: 78/77, 100/99, 144/143, 676/675

POTE generator: ~6/5 = 317.118

Map: [<1 0 1 -3 4 0|, <0 6 5 22 -2 14|]

EDOs: 19, 34d, 53e, 87de

Badness: 0.0273

Countercata

Commas: 5120/5103, 15625/15552

POTE generator: 317.121

Map: [<1 0 1 11|, <0 6 5 -31|]

Wedgie: <<6 5 -31 -6 -66 -86||

EDOs: 19d, 34, 53, 87, 140, 333, 473, 806b

Badness: 0.0521

11-limit

Commas: 385/384, 2200/2187, 3388/3375

POTE generator: ~6/5 = 317.162

Map: [<1 0 1 11 -5|, <0 6 5 -31 32|]

EDOs: 34, 53, 87, 140, 227

Badness: 0.0398

13-limit

Commas: 325/324, 352/351, 385/384, 625/624

POTE generator: ~6/5 = 317.162

Map: [<1 0 1 11 -5 0|, <0 6 5 -31 32 14|]

EDOs: 34, 53, 87, 140, 227, 367e, 507e

Badness: 0.0202

Metakleismic

Commas: 15625/15552, 179200/177147

POTE generator: ~6/5 = 317.314

Map: [<1 0 1 -12|, <0 6 5 56|]

Wedgie: <<6 5 56 -6 72 116||

EDOs: 34d, 53d, 87, 121, 208

Badness: 0.1635

11-limit

Commas: 896/891, 2200/2187, 14700/14641

POTE generator: ~6/5 = 317.311

Map: [<1 0 1 -12 -5|, <0 6 5 56 32|]

EDOs: 34d, 53d, 87, 121, 208

Badness: 0.0486

13-limit

Commas: 325/324, 352/351, 364/363, 625/624

POTE generator: ~6/5 = 317.311

Map: [<1 0 1 -12 -5 0|, <0 6 5 56 32 14|]

EDOs: 34d, 53d, 87, 121, 208

Badness: 0.0244

Hemikleismic

Commas: 4000/3969, 6144/6125

POTE generator: 158.649

Map: [<1 0 1 4|, <0 12 10 -9|]

EDOs: 15, 53, 121

Badness: 0.0521

11-limit

Commas: 121/120, 176/175, 4000/3969

POTE generator: ~11/10 = 158.677

Map: [<1 0 1 4 2|, <0 12 10 -9 11|]

EDOs: 15, 38, 53, 68, 121e

Badness: 0.0380

13-limit

Commas: 121/120, 176/175, 275/273, 325/324

POTE generator: ~11/10 = 158.655

Map: [<1 0 1 4 2 0|, <0 12 10 -9 11 28|]

EDOs: 15, 53, 121e

Badness: 0.0260

Clyde

Commas: 245/243, 3136/3125

7 and 9 limit minimax

[|1 0 0 0>, |6/25 0 0 12/25>, |6/5 0 0 2/5>, |0 0 0 1>]

Eigenmonzos: 2, 7

POTE generator: ~9/7 = 441.335

Algebraic generator: real root of 5x^3-6x-3, the Poussami generator. Approximately 441.309 cents. Associated recurrence relationship quickly converges.

Map: [<1 6 6 12|, <0 -12 -10 -25|]

Generators: 2, 9/7

EDOs: 19, 49, 68, 87, 155

Badness: 0.0473

11-limit

Commas: 245/243, 3136/3125, 385/384

POTE generator: ~9/7 = 441.355

Map: [<1 6 6 12 -5|, <0 -12 -10 -25 23|]

EDOs: 19, 68, 87, 329bd, 419bd, 503bd, 590bd

Badness: 0.0474

13-limit

Commas: 196/195, 245/243, 385/384, 625/624

POTE generator: ~9/7 = 441.363

Map: [<1 6 6 12 -5 14|, <0 -12 -10 -25 23 -28|]

EDOs: 19, 68, 87, 503bdf, 590bdf

Badness: 0.0268

Bikleismic

Commas: 225/224, 243/242, 4375/4356

POTE generator: ~6/5 = 316.721

Map: [<2 0 2 -6 -1|, <0 6 5 22 15|]

EDOs: 72, 106, 178, 250, 322c, 394c, 466bc, 538bc, 610bc

Badness: 0.0293

13-limit

Commas: 169/168, 225/224, 243/242, 325/324

POTE generator: ~6/5 = 316.726

Map: [<2 0 2 -6 -1 0|, <0 6 5 22 15 14|]

EDOs: 72, 106, 322cff, 394cff, 466bcff, 538bcfff

Badness: 0.0218

Tritikleismic

See also: Gamelismic clan § Tritikleismic

Commas: 1029/1024, 15625/15552

POTE generator: 316.872

Map: [<3 0 3 10|, <0 6 5 -2|]

Wedgie: <<18 15 -6 -18 -60 -56||

EDOs: 15, 57, 72, 159, 231

Badness: 0.0563

11-limit

Commas: 385/384, 441/440, 4000/3993

POTE generator: 316.881

Map: [<3 0 3 10 8|, <0 6 5 -2 3|]

EDOs: 15, 57, 72, 159, 231

Badness: 0.0193

13-limit

Commas: 325/324, 364/363, 441/440, 625/624

POTE generator: 316.959

Map: [<3 0 3 10 8 0|, <0 6 5 -2 3 14|]

EDOs: 15, 72, 87, 159, 867, 1026

Badness: 0.0157

Quadritikleismic

Commas: 2401/2400, 15625/15552

POTE generator: 316.9999

Map: [<4 0 4 7|, <0 6 5 4|]

Wedgie: <<24 20 16 -24 -42 -19||

EDOs: 68, 72, 140, 212, 1200

Badness: 0.0392

11-limit

Commas: 385/384, 1375/1372, 6250/6237

POTE generator: 316.925

Map: [<4 0 4 7 17|, <0 6 5 4 -3|]

EDOs: 68, 72, 140, 212, 284, 496, 780

Badness: 0.0234

13-limit

Commas: 325/324, 385/384, 625/624, 1573/1568

POTE generator: 316.989

Map: [<4 0 4 7 17 0|, <0 6 5 4 -3 14|]

EDOs: 68, 72, 140, 212

Badness: 0.0187

Kleiboh

Commas: 1728/1715, 3125/3087

POTE generator: ~25/21 = 294.303

Map: [<1 6 6 6|, <0 -18 -15 -13|]

Wedgie: <<18 15 13 -18 -30 -12||

EDOs: 49, 53, 314d

Badness: 0.0765

11-limit

Commas: 176/175, 540/539, 3125/3087

POTE generator: ~25/21 = 294.181

Map: [<1 6 6 6 14|, <0 -18 -15 -13 -43|]

EDOs: 49, 53, 102d, 155d

Badness: 0.0528

13-limit

Commas: 176/175, 275/273, 325/324, 540/539

POTE generator: ~13/11 = 294.187

Map: [<1 6 6 6 14 14|, <0 -18 -15 -13 -43 -42|]

EDOs: 53, 102df, 155d

Badness: 0.0311

Novemkleismic

Commas: 15625/15552, 40353607/40310784

POTE generator: ~6/5 = 317.005

Map: [<9 0 9 11|, <0 6 5 6|]

Wedgie: <<54 45 54 -54 -66 -1||

EDOs: 72, 261, 333, 405, 477c, 882c

Badness: 0.1934

11-limit

Commas: 1375/1372, 4000/3993, 15625/15552

POTE generator: ~6/5 = 317.010

Map: [<9 0 9 11 24|, <0 6 5 6 3|]

EDOs: 72, 261, 333, 405, 882c

Badness: 0.05172

13-limit

Commas: 325/324, 625/624, 1375/1372, 4000/3993

POTE generator: ~6/5 = 317.086

Map: [<9 0 9 11 24 0|, <0 6 5 6 3 14|]

EDOs: 72, 261, 333, 738cf, 1071bcf

Badness: 0.0391

Sqrtphi

Commas: 15625/15552, 16875/16807

POTE generator: ~125/98 = 416.603 cents

Sqrt(phi) = 416.545 cents

Map: [<1 12 11 16|, <0 -30 -25 -38|]

EDOs: 49, 72, 193, 265

Badness: 0.0704

11-limit

Commas: 540/539, 1375/1372, 4375/4356

POTE generator: ~14/11 = 416.604

Map: [<1 12 11 16 17|, <0 -30 -25 -38 -39|]

EDOs: 49, 72, 193, 265

Badness: 0.0255

13-limit

Commas: 325/324, 364/363, 625/624, 1375/1372

POTE generator: ~14/11 = 416.585

Map: [<1 12 11 16 17 28|, <0 -30 -25 -38 -39 -70|]

EDOs: 72, 121, 193

Badness: 0.0200

17-limit

Commas: 325/324, 364/363, 375/374, 540/539, 595/594

POTE generator: ~14/11 = 416.585

Map: [<1 12 11 16 17 28 27|, <0 -30 -25 -38 -39 -70 -66|]

EDOs: 72, 121, 193

Badness: 0.0130

19-limit

Commas: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594

POTE generator: ~14/11 = 416.580

Map: [<1 12 11 16 17 28 27 -2|, <0 -30 -25 -38 -39 -70 -66 18|]

EDOs: 72, 121, 193

Badness: 0.0147

Scales

Music

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