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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
Retrieved from "https://en.xen.wiki/w/Kleismic_family"