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Kleismic family: Difference between revisions

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Line 1: Line 1:
<span style="display: block; text-align: right;">
<span style="display: block; text-align: right;">
__FORCETOC__
[[de:Hanson_Kleismisch]]</span>
[[de:Hanson_Kleismisch]]</span>


The [[5-limit|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|72edo]], [[87edo|87edo]] and [[140edo|140edo]].
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]].


[[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)


nice range: [315.641, 317.263]
nice range: [315.641, 317.263]
Line 11: Line 10:
strict range: [315.641, 317.263]
strict range: [315.641, 317.263]


[[POTE_tuning|POTE generator]]: 317.007
[[POTE tuning|POTE generator]]: 317.007


Map: [&lt;1 0 1|, &lt;0 6 5|]
Map: [&lt;1 0 1|, &lt;0 6 5|]
Line 19: Line 18:
Music:
Music:


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Parizek/Hanson%20%20Improv.mp3 Hanson Improv] by [[Petr_Parizek|Petr Parizek]]
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Parizek/Hanson%20%20Improv.mp3 Hanson Improv] by [[Petr Parizek]]


[http://clones.soonlabel.com/public/micro/Hanson/daily20110127-in-hanson11.mp3 In Hanson11] by [[Chris_Vaisvil|Chris Vaisvil]]
[http://clones.soonlabel.com/public/micro/Hanson/daily20110127-in-hanson11.mp3 In Hanson11] by [[Chris Vaisvil]]


Analysis and diagrams:
Analysis and diagrams:
Line 27: Line 26:
<span style="">[http://dkeenan.com/Music/ChainOfMinor3rds.htm 11 note chain-of-minor-thirds scale], by David Keenan</span>
<span style="">[http://dkeenan.com/Music/ChainOfMinor3rds.htm 11 note chain-of-minor-thirds scale], by David Keenan</span>


=Seven limit children=
== Seven limit children ==
The second comma of the [[Normal_lists|normal comma list]] defines which [[7-limit|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.
The second comma of the [[Normal lists|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=
== Keemun ==
[[Comma]]s: 49/48, 126/125
[[Comma]]s: 49/48, 126/125


Line 39: Line 38:
strict range: [308.744, 322.942]
strict range: [308.744, 322.942]


[[POTE_tuning|POTE generator]]: ~6/5 = 316.473
[[POTE tuning|POTE generator]]: ~6/5 = 316.473


Map: [&lt;1 0 1 2|, &lt;0 6 5 3|]
Map: [&lt;1 0 1 2|, &lt;0 6 5 3|]
Line 49: Line 48:
[[Badness]]: 0.0274
[[Badness]]: 0.0274


==11-limit==
=== 11-limit ===
Commas: 49/48, 56/55, 100/99
Commas: 49/48, 56/55, 100/99


Line 66: Line 65:
Badness: 0.0274
Badness: 0.0274


==13-limit==
=== 13-limit ===
Commas: 49/48, 56/55, 78/77, 100/99
Commas: 49/48, 56/55, 78/77, 100/99


Line 83: Line 82:
Badness: 0.0297
Badness: 0.0297


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


Line 100: Line 99:
Badness: 0.0227
Badness: 0.0227


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


Line 111: Line 110:
Badness: 0.0316
Badness: 0.0316


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


Line 122: Line 121:
Badness: 0.0401
Badness: 0.0401


===13-limit===
==== 13-limit ====
Commas: 45/44, 49/48, 78/77, 126/125
Commas: 45/44, 49/48, 78/77, 126/125


Line 133: Line 132:
Badness: 0.0294
Badness: 0.0294


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


Line 144: Line 143:
Badness: 0.0276
Badness: 0.0276


===13-limit===
==== 13-limit ====
Commas: 49/48, 55/54, 66/65, 77/75
Commas: 49/48, 55/54, 66/65, 77/75


Line 155: Line 154:
Badness: 0.02144
Badness: 0.02144


=Catalan=
== Catalan ==
Commas: 64/63, 15625/15552
Commas: 64/63, 15625/15552


Line 168: Line 167:
Badness: 0.0949
Badness: 0.0949


==11-limit==
=== 11-limit ===
Commas: 64/63, 100/99, 1331/1323
Commas: 64/63, 100/99, 1331/1323


Line 179: Line 178:
Badness: 0.0369
Badness: 0.0369


=Catakleismic=
== Catakleismic ==
[[Comma|Commas]]: 225/224, 4375/4374
[[Comma|Commas]]: 225/224, 4375/4374


Line 188: Line 187:
strict range: [315.789, 317.263]
strict range: [315.789, 317.263]


[[POTE_tuning|POTE generator]]: 316.732
[[POTE tuning|POTE generator]]: 316.732


Map: [&lt;1 0 1 -3|, &lt;0 6 5 22|]
Map: [&lt;1 0 1 -3|, &lt;0 6 5 22|]
Line 198: Line 197:
[[Badness]]: 0.0215
[[Badness]]: 0.0215


==11-limit==
=== 11-limit ===
[[Comma|Commas]]: 225/224, 385/384, 4375/4374
[[Comma|Commas]]: 225/224, 385/384, 4375/4374


Line 207: Line 206:
strict range: [315.789, 316.981]
strict range: [315.789, 316.981]


[[POTE_tuning|POTE generator]]: 316.719
[[POTE tuning|POTE generator]]: 316.719


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|]
Line 215: Line 214:
[[Badness]]: 0.0218
[[Badness]]: 0.0218


==13-limit==
=== 13-limit ===
[[Comma|Commas]]: 169/168, 225/224, 325/324, 540/539
[[Comma|Commas]]: 169/168, 225/224, 325/324, 540/539


Line 224: Line 223:
strict range: [315.789, 316.981]
strict range: [315.789, 316.981]


[[POTE_tuning|POTE generator]]: 316.738
[[POTE tuning|POTE generator]]: 316.738


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|]
Line 232: Line 231:
[[Badness]]: 0.0169
[[Badness]]: 0.0169


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


Line 243: Line 242:
Badness: 0.0400
Badness: 0.0400


==13-limit==
=== 13-limit ===
Commas: 99/98, 169/168, 176/175, 275/273
Commas: 99/98, 169/168, 176/175, 275/273


Line 254: Line 253:
Badness: 0.0226
Badness: 0.0226


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


Line 265: Line 264:
Badness: 0.0304
Badness: 0.0304


==13-limit==
=== 13-limit ===
Commas: 169/168 225/224 325/324 1716/1715
Commas: 169/168 225/224 325/324 1716/1715


Line 276: Line 275:
Badness: 0.0223
Badness: 0.0223


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


Line 287: Line 286:
Badness: 0.0443
Badness: 0.0443


==13-limit==
=== 13-limit ===
Commas: 78/77, 100/99, 144/143, 676/675
Commas: 78/77, 100/99, 144/143, 676/675


Line 298: Line 297:
Badness: 0.0273
Badness: 0.0273


=Countercata=
== Countercata ==
[[Comma|Commas]]: 15625/15552, 5120/5103
[[Comma|Commas]]: 15625/15552, 5120/5103


[[POTE_tuning|POTE generator]]: 317.121
[[POTE tuning|POTE generator]]: 317.121


Map: [&lt;1 0 1 11|, &lt;0 6 5 -31|]
Map: [&lt;1 0 1 11|, &lt;0 6 5 -31|]
Line 311: Line 310:
[[Badness]]: 0.0521
[[Badness]]: 0.0521


==11-limit==
=== 11-limit ===
[[Comma]]s: 385/384, 2200/2187, 3388/3375
[[Comma]]s: 385/384, 2200/2187, 3388/3375


Line 322: Line 321:
[[Badness]]: 0.0398
[[Badness]]: 0.0398


==13-limit==
=== 13-limit ===
[[Comma]]s: 325/324, 352/351, 385/384, 625/624
[[Comma]]s: 325/324, 352/351, 385/384, 625/624


Line 333: Line 332:
[[Badness]]: 0.0202
[[Badness]]: 0.0202


=Metakleismic=
== Metakleismic ==
[[Comma]]s: 15625/15552, 179200/177147
[[Comma]]s: 15625/15552, 179200/177147


Line 346: Line 345:
[[Badness]]: 0.1635
[[Badness]]: 0.1635


==11-limit==
=== 11-limit ===
[[Comma]]s: 896/891, 2200/2187, 14700/14641
[[Comma]]s: 896/891, 2200/2187, 14700/14641


Line 357: Line 356:
[[Badness]]: 0.0486
[[Badness]]: 0.0486


==13-limit==
=== 13-limit ===
[[Comma|Comma]]s: 325/324, 352/351, 364/363, 625/624
[[Comma]]s: 325/324, 352/351, 364/363, 625/624


POTE generator: ~6/5 = 317.311
POTE generator: ~6/5 = 317.311
Line 368: Line 367:
[[Badness]]: 0.0244
[[Badness]]: 0.0244


=Hemikleismic=
== Hemikleismic ==
Commas: 4000/3969, 6144/6125
Commas: 4000/3969, 6144/6125


[[POTE_tuning|POTE generator]]: 158.649
[[POTE tuning|POTE generator]]: 158.649


Map: [&lt;1 0 1 4|, &lt;0 12 10 -9|]
Map: [&lt;1 0 1 4|, &lt;0 12 10 -9|]
Line 379: Line 378:
Badness: 0.0521
Badness: 0.0521


==11-limit==
=== 11-limit ===
Commas: 121/120, 176/175, 4000/3969
Commas: 121/120, 176/175, 4000/3969


Line 390: Line 389:
Badness: 0.0380
Badness: 0.0380


==13-limit==
=== 13-limit ===
Commas: 121/120, 176/175, 275/273, 325/324
Commas: 121/120, 176/175, 275/273, 325/324


Line 401: Line 400:
Badness: 0.0260
Badness: 0.0260


=Clyde=
== Clyde ==
[[Comma|Commas]]: 245/243, 3136/3125
[[Comma|Commas]]: 245/243, 3136/3125


Line 410: Line 409:
[[Eigenmonzo|Eigenmonzos]]: 2, 7
[[Eigenmonzo|Eigenmonzos]]: 2, 7


[[POTE_tuning|POTE generator]]: ~9/7 = 441.335
[[POTE tuning|POTE generator]]: ~9/7 = 441.335


Algebraic generator: real root of 5x^3-6x-3, the Poussami generator. Approximately 441.309 [[cent|cents]]. Associated recurrence relationship quickly converges.
Algebraic generator: real root of 5x^3-6x-3, the Poussami generator. Approximately 441.309 [[cent|cents]]. Associated recurrence relationship quickly converges.
Line 422: Line 421:
Badness: 0.0473
Badness: 0.0473


==11-limit==
=== 11-limit ===
Commas: 245/243, 3136/3125, 385/384
Commas: 245/243, 3136/3125, 385/384


Line 433: Line 432:
Badness: 0.0474
Badness: 0.0474


==13-limit==
=== 13-limit ===
Commas: 196/195, 245/243, 385/384, 625/624
Commas: 196/195, 245/243, 385/384, 625/624


Line 444: Line 443:
Badness: 0.0268
Badness: 0.0268


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


Line 455: Line 454:
Badness: 0.0293
Badness: 0.0293


==13-limit==
=== 13-limit ===
Commas: 169/168, 225/224, 243/242, 325/324
Commas: 169/168, 225/224, 243/242, 325/324


Line 466: Line 465:
Badness: 0.0218
Badness: 0.0218


=Tritikleismic=
== Tritikleismic ==
[[Comma]]s: 15625/15552, 1029/1024
[[Comma]]s: 15625/15552, 1029/1024


[[POTE_tuning|POTE generator]]: 316.872
[[POTE tuning|POTE generator]]: 316.872


Map: [&lt;3 0 3 10|, &lt;0 6 5 -2|]
Map: [&lt;3 0 3 10|, &lt;0 6 5 -2|]
Line 479: Line 478:
[[Badness]]: 0.0563
[[Badness]]: 0.0563


==11-limit==
=== 11-limit ===
[[Comma]]s: 385/384, 441/440, 4000/3993
[[Comma]]s: 385/384, 441/440, 4000/3993


[[POTE_tuning|POTE generator]]: 316.881
[[POTE tuning|POTE generator]]: 316.881


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|]
Line 490: Line 489:
[[Badness]]: 0.0193
[[Badness]]: 0.0193


==13-limit==
=== 13-limit ===
[[Comma]]s: 325/324, 364/363, 441/440, 625/624
[[Comma]]s: 325/324, 364/363, 441/440, 625/624


[[POTE_tuning|POTE generator]]: 316.959
[[POTE tuning|POTE generator]]: 316.959


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|]
Line 501: Line 500:
[[Badness]]: 0.0157
[[Badness]]: 0.0157


=Quadritikleismic=
== Quadritikleismic ==
[[Comma]]s: 15625/15552, 2401/2400
[[Comma]]s: 15625/15552, 2401/2400


[[POTE_tuning|POTE generator]]: 316.9999
[[POTE tuning|POTE generator]]: 316.9999


Map: [&lt;4 0 4 7|, &lt;0 6 5 4|]
Map: [&lt;4 0 4 7|, &lt;0 6 5 4|]
Line 514: Line 513:
[[Badness]]: 0.0392
[[Badness]]: 0.0392


==11-limit==
=== 11-limit ===
Commas: 385/384, 1375/1372, 6250/6237
Commas: 385/384, 1375/1372, 6250/6237


[[POTE_tuning|POTE generator]]: 316.925
[[POTE tuning|POTE generator]]: 316.925


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|]
Line 525: Line 524:
[[Badness]]: 0.0234
[[Badness]]: 0.0234


==13-limit==
=== 13-limit ===
Commas: 325/324, 385/384, 625/624, 1573/1568
Commas: 325/324, 385/384, 625/624, 1573/1568


[[POTE_tuning|POTE generator]]: 316.989
[[POTE tuning|POTE generator]]: 316.989


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|]
Line 536: Line 535:
[[Badness]]: 0.0187
[[Badness]]: 0.0187


=Kleiboh=
== Kleiboh ==
Commas: 1728/1715, 3125/3087
Commas: 1728/1715, 3125/3087


Line 549: Line 548:
Badness: 0.0765
Badness: 0.0765


==11-limit==
=== 11-limit ===
Commas: 176/175, 540/539, 3125/3087
Commas: 176/175, 540/539, 3125/3087


Line 560: Line 559:
Badness: 0.0528
Badness: 0.0528


==13-limit==
=== 13-limit ===
Commas: 176/175, 275/273, 325/324, 540/539
Commas: 176/175, 275/273, 325/324, 540/539


Line 571: Line 570:
Badness: 0.0311
Badness: 0.0311


=Novemkleismic=
== Novemkleismic ==
Commas: 15625/15552, 40353607/40310784
Commas: 15625/15552, 40353607/40310784


Line 584: Line 583:
Badness: 0.1934
Badness: 0.1934


==11-limit==
=== 11-limit ===
Commas: 1375/1372, 4000/3993, 15625/15552
Commas: 1375/1372, 4000/3993, 15625/15552


Line 595: Line 594:
Badness: 0.05172
Badness: 0.05172


==13-limit==
=== 13-limit ===
Commas: 325/324, 625/624, 1375/1372, 4000/3993
Commas: 325/324, 625/624, 1375/1372, 4000/3993


Line 606: Line 605:
Badness: 0.0391
Badness: 0.0391


=Sqrtphi=
== Sqrtphi ==
Commas: 15625/15552, 16875/16807
Commas: 15625/15552, 16875/16807


[[POTE_tuning|POTE generator]]: ~125/98 = 416.603 cents
[[POTE tuning|POTE generator]]: ~125/98 = 416.603 cents


Sqrt(phi) = 416.545 cents
Sqrt(phi) = 416.545 cents
Line 619: Line 618:
Badness: 0.0704
Badness: 0.0704


==11-limit==
=== 11-limit ===
Commas: 540/539, 1375/1372, 4375/4356
Commas: 540/539, 1375/1372, 4375/4356


Line 630: Line 629:
Badness: 0.0255
Badness: 0.0255


==13-limit==
=== 13-limit ===
Commas: 325/324, 364/363, 625/624, 1375/1372
Commas: 325/324, 364/363, 625/624, 1375/1372


Line 641: Line 640:
Badness: 0.0200
Badness: 0.0200


==17-limit==
=== 17-limit ===
Commas: 325/324, 364/363, 375/374, 540/539, 595/594
Commas: 325/324, 364/363, 375/374, 540/539, 595/594


Line 652: Line 651:
Badness: 0.0130
Badness: 0.0130


==19-limit==
=== 19-limit ===
Commas: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594
Commas: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594


Line 663: Line 662:
Badness: 0.0147
Badness: 0.0147


==Scales==
=== Scales ===
[[sqrtphi17|sqrtphi17]]
* [[sqrtphi17]]


[[sqrtphi23|sqrtphi23]]
* [[sqrtphi23]]


[[sqrtphi49|sqrtphi49]]
* [[sqrtphi49]]


==Music==
=== Music ===
[http://micro.soonlabel.com/sqrt_phi/daily20111123a-sqrt-phi-17.mp3 Prelude for Piano in Square root of Phi Tuning] by [[Chris_Vaisvil|Chris Vaisvil]]
* [http://micro.soonlabel.com/sqrt_phi/daily20111123a-sqrt-phi-17.mp3 Prelude for Piano in Square root of Phi Tuning] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3 A Fight for Phi] by [[Vito Sicurella]]


[http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3 A Fight for Phi] by Vito Sicurella
[[Category:Theory]]
[[Category:family]]
[[Category:Family]]
[[Category:kleismic]]
[[Category:Kleismic]]
[[Category:list]]
[[Category:Listen]]
[[Category:listen]]
[[Category:overview]]
[[Category:theory]]
[[Category:todo:add_definition]]
[[Category:todo:add_definition]]
Retrieved from "https://en.xen.wiki/w/Kleismic_family"