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Kleismic family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
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| en = Kleismic family
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-20 18:57:39 UTC</tt>.<br>
| de = Hanson-Kleismisch
: The original revision id was <tt>203543908</tt>.<br>
| es =
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<h4>Original Wikitext content:</h4>
{{Technical data page}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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. 14/53 is about perfect as a generator, though 9/34 also makes sense and using [[19edo]] is possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].
The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5|classical minor third]]s falls short of the [[3/1|3rd]] [[harmonic]].  


[[POTE tuning|POTE generator]]: 317.007
== Kleismic a.k.a. hanson ==
{{Main| Kleismic }}


Map: [&lt;1 0 1|, &lt;0 6 5|]
The [[generator]] of kleismic is a [[6/5|classical minor third]], and to get to the interval class of [[5/4|major thirds]] requires five of these, and so to get to [[3/2|fifths]] requires six. In fact, (6/5)<sup>5</sup> = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a [[microtemperament]]) is sometimes called ''hanson'', and [[53edo|14\53]] is about perfect as a generator, though [[34edo|9\34]] also makes sense, and [[19edo|5\19]] and [[15edo|4\15]] are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].


==Seven limit children==
[[Subgroup]]: 2.3.5
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==
[[Comma list]]: 15625/15552
[[Comma|Commas]]: 49/48, 126/125


[[POTE tuning|POTE generator]]: 316.473
{{Mapping|legend=1| 1 0 1 | 0 6 5 }}
: mapping generators: ~2, ~6/5


Map: [&lt;1 0 1 2|, &lt;0 6 5 3|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;6 5 3 -6 -12 -7||
* [[WE]]: ~2 = 1200.1659{{c}}, ~6/5 = 317.0504{{c}}
EDOs: [[15edo|15]], [[19edo|19]], [[91edo|91]]
: [[error map]]: {{val| +0.166 +0.347 -0.896 }}
Badness: 0.0274
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.0308{{c}}
: error map: {{val| 0.000 +0.230 -1.160 }}


==Catakleismic==
[[Tuning ranges]]:
[[Comma|Commas]]: 225/224, 4375/4374
* [[5-odd-limit]] [[diamond monotone]]: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
* 5-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263] (untempered to 1/5-comma)


[[POTE tuning|POTE generator]]: 316.732
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 458, 511c, …, 829c, 882c }}


Map: [&lt;1 0 1 -3|, &lt;0 6 5 22|]
[[Badness]] (Sintel): 0.310
Wedgie: &lt;&lt;6 5 22 -6 18 37||
EDOs: [[19edo|19]], [[53edo|15]], [[72edo|72]], [[197edo|197]], [[269edo|269]]
Badness: 0.0215


===11-limit===
=== Overview to extensions ===
[[Comma|Commas]]: 225/224, 385/384, 4375/4374
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], hemifamity, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan.


[[POTE tuning|POTE generator]]: 316.719
[[6144/6125]], the porwell comma, gives [[#Hemikleismic|hemikleismic]]. [[245/243]], sensamagic, gives [[#Clyde|clyde]]. [[1029/1024]], the gamelisma, gives [[#Tritikleismic|tritikleismic]]. [[10976/10935]], hemimage, gives [[#Marfifths|marfifths]]. [[1728/1715]], the orwellismia, gives [[#Kleiboh|kleiboh]]. [[2401/2400]], the breedsma, gives [[#Quadritikleismic|quadritikleismic]]. [[2460375/2458624]], the breeze comma, gives [[#Marthirds|marthirds]]. 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. Marfifths splits the 12/5 into three. Kleiboh splits the 24/5 into three. Marthirds splits the 12/5 into four. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.


Map: [&lt;1 0 1 -3 9|, &lt;0 6 5 22 -21|]
Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts.
EDOs: [[15edo|15]], [[19edo|19]], [[53edo|53]], [[72edo|72]], [[269edo|269]], [[341edo|341]]
Badness: 0.0218


===13-limit===
The kleismic family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], which has further extensions with higher primes. These are listed at the bottom of this page, in [[#Subgroup extensions]].
[[Comma|Commas]]: 169/168, 225/224, 325/324, 540/539


[[POTE tuning|POTE generator]]: 316.738
== Catakleismic ==
{{Main| Catakleismic }}


Map: [&lt;1 0 1 -3 9 0|, &lt;0 6 5 22 -21 14|]
Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings.  
EDOs: [[15edo|15]], [[19edo|19]], [[53edo|53]], [[72edo|72]], [[197edo|197]], [[269edo|269]], [[466edo|466]]
Badness: 0.0169


==Countercata==
Catakleismic extends easily with [[prime interval|prime]] [[13/1|13]]. The [[S-expression]]-based comma list of this extension is {[[169/168|S13]], [[225/224|S15 = S25⋅S26⋅S27]], [[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}.
[[Comma|Commas]]: 15625/15552, 5120/5103


[[POTE tuning|POTE generator]]: 317.121
=== 7-limit ===
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 1 11|, &lt;0 6 5 -31|]
[[Comma list]]: 225/224, 4375/4374
Wedgie: &lt;&lt;6 5 -31 -6 -66 -86||
EDOs: 15, 19, 34, 53, 87, 140, 333, 473, 806
Badness: 0.0521


===11-limit===
{{Mapping|legend=1| 1 0 1 -3 | 0 6 5 22 }}
Commas: 385/384, 2200/2187, 3388/3375


POTE generaor: ~6/5 = 317.162
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5965{{c}}, ~6/5 = 316.8893{{c}}
: [[error map]]: {{val| +0.596 -0.619 -1.271 +0.948 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7705{{c}}
: error map: {{val| 0.000 -1.332 -2.461 +0.126 }}


Map: [&lt;1 0 1 11 -5|, &lt;0 6 5 -31 32|]
[[Tuning ranges]]:  
EDOs: 34, 53, 87, 140, 227
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [315.789, 317.647] (5\19 to 9\34)
Badness: 0.0398
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]


==Hemikleismic==
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197, 269c }}
Commas: 4000/3969, 6144/6125


[[POTE tuning|POTE generator]]: 158.649
[[Badness]] (Sintel): 0.544


Map: [&lt;1 0 1 4|, &lt;0 12 10 -9|]
==== 2.3.5.7.13 subgroup ====
EDOs: 53, 121
Subgroup: 2.3.5.7.13


==Clyde==
Comma list: 169/168, 225/224, 325/324
[[Comma|Commas]]: 245/243, 3136/3125


7 and 9 limit minimax
Subgroup-val mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}
[|1 0 0 0&gt;, |6/25 0 0 12/25&gt;, |6/5 0 0 2/5&gt;, |0 0 0 1&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 7


[[POTE tuning|POTE generator]]: 441.335
Optimal tunings:  
* WE: ~2 = 1200.7838{{c}}, ~6/5 = 316.9478{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}


Algebraic generator: real root of 5x^3-6x-3, the Poussami generator. Approximately 441.309 [[Cent|cents]]. Associated recurrence relationship quickly converges.
{{Optimal ET sequence|legend=0| 19, 34d, 53, 72, 125f, 197f }}


Map: [&lt;1 6 6 12|, &lt;0 -12 -10 -25|]
Badness (Sintel): 0.410
[[Generator|Generators]]: 2, 9/7
[[Edo|Edos]]: [[19edo|19]], [[49edo|49]], [[68edo|68]], [[87edo|87]], [[155edo|155]]


==Tritikleismic==
=== 11-limit ===
Commas: 15625/15552, 1029/1024
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE generator]]: 316.872
Comma list: 225/224, 385/384, 4375/4374


Map: [&lt;3 0 3 10|, &lt;0 6 5 -2|]
Mapping: {{mapping| 1 0 1 -3 9 | 0 6 5 22 -21 }}
Wedgie: &lt;&lt;18 15 -6 -18 -60 -56||
EDOs: 12, 15, 57, 72, 159, 231
Badness: 0.0563


===11-limit===
Optimal tunings:
Commas: 385/384, 441/440, 4000/3993
* WE: ~2 = 1200.6524{{c}}, ~6/5 = 316.8911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7267{{c}}


[[POTE tuning|POTE generator]]: 316.881
Tuning ranges:
* 11-odd-limit diamond monotone range: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
* 11-odd-limit diamond tradeoff range: ~6/5 = [315.641, 317.263]


Map: [&lt;3 0 3 10 8|, &lt;0 6 5 -2 3|]
{{Optimal ET sequence|legend=0| 19, 53, 72, 197e, 269ce, 341ce }}
EDOs: 12, 15, 57, 72, 159, 231
Badness: 0.0193


===13-limit===
Badness (Sintel): 0.722
Commas: 325/324, 364/363, 441/440, 625/624


[[POTE tuning|POTE generator]]: 316.959
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;3 0 3 10 8 0|, &lt;0 6 5 -2 3 14|]
Comma list: 169/168, 225/224, 325/324, 385/384
EDOs: 12, 15, 72, 87, 159, 867, 1026
Badness: 0.0157


==Quadritikleismic==
Mapping: {{mapping| 1 0 1 -3 9 0 | 0 6 5 22 -21 14 }}
Commas: 15625/15552, 2401/2400


[[POTE tuning|POTE generator]]: 316.9999
Optimal tunings:  
* WE: ~2 = 1200.7982{{c}}, ~6/5 = 316.9482{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7491{{c}}


Map: [&lt;4 0 4 7|, &lt;0 6 5 4|]
Tuning ranges:  
Wedgie: &lt;&lt;24 20 16 -24 -42 -19||
* 13- and 15-odd-limit diamond monotone: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
EDOs: [[68edo|68]], [[72edo|72]], [[140edo|140]], [[212edo|212]], [[1200edo|1200]]
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
Badness: 0.0392


===11-limit===
{{Optimal ET sequence|legend=0| 19, 53, 72, 125f, 197ef }}
Commas: 385/384, 1375/1372, 6250/6237


[[POTE tuning|POTE generator]]: 316.925
Badness (Sintel): 0.698


Map: [&lt;4 0 4 7 17|, &lt;0 6 5 4 -3|]
=== Cataclysmic ===
EDOs: [[68edo|68]], [[72edo|72]], [[140edo|140]], [[212edo|212]], [[284edo|284]], [[496edo|496]], [[780edo|780]]
Subgroup: 2.3.5.7.11
Badness: 0.0234


===13-limit===
Comma list: 99/98, 176/175, 2200/2187
Commas: 325/324, 385/384, 625/624, 1573/1568


[[POTE tuning|POTE generator]]: 316.989
Mapping: {{mapping| 1 0 1 -3 -5 | 0 6 5 22 32 }}


Map: [&lt;4 0 4 7 17 0|, &lt;0 6 5 4 -3 14|]
Optimal tunings:
EDOs: 68, 72, 140, 212
* WE: ~2 = 1199.9590{{c}}, ~6/5 = 317.0315{{c}}
Badness: 0.0187</pre></div>
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0403{{c}}
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision
{{Optimal ET sequence|legend=0| 19e, 34d, 53 }}
 
Badness (Sintel): 1.32
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 99/98, 169/168, 176/175, 275/273
 
Mapping: {{mapping| 1 0 1 -3 -5 0 | 0 6 5 22 32 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.0797{{c}}, ~6/5 = 317.0571{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0400{{c}}
 
{{Optimal ET sequence|legend=0| 19e, 34d, 53 }}
 
Badness (Sintel): 0.932
 
=== Catalytic ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 441/440, 4375/4374
 
Mapping: {{mapping| 1 0 1 -3 -10 | 0 6 5 22 51 }}
 
Optimal tunings:
* WE: ~2 = 1200.8102{{c}}, ~6/5 = 316.8669{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6768{{c}}
 
{{Optimal ET sequence|legend=0| 19e, 53e, 72 }}
 
Badness (Sintel): 1.01
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 325/324, 1716/1715
 
Mapping: {{mapping| 1 0 1 -3 -10 0 | 0 6 5 22 51 14 }}
 
Optimal tunings:
* WE: ~2 = 1201.0807{{c}}, ~6/5 = 316.9246{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6700{{c}}
 
{{Optimal ET sequence|legend=0| 19e, 53e, 72, 307bcdeeffff }}
 
Badness (Sintel): 0.923
 
=== Cataleptic ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 225/224, 864/847
 
Mapping: {{mapping| 1 0 1 -3 4 | 0 6 5 22 -2 }}
 
Optimal tunings:
* WE: ~2 = 1198.6575{{c}}, ~6/5 = 316.7282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0257{{c}}
 
{{Optimal ET sequence|legend=0| 19, 34d, 53e }}
 
Badness (Sintel): 1.47
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 100/99, 144/143, 676/675
 
Mapping: {{mapping| 1 0 1 -3 4 0 | 0 6 5 22 -2 14 }}
 
Optimal tunings:
* WE: ~2 = 1198.8403{{c}}, ~6/5 = 316.8111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0652{{c}}
 
{{Optimal ET sequence|legend=0| 19, 34d, 53e }}
 
Badness (Sintel): 1.13
 
=== Bikleismic ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 243/242, 4375/4356
 
Mapping: {{mapping| 2 0 2 -6 -1 | 0 6 5 22 15 }}
: mapping generators: ~99/70, ~6/5
 
Optimal tunings:
* WE: ~99/70 = 600.2674{{c}}, ~6/5 = 316.8624{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 316.7575{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 72, 322c, 394c }}
 
Badness (Sintel): 0.969
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 243/242, 325/324
 
Mapping: {{mapping| 2 0 2 -6 -1 0 | 0 6 5 22 15 14 }}
 
Optimal tunings:
* WE: ~55/39 = 600.3582{{c}}, ~
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