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Kleismic family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| en = Kleismic family
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-08-21 20:22:16 UTC</tt>.<br>
| de = Hanson-Kleismisch
: The original revision id was <tt>519242608</tt>.<br>
| es =
: The revision comment was: <tt></tt><br>
| ja =  
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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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">&lt;span style="display: block; text-align: right;"&gt;[[toc]]
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]].  
Other languages: [[xenharmonie/Hanson Kleismisch|Deutsch]]
&lt;/span&gt;
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)
== Kleismic a.k.a. hanson ==
nice range: [315.641, 317.263]
{{Main| Kleismic }}
strict range: [315.641, 317.263]


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


Map: [&lt;1 0 1|, &lt;0 6 5|]
[[Subgroup]]: 2.3.5
EDOs: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[53edo|53]], [[458edo|458]], [[882edo|882c]]


Music:
[[Comma list]]: 15625/15552
[[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]]


Analysis and diagrams:
{{Mapping|legend=1| 1 0 1 | 0 6 5 }}
&lt;span class="wiki_link_ext"&gt;[[@http://dkeenan.com/Music/ChainOfMinor3rds.htm|11 note chain-of-minor-thirds scale]], by David Keenan&lt;/span&gt;
: mapping generators: ~2, ~6/5


[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1659{{c}}, ~6/5 = 317.0504{{c}}
: [[error map]]: {{val| +0.166 +0.347 -0.896 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.0308{{c}}
: error map: {{val| 0.000 +0.230 -1.160 }}


=Seven limit children=
[[Tuning ranges]]:
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.
* [[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)


=Keemun=
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 458, 511c, …, 829c, 882c }}
[[Comma|Commas]]: 49/48, 126/125


valid range: [300.000, 327.273] (4 to 11b)
[[Badness]] (Sintel): 0.310
nice range: [308.744, 322.942]
strict range: [308.744, 322.942]


[[POTE tuning|POTE generator]]: ~6/5 = 316.473
=== Overview to extensions ===
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.  


Map: [&lt;1 0 1 2|, &lt;0 6 5 3|]
[[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.  
[[Wedgie]]: &lt;&lt;6 5 3 -6 -12 -7||
EDOs: [[15edo|15]], [[19edo|19]], [[53edo|53d]], [[72edo|72d]], [[91edo|91d]]
[[Badness]]: 0.0274


==11-limit==
Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts.
[[Comma]]s: 49/48, 56/55, 100/99


valid range: [315.789, 320.000] (19 to 15)
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]].  
nice range: [308.744, 324.341]
strict range: [315.789, 320.000]


[[POTE tuning|POTE generator]]: ~6/5 = 317.576
== Catakleismic ==
{{Main| Catakleismic }}


Map: [&lt;1 0 1 2 4|, &lt;0 6 5 3 -2|]
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: [[4edo|4]], [[15edo|15]], [[19edo|19]], [[34edo|34]], [[102edo|102de]]
[[Badness]]: 0.0274


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


valid range: 315.789 (19)
=== 7-limit ===
nice range: [303.597, 324.341]
[[Subgroup]]: 2.3.5.7
strict range: 315.789


[[POTE tuning|POTE generator]]: ~6/5 = 316.611
[[Comma list]]: 225/224, 4375/4374


Map: [&lt;1 0 1 2 4 5|, &lt;0 6 5 3 -2 -5|]
{{Mapping|legend=1| 1 0 1 -3 | 0 6 5 22 }}
EDOs: [[4edo|4]], [[15edo|15]], [[19edo|19]], [[53def]], [[72edo|72def]]
[[Badness]]: 0.0297


===Kema===  
[[Optimal tuning]]s:
Commas: 49/48, 56/55, 91/90, 100/99
* [[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 }}


valid range: [315.789, 320.000] (19 to 15)
[[Tuning ranges]]:
nice range: [308.744, 324.341]
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [315.789, 317.647] (5\19 to 9\34)
strict range: [315.789, 320.000]  
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]


POTE generator: ~6/5 = 317.423
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197, 269c }}


Map: [&lt;1 0 1 2 4 0|, &lt;0 6 5 3 -2 14|]
[[Badness]] (Sintel): 0.544
EDOs: 15, 19, 34, 87de
Badness: 0.0227


===Kumbaya===  
==== 2.3.5.7.13 subgroup ====
Commas: 40/39, 49/48, 56/55, 66/65
Subgroup: 2.3.5.7.13


POTE generator: ~6/5 = 318.595
Comma list: 169/168, 225/224, 325/324


Map: [&lt;1 0 1 2 4 4|, &lt;0 6 5 3 -2 -1|]
Subgroup-val mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}
EDOs: 15, 34f
Badness: 0.0316


==Qeema==
Optimal tunings:
Commas: 45/44, 49/48, 126/125
* WE: ~2 = 1200.7838{{c}}, ~6/5 = 316.9478{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}


POTE generator: ~6/5 = 314.730
{{Optimal ET sequence|legend=0| 19, 34d, 53, 72, 125f, 197f }}


Map: [&lt;1 0 1 2 -1|, &lt;0 6 5 3 17|]
Badness (Sintel): 0.410
EDOs: 19, 42bcd, 61bcd
Badness: 0.0401


===13-limit===  
=== 11-limit ===
Commas: 45/44, 49/48, 78/77, 126/125
Subgroup: 2.3.5.7.11


POTE generator: ~6/5 = 315.044
Comma list: 225/224, 385/384, 4375/4374


Map: [&lt;1 0 1 2 -1 0|, &lt;0 6 5 3 17 14|]
Mapping: {{mapping| 1 0 1 -3 9 | 0 6 5 22 -21 }}
EDOs: 19
Badness: 0.0294


==Darjeeling==
Optimal tunings:
Commas: 49/48, 55/54, 77/75
* WE: ~2 = 1200.6524{{c}}, ~6/5 = 316.8911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7267{{c}}


POTE generator: ~6/5 = 317.656
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;1 0 1 2 0|, &lt;0 6 5 3 13|]
{{Optimal ET sequence|legend=0| 19, 53, 72, 197e, 269ce, 341ce }}
EDOs: 15, 19e, 34e
Badness: 0.0276


===13-limit Darjeeling===
Badness (Sintel): 0.722
Commas: 49/48, 55/54, 66/65, 77/75


POTE generator: ~6/5 = 317.298
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 1 2 0 0|, &lt;0 6 5 3 13 14|]
Comma list: 169/168, 225/224, 325/324, 385/384
EDOs: 15, 19e, 34e, 53de
Badness: 0.02144


=Catalan=
Mapping: {{mapping| 1 0 1 -3 9 0 | 0 6 5 22 -21 14 }}
Commas: 64/63, 15625/15552


POTE generator: ~6/5 = 318.267
Optimal tunings:  
* WE: ~2 = 1200.7982{{c}}, ~6/5 = 316.9482{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7491{{c}}


Map: [&lt;1 0 1 6|, &lt;0 6 5 -12|]
Tuning ranges:  
Wedgie: &lt;&lt;6 5 -12 -6 -36 -42||
* 13- and 15-odd-limit diamond monotone: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
EDOs: 15, 34d, 49, 132bcd
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
Badness: 0.0949


==11-limit==
{{Optimal ET sequence|legend=0| 19, 53, 72, 125f, 197ef }}
Commas: 64/63, 100/99, 1331/1323


POTE generator: ~6/5 = 318.282
Badness (Sintel): 0.698


Map: [&lt;1 0 1 6 4|, &lt;0 6 5 -12 -2|]
=== Cataclysmic ===
EDOs: 15, 34d, 49
Subgroup: 2.3.5.7.11
Badness: 0.0369


=Catakleismic=
Comma list: 99/98, 176/175, 2200/2187
[[Comma|Commas]]: 225/224, 4375/4374


valid range: [315.789, 317.647] (19 to 34)
Mapping: {{mapping| 1 0 1 -3 -5 | 0 6 5 22 32 }}
nice range: [315.641, 317.263]
strict range: [315.789, 317.263]


[[POTE tuning|POTE generator]]: 316.732
Optimal tunings:  
* WE: ~2 = 1199.9590{{c}}, ~6/5 = 317.0315{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0403{{c}}


Map: [&lt;1 0 1 -3|, &lt;0 6 5 22|]
{{Optimal ET sequence|legend=0| 19e, 34d, 53 }}
[[Wedgie]]: &lt;&lt;6 5 22 -6 18 37||
EDOs: [[19edo|19]], [[53edo|15]], [[72edo|72]], [[197edo|197]], [[269edo|269c]]
[[Badness]]: 0.0215


==11-limit==
Badness (Sintel): 1.32
[[Comma|Commas]]: 225/224, 385/384, 4375/4374


valid range: [315.789, 316.981] (19 to 53)
==== 13-limit ====
nice range: [315.641, 317.263]
Subgroup: 2.3.5.7.11.13
strict range: [315.789, 316.981]


[[POTE tuning|POTE generator]]: 316.719
Comma list: 99/98, 169/168, 176/175, 275/273


Map: [&lt;1 0 1 -3 9|, &lt;0 6 5 22 -21|]
Mapping: {{mapping| 1 0 1 -3 -5 0 | 0 6 5 22 32 14 }}
EDOs: [[19edo|19]], [[53edo|53]], [[72edo|72]], [[197edo|197e]], [[269edo|269ce]], [[341edo|341ce]], [[610edo|610bce]]
[[Badness]]: 0.0218


==13-limit==
Optimal tunings:
[[Comma|Commas]]: 169/168, 225/224, 325/324, 540/539
* WE: ~2 = 1200.0797{{c}}, ~6/5 = 317.0571{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0400{{c}}


valid range: [315.789, 316.981] (19 to 53)
{{Optimal ET sequence|legend=0| 19e, 34d, 53 }}
nice range: [315.641, 318.309]
strict range: [315.789, 316.981]


[[POTE tuning|POTE generator]]: 316.738
Badness (Sintel): 0.932


Map: [&lt;1 0 1 -3 9 0|, &lt;0 6 5 22 -21 14|]
=== Catalytic ===
EDOs: [[15edo|15]], [[19edo|19]], [[53edo|53]], [[72edo|72]], [[125edo|125f]], [[197edo|197ef]], [[269edo|269cef]]
Subgroup: 2.3.5.7.11
[[Badness]]: 0.0169


=Cataclysmic=
Comma list: 225/224, 441/440, 4375/4374
Commas: 99/98, 176/175, 2200/2187


POTE generator: ~6/5 = 317.042
Mapping: {{mapping| 1 0 1 -3 -10 | 0 6 5 22 51 }}


Map: [&lt;1 0 1 -3 -5|, &lt;0 6 5 22 32|]
Optimal tunings:  
EDOs: 53, 87d, 140d, 171de, 181de, 193de, 224de, 246de, 277de
* WE: ~2 = 1200.8102{{c}}, ~6/5 = 316.8669{{c}}
Badness: 0.0400
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6768{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 19e, 53e, 72 }}
Commas: 99/98, 169/168, 176/175, 275/273


POTE generator: ~6/5 = 317.036
Badness (Sintel): 1.01


Map: [&lt;1 0 1 -3 -5 0|, &lt;0 6 5 22 32 14|]
==== 13-limit ====
EDOs: 53, 87d, 140d, 193de, 246de
Subgroup: 2.3.5.7.11.13
Badness: 0.0226


=Catalytic=
Comma list: 169/168, 225/224, 325/324, 1716/1715
Commas: 225/224, 441/440, 4375/4374


POTE generator: ~6/5 = 316.653
Mapping: {{mapping| 1 0 1 -3 -10 0 | 0 6 5 22 51 14 }}


Map: [&lt;1 0 1 -3 -10|, &lt;0 6 5 22 51|]
Optimal tunings:  
EDOs: 53e, 72
* WE: ~2 = 1201.0807{{c}}, ~6/5 = 316.9246{{c}}
Badness: 0.0304
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6700{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 19e, 53e, 72, 307bcdeeffff }}
Commas: 169/168 225/224 325/324 1716/1715


POTE generator: ~6/5 = 316.639
Badness (Sintel): 0.923


Map: [&lt;1 0 1 -3 -10 0|, &lt;0 6 5 22 51 14|]
=== Cataleptic ===
EDOs: 19e, 53e, 72
Subgroup: 2.3.5.7.11
Badness: 0.0223


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


POTE generator: ~6/5 = 317.083
Mapping: {{mapping| 1 0 1 -3 4 | 0 6 5 22 -2 }}


Map: [&lt;1 0 1 -3 4|, &lt;0 6 5 22 -2|]
Optimal tunings:  
EDOs: 19, 34d, 53e
* WE: ~2 = 1198.6575{{c}}, ~6/5 = 316.7282{{c}}
Badness: 0.0443
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0257{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 19, 34d, 53e }}
Commas: 78/77, 100/99, 144/143, 676/675


POTE generator: ~6/5 = 317.118
Badness (Sintel): 1.47


Map: [&lt;1 0 1 -3 4 0|, &lt;0 6 5 22 -2 14|]
==== 13-limit ====
EDOs: 19, 34d, 53e, 87de
Subgroup: 2.3.5.7.11.13
Badness: 0.0273


=Countercata=
Comma list: 78/77, 100/99, 144/143, 676/675
[[Comma|Commas]]: 15625/15552, 5120/5103


[[POTE tuning|POTE generator]]: 317.121
Mapping: {{mapping| 1 0 1 -3 4 0 | 0 6 5 22 -2 14 }}


Map: [&lt;1 0 1 11|, &lt;0 6 5 -31|]
Optimal tunings:  
[[Wedgie]]: &lt;&lt;6 5 -31 -6 -66 -86||
* WE: ~2 = 1198.8403{{c}}, ~6/5 = 316.8111{{c}}
EDOs: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[53edo|53]], [[87edo|87]], [[140edo|140]], [[333edo|333]], [[473edo|473]], [[806edo|806b]]
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0652{{c}}
[[Badness]]: 0.0521


==11-limit==
{{Optimal ET sequence|legend=0| 19, 34d, 53e }}
[[Comma]]s: 385/384, 2200/2187, 3388/3375


POTE generator: ~6/5 = 317.162
Badness (Sintel): 1.13


Map: [&lt;1 0 1 11 -5|, &lt;0 6 5 -31 32|]
=== Bikleismic ===
EDOs: [[34edo|34]], [[53edo|53]], [[87edo|87]], [[140edo|140]], [[227edo|227]]
Subgroup: 2.3.5.7.11
[[Badness]]: 0.0398


==13-limit==
Comma list: 225/224, 243/242, 4375/4356
[[Comma]]s: 325/324, 352/351, 385/384, 625/624


POTE generator: ~6/5 = 317.162
Mapping: {{mapping| 2 0 2 -6 -1 | 0 6 5 22 15 }}
: mapping generators: ~99/70, ~6/5


Map: [&lt;1 0 1 11 -5 0|, &lt;0 6 5 -31 32 14|]
Optimal tunings:  
EDOs: [[34edo|34]], [[53edo|53]], [[87edo|87]], [[140edo|140]], [[227edo|227]], [[367edo|367e]], [[507edo|507e]]
* WE: ~99/70 = 600.2674{{c}}, ~6/5 = 316.8624{{c}}
[[Badness]]: 0.0202
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 316.7575{{c}}


=Metakleismic=
{{Optimal ET sequence|legend=0| 34d, 72, 322c, 394c }}
[[Comma]]s: 15625/15552, 179200/177147


POTE generator: ~6/5 = 317.314
Badness (Sintel): 0.969


Map: [&lt;1 0 1 -12|, &lt;0 6 5 56|]
==== 13-limit ====
Wedgie: &lt;&lt;6 5 56 -6 72 116||
Subgroup: 2.3.5.7.11.13
EDOs: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[87edo|87]], [[121edo|121]], [[208edo|208]]
[[Badness]]: 0.1635


==11-limit==
Comma list: 169/168, 225/224, 243/242, 325/324
[[Comma]]s: 896/891, 2200/2187, 14700/14641


POTE generator: ~6/5 = 317.311
Mapping: {{mapping| 2 0 2 -6 -1 0 | 0 6 5 22 15 14 }}


Map: [&lt;1 0 1 -12 -5|, &lt;0 6 5 56 32|]
Optimal tunings:  
EDOs: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[87edo|87]], [[121edo|121]], [[208edo|208]]
* WE: ~55/39 = 600.3582{{c}}, ~6/5 = 316.9152{{c}}
[[Badness]]: 0.0486
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 316.7759{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 34d, 72 }}
[[Comma]]s: 325/324, 352/351, 364/363, 625/624


POTE generator: ~6/5 = 317.311
Badness (Sintel): 0.901


Map: [&lt;1 0 1 -12 -5 0|, &lt;0 6 5 56 32 14|]
==== 17-limit ====
EDOs: [[15edo|15]], [[19edo|19]], [[34edo|34]], [[87edo|87]], [[121edo|121]], [[208edo|208]]
Subgroup: 2.3.5.7.11.13.17
[[Badness]]: 0.0244


=Hemikleismic=
Comma list: 169/168, 221/220, 225/224, 243/242, 325/324
Commas: 4000/3969, 6144/6125


[[POTE tuning|POTE generator]]: 158.649
Mapping: {{mapping| 2 0 2 -6 -1 0 5 | 0 6 5 22 15 14 6 }}


Map: [&lt;1 0 1 4|, &lt;0 12 10 -9|]
Optimal tunings:  
EDOs: [[53edo|53]], [[121edo|121]]
* WE: ~17/12 = 600.4210{{c}}, ~6/5 = 316.9282{{c}}
Badness: 0.0521
* CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7578{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 34d, 38df, 72 }}
Commas: 121/120, 176/175, 4000/3969


POTE generator: ~11/10 = 158.677
Badness (Sintel): 0.798


Map: [&lt;1 0 1 4 2|, &lt;0 12 10 -9 11|]
==== 19-limit ====
EDOs: 15, 38, 53, 68, 121e
Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.0380


==13-limit==
Comma list: 153/152, 169/168, 221/220, 225/224, 243/242, 325/324
Commas: 121/120, 176/175, 275/273, 325/324


POTE generator: ~11/10 = 158.655
Mapping: {{mapping| 2 0 2 -6 -1 0 5 -1 | 0 6 5 22 15 14 6 18 }}


Map: [&lt;1 0 1 4 2 0|, &lt;0 12 10 -9 11 28|]
Optimal tunings:  
EDOs: 15, 53, 121e
* WE: ~17/12 = 600.3763{{c}}, ~6/5 = 316.8720{{c}}
Badness: 0.0260
* CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7205{{c}}


=Clyde=
{{Optimal ET sequence|legend=0| 34dh, 38df, 72 }}
[[Comma|Commas]]: 245/243, 3136/3125


7 and 9 limit minimax
Badness (Sintel): 0.959
[|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]]: ~9/7 = 441.335
== Keemun ==
{{Main| Keemun }}


Algebraic generator: real root of 5x^3-6x-3, the Poussami generator. Approximately 441.309 [[Cent|cents]]. Associated recurrence relationship quickly converges.
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 6 6 12|, &lt;0 -12 -10 -25|]
[[Comma list]]: 49/48, 126/125
[[Generator|Generators]]: 2, 9/7
[[Edo|Edos]]: [[19edo|19]], [[49edo|49]], [[68edo|68]], [[87edo|87]], [[155edo|155]]
Badness: 0.0473


==11-limit==
{{Mapping|legend=1| 1 0 1 2 | 0 6 5 3 }}
Commas: 245/243, 3136/3125, 385/384


POTE generator: ~9/7 = 441.355
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1202.6235{{c}}, ~6/5 = 317.1646{{c}}
: [[error map]]: {{val| +2.624 +1.033 +2.133 -12.085 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.8293{{c}}
: error map: {{val| 0.000 -0.979 -2.167 -18.388 }}


Map: [&lt;1 6 6 12 -5|, &lt;0 -12 -10 -25 23|]
[[Tuning ranges]]:
EDOs: 19, 68, 87, 329bd, 419bd, 503bd, 590bd
* 7-odd-limit [[diamond monotone]]: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
Badness: 0.0474
* 9-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [308.744, 322.942]


==13-limit==
{{Optimal ET sequence|legend=1| 15, 19, 53d, 72dd }}
Commas: 196/195, 245/243, 385/384, 625/624


POTE generator: ~9/7 = 441.363
[[Badness]] (Sintel): 0.694


Map: [&lt;1 6 6 12 -5 14|, &lt;0 -12 -10 -25 23 -28|]
=== 11-limit ===
EDOs: 19, 68, 87, 503bdf, 590bdf
Subgroup: 2.3.5.7.11
Badness: 0.0268


=Bikleismic=
Comma list: 49/48, 56/55, 100/99
Commas: 225/224, 243/242, 4375/4356


POTE generator: ~6/5 = 316.721
Mapping: {{mapping| 1 0 1 2 4 | 0 6 5 3 -2 }}


Map: [&lt;2 0 2 -6 -1|, &lt;0 6 5 22 15|]
Optimal tunings:  
EDOs: 72, 106, 178, 250, 322c, 394c, 466bc, 538bc, 610bc
* WE: ~2 = 1199.7353{{c}}, ~6/5 = 317.5055{{c}}
Badness: 0.0293
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5546{{c}}


==13-limit==
Tuning ranges:
Commas: 169/168, 225/224, 243/242, 325/324
* 11-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]


POTE generator: ~6/5 = 316.726
{{Optimal ET sequence|legend=0| 15, 19, 34 }}


Map: [&lt;2 0 2 -6 -1 0|, &lt;0 6 5 22 15 14|]
Badness (Sintel): 0.906
EDOs: 72, 106, 322cff, 394cff, 466bcff, 538bcfff
Badness: 0.0218


=Tritikleismic=  
==== 13-limit ====
[[Comma]]s: 15625/15552, 1029/1024
Subgroup: 2.3.5.7.11.13


[[POTE tuning|POTE generator]]: 316.872
Comma list: 49/48, 56/55, 65/64, 100/99


Map: [&lt;3 0 3 10|, &lt;0 6 5 -2|]
Mapping: {{mapping| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}
[[Wedgie]]: &lt;&lt;18 15 -6 -18 -60 -56||
EDOs: [[12edo|12]], [[15edo|15]], [[57edo|57]], [[72edo|72]], [[159edo|159]], [[231edo|231]]
[[Badness]]: 0.0563


==11-limit==
Optimal tunings:
[[Comma]]s: 385/384, 441/440, 4000/3993
* WE: ~2 = 1201.8360{{c}}, ~6/5 = 317.0958{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6829{{c}}


[[POTE tuning|POTE generator]]: 316.881
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]


Map: [&lt;3 0 3 10 8|, &lt;0 6 5 -2 3|]
{{Optimal ET sequence|legend=0| 4, 15f, 19 }}
EDOs: [[12edo|12]], [[15edo|15]], [[57edo|57]], [[72edo|72]], [[159edo|159]], [[231edo|231]]
[[Badness]]: 0.0193


==13-limit==
Badness (Sintel): 1.23
[[Comma]]s: 325/324, 364/363, 441/440, 625/624


[[POTE tuning|POTE generator]]: 316.959
==== Kema ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;3 0 3 10 8 0|, &lt;0 6 5 -2 3 14|]
Comma list: 49/48, 56/55, 91/90, 100/99
EDOs: [[12edo|12]], [[15edo|15]], [[72edo|72]], 87, 159, 867, 1026
[[Badness]]: 0.0157


=Quadritikleismic=
Mapping: {{mapping| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}
[[Comma]]s: 15625/15552, 2401/2400


[[POTE tuning|POTE generator]]: 316.9999
Optimal tunings:  
* WE: ~2 = 1199.7816{{c}}, ~6/5 = 317.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.4070{{c}}


Map: [&lt;4 0 4 7|, &lt;0 6 5 4|]
Tuning ranges:  
Wedgie: &lt;&lt;24 20 16 -24 -42 -19||
* 13-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
EDOs: [[68edo|68]], [[72edo|72]], [[140edo|140]], [[212edo|212]], [[1200edo|1200]]
* 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
[[Badness]]: 0.0392
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]


==11-limit==
{{Optimal ET sequence|legend=0| 15, 19, 34 }}
Commas: 385/384, 1375/1372, 6250/6237


[[POTE tuning|POTE generator]]: 316.925
Badness (Sintel): 0.940


Map: [&lt;4 0 4 7 17|, &lt;0 6 5 4 -3|]
==== Kumbaya ====
EDOs: [[68edo|68]], [[72edo|72]], [[140edo|140]], [[212edo|212]], [[284edo|284]], [[496edo|496]], [[780edo|780]]
Subgroup: 2.3.5.7.11.13
[[Badness]]: 0.0234


==13-limit==
Comma list: 40/39, 49/48, 56/55, 66/65
Commas: 325/324, 385/384, 625/624, 1573/1568


[[POTE tuning|POTE generator]]: 316.989
Mapping: {{mapping| 1 0 1 2 4 4 | 0 6 5 3 -2 -1 }}


Map: [&lt;4 0 4 7 17 0|, &lt;0 6 5 4 -3 14|]
Optimal tunings:  
EDOs: [[68edo|68]], [[72edo|72]], [[140edo|140]], [[212edo|212]]
* WE: ~2 = 1196.7615{{c}}, ~6/5 = 317.7353{{c}}
[[Badness]]: 0.0187
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.4059{{c}}


=Kleiboh=
{{Optimal ET sequence|legend=0| 4, 11b, 15 }}
Commas: 1728/1715, 3125/3087


POTE generator: ~25/21 = 294.303
Badness (Sintel): 1.31


Map: [&lt;1 6 6 6|, &lt;0 -18 -15 -13|]
=== Qeema ===
Wedgie: &lt;&lt;18 15 13 -18 -30 -12||
Subgroup: 2.3.5.7.11
EDOs: 49, 53, 314d
Badness: 0.0765


==11-limit==
Comma list: 45/44, 49/48, 126/125
Commas: 176/175, 540/539, 3125/3087


POTE generator: ~25/21 = 294.181
Mapping: {{mapping| 1 0 1 2 -1 | 0 6 5 3 17 }}


Map: [&lt;1 6 6 6 14|, &lt;0 -18 -15 -13 -43|]
Optimal tunings:  
EDOs: 49, 53, 102d, 155d
* WE: ~2 = 1204.5534{{c}}, ~6/5 = 315.9247{{c}}
Badness: 0.0528
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 315.1686{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 4e, 19, 42bcd }}
Commas: 176/175, 275/273, 325/324, 540/539


POTE generator: ~13/11 = 294.187
Badness (Sintel): 1.32


Map: [&lt;1 6 6 6 14 14|, &lt;0 -18 -15 -13 -43 -42|]
==== 13-limit ====
EDOs: 53, 102df, 155d
Subgroup: 2.3.5.7.11.13
Badness: 0.0311


=Novemkleismic=
Comma list: 45/44, 49/48, 78/77, 126/125
Commas: 15625/15552, 40353607/40310784


POTE generator: ~6/5 = 317.005
Mapping: {{mapping| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}


Map: [&lt;9 0 9 11|, &lt;0 6 5 6|]
Optimal tunings:  
Wedgie: &lt;&lt;54 45 54 -54 -66 -1||
* WE: ~2 = 1204.4937{{c}}, ~6/5 = 316.2241{{c}}
EDOs: 72, 261, 333, 405, 477c, 882c
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 315.4748{{c}}
Badness: 0.1934


==11-limit==
{{Optimal ET sequence|legend=0| 4ef, 19 }}
Commas: 1375/1372, 4000/3993, 15625/15552


POTE generator: ~6/5 = 317.010
Badness (Sintel): 1.22


Map: [&lt;9 0 9 11 24|, &lt;0 6 5 6 3|]
=== Darjeeling ===
EDOs: 72, 261, 333, 405, 882c
Subgroup: 2.3.5.7.11
Badness: 0.05172


==13-limit==
Comma list: 49/48, 55/54, 77/75
Commas: 325/324, 625/624, 1375/1372, 4000/3993


POTE generator: ~6/5 = 317.086
Mapping: {{mapping| 1 0 1 2 0 | 0 6 5 3 13 }}


Map: [&lt;9 0 9 11 24 0|, &lt;0 6 5 6 3 14|]
Optimal tunings:  
EDOs: 72, 261, 333, 738cf, 1071bcf
* WE: ~2 = 1201.6569{{c}}, ~6/5 = 318.0942{{c}}
Badness: 0.0391
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.8547{{c}}


=Sqrtphi=
{{Optimal ET sequence|legend=0| 15, 19e, 34e }}
Commas: 15625/15552, 16875/16807


[[POTE tuning|POTE generator]]: ~125/98 = 416.603 cents
Badness (Sintel): 0.914


Sqrt(phi) = 416.545 cents
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 12 11 16|, &lt;0 -30 -25 -38|]
Comma list: 49/48, 55/54, 66/65, 77/75
EDOs: [[49edo|49]], 72, [[193edo|193]], [[265edo|265]]
Badness: 0.0704


==11-limit==
Mapping: {{mapping| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}
Commas: 540/539, 1375/1372, 4375/4356


POTE generator: ~14/11 = 416.604
Optimal tunings:  
* WE: ~2 = 1201.9324{{c}}, ~6/5 = 317.8090{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5170{{c}}


Map: [&lt;1 12 11 16 17|, &lt;0 -30 -25 -38 -39|]
{{Optimal ET sequence|legend=0| 15, 19e, 34e }}
EDOs: 49, 72, 193, 265
Badness: 0.0255


==13-limit==
Badness (Sintel): 0.886
Commas: 325/324, 364/363, 625/624, 1375/1372


POTE generator: ~14/11 = 416.585
== Catalan ==
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 12 11 16 17 28|, &lt;0 -30 -25 -38 -39 -70|]
[[Comma list]]: 64/63, 15625/15552
EDOs: 72, 121, 193
Badness: 0.0200


==17-limit==
{{Mapping|legend=1| 1 0 1 6 | 0 6 5 -12 }}
Commas: 325/324, 364/363, 375/374, 540/539, 595/594


POTE generator: ~14/11 = 416.585
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1197.1789{{c}}, ~6/5 = 317.5185{{c}}
: [[error map]]: {{val| -2.821 +3.156 -1.542 +4.025 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 318.2411{{c}}
: error map: {{val| 0.000 +7.492 +4.892 +12.281 }}


Map: [&lt;1 12 11 16 17 28 27|, &lt;0 -30 -25 -38 -39 -70 -66|]
[[Tuning ranges]]:  
EDOs: 72, 121, 193
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
Badness: 0.0130
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 319.265]


==19-limit==
{{Optimal ET sequence|legend=1| 15, 34d, 49, 132bcdd, 181bbcddd }}
Commas: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594


POTE generator: ~14/11 = 416.580
[[Badness]] (Sintel): 2.40


Map: [&lt;1 12 11 16 17 28 27 -2|, &lt;0 -30 -25 -38 -39 -70 -66 18|]
=== 11-limit ===
EDOs: 72, 121, 193
Subgroup: 2.3.5.7.11
Badness: 0.0147


==Scales==
Comma list: 64/63, 100/99, 1331/1323
[[sqrtphi17]]
[[sqrtphi23]]
[[sqrtphi49]]


==Music==
Mapping: {{mapping| 1 0 1 6 4 | 0 6 5 -12 -2 }}
[[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</pre></div>
Optimal tunings:
<h4>Original HTML content:</h4>
* WE: ~2 = 1197.0368{{c}}, ~6/5 = 317.4956{{c}}
<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-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Kleismic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;&lt;!-- ws:start:WikiTextTocRule:108:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&lt;!-- ws:start:WikiTextTocRule:109: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Seven limit children"&gt;Seven limit children&lt;/a&gt;&lt;/div&gt;
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 318.2672{{c}}
&lt;!-- ws:end:WikiTextTocRule:109 --&gt;&lt;!-- ws:start:WikiTextTocRule:110: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Keemun"&gt;Keemun&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:110 -
Tuning ranges:
* 11-odd-limit diamond monotone: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
* 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 324.341]
 
{{Optimal ET sequence|legend=0| 15, 34d, 49, 181bbcdddeee }}
 
Badness (Sintel): 1.22
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 64/63, 100/99, 144/143, 275/273
 
Mapping: {{mapping| 1 0 1 6 4 0 | 0 6 5 -12 -2 14 }}
 
Optimal tunings:
* WE: ~2 = 1196.8961{{c}}, ~6/5 = 317.3837{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 318.1621{{c}}
 
{{Optimal ET sequence|legend=0| 15, 34d, 49f, 83def, 132bcddeefff }}
 
Badness (Sintel): 1.09
 
== Countercata ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 15625/15552
 
{{Mapping|legend=1| 1 0 1 11 | 0 6 5 -31 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9172{{c}}, ~6/5 = 317.0995{{c}}
: [[error map]]: {{val| -0.083 +0.642 -0.899 +0.178 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.1220{{c}}
: error map: {{val| 0.000 +0.777 -0.704 +0.391 }}
 
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [316.667, 317.647] (19\72 to 9\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]
 
{{Optimal ET sequence|legend=1| 19d, 34, 53, 87, 140, 333, 473 }}
 
[[Badness]] (Sintel): 1.32
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 2200/2187, 3388/3375
 
Mapping: {{mapping| 1 0 1 11 -5 | 0 6 5 -31 32 }}
 
Optimal tunings:
* WE: ~2 = 1200.0980{{c}}, ~6/5 = 317.1879{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1623{{c}}
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
* 11-odd-limit diamond tradeoff:
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