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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>2010-12-10 13:39:02 UTC</tt>.<br>
| de = Hanson-Kleismisch
: The original revision id was <tt>187170975</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, 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 and catakleismic both have octave period and use the minor third as a generator; catakleismic defines 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:
* [[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 }}


===Clyde===
[[Tuning ranges]]:
[[Comma|Commas]]: 245/243, 3136/3125
* [[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)


7 and 9 limit minimax
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 458, 511c, …, 829c, 882c }}
[|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
[[Badness]] (Sintel): 0.310


Algebraic generator: real root of 5x^3-6x-3, the Poussami generator. Approximately 441.309 [[Cent|cents]]. Associated recurrence relationship quickly converges.
=== 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 6 6 12|, &lt;0 -12 -10 -25|]
[[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.
[[Generator|Generators]]: 2, 9/7
 
[[Edo|Edos]]: [[19edo|19]], [[49edo|49]], [[68edo|68]], [[87edo|87]], [[155edo|155]]</pre></div>
Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts.
<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-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Kleismic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 5-limit parent comma for the kleismic family is 15625/15552, the kleisma. Its monzo is |-6 -5 6&amp;gt;, and flipping that yields &amp;lt;&amp;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 &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt; is possible. Other tunings include &lt;a class="wiki_link" href="/72edo"&gt;72edo&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87edo&lt;/a&gt; and &lt;a class="wiki_link" href="/140edo"&gt;140edo&lt;/a&gt;.&lt;br /&gt;
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]].
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 317.007&lt;br /&gt;
== Catakleismic ==
&lt;br /&gt;
{{Main| Catakleismic }}
Map: [&amp;lt;1 0 1|, &amp;lt;0 6 5|]&lt;br /&gt;
 
&lt;br /&gt;
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.
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Seven limit children&lt;/h2&gt;
 
The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives keemun, 4375/4374, the ragisma, gives catakleismic, 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 and catakleismic both have octave period and use the minor third as a generator; catakleismic defines 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.&lt;br /&gt;
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]])}.  
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Seven limit children-Keemun"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Keemun&lt;/h3&gt;
=== 7-limit ===
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 49/48, 126/125&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 316.473&lt;br /&gt;
[[Comma list]]: 225/224, 4375/4374
&lt;br /&gt;
 
Map: [&amp;lt;1 0 1 2|, &amp;lt;0 6 5 3|]&lt;br /&gt;
{{Mapping|legend=1| 1 0 1 -3 | 0 6 5 22 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Seven limit children-Clyde"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Clyde&lt;/h3&gt;
[[Optimal tuning]]s:
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 245/243, 3136/3125&lt;br /&gt;
* [[WE]]: ~2 = 1200.5965{{c}}, ~6/5 = 316.8893{{c}}
&lt;br /&gt;
: [[error map]]: {{val| +0.596 -0.619 -1.271 +0.948 }}
7 and 9 limit minimax&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.7705{{c}}
[|1 0 0 0&amp;gt;, |6/25 0 0 12/25&amp;gt;, |6/5 0 0 2/5&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
: error map: {{val| 0.000 -1.332 -2.461 +0.126 }}
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7&lt;br /&gt;
 
&lt;br /&gt;
[[Tuning ranges]]:
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 441.335&lt;br /&gt;
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [315.789, 317.647] (5\19 to 9\34)
&lt;br /&gt;
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 317.263]
Algebraic generator: real root of 5x^3-6x-3, the Poussami generator. Approximately 441.309 &lt;a class="wiki_link" href="/Cent"&gt;cents&lt;/a&gt;. Associated recurrence relationship quickly converges.&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 19, 34d, 53, 72, 197, 269c }}
Map: [&amp;lt;1 6 6 12|, &amp;lt;0 -12 -10 -25|]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 9/7&lt;br /&gt;
[[Badness]] (Sintel): 0.544
&lt;a class="wiki_link" href="/Edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/49edo"&gt;49&lt;/a&gt;, &lt;a class="wiki_link" href="/68edo"&gt;68&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87&lt;/a&gt;, &lt;a class="wiki_link" href="/155edo"&gt;155&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
 
==== 2.3.5.7.13 subgroup ====
Subgroup: 2.3.5.7.13
 
Comma list: 169/168, 225/224, 325/324
 
Subgroup-val mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}
 
Optimal tunings:  
* WE: ~2 = 1200.7838{{c}}, ~6/5 = 316.9478{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7939{{c}}
 
{{Optimal ET sequence|legend=0| 19, 34d, 53, 72, 125f, 197f }}
 
Badness (Sintel): 0.410
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 4375/4374
 
Mapping: {{mapping| 1 0 1 -3 9 | 0 6 5 22 -21 }}
 
Optimal tunings:
* WE: ~2 = 1200.6524{{c}}, ~6/5 = 316.8911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7267{{c}}
 
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]
 
{{Optimal ET sequence|legend=0| 19, 53, 72, 197e, 269ce, 341ce }}
 
Badness (Sintel): 0.722
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 225/224, 325/324, 385/384
 
Mapping: {{mapping| 1 0 1 -3 9 0 | 0 6 5 22 -21 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.7982{{c}}, ~6/5 = 316.9482{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.7491{{c}}
 
Tuning ranges:  
* 13- and 15-odd-limit diamond monotone: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
 
{{Optimal ET sequence|legend=0| 19, 53, 72, 125f, 197ef }}
 
Badness (Sintel): 0.698
 
=== Cataclysmic ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 2200/2187
 
Mapping: {{mapping| 1 0 1 -3 -5 | 0 6 5 22 32 }}
 
Optimal tunings:
* WE: ~2 = 1199.9590{{c}}, ~6/5 = 317.0315{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0403{{c}}
 
{{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}}, ~6/5 = 316.9152{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 316.7759{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 72 }}
 
Badness (Sintel): 0.901
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 225/224, 243/242, 325/324
 
Mapping: {{mapping| 2 0 2 -6 -1 0 5 | 0 6 5 22 15 14 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.4210{{c}}, ~6/5 = 316.9282{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7578{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 38df, 72 }}
 
Badness (Sintel): 0.798
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 153/152, 169/168, 221/220, 225/224, 243/242, 325/324
 
Mapping: {{mapping| 2 0 2 -6 -1 0 5 -1 | 0 6 5 22 15 14 6 18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.3763{{c}}, ~6/5 = 316.8720{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~6/5 = 316.7205{{c}}
 
{{Optimal ET sequence|legend=0| 34dh, 38df, 72 }}
 
Badness (Sintel): 0.959
 
== Keemun ==
{{Main| Keemun }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 49/48, 126/125
 
{{Mapping|legend=1| 1 0 1 2 | 0 6 5 3 }}
 
[[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 }}
 
[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
* 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]
 
{{Optimal ET sequence|legend=1| 15, 19, 53d, 72dd }}
 
[[Badness]] (Sintel): 0.694
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 56/55, 100/99
 
Mapping: {{mapping| 1 0 1 2 4 | 0 6 5 3 -2 }}
 
Optimal tunings:
* WE: ~2 = 1199.7353{{c}}, ~6/5 = 317.5055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5546{{c}}
 
Tuning ranges:
* 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]
 
{{Optimal ET sequence|legend=0| 15, 19, 34 }}
 
Badness (Sintel): 0.906
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 56/55, 65/64, 100/99
 
Mapping: {{mapping| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}
 
Optimal tunings:
* WE: ~2 = 1201.8360{{c}}, ~6/5 = 317.0958{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6829{{c}}
 
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]
 
{{Optimal ET sequence|legend=0| 4, 15f, 19 }}
 
Badness (Sintel): 1.23
 
==== Kema ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 56/55, 91/90, 100/99
 
Mapping: {{mapping| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}
 
Optimal tunings:
* WE: ~2 = 1199.7816{{c}}, ~6/5 = 317.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.4070{{c}}
 
Tuning ranges:
* 13-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
 
{{Optimal ET sequence|legend=0| 15, 19, 34 }}
 
Badness (Sintel): 0.940
 
==== Kumbaya ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 49/48, 56/55, 66/65
 
Mapping: {{mapping| 1 0 1 2 4 4 | 0 6 5 3 -2 -1 }}
 
Optimal tunings:
* WE: ~2 = 1196.7615{{c}}, ~6/5 = 317.7353{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.4059{{c}}
 
{{Optimal ET sequence|legend=0| 4, 11b, 15 }}
 
Badness (Sintel): 1.31
 
=== Qeema ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 49/48, 126/125
 
Mapping: {{mapping| 1 0 1 2 -1 | 0 6 5 3 17 }}
 
Optimal tunings:
* WE: ~2 = 1204.5534{{c}}, ~6/5 = 315.9247{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 315.1686{{c}}
 
{{Optimal ET sequence|legend=0| 4e, 19, 42bcd }}
 
Badness (Sintel): 1.32
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 45/44, 49/48, 78/77, 126/125
 
Mapping: {{mapping| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}
 
Optimal tunings:
* WE: ~2 = 1204.4937{{c}}, ~6/5 = 316.2241{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 315.4748{{c}}
 
{{Optimal ET sequence|legend=0| 4ef, 19 }}
 
Badness (Sintel): 1.22
 
=== Darjeeling ===
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 55/54, 77/75
 
Mapping: {{mapping| 1 0 1 2 0 | 0 6 5 3 13 }}
 
Optimal tunings:
* WE: ~2 = 1201.6569{{c}}, ~6/5 = 318.0942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.8547{{c}}
 
{{Optimal ET sequence|legend=0| 15, 19e, 34e }}
 
Badness (Sintel): 0.914
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 55/54, 66/65, 77/75
 
Mapping: {{mapping| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}
 
Optimal tunings:
* WE: ~2 = 1201.9324{{c}}, ~6/5 = 317.8090{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.5170{{c}}
 
{{Optimal ET sequence|legend=0| 15, 19e, 34e }}
 
Badness (Sintel): 0.886
 
== Catalan ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 64/63, 15625/15552
 
{{Mapping|legend=1| 1 0 1 6 | 0 6 5 -12 }}
 
[[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 }}
 
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 319.265]
 
{{Optimal ET sequence|legend=1| 15, 34d, 49, 132bcdd, 181bbcddd }}
 
[[Badness]] (Sintel): 2.40
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 64/63, 100/99, 1331/1323
 
Mapping: {{mapping| 1 0 1 6 4 | 0 6 5 -12 -2 }}
 
Optimal tunings:
* WE: ~2 = 1197.0368{{c}}, ~6/5 = 317.4956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 318.2672{{c}}
 
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: ~6/5 = [315.641, 317.370]
 
{{Optimal ET sequence|legend=0| 34, 53, 87, 140, 227, 367e }}
 
Badness (Sintel): 1.31
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 385/384, 625/624
 
Mapping: {{mapping| 1 0 1 11 -5 0 | 0 6 5 -31 32 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.0936{{c}}, ~6/5 = 317.1864{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1622{{c}}
 
Tuning ranges:
* 13-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
* 15-odd-limit diamond monotone: ~6/5 = [316.981, 317.241] (14\53 to 23\87)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]
 
{{Optimal ET sequence|legend=0| 34, 53, 87, 140, 367e, 507e }}
 
Badness (Sintel): 0.833
 
== Metakleismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 15625/15552, 179200/177147
 
{{Mapping|legend=1| 1 0 1 -12 | 0 6 5 56 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5969{{c}}, ~6/5 = 317.2079{{c}}
: [[error map]]: {{val| -0.403 +1.292 -0.678 -0.349 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 317.3071{{c}}
: error map: {{val| 0.000 +1.887 +0.222 +0.370 }}
 
{{Optimal ET sequence|legend=1| 34d, 87, 121, 208, 537b }}
 
[[Badness]] (Sintel): 4.14
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 896/891, 2200/2187, 14700/14641
 
Mapping: {{mapping| 1 0 1 -12 -5 | 0 6 5 56 32 }}
 
Optimal tunings:
* WE: ~2 = 1199.5425{{c}}, ~6/5 = 317.1901{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.3020{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}
 
Badness (Sintel): 1.61
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 364/363, 625/624
 
Mapping: {{mapping| 1 0 1 -12 -5 0 | 0 6 5 56 32 14 }}
 
Optimal tunings:
* WE: ~2 = 1199.5339{{c}}, ~6/5 = 317.1882{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.3028{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 53d, 87, 121, 208 }}
 
Badness (Sintel): 1.01
 
== Hemikleismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4000/3969, 6144/6125
 
{{Mapping|legend=1| 1 0 1 4 | 0 12 10 -9 }}
: mapping generators: ~2, ~35/32
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3950{{c}}, ~35/32 = 158.5686{{c}}
: [[error map]]: {{val| -0.605 +0.868 -1.233 +1.637 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 158.6338{{c}}
: error map: {{val| 0.000 +1.651 +0.024 +3.470 }}
 
{{Optimal ET sequence|legend=1| 15, 38, 53, 121, 174d, 295d }}
 
[[Badness]] (Sintel): 1.32
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 4000/3969
 
Mapping: {{mapping| 1 0 1 4 2 | 0 12 10 -9 11 }}
 
Optimal tunings:
* WE: ~2 = 1199.8009{{c}}, ~11/10 = 158.6508{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 158.6717{{c}}
 
{{Optimal ET sequence|legend=0| 15, 38, 53, 68, 121e }}
 
Badness (Sintel): 1.26
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 176/175, 275/273, 325/324
 
Mapping: {{mapping| 1 0 1 4 2 0 | 0 12 10 -9 11 28 }}
 
Optimal tunings:
* WE: ~2 = 1199.7952{{c}}, ~11/10 = 158.6279{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 158.6493{{c}}
 
{{Optimal ET sequence|legend=0| 15, 38f, 53, 121e }}
 
Badness (Sintel): 1.07
 
== Clyde ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 245/243, 3136/3125
 
{{Mapping|legend=1| 1 -6 -4 -13 | 0 12 10 25 }}
: mapping generators: ~2, ~14/9
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8369{{c}}, ~14/9 = 758.5621{{c}}
: [[error map]]: {{val| -0.163 +1.769 -0.040 -2.652 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 758.6554{{c}}
: error map: {{val| 0.000 +1.910 +0.240 -2.441 }}
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~14/9 = {{monzo| 13/25 0 0 1/25 }}
: {{monzo list| 1 0 0 0 | 6/25 0 0 12/25 | 6/5 0 0 2/5 | 0 0 0 1 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
 
[[Algebraic generator]]: real root of 5''x''<sup>3</sup> - 6''x'' - 3, the Poussami generator. Approximately 441.309 [[cent]]s. Associated recurrence relationship quickly converges.
 
{{Optimal ET sequence|legend=1| 19, 49, 68, 87, 155, 242 }}
 
[[Badness]] (Sintel): 1.20
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/243, 385/384, 3136/3125
 
Mapping: {{mapping| 1 -6 -4 -13 18 | 0 12 10 25 -23 }}
 
Optimal tunings:
* WE: ~2 = 1199.9620{{c}}, ~14/9 = 758.6210{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 758.6445{{c}}
 
{{Optimal ET sequence|legend=0| 19, 49e, 68, 87 }}
 
Badness (Sintel): 1.57
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 245/243, 385/384, 625/624
 
Mapping: {{mapping| 1 -6 -4 -13 18 -14 | 0 12 10 25 -23 28 }}
 
Optimal tunings:
* WE: ~2 = 1199.9292{{c}}, ~14/9 = 758.5919{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 758.6355{{c}}
 
{{Optimal ET sequence|legend=0| 19, 68, 87 }}
 
Badness (Sintel): 1.11
 
== Tritikleismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1029/1024, 15625/15552
 
{{Mapping|legend=1| 3 0 3 10 | 0 6 5 -2 }}
: mapping generators: ~63/50, ~6/5
 
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.1845{{c}}, ~6/5 = 317.0178{{c}} (~21/20 = 83.1667{{c}})
: [[error map]]: {{val| +0.553 +0.152 -0.671 -1.017 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~6/5 = 316.9129{{c}} (~21/20 = 83.0871{{c}})
: error map: {{val| 0.000 -0.478 -1.749 -2.652 }}
 
[[Minimax tuning]]:
* [[7-odd-limit]]: ~6/5 = {{monzo| 1/3 0 1/7 -1/7 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 2 0 6/7 -6/7 }}, {{monzo| 8/3 0 5/7 -5/7 }}, {{monzo| 8/3 0 -2/7 2/7 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~6/5 = {{monzo| 5/21 1/7 0 -1/14 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 46/21 5/7 0 -5/14 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
 
{{Optimal ET sequence|legend=1| 15, 42bc, 57, 72, 159, 231, 765ccddd }}
 
[[Badness]] (Sintel): 1.43
 
; Music
* [https://www.youtube.com/watch?v=vdjhC9i5KF4 ''Four Short Experiments in Octave Stretched 42edo''] (2024) by [[Budjarn Lambeth]]
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 4000/3993
 
Mapping: {{mapping| 3 0 3 10 8 | 0 6 5 -2 3 }}
 
Optimal tunings:
* WE: ~44/35 = 400.1571{{c}}, ~6/5 = 317.0058{{c}} (~21/20 = 83.1514{{c}})
* CWE: ~44/35 = 400.0000{{c}}, ~6/5 = 316.9154{{c}} (~21/20 = 83.0846{{c}})
 
Minimax tuning:
* 11-odd-limit: ~6/5 = {{monzo| 5/21 1/7 0 -1/14 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 46/21 5/7 0 -5/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 71/21 3/7 0 -3/14 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.9/7
 
{{Optimal ET sequence|legend=0| 15, 42bc, 57, 72, 159, 231 }}
 
Badness (Sintel): 0.639
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 364/363, 385/384, 625/624
 
Mapping: {{mapping| 3 0 3 10 8 0 | 0 6 5 -2 3 14 }}
 
Optimal tunings:
* WE: ~44/35 = 400.1514{{c}}, ~6/5 = 317.0785{{c}} (~21/20 = 83.0729{{c}})
* CWE: ~44/35 = 400.0000{{c}}, ~6/5 = 316.9896{{c}} (~21/20 = 83.0104{{c}})
 
{{Optimal ET sequence|legend=0| 15, 57f, 72, 87, 159 }}
 
Badness (Sintel): 0.647
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 273/272, 325/324, 364/363, 375/374, 385/384
 
Mapping: {{mapping| 3 0 3 10 8 0 -2 | 0 6 5 -2 3 14 18 }}
 
Optimal tunings:
* WE: ~34/27 = 400.1604{{c}}, ~6/5 = 317.0353{{c}} (~21/20 = 83.1251{{c}})
* CWE: ~34/27 = 400.0000{{c}}, ~6/5 = 316.9384{{c}} (~21/20 = 83.0616{{c}})
 
{{Optimal ET sequence|legend=0| 15g, 57fg, 72, 159, 231f }}
 
Badness (Sintel): 0.690
 
== Marfifths ==
Named by [[Xenllium]] in 2021, marfifths tempers out the 10976/10935, the [[hemimage comma]], and may be described as the {{nowrap| 19 & 140 }} temperament. It is generated by a marvel fourth of [[75/56]] (or a marvel fifth of [[112/75]]), three of which minus an octave make the hanson generator of ~6/5. Its [[ploidacot]] is zeta-18-cot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 15625/15552
 
{{Mapping|legend=1| 1 -6 -4 -17 | 0 18 15 47 }}
: mapping generators: ~2, ~75/56
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0223{{c}}, ~75/56 = 505.7147{{c}}
: [[error map]]: {{val| +0.022 +0.775 -0.683 -0.615 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/56 = 505.7060{{c}}
: error map: {{val| 0.000 +0.753 -0.724 -0.643 }}
 
{{Optimal ET sequence|legend=1| 19, …, 121, 140, 579, 719 }}
 
[[Badness]] (Sintel): 1.61
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 6250/6237, 10976/10935
 
Mapping: {{mapping| 1 -6 -4 -17 22 | 0 18 15 47 -44 }}
 
Optimal tunings:
* WE: ~2 = 1200.2484{{c}}, ~75/56 = 505.7882{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.6853{{c}}
 
{{Optimal ET sequence|legend=0| 19, 121e, 140, 159, 299 }}
 
Badness (Sintel): 1.95
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 385/384, 625/624, 10976/10935
 
Mapping: {{mapping| 1 -6 -4 -17 22 -14 | 0 18 15 47 -44 42 }}
 
Optimal tunings:
* WE: ~2 = 1200.2747{{c}}, ~75/56 = 505.8019{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.6883{{c}}
 
{{Optimal ET sequence|legend=0| 19, 121e, 140, 159, 299 }}
 
Badness (Sintel): 1.24
 
=== Diatessic ===
Diatessic may be described as {{nowrap| 121 & 140 }} and is closely related to the Diatess tuning (generator: 505.727281 cents).
 
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 2200/2187, 5632/5625
 
Mapping: {{mapping| 1 -6 -4 -17 -37 | 0 18 15 47 96 }}
 
Optimal tunings:
* WE: ~2 = 1199.7886{{c}}, ~75/56 = 505.6513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7366{{c}}
 
{{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}
 
Badness (Sintel): 2.02
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 625/624, 1375/1372
 
Mapping: {{mapping| 1 -6 -4 -17 -37 -14 | 0 18 15 47 96 42 }}
 
Optimal tunings:
* WE: ~2 = 1199.7996{{c}}, ~75/56 = 505.6558{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7366{{c}}
 
{{Optimal ET sequence|legend=0| 19e, …, 121, 140, 261, 401 }}
 
Badness (Sintel): 1.18
 
=== Marf ===
Marf may be described as {{nowrap| 19 & 121 }}. It has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.
 
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 896/891, 15625/15552
 
Mapping: {{mapping| 1 -6 -4 -17 14 | 0 18 15 47 -25 }}
 
Optimal tunings:
* WE: ~2 = 1199.3198{{c}}, ~75/56 = 505.4822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7607{{c}}
 
{{Optimal ET sequence|legend=0| 19, 102d, 121 }}
 
Badness (Sintel): 2.48
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 540/539, 625/624, 896/891
 
Mapping: {{mapping| 1 -6 -4 -17 14 -14 | 0 18 15 47 -25 42 }}
 
Optimal tunings:
* WE: ~2 = 1199.3368{{c}}, ~75/56 = 505.4919{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/56 = 505.7627{{c}}
 
{{Optimal ET sequence|legend=0| 19, 102df, 121 }}
 
Badness (Sintel): 1.58
 
== Kleiboh ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 3125/3087
 
{{Mapping|legend=1| 1 -12 -9 -7 | 0 18 15 13 }}
: mapping generators: ~2, ~42/25
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5290{{c}}, ~42/25 = 905.3417{{c}}
: [[error map]]: {{val| -0.471 -0.152 -1.949 +3.914 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 905.6741{{c}}
: error map: {{val| 0.000 +0.178 -1.203 +4.937 }}
 
{{Optimal ET sequence|legend=1| 49, 53 }}
 
[[Badness]] (Sintel): 1.93
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 540/539, 3125/3087
 
Mapping: {{mapping| 1 -12 -9 -7 -29 | 0 18 15 13 43 }}
 
Optimal tunings:
* WE: ~2 = 1199.1389{{c}}, ~42/25 = 905.1688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 905.7840{{c}}
 
{{Optimal ET sequence|legend=0| 49, 53, 102d }}
 
Badness (Sintel): 1.75
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 275/273, 325/324, 540/539
 
Mapping: {{mapping| 1 -12 -9 -7 -29 -28 | 0 18 15 13 43 42 }}
 
Optimal tunings:
* WE: ~2 = 1199.1517{{c}}, ~22/13 = 905.1727{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 905.7801{{c}}
 
{{Optimal ET sequence|legend=0| 49f, 53, 102df }}
 
Badness (Sintel): 1.28
 
== Quadritikleismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 15625/15552
 
{{Mapping|legend=1| 4 0 4 7 | 0 6 5 4 }}
: mapping generators: ~25/21, ~6/5
 
[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 300.0520{{c}}, ~6/5 = 317.0548{{c}} (~126/125 = 17.0029{{c}})
: [[error map]]: {{val| +0.208 +0.374 -0.832 -0.243 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0301{{c}} (~126/125 = 17.0301{{c}})
: error map: {{val| 0.000 +0.225 -1.163 -0.706 }}
 
{{Optimal ET sequence|legend=1| 68, 72, 140, 212, 776cd, 988ccd, 1200ccd }}
 
[[Badness]] (Sintel): 0.993
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1375/1372, 6250/6237
 
Mapping: {{mapping| 4 0 4 7 17 | 0 6 5 4 -3 }}
 
Optimal tunings:
* WE: ~25/21 = 300.0995{{c}}, ~6/5 = 317.0298{{c}} (~100/99 = 16.9303{{c}})
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 316.9540{{c}} (~100/99 = 16.9540{{c}})
 
{{Optimal ET sequence|legend=0| 68, 72, 140, 212, 284, 496ce, 780ccdee }}
 
Badness (Sintel): 0.774
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 385/384, 625/624, 1375/1372
 
Mapping: {{mapping| 4 0 4 7 17 0 | 0 6 5 4 -3 14 }}
 
Optimal tunings:
* WE: ~25/21 = 300.0985{{c}}, ~6/5 = 317.0899{{c}} (~100/99 = 16.9941{{c}})
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
 
{{Optimal ET sequence|legend=0| 68, 72, 140, 212 }}
 
Badness (Sintel): 0.774
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 325/324, 385/384, 442/441, 625/624
 
Mapping: {{mapping| 4 0 4 7 17 0 10 | 0 6 5 4 -3 14 6 }}
 
Optimal tunings:
* WE: ~25/21 = 300.1102{{c}}, ~6/5 = 317.1011{{c}} (~100/99 = 16.9909{{c}})
* CWE: ~25/21 = 300.0000{{c}}, ~6/5 = 317.0155{{c}} (~100/99 = 17.0155{{c}})
 
{{Optimal ET sequence|legend=0| 68, 72, 140, 212g }}
 
Badness (Sintel): 0.651
 
== Marthirds ==
Named by [[Xenllium]] in 2021, marthirds tempers out 2460375/2458624, the [[breeze comma]], and may be described as the {{nowrap| 19 & 193 }} temperament. It is generated by a marvel-comma-flat classical major third, [[56/45]], four of which minus an octave make the hanson generator of [[6/5]]. Its [[ploidacot]] is zeta-24-cot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 15625/15552, 2460375/2458624
 
{{Mapping|legend=1| 1 -6 -4 -19 | 0 24 20 69 }}
: mapping generators: ~2, ~56/45
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1662{{c}}, ~56/45 = 379.3041{{c}}
: [[error map]]: {{val| +0.166 +0.347 -0.896 +0.000 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 379.2552{{c}}
: error map: {{val| 0.000 +0.171 -1.209 -0.214 }}
 
{{Optimal ET sequence|legend=1| 19, …, 193, 212, 617c, 829c }}
 
[[Badness]] (Sintel): 2.64
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 15625/15552, 19712/19683
 
Mapping: {{mapping| 1 -6 -4 -19 -43 | 0 24 20 69 147 }}
 
Optimal tunings:
* WE: ~2 = 1200.1189{{c}}, ~56/45 = 379.2942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}
 
{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405, 617c }}
 
Badness (Sintel): 2.50
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 625/624, 1375/1372, 19712/19683
 
Mapping: {{mapping| 1 -6 -4 -19 -43 -14 | 0 24 20 69 147 56 }}
 
Optimal tunings:
* WE: ~2 = 1200.2154{{c}}, ~56/45 = 379.3236{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 379.2580{{c}}
 
{{Optimal ET sequence|legend=0| 19e, …, 193, 212, 405f, 617cff }}
 
Badness (Sintel): 1.81
 
== Sqrtphi ==
{{Main| Sqrtphi }}
 
Sqrtphi tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 49 & 72 }} temperament. The just value of sqrt(φ) is 416.545 cents, and this temperament gives a close approximation of it.
 
Note that in the data below, the generator is given as its [[octave complement]], which stands in for [[~]][[11/7]] from the [[11-limit]] onwards. Five generators octave reduced make the hanson generator of ~[[6/5]]. The [[ploidacot]] for this temperament is 19-sheared 30-cot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 15625/15552, 16875/16807
 
{{Mapping|legend=1| 1 -18 -14 -22 | 0 30 25 38 }}
: mapping generators: ~2, 196/125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1357{{c}}, ~196/125 = 783.4853{{c}}
: [[error map]]: {{val| +0.136 +0.163 -1.080 +0.632 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/125 = 783.4009{{c}}
: error map: {{val| 0.000 +0.072 -1.291 +0.408 }}
 
{{Optimal ET sequence|legend=1| 23d, 49, 72, 193, 265 }}
 
[[Badness]] (Sintel): 1.78
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 4375/4356
 
Mapping: {{mapping| 1 -18 -14 -22 -22 | 0 30 25 38 39 }}
 
Optimal tunings:
* WE: ~2 = 1200.0514{{c}}, ~11/7 = 783.4294{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.3975{{c}}
 
{{Optimal ET sequence|legend=0| 23de, 49, 72, 193, 265 }}
 
Badness (Sintel): 0.844
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 364/363, 625/624, 1375/1372
 
Mapping: {{mapping| 1 -18 -14 -22 -22 -42 | 0 30 25 38 39 70 }}
 
Optimal tunings:
* WE: ~2 = 1199.9314{{c}}, ~11/7 = 783.3705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4134{{c}}
 
{{Optimal ET sequence|legend=0| 23deff, 49f, 72, 121, 193 }}
 
Badness (Sintel): 0.828
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 325/324, 364/363, 375/374, 540/539, 595/594
 
Mapping: {{mapping| 1 -18 -14 -22 -22 -42 -39 | 0 30 25 38 39 70 66 }}
 
Optimal tunings:
* WE: ~2 = 1199.9324{{c}}, ~11/7 = 783.3706{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4129{{c}}
 
{{Optimal ET sequence|legend=0| 23deffgg, 49fg, 72, 121, 193 }}
 
Badness (Sintel): 0.664
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594
 
Mapping: {{mapping| 1 -18 -14 -22 -22 -42 -39 16 | 0 30 25 38 39 70 66 -18 }}
 
Optimal tunings:
* WE: ~2 = 1199.8567{{c}}, ~11/7 = 783.3262{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 783.4176{{c}}
 
{{Optimal ET sequence|legend=0| 49fg, 72, 121, 193 }}
 
Badness (Sintel): 0.897
 
== Quartkeenlig ==
Named by [[Eliora]] in 2022, quartkeenlig uses a generator that is a quartertone of [[33/32]][[~]][[36/35]] tempered together in the [[11-limit]], and is called so because it tempers out the [[quartisma]] by virtue of five 33/32's being with [[7/6]], keenanisma, [[385/384]], tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). As six quartertones make the hanson generator of ~[[6/5]], its [[ploidacot]] is alpha-36-cot. It can also be viewed as a regular temperament interpretation of [[23edo and octave stretching|stretched 23edo]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 15625/15552, 117649/116640
 
{{Mapping|legend=1| 1 0 1 1 | 0 36 30 41 }}
: mapping generator: ~2, ~36/35
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2825{{c}}, ~36/35 = 52.8528{{c}}
: [[error map]]: {{val| +0.282 +0.745 -0.448 -1.579 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 52.8476{{c}}
: error map: {{val| 0.000 +0.558 -0.886 -2.074 }}
 
{{Optimal ET sequence|legend=1| 68, 91, 159, 386d, 545dd }}
 
[[Badness]] (Sintel): 3.69
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 6250/6237, 67228/66825
 
Mapping: {{mapping| 1 0 1 1 5 | 0 36 30 41 -35 }}
 
Optimal tunings:
* WE: ~2 = 1200.2526{{c}}, ~36/35 = 52.8534{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8446{{c}}
 
{{Optimal ET sequence|legend=0| 68, 91, 159, 386d, 545dd }}
 
Badness (Sintel): 2.86
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 385/384, 625/624, 16807/16731
 
Mapping: {{mapping| 1 0 1 1 5 0 | 0 36 30 41 -35 84 }}
 
Optimal tunings:
* WE: ~2 = 1200.2564{{c}}, ~36/35 = 52.8568{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 52.8479{{c}}
 
{{Optimal ET sequence|legend=0| 68, 159, 386d, 545ddf }}
 
Badness (Sintel): 1.97
 
== Novemkleismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 15625/15552, 40353607/40310784
 
{{Mapping|legend=1| 9 0 9 11 | 0 6 5 6 }}
: mapping generators: ~2592/2401, ~6/5
 
[[Optimal tuning]]s:
* [[WE]]: ~2592/2401 = 133.3488{{c}}, ~6/5 = 317.0413{{c}} (~36/35 = 50.3437{{c}})
: [[error map]]: {{val| +0.139 +0.293 -0.968 +0.259 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~6/5 = 317.0260{{c}} (~36/35 = 50.3593{{c}})
: error map: {{val| 0.000 +0.201 -1.184 -0.003 }}
 
{{Optimal ET sequence|legend=1| 72, 261, 333, 405, 477c, 882c }}
 
[[Badness]] (Sintel): 4.90
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 4000/3993, 15625/15552
 
Mapping: {{mapping| 9 0 9 11 24 | 0 6 5 6 3 }}
 
Optimal tunings:
* WE: ~250/231 = 133.3465{{c}}, ~6/5 = 317.0416{{c}} (~36/35 = 50.3486{{c}})
* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0264{{c}} (~36/35 = 50.3597{{c}})
 
{{Optimal ET sequence|legend=0| 72, 261, 333, 405, 882c }}
 
Badness (Sintel): 1.71
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 625/624, 1375/1372, 4000/3993
 
Mapping: {{mapping| 9 0 9 11 24 0 | 0 6 5 6 3 14 }}
 
Optimal tunings:
* WE: ~250/231 = 133.3385{{c}}, ~6/5 = 317.0978{{c}} (~36/35 = 50.4208{{c}})
* CWE: ~250/231 = 133.3333{{c}}, ~6/5 = 317.0910{{c}} (~36/35 = 50.4243{{c}})
 
{{Optimal ET sequence|legend=0| 72, 189f, 261, 333, 738cf }}
 
Badness (Sintel): 1.61
 
== Subgroup extensions ==
=== Kleismic (2.3.5.13) a.k.a. cata ===
Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]]), [[676/675|S13/S15 = S26]]}. For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].
 
Subgroup: 2.3.5.13
 
Comma list: 325/324, 625/624
 
Subgroup-val mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}
 
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}
 
Badness (Sintel): 0.131
 
==== 2.3.5.13.37 subgroup ====
Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy.
 
Subgroup: 2.3.5.13.37.41
 
Comma list: 325/324, 481/480, 625/624
 
Subgroup-val mapping: {{mapping| 1 0 1 0 6 | 0 6 5 14 -3 }}
 
Optimal tunings:
* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0452{{c}}
 
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}
 
Badness (Sintel): 0.167
 
==== 2.3.5.13.37.41 subgroup ====
Subgroup: 2.3.5.13.37.41
 
Comma list: 325/324, 481/480, 625/624, 1025/1024
 
Subgroup-val mapping: {{mapping| 1 0 1 0 6 8 | 0 6 5 14 -3 -10 }}
 
Optimal tunings:
* WE: ~2 = 1200.1651{{c}}, ~6/5 = 317.1126{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0748{{c}}
 
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246l }}
 
Badness (Sintel): 0.223
 
[[Category:Temperament families]]
[[Category:Kleismic family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Kleismic_family"