Stearnsmic clan: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''stearnsmic clan''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[stearnsma]] ({{monzo|legend=1| 1 10 0 -6 }}, [[ratio]]: 118098/117649), a no-5's comma of about 6.59 [[cent]]s.  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-05 06:07:49 UTC</tt>.<br>
: The original revision id was <tt>307784890</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">Stearnsmic temperaments temper out the stearnsma, the no-fives comma |1 10 0 -6&gt; = 118098/117649. Sternismic temperaments not listed include [[Gravity family#Harry|harry]], [[Marvel temperaments#Wizard|wizard]], and [[Porcupine family#Hedgehog|hedgehog]].


=Sternscape=
== No-5 stearnsmic ==
Commas: 118098/117649, 250047/250000
This temperament is generated by a very slightly flat [[~]][[9/7]] major third, three of which minus a [[semi-octave]] period give the [[3/2|perfect fifth]]. Its [[ploidacot]] is diploid alpha-tricot.


POTE generator: ~245/243 = 16.942
[[Subgroup]]: 2.3.7


Map: [&lt;6 3 2 6|, &lt;0 6 11 10|]
[[Comma list]]: 118098/117649
Wedgie: &lt;&lt;36 66 60 21 -6 -46||
EDOs: 72, 210, 282, 354
Badness: 0.0905


==11-limit==
{{Mapping|legend=2| 2 1 2 | 0 3 5 }}
Commas: 540/539, 4000/3993, 137781/137500


POTE generator: ~245/243 = 16.950
{{Mapping|legend=3| 2 1 0 2 | 0 3 0 5 }}
: mapping generators: ~343/243, ~9/7


Map: [&lt;6 3 2 6 11|, &lt;0 6 11 10 9|]
[[Optimal tuning]]s:  
EDOs: 72, 282, 354
* [[WE]]: ~343/243 = 599.994{{c}}, ~9/7 = 433.884{{c}}
Badness: 0.0321</pre></div>
: [[error map]]: {{val| -0.012 -0.309 +0.582 }}
<h4>Original HTML content:</h4>
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 433.885{{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;Stearnsmic temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Stearnsmic temperaments temper out the stearnsma, the no-fives comma |1 10 0 -6&amp;gt; = 118098/117649. Sternismic temperaments not listed include &lt;a class="wiki_link" href="/Gravity%20family#Harry"&gt;harry&lt;/a&gt;, &lt;a class="wiki_link" href="/Marvel%20temperaments#Wizard"&gt;wizard&lt;/a&gt;, and &lt;a class="wiki_link" href="/Porcupine%20family#Hedgehog"&gt;hedgehog&lt;/a&gt;.&lt;br /&gt;
: error map: {{val| 0.000 -0.300 +0.600 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Sternscape"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Sternscape&lt;/h1&gt;
{{Optimal ET sequence|legend=1| 14, 22, 36, 94, 130, 224, 354, 484, 838, 1322d }}
Commas: 118098/117649, 250047/250000&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 0.382
POTE generator: ~245/243 = 16.942&lt;br /&gt;
 
&lt;br /&gt;
=== Overview to extensions ===
Map: [&amp;lt;6 3 2 6|, &amp;lt;0 6 11 10|]&lt;br /&gt;
The second comma in the comma list determines how we extend it to include the [[harmonic]] [[5/1|5]]. Pogo (94 & 130) adds [[32805/32768]], supers (58 & 94) adds [[5120/5103]], echidna (22 & 58) adds [[1728/1715]], and hedgehog (14c & 22) adds [[50/49]]. Those are the [[strong extension]]s.
Wedgie: &amp;lt;&amp;lt;36 66 60 21 -6 -46||&lt;br /&gt;
 
EDOs: 72, 210, 282, 354&lt;br /&gt;
Wizard adds [[225/224]]. Harry adds [[2401/2400]]. Those split the generator in two. Septisuperfourth adds [[6144/6125]] and splits the generator in three. Stearnscape adds [[250047/250000]] and splits the period in three. Octoid adds [[4375/4374]] and splits the period in four. Decistearn adds [[3136/3125]] splits the period in five.  
Badness: 0.0905&lt;br /&gt;
 
&lt;br /&gt;
All of them extend naturally to the 11-limit via tempering out both [[540/539]] and [[4000/3993]], as well as [[9801/9800]], so that ~[[11/10]] and ~[[9/7]] add up to the semi-octave period. For strong extensions, the generator can be taken to be ~11/10, and three ~11/10's then make a ~4/3 as a result of tempering out 4000/3993, making [[10/9]] and [[12/11]] equidistant from 11/10.
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Sternscape-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
 
Commas: 540/539, 4000/3993, 137781/137500&lt;br /&gt;
Temperaments discussed elsewhere are:  
&lt;br /&gt;
* [[Harry]] (+2401/2400 or 19683/19600) → [[Gravity family #Harry|Gravity family]]
POTE generator: ~245/243 = 16.950&lt;br /&gt;
* ''[[Septisuperfourth]]'' (+6144/6125) → [[Escapade family #Septisuperfourth|Escapade family]]
&lt;br /&gt;
* ''[[Decistearn]]'' (+3136/3125) → [[Trisedodge family #Decistearn|Trisedodge family]]
Map: [&amp;lt;6 3 2 6 11|, &amp;lt;0 6 11 10 9|]&lt;br /&gt;
* ''[[Garistearn]]'' (+33554432/33480783) → [[94th-octave temperaments #Garistearn|94th-octave temperaments]]
EDOs: 72, 282, 354&lt;br /&gt;
 
Badness: 0.0321&lt;/body&gt;&lt;/html&gt;</pre></div>
The rest are considered below.
 
== Pogo ==
The pogo temperament tempers out the [[schisma]], whose amount of tempering of the fifth is just about right for the stearnsma and vice versa. It may be described as 94 & 130, and is the temperament that reconciles the difference between [[tertiaschis]] and [[countertertiaschis]], using its semi-octave period.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 32805/32768, 118098/117649
 
{{Mapping|legend=1| 2 1 22 2 | 0 3 -24 5 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.9860{{c}}, ~9/7 = 433.8905{{c}}
: [[error map]]: {{val| -0.028 -0.298 +0.005 +0.598 }}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 433.9010{{c}}
: error map: {{val| 0.000 -0.252 +0.062 +0.679 }}
 
{{Optimal ET sequence|legend=1| 36, 94, 130, 224, 354 }}
 
[[Badness]] (Sintel): 2.02
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4000/3993, 32805/32768
 
Mapping: {{mapping| 2 1 22 2 25 | 0 3 -24 5 -25 }}
 
Optimal tunings:
* WE: ~99/70 = 599.9700{{c}}, ~9/7 = 433.8898{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 433.9125{{c}}
 
{{Optimal ET sequence|legend=0| 36, 58ce, 94, 130, 224, 354, 578 }}
 
Badness (Sintel): 1.05
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1575/1573, 4096/4095
 
Mapping: {{mapping| 2 1 22 2 25 -2 | 0 3 -24 5 -25 13 }}
 
Optimal tunings:
* WE: ~99/70 = 599.9670{{c}}, ~9/7 = 433.8867{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 433.9112{{c}}
 
{{Optimal ET sequence|legend=0| 36, 94, 130, 224, 354, 578 }}
 
Badness (Sintel): 0.724
 
== Supers ==
Supers tempers out 5120/5103, the [[aberschisma]], and may be described as the {{nowrap| 58 & 94 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 118098/117649
 
{{Mapping|legend=1| 2 1 -12 2 | 0 3 23 5 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8062{{c}}, ~9/7 = 434.0778{{c}}
: [[error map]]: {{val| -0.388 +0.085 -0.199 +1.175 }}
* [[CWE]]: ~343/243 = 600.000{{c}}, ~9/7 = 434.2025{{c}}
: error map: {{val| 0.000 +0.653 +0.335 +2.187 }}
 
{{Optimal ET sequence|legend=1| 36c, 58, 94, 152 }}
 
[[Badness]] (Sintel): 2.35
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4000/3993, 5120/5103
 
Mapping: {{mapping| 2 1 -12 2 -9 | 0 3 23 5 22 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8051{{c}}, ~9/7 = 434.0755{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 434.2002{{c}}
 
{{Optimal ET sequence|legend=0| 36ce, 58, 94, 152 }}
 
Badness (Sintel): 0.934
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 729/728, 1575/1573
 
Mapping: {{mapping| 2 1 -12 2 -9 -2 | 0 3 23 5 22 13 }}
 
Optimal tunings:
* WE: ~99/70 = 599.7318{{c}}, ~9/7 = 434.0271{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 434.1985{{c}}
 
{{Optimal ET sequence|legend=0| 36ce, 58, 94, 152f }}
 
Badness (Sintel): 0.894
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 170/169, 289/288, 352/351, 442/441, 561/560
 
Mapping: {{mapping| 2 1 -12 2 -9 -2 6 | 0 3 23 5 22 13 3 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8786{{c}}, ~9/7 = 434.0934{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 434.1726{{c}}
 
{{Optimal ET sequence|legend=0| 36ce, 58, 94, 152f }}
 
Badness (Sintel): 1.09
 
== Echidna ==
{{See also| Chords of echidna }}
 
Echidna adds 1728/1715, the [[orwellisma]], and 2048/2025, the [[diaschisma]], to the commas. It may be described as the {{nowrap| 22 & 58 }} temperament. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp [[7/4]] which some prefer.
 
Echidna becomes more interesting when extended to be an 11-limit temperament by adding [[176/175]], [[540/539]] or [[896/891]] to the commas, where the same tunings can be used as before. It then is able to represent the entire [[11-odd-limit]] [[tonality diamond|diamond]] to within about 6 cents of error within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note 2mos much more. Better yet, it relates three important 11-limit edos: 22edo is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[#Hedgehog|hedgehog]]; 58edo is the smallest tuning that is distinctly consistent in the 11-odd-limit, and 80edo is the third smallest distinctly consistent in the 11-odd-limit.
 
Like most [[diaschismic family #Diaschismic|diaschismic]] extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[diaschismic family #srutal archagall (2.3.5.17)|srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, the last two of which are consistent in the [[17-odd-limit]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 2048/2025
 
{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
: mapping generators: ~45/32, ~9/7
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.3056{{c}}, ~9/7 = 434.3524{{c}}
: [[error map]]: {{val| -1.389 +0.408 +1.322 +1.547 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8327{{c}}
: error map: {{val| 0.000 +2.543 +4.690 +5.338 }}
 
{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}
 
[[Badness]] (Sintel): 1.47
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 540/539, 896/891
 
Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}
 
Optimal tunings:
* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}
 
Minimax tuning:
* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7
 
{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}
 
Badness (Sintel): 0.859
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 364/363, 540/539
 
Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}
 
Optimal tunings:
* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}
 
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}
 
Badness (Sintel): 0.978
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
 
Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}
 
Optimal tunings:
* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}
 
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}
 
Badness (Sintel): 1.03
 
== Hedgehog ==
{{See also| Sensamagic clan }}
 
Hedgehog is a relatively low-accuracy temperament which tempers out 50/49 ([[jubilisma]], 245/243 ([[sensamagic comma]]), 250/243 ([[porcupine comma]]), and 2430/2401 ([[nuwell comma]]). It is also a strong extension of [[BPS]].
 
22edo provides an obvious tuning, which happens to be the only [[patent val|patent-val]] edo tuning, but if you are looking for an alternative you could try the {{val| 146 232 338 411 }} (146bccdd) val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14-note [[mos]] gives scope for harmony while stopping well short of 22. A related temperament is [[#Echidna|echidna]], which offers much more accuracy. They merge on 22edo.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 50/49, 245/243
 
{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}
 
: mapping generators: ~7/5, ~9/7
 
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 599.6061{{c}}, ~9/7 = 435.3620{{c}}
: [[error map]]: {{val| -0.788 +3.737 -9.897 +7.197 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4483{{c}}
: error map: {{val| 0.000 +4.390 -9.072 +8.416 }}
 
{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}
 
[[Badness]] (Sintel): 1.11
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 55/54, 99/98
 
Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}
 
Optimal tunings:
* WE: ~7/5 = 600.1133{{c}}, ~9/7 = 435.4680{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4431{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}
 
Badness (Sintel): 0.764
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 55/54, 65/63, 99/98
 
Mapping: {{mapping| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}
 
Optimal tunings:
* WE: ~7/5 = 600.3651{{c}}, ~9/7 = 436.1258{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.0483{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}
 
Badness (Sintel): 0.889
 
==== Urchin ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 50/49, 55/54, 66/65
 
Mapping: {{mapping| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}
 
Optimal tunings:
* WE: ~7/5 = 598.3303{{c}}, ~9/7 = 435.8617{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.3485{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 22f }}
 
Badness (Sintel): 1.04
 
=== Hedgepig ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 245/243, 385/384
 
Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}
 
Optimal tunings:
* WE: ~7/5 = 599.7917{{c}}, ~9/7 = 435.2737{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4047{{c}}
 
{{Optimal ET sequence|legend=0| 22 }}
 
Badness (Sintel): 2.26
 
; Music
* [https://web.archive.org/web/20240624173512/http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] – in [[hedgehog14|Hedgehog[14]]], 22edo tuning.
 
== Wizard ==
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''
 
Wizard tempers out the [[marvel comma]] and may be described as {{nowrap| 22 & 72 }}. It splits the ~9/7 in two parts, which can be treated as ~[[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. The [[ploidacot]] signature of wizard is diploid alpha-hexacot. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 118098/117649
 
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216
 
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}
 
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}
 
[[Badness]] (Sintel): 1.03
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 385/384, 4000/3993
 
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
 
Optimal tunings:
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
 
Badness (Sintel): 0.613
 
==== Lizard ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 351/350, 364/363, 385/384
 
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
 
Optimal tunings:
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
 
Badness (Sintel): 0.900
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
 
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50, 72 }}
 
Badness (Sintel): 0.741
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
 
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}
 
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
 
Badness (Sintel): 0.955
 
==== Gizzard ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 325/324, 385/384, 1573/1568
 
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
 
Optimal tunings:
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
 
Badness (Sintel): 0.837
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
 
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
 
Optimal tunings:
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
 
Badness (Sintel): 0.694
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
 
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
 
Optimal tunings:
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}
 
{{Optimal ET sequence|legend=0| 72, 94, 166g }}
 
Badness (Sintel): 0.901
 
=== Mage ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 176/175, 1331/1296
 
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
 
Optimal tunings:
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}
 
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
 
Badness (Sintel): 1.91
 
== Stearnscape ==
Stearnscape tempers out 250047/250000, the [[landscape comma]], with a period of 1/6 octave, and may be described as the {{nowrap| 72 & 282 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 118098/117649, 250047/250000
 
{{Mapping|legend=1| 6 3 2 6 | 0 6 11 10 }}
: mapping generators: ~2450/2187, ~567/500
 
[[Optimal tuning]]s:
* [[WE]]: ~2450/2187 = 199.9984{{c}}, ~567/500 = 216.9400{{c}}
: [[error map]]: {{val| -0.010 -0.320 +0.023 +0.564 }}
* [[CWE]]: ~2450/2187 = 200.0000{{c}}, ~567/500 = 216.9406{{c}}
: error map: {{val| 0.000 -0.312 +0.033 +0.580 }}
 
{{Optimal ET sequence|legend=1| 72, 210, 282, 354 }}
 
[[Badness]] (Sintel): 2.29
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4000/3993, 137781/137500
 
Mapping: {{mapping| 6 3 2 6 11 | 0 6 11 10 9 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9835{{c}}, ~567/500 = 216.9321{{c}} (~100/99 = 16.9487{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~567/500 = 216.9381{{c}} (~100/99 = 16.9381{{c}})
 
{{Optimal ET sequence|legend=0| 72, 210e, 282, 354 }}
 
Badness (Sintel): 1.06
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 729/728, 1575/1573, 34398/34375
 
Mapping: {{mapping| 6 3 2 6 11 -6 | 0 6 11 10 9 26 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9810{{c}}, ~312/275 = 216.9360{{c}} (~105/104 = 16.9550{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~312/275 = 216.9474{{c}} (~105/104 = 16.9474{{c}})
 
{{Optimal ET sequence|legend=0| 72, 210ef, 282, 354 }}
 
Badness (Sintel): 1.06
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 540/539, 729/728, 936/935, 1156/1155, 1575/1573
 
Mapping: {{mapping| 6 3 2 6 11 -6 5 | 0 6 11 10 9 26 18 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9814{{c}}, ~17/15 = 216.9376{{c}} (~105/104 = 16.9563{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~17/15 = 216.9487{{c}} (~105/104 = 16.9345{{c}})
 
{{Optimal ET sequence|legend=0| 72, 210efg, 282, 354 }}
 
Badness (Sintel): 0.782
 
== Octoid ==
: {{Main| Octoid }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Octoid]].''
 
The octoid temperament has a period of 1/8 octave and tempers out 4375/4374, the [[ragisma]], and 16875/16807, the [[canopic comma]]. In the 11-limit, it tempers out [[540/539]], [[1375/1372]], and [[6250/6237]]. In this temperament, one period gives ~[[12/11]], two give ~[[25/21]], three give ~[[35/27]], and four give [[99/70]]~[[140/99]].
 
The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimizing the average damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out [[324/323]], [[375/374]], [[495/494]], [[625/624]], [[715/714]] or [[729/728]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 16875/16807
 
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
: mapping generators: ~49/45, ~7/5
 
[[Optimal tuning]]s:
* [[WE]]: ~49/45 = 150.0003{{c}}, ~7/5 = 583.9416{{c}}
: [[error map]]: {{val| +0.002 -0.130 -0.547 +0.883 }}
* [[CWE]]: ~49/45 = 150.0000{{c}}, ~7/5 = 583.9411{{c}}
: error map: {{val| 0.000 -0.132 -0.549 +0.880 }}
 
[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
 
{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}
 
[[Badness]] (Sintel): 1.08
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 4000/3993
 
Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
 
Optimal tunings:
* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}}
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
 
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }}
 
Badness (Sintel): 0.466
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 625/624, 729/728, 1375/1372
 
Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
 
Optimal tunings:
* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}
 
{{Optimal ET sequence|legend=0| 72, 152f, 224 }}
 
Badness (Sintel): 0.631
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
 
Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}
 
Optimal tunings:
* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}
 
{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}
 
Badness (Sintel): 0.729
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
 
Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
 
Optimal tunings:
* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}
 
{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}
 
Badness (Sintel): 0.975
 
==== Octopus ====
A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4.
 
(It is perhaps worth noting, for the purpose of the 56-note MOS scale (the smallest to reach [[9/8]], thus going one generator further than required to reach every prime), that if we were to use [[56edo]] to support it, we would have to use the val 56bccddeffg (using [[wart notation]]), corresponding to flattening every prime in the 19-limit from its [[patent val]] mapping (closest [[direct approximation]]) except for 2 and 19, corresponding to these only using periods to reach.)
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 325/324, 364/363, 540/539
 
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
 
Optimal tunings:
* WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}
 
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}
 
Badness (Sintel): 0.896
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
 
Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}
 
Optimal tunings:
* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}
 
{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}
 
Badness (Sintel): 0.795
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
 
Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}
 
Optimal tunings:
* WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 72, 152 }}
 
Badness (Sintel): 0.993
 
Scales: [[Octoid72]], [[Octoid80]]
 
==== Hexadecoid ====
Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
 
Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
: mapping generators: ~448/429, ~7/5
 
Optimal tunings:
* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}
* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}
 
{{Optimal ET sequence|legend=0| 80, 144, 224 }}
 
Badness (Sintel): 1.27
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
 
Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
 
Optimal tunings:
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}}
 
{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}
 
Badness (Sintel): 1.46
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
 
Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}
 
Optimal tunings:
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}
 
{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}
 
Badness (Sintel): 1.44
 
[[Category:Stearnsmic clan| ]] <!-- main article -->
[[Category:Hades]] <!-- key article -->
[[Category:Temperament clans]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 13:32, 16 September 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The stearnsmic clan of temperaments tempers out the stearnsma (monzo: [1 10 0 -6⟩, ratio: 118098/117649), a no-5's comma of about 6.59 cents.

No-5 stearnsmic

This temperament is generated by a very slightly flat ~9/7 major third, three of which minus a semi-octave period give the perfect fifth. Its ploidacot is diploid alpha-tricot.

Subgroup: 2.3.7

Comma list: 118098/117649

Subgroup-val mapping: [⟨2 1 2], ⟨0 3 5]]

Gencom mapping: [⟨2 1 0 2], ⟨0 3 0 5]]

mapping generators: ~343/243, ~9/7

Optimal tunings:

  • WE: ~343/243 = 599.994 ¢, ~9/7 = 433.884 ¢
error map: ⟨-0.012 -0.309 +0.582]
  • CWE: ~343/243 = 600.000 ¢, ~9/7 = 433.885 ¢
error map: ⟨0.000 -0.300 +0.600]

Optimal ET sequence: 14, 22, 36, 94, 130, 224, 354, 484, 838, 1322d

Badness (Sintel): 0.382

Overview to extensions

The second comma in the comma list determines how we extend it to include the harmonic 5. Pogo (94 & 130) adds 32805/32768, supers (58 & 94) adds 5120/5103, echidna (22 & 58) adds 1728/1715, and hedgehog (14c & 22) adds 50/49. Those are the strong extensions.

Wizard adds 225/224. Harry adds 2401/2400. Those split the generator in two. Septisuperfourth adds 6144/6125 and splits the generator in three. Stearnscape adds 250047/250000 and splits the period in three. Octoid adds 4375/4374 and splits the period in four. Decistearn adds 3136/3125 splits the period in five.

All of them extend naturally to the 11-limit via tempering out both 540/539 and 4000/3993, as well as 9801/9800, so that ~11/10 and ~9/7 add up to the semi-octave period. For strong extensions, the generator can be taken to be ~11/10, and three ~11/10's then make a ~4/3 as a result of tempering out 4000/3993, making 10/9 and 12/11 equidistant from 11/10.

Temperaments discussed elsewhere are:

The rest are considered below.

Pogo

The pogo temperament tempers out the schisma, whose amount of tempering of the fifth is just about right for the stearnsma and vice versa. It may be described as 94 & 130, and is the temperament that reconciles the difference between tertiaschis and countertertiaschis, using its semi-octave period.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 118098/117649

Mapping: [⟨2 1 22 2], ⟨0 3 -24 5]]

Optimal tunings:

  • WE: ~343/243 = 599.9860 ¢, ~9/7 = 433.8905 ¢
error map: ⟨-0.028 -0.298 +0.005 +0.598]
  • CWE: ~343/243 = 600.000 ¢, ~9/7 = 433.9010 ¢
error map: ⟨0.000 -0.252 +0.062 +0.679]

Optimal ET sequence: 36, 94, 130, 224, 354

Badness (Sintel): 2.02

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 32805/32768

Mapping: [⟨2 1 22 2 25], ⟨0 3 -24 5 -25]]

Optimal tunings:

  • WE: ~99/70 = 599.9700 ¢, ~9/7 = 433.8898 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~9/7 = 433.9125 ¢

Optimal ET sequence: 36, 58ce, 94, 130, 224, 354, 578

Badness (Sintel): 1.05

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1575/1573, 4096/4095

Mapping: [⟨2 1 22 2 25 -2], ⟨0 3 -24 5 -25 13]]

Optimal tunings:

  • WE: ~99/70 = 599.9670 ¢, ~9/7 = 433.8867 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~9/7 = 433.9112 ¢

Optimal ET sequence: 36, 94, 130, 224, 354, 578

Badness (Sintel): 0.724

Supers

Supers tempers out 5120/5103, the aberschisma, and may be described as the 58 & 94 temperament.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 118098/117649

Mapping: [⟨2 1 -12 2], ⟨0 3 23 5]]

Optimal tunings:

  • WE: ~343/243 = 599.8062 ¢, ~9/7 = 434.0778 ¢
error map: ⟨-0.388 +0.085 -0.199 +1.175]
  • CWE: ~343/243 = 600.000 ¢, ~9/7 = 434.2025 ¢
error map: ⟨0.000 +0.653 +0.335 +2.187]

Optimal ET sequence: 36c, 58, 94, 152

Badness (Sintel): 2.35

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 5120/5103

Mapping: [⟨2 1 -12 2 -9], ⟨0 3 23 5 22]]

Optimal tunings:

  • WE: ~99/70 = 599.8051 ¢, ~9/7 = 434.0755 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~9/7 = 434.2002 ¢

Optimal ET sequence: 36ce, 58, 94, 152

Badness (Sintel): 0.934

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1575/1573

Mapping: [⟨2 1 -12 2 -9 -2], ⟨0 3 23 5 22 13]]

Optimal tunings:

  • WE: ~99/70 = 599.7318 ¢, ~9/7 = 434.0271 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~9/7 = 434.1985 ¢

Optimal ET sequence: 36ce, 58, 94, 152f

Badness (Sintel): 0.894

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 289/288, 352/351, 442/441, 561/560

Mapping: [⟨2 1 -12 2 -9 -2 6], ⟨0 3 23 5 22 13 3]]

Optimal tunings:

  • WE: ~99/70 = 599.8786 ¢, ~9/7 = 434.0934 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~9/7 = 434.1726 ¢

Optimal ET sequence: 36ce, 58, 94, 152f

Badness (Sintel): 1.09

Echidna

See also: Chords of echidna

Echidna adds 1728/1715, the orwellisma, and 2048/2025, the diaschisma, to the commas. It may be described as the 22 & 58 temperament. 58edo or 80edo make for good tunings, or their vals can be added to ⟨138 219 321 388] (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.

Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about 6 cents of error within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note 2mos much more. Better yet, it relates three important 11-limit edos: 22edo is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with hedgehog; 58edo is the smallest tuning that is distinctly consistent in the 11-odd-limit, and 80edo is the third smallest distinctly consistent in the 11-odd-limit.

Like most diaschismic extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out 176/175, tempering out 351/350 and 352/351 which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with srutal archagall, leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, the last two of which are consistent in the 17-odd-limit.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 2048/2025

Mapping: [⟨2 1 9 2], ⟨0 3 -6 5]]

mapping generators: ~45/32, ~9/7

Optimal tunings:

  • WE: ~45/32 = 599.3056 ¢, ~9/7 = 434.3524 ¢
error map: ⟨-1.389 +0.408 +1.322 +1.547]
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8327 ¢
error map: ⟨0.000 +2.543 +4.690 +5.338]

Optimal ET sequence: 22, 58, 80, 138cd, 218cd

Badness (Sintel): 1.47

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 896/891

Mapping: [⟨2 1 9 2 12], ⟨0 3 -6 5 -7]]

Optimal tunings:

  • WE: ~45/32 = 599.3085 ¢, ~9/7 = 434.3511 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8647 ¢

Minimax tuning:

  • 11-odd-limit: ~9/7 = [5/12 0 0 1/12 -1/12⟩
[[1 0 0 0 0⟩, [7/4 0 0 1/4 -1/4⟩, [2 0 0 -1/2 1/2⟩, [37/12 0 0 5/12 -5/12⟩, [37/12 0 0 -7/12 7/12⟩]
unchanged-interval (eigenmonzo) basis: 2.11/7

Optimal ET sequence: 22, 58, 80, 138cde, 218cde

Badness (Sintel): 0.859

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 364/363, 540/539

Mapping: [⟨2 1 9 2 12 19], ⟨0 3 -6 5 -7 -16]]

Optimal tunings:

  • WE: ~45/32 = 599.3397 ¢, ~9/7 = 434.2772 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.7864 ¢

Optimal ET sequence: 22, 36f, 58, 80, 138cde

Badness (Sintel): 0.978

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 221/220, 256/255, 540/539

Mapping: [⟨2 1 9 2 12 19 6], ⟨0 3 -6 5 -7 -16 3]]

Optimal tunings:

  • WE: ~45/32 = 599.4645 ¢, ~9/7 = 434.4282 ¢
  • CWE: ~45/32 = 600.0000 ¢, ~9/7 = 434.8340 ¢

Optimal ET sequence: 22, 36f, 58, 80, 138cde

Badness (Sintel): 1.03

Hedgehog

See also: Sensamagic clan

Hedgehog is a relatively low-accuracy temperament which tempers out 50/49 (jubilisma, 245/243 (sensamagic comma), 250/243 (porcupine comma), and 2430/2401 (nuwell comma). It is also a strong extension of BPS.

22edo provides an obvious tuning, which happens to be the only patent-val edo tuning, but if you are looking for an alternative you could try the ⟨146 232 338 411] (146bccdd) val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14-note mos gives scope for harmony while stopping well short of 22. A related temperament is echidna, which offers much more accuracy. They merge on 22edo.

Subgroup: 2.3.5.7

Comma list: 50/49, 245/243

Mapping: [⟨2 1 1 2], ⟨0 3 5 5]]

mapping generators: ~7/5, ~9/7

Optimal tunings:

  • WE: ~7/5 = 599.6061 ¢, ~9/7 = 435.3620 ¢
error map: ⟨-0.788 +3.737 -9.897 +7.197]
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 435.4483 ¢
error map: ⟨0.000 +4.390 -9.072 +8.416]

Optimal ET sequence: 8d, 14c, 22

Badness (Sintel): 1.11

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 55/54, 99/98

Mapping: [⟨2 1 1 2 4], ⟨0 3 5 5 4]]

Optimal tunings:

  • WE: ~7/5 = 600.1133 ¢, ~9/7 = 435.4680 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 435.4431 ¢

Optimal ET sequence: 8d, 14c, 22, 58ce

Badness (Sintel): 0.764

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 55/54, 65/63, 99/98

Mapping: [⟨2 1 1 2 4 3], ⟨0 3 5 5 4 6]]

Optimal tunings:

  • WE: ~7/5 = 600.3651 ¢, ~9/7 = 436.1258 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 436.0483 ¢

Optimal ET sequence: 8d, 14cf, 22

Badness (Sintel): 0.889

Urchin

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 50/49, 55/54, 66/65

Mapping: [⟨2 1 1 2 4 6], ⟨0 3 5 5 4 2]]

Optimal tunings:

  • WE: ~7/5 = 598.3303 ¢, ~9/7 = 435.8617 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 436.3485 ¢

Optimal ET sequence: 14c, 22f

Badness (Sintel): 1.04

Hedgepig

Subgroup: 2.3.5.7.11

Comma list: 50/49, 245/243, 385/384

Mapping: [⟨2 1 1 2 12], ⟨0 3 5 5 -7]]

Optimal tunings:

  • WE: ~7/5 = 599.7917 ¢, ~9/7 = 435.2737 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 435.4047 ¢

Optimal ET sequence: 22

Badness (Sintel): 2.26

Music

Wizard

Main article: Wizard
For the 5-limit version, see Miscellaneous 5-limit temperaments #Wizard.

Wizard tempers out the marvel comma and may be described as 22 & 72. It splits the ~9/7 in two parts, which can be treated as ~17/15. The semi-octave complement of this interval is ~5/4. The ploidacot signature of wizard is diploid alpha-hexacot. 72edo, 94edo, and especially 166edo are good tunings for it.

Subgroup: 2.3.5.7

Comma list: 225/224, 118098/117649

Mapping: [⟨2 1 5 2], ⟨0 6 -1 10]]

mapping generators: ~1225/864, ~245/216

Optimal tunings:

  • WE: ~1225/864 = 600.3438 ¢, ~245/216 = 216.8680 ¢
error map: ⟨+0.688 -0.403 -1.463 +0.541]
  • CWE: ~1225/864 = 600.0000 ¢, ~245/216 = 216.7977 ¢
error map: ⟨0.000 -1.169 -3.111 -0.849]

Optimal ET sequence: 22, 50, 72, 238c, 310c, 382c, 454bccd

Badness (Sintel): 1.03

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 4000/3993

Mapping: [⟨2 1 5 2 8], ⟨0 6 -1 10 -3]]

Optimal tunings:

  • WE: ~99/70 = 600.3051 ¢, ~25/22 = 216.8782 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~25/22 = 216.7961 ¢

Optimal ET sequence: 22, 50, 72, 166, 238c, 310c

Badness (Sintel): 0.613

Lizard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 364/363, 385/384

Mapping: [⟨2 1 5 2 8 11], ⟨0 6 -1 10 -3 -10]]

Optimal tunings:

  • WE: ~55/39 = 600.4824 ¢, ~25/22 = 216.7852 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~25/22 = 216.6247 ¢

Optimal ET sequence: 22, 50, 72

Badness (Sintel): 0.900

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 273/272, 289/288, 351/350, 375/374

Mapping: [⟨2 1 5 2 8 11 6], ⟨0 6 -1 10 -3 -10 6]]

Optimal tunings:

  • WE: ~17/12 = 600.5032 ¢, ~17/15 = 216.8002 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.6361 ¢

Optimal ET sequence: 22, 50, 72

Badness (Sintel): 0.741

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342

Mapping: [⟨2 1 5 2 8 11 6 2], ⟨0 6 -1 10 -3 -10 6 18]]

Optimal tunings:

  • WE: ~17/12 = 600.4698 ¢, ~17/15 = 216.6925 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.5434 ¢

Optimal ET sequence: 22h, 50, 72, 122g, 194dfg

Badness (Sintel): 0.955

Gizzard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 1573/1568

Mapping: [⟨2 1 5 2 8 -2], ⟨0 6 -1 10 -3 26]]

Optimal tunings:

  • WE: ~99/70 = 600.2896 ¢, ~25/22 = 216.9343 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~25/22 = 216.8501 ¢

Optimal ET sequence: 22f, 72, 166, 238cf

Badness (Sintel): 0.837

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 289/288, 325/324, 375/374, 385/384

Mapping: [⟨2 1 5 2 8 -2 6], ⟨0 6 -1 10 -3 26 6]]

Optimal tunings:

  • WE: ~17/12 = 600.3227 ¢, ~17/15 = 216.9414 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.8469 ¢

Optimal ET sequence: 22f, 72, 166g, 238cfg

Badness (Sintel): 0.694

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594

Mapping: [⟨2 1 5 2 8 -2 6 15], ⟨0 6 -1 10 -3 26 6 -18]]

Optimal tunings:

  • WE: ~17/12 = 600.2637 ¢, ~17/15 = 216.9570 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.8687 ¢

Optimal ET sequence: 72, 94, 166g

Badness (Sintel): 0.901

Mage

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1331/1296

Mapping: [⟨2 1 5 2 4], ⟨0 6 -1 10 8]]

Optimal tunings:

  • WE: ~77/54 = 600.6486 ¢, ~55/48 = 217.1099 ¢
  • CWE: ~77/54 = 600.0000 ¢, ~55/48 = 216.9841 ¢

Optimal ET sequence: 22, 50e, 72ee

Badness (Sintel): 1.91

Stearnscape

Stearnscape tempers out 250047/250000, the landscape comma, with a period of 1/6 octave, and may be described as the 72 & 282 temperament.

Subgroup: 2.3.5.7

Comma list: 118098/117649, 250047/250000

Mapping: [⟨6 3 2 6], ⟨0 6 11 10]]

mapping generators: ~2450/2187, ~567/500

Optimal tunings:

  • WE: ~2450/2187 = 199.9984 ¢, ~567/500 = 216.9400 ¢
error map: ⟨-0.010 -0.320 +0.023 +0.564]
  • CWE: ~2450/2187 = 200.0000 ¢, ~567/500 = 216.9406 ¢
error map: ⟨0.000 -0.312 +0.033 +0.580]

Optimal ET sequence: 72, 210, 282, 354

Badness (Sintel): 2.29

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 137781/137500

Mapping: [⟨6 3 2 6 11], ⟨0 6 11 10 9]]

Optimal tunings:

  • WE: ~55/49 = 199.9835 ¢, ~567/500 = 216.9321 ¢ (~100/99 = 16.9487 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~567/500 = 216.9381 ¢ (~100/99 = 16.9381 ¢)

Optimal ET sequence: 72, 210e, 282, 354

Badness (Sintel): 1.06

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1575/1573, 34398/34375

Mapping: [⟨6 3 2 6 11 -6], ⟨0 6 11 10 9 26]]

Optimal tunings:

  • WE: ~55/49 = 199.9810 ¢, ~312/275 = 216.9360 ¢ (~105/104 = 16.9550 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~312/275 = 216.9474 ¢ (~105/104 = 16.9474 ¢)

Optimal ET sequence: 72, 210ef, 282, 354

Badness (Sintel): 1.06

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 729/728, 936/935, 1156/1155, 1575/1573

Mapping: [⟨6 3 2 6 11 -6 5], ⟨0 6 11 10 9 26 18]]

Optimal tunings:

  • WE: ~55/49 = 199.9814 ¢, ~17/15 = 216.9376 ¢ (~105/104 = 16.9563 ¢)
  • CWE: ~55/49 = 200.0000 ¢, ~17/15 = 216.9487 ¢ (~105/104 = 16.9345 ¢)

Optimal ET sequence: 72, 210efg, 282, 354

Badness (Sintel): 0.782

Octoid

Main article: Octoid
For the 5-limit version, see Miscellaneous 5-limit temperaments #Octoid.

The octoid temperament has a period of 1/8 octave and tempers out 4375/4374, the ragisma, and 16875/16807, the canopic comma. In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives ~12/11, two give ~25/21, three give ~35/27, and four give 99/70~140/99.

The 11-limit is the last place where all the extensions of octoid shown here agree in the mappings of primes. 80edo is an alternative tuning for octoid in the 11-limit; though 72edo does better for minimizing the average damage on the 11-odd-limit, 80edo damages prime 7 in favor of practically-just 17/16's, 11/10's and 9/7's. In higher limits, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 16875/16807

Mapping: [⟨8 1 3 3], ⟨0 3 4 5]]

mapping generators: ~49/45, ~7/5

Optimal tunings:

  • WE: ~49/45 = 150.0003 ¢, ~7/5 = 583.9416 ¢
error map: ⟨+0.002 -0.130 -0.547 +0.883]
  • CWE: ~49/45 = 150.0000 ¢, ~7/5 = 583.9411 ¢
error map: ⟨0.000 -0.132 -0.549 +0.880]

Tuning ranges:

  • 7-odd-limit diamond monotone: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
  • 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
  • 7-odd-limit diamond tradeoff: ~7/5 = [582.512, 584.359]
  • 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

Optimal ET sequence: 8d, …, 72, 152, 224

Badness (Sintel): 1.08

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4000/3993

Mapping: [⟨8 1 3 3 16], ⟨0 3 4 5 3]]

Optimal tunings:

  • WE: ~12/11 = 149.9932 ¢, ~7/5 = 583.9356 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9477 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
  • 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

Optimal ET sequence: 8d, …, 72, 152, 224, 824d

Badness (Sintel): 0.466

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 1375/1372

Mapping: [⟨8 1 3 3 16 -21], ⟨0 3 4 5 3 13]]

Optimal tunings:

  • WE: ~12/11 = 150.0005 ¢, ~7/5 = 583.9066 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9052 ¢

Optimal ET sequence: 72, 152f, 224

Badness (Sintel): 0.631

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 715/714, 729/728

Mapping: [⟨8 1 3 3 16 -21 -14], ⟨0 3 4 5 3 13 12]]

Optimal tunings:

  • WE: ~12/11 = 150.0064 ¢, ~7/5 = 583.8666 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.8489 ¢

Optimal ET sequence: 72, 152fg, 224, 296, 520g

Badness (Sintel): 0.729

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714

Mapping: [⟨8 1 3 3 16 -21 -14 34], ⟨0 3 4 5 3 13 12 0]]

Optimal tunings:

  • WE: ~12/11 = 149.9785 ¢, ~7/5 = 583.8482 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9138 ¢

Optimal ET sequence: 72, 152fg, 224

Badness (Sintel): 0.975

Octopus

A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is 80edo, which has a strong sharp tendency that can be thought of as matching the sharpness of mapping 19/16 to 1\4.

(It is perhaps worth noting, for the purpose of the 56-note MOS scale (the smallest to reach 9/8, thus going one generator further than required to reach every prime), that if we were to use 56edo to support it, we would have to use the val 56bccddeffg (using wart notation), corresponding to flattening every prime in the 19-limit from its patent val mapping (closest direct approximation) except for 2 and 19, corresponding to these only using periods to reach.)

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 540/539

Mapping: [⟨8 1 3 3 16 14], ⟨0 3 4 5 3 4]]

Optimal tunings:

  • WE: ~12/11 = 150.0313 ¢, ~7/5 = 584.0134 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9583 ¢

Optimal ET sequence: 8d, …, 72, 152, 224f

Badness (Sintel): 0.896

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 289/288, 325/324, 540/539

Mapping: [⟨8 1 3 3 16 14 21], ⟨0 3 4 5 3 4 3]]

Optimal tunings:

  • WE: ~12/11 = 150.0528 ¢, ~7/5 = 584.0161 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9166 ¢

Optimal ET sequence: 8d, …, 72, 152, 224fg, 296ffg

Badness (Sintel): 0.795

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399

Mapping: [⟨8 1 3 3 16 14 21 34], ⟨0 3 4 5 3 4 3 0]]

Optimal tunings:

  • WE: ~12/11 = 150.0049 ¢, ~7/5 = 584.0833 ¢
  • CWE: ~12/11 = 150.0000 ¢, ~7/5 = 584.0712 ¢

Optimal ET sequence: 8d, 72, 152

Badness (Sintel): 0.993

Scales: Octoid72, Octoid80

Hexadecoid

Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224

Mapping: [⟨16 2 6 6 32 67], ⟨0 3 4 5 3 -1]]

mapping generators: ~448/429, ~7/5

Optimal tunings:

  • WE: ~448/429 = 74.9943 ¢, ~7/5 = 583.9408 ¢
  • CWE: ~448/429 = 75.0000 ¢, ~7/5 = 583.9709 ¢

Optimal ET sequence: 80, 144, 224

Badness (Sintel): 1.27

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224

Mapping: [⟨16 2 6 6 32 67 81], ⟨0 3 4 5 3 -1 -2]]

Optimal tunings:

  • WE: ~117/112 = 74.9865 ¢, ~7/5 = 583.9626 ¢
  • CWE: ~117/112 = 75.0000 ¢, ~7/5 = 584.0463 ¢

Optimal ET sequence: 80, 144, 224, 528dg

Badness (Sintel): 1.46

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444

Mapping: [⟨16 2 6 6 32 67 81 68], ⟨0 3 4 5 3 -1 -2 0]]

Optimal tunings:

  • WE: ~117/112 = 74.9865 ¢, ~7/5 = 583.9642 ¢
  • CWE: ~117/112 = 75.0000 ¢, ~7/5 = 584.0803 ¢

Optimal ET sequence: 80, 144, 224, 304dh, 528dghh

Badness (Sintel): 1.44

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