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Diaschismic family: 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 '''diaschismic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diaschisma, [[2048/2025]].  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-02 22:31:00 UTC</tt>.<br>
: The original revision id was <tt>146642457</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">The 5-limit parent comma for the diaschismic family is 2048/2025, the diaschisma. Its monzo is |11 -4 -2&gt;, and flipping that yields &lt;&lt;2 -4 -11|| for the wedgie. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]] or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities.


==Seven limit children==
== Diaschismic ==
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, and shrutar 245/243, the sensamagic comma. The other temperaments all keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as 36/35, the septimal quarter-tone.) </pre></div>
{{Main| Diaschismic }}
<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;Diaschismic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 5-limit parent comma for the diaschismic family is 2048/2025, the diaschisma. Its monzo is |11 -4 -2&amp;gt;, and flipping that yields &amp;lt;&amp;lt;2 -4 -11|| for the wedgie. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt; is a good tuning choice, with &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt;, &lt;a class="wiki_link" href="/56edo"&gt;56edo&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58edo&lt;/a&gt; or &lt;a class="wiki_link" href="/80edo"&gt;80edo&lt;/a&gt; being other possibilities. Both &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; and &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; support it, and retuning them to a MOS of diaschismic gives two scale possibilities.&lt;br /&gt;
The [[period]] of diaschismic is half an [[octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.
&lt;br /&gt;
 
&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;
This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.
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. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, and shrutar 245/243, the sensamagic comma. The other temperaments all keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as 36/35, the septimal quarter-tone.)&lt;/body&gt;&lt;/html&gt;</pre></div>
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 2048/2025
 
{{Mapping|legend=1| 2 0 11 | 0 1 -2 }}
: mapping generators: ~45/32, ~3
 
[[Optimal tuning]]s:  
* [[WE]]: ~45/32 = 599.4107{{c}}, ~3/2 = 704.2059{{c}}
: [[error map]]: {{val| -1.179 +1.072 +1.150 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.9585{{c}}
: error map: {{val| 0.000 +3.003 +3.769 }}
 
[[Tuning ranges]]:
* [[5-odd-limit]] [[diamond monotone]]: ~3/2 = [600.000 to 720.000] (1\2 to 6\10)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
 
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}
 
[[Badness]] (Sintel): 0.467
 
=== Overview to extensions ===
==== 7-limit extensions ====
To get the 7-limit extensions, we add another comma:
* Septimal diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.  
* Pajara adds [[50/49]] or [[64/63]] and is a popular and well-known choice.
* Srutal adds [[4375/4374]], the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.
* Keen adds [[875/864]].
 
Those all keep the same half-octave period and fifth generator.
 
Bidia adds [[3136/3125]], the hemimean comma, with a 1/4-octave period. Shrutar adds [[245/243]] and shru adds [[392/375]], with a quartertone generator. Sruti adds [[19683/19600]] and anguirus adds [[49/48]], with a neutral third or hemitwelfth generator. Those split the original generator in two. Echidna adds [[1728/1715]], the orwellisma, with a ~9/7 generator. Echidnic adds [[686/675]], the senga, with a ~8/7 generator. Those split the original generator in three. Finally, quadrasruta adds [[2401/2400]] and splits the original generator in four.  
 
Temperaments discussed elsewhere include [[stearnsmic clan #Echidna|echidna]]. The rest are considered below.
 
==== Subgroup extensions ====
Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', considered in [[#Subgroup extensions_2|#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.
 
More generally, one may note that since the fifth of diaschismic is sharp in any good tuning, it is compatible with the logic of [[parapyth]] in making ~13/11 a minor third and ~14/11 a major third, leading to the rank-3 temperament [[varda]], which multiple temps here are rank-2 tunings of, such as 17-limit diaschismic and srutal (and their obvious extensions to prime 23, na"naa' and srutaloo, both by mapping ~23/16 as an augmented fourth aka. tritone).
 
== Septimal diaschismic ==
{{Main| Diaschismic }}
{{See also| Srutal vs diaschismic }}
 
A simpler characterization than the one given by the normal comma list is that septimal diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, septimal diaschismic has a 1/2-octave period and a sharp fifth generator like the 5-limit version, but not so sharp, giving a more accurate but more complex temperament. [[104edo]] with the 104c [[val]] provides an excellent tuning, which is close to tuning [[7/4]] just by making the fifth 703.897 cents.
 
Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. This mapping can also be rationalized by [[parapyth]], which makes sense due to the sharp fifth, and prime 17 is found as in srutal archagall. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] scales of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 2048/2025
 
{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4449{{c}}, ~3/2 = 703.0299{{c}}
: [[error map]]: {{val| -1.110 -0.035 +3.740 -1.391 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7739{{c}}
: error map: {{val| 0.000 +1.819 +6.138 +0.983 }}
 
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 705.882] (7\12 to 20\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
 
{{Optimal ET sequence|legend=1| 12, 34, 46, 58, 104c, 162c }}
 
[[Badness]] (Sintel): 0.959
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 176/175, 896/891
 
{{Mapping|legend=0| 2 0 11 31 45 | 0 1 -2 -8 -12 }}
 
Optimal tunings:
* WE: ~45/32 = 599.4471{{c}}, ~3/2 = 703.0657{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7996{{c}}
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 704.348] (7\12 to 27\46)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
 
{{Optimal ET sequence|legend=0| 12, 34e, 46, 58, 104c, 162ce }}
 
Badness (Sintel): 0.828
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 176/175, 196/195, 364/363
 
{{Mapping|legend=0| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}
 
Optimal tunings:
* WE: ~45/32 = 599.4451{{c}}, ~3/2 = 703.0528{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7813{{c}}
 
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
 
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c, 162cef }}
 
Badness (Sintel): 0.782
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255
 
{{Mapping|legend=0| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}
 
Optimal tunings:
* WE: ~17/12 = 599.6253{{c}}, ~3/2 = 703.3726{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.8520{{c}}
 
Tuning ranges:
* 17-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
 
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c }}
 
Badness (Sintel): 0.837
 
=== 2.3.5.7.11.13.17.23 subgroup (Na"Naa') ===
<b>Na"Naa'</b> is a remarkable subgroup temperament of {{nowrap| 46 & 58 }} with a prime harmonic of 23. It is yet to be found why it got this strange name.
 
Subgroup: 2.3.5.7.11.13.17.23
 
Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255
 
{{Mapping|legend=2| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}
 
Optimal tunings:
* WE: ~17/12 = 599.6272{{c}}, ~3/2 = 703.4326{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.9093{{c}}
 
{{Optimal ET sequence|legend=0| 12i, 34efi, 46, 58i, 104ci }}
 
Badness (Sintel): 0.882
 
== Pajara ==
{{Main| Pajara }}
 
Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} (34d) and 56 with the val {{val| 56 89 130 158 }} (56d) are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
 
Pajara extends nicely to an 11-limit version, for which the 56edo tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 50/49, 64/63
 
{{Mapping|legend=1| 2 0 11 12 | 0 1 -2 -2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 598.8483{{c}}, ~3/2 = 705.6906{{c}}
: [[error map]]: {{val| -2.303 +1.432 -5.756 +10.580 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 707.3438{{c}}
: error map: {{val| 0.000 +5.389 -1.001 +16.487 }}
 
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 6\10)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 715.587]
 
{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}
 
[[Badness]] (Sintel): 0.507
 
=== 2.3.5.7.17 subgroup ===
Subgroup: 2.3.5.7.17
 
Comma list: 50/49, 64/63, 85/84
 
{{Mapping|legend=0| 2 0 11 12 5 | 0 1 -2 -2 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.053{{c}}, ~3/2 = 706.355{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.607{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12, 22, 56d }}
 
Badness (Sintel): 0.438
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 64/63, 99/98
 
{{Mapping|legend=0| 2 0 11 12 26 | 0 1 -2 -2 -6 }}
 
Optimal tunings:
* WE: ~7/5 = 598.8485{{c}}, ~3/2 = 705.5285{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.1826{{c}}
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 709.091] (7\12 to 13\22)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.587]
 
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}
 
Badness (Sintel): 0.673
 
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.17
 
Comma list: 50/49, 64/63, 85/84, 99/98
 
{{Mapping|legend=0| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.062{{c}}, ~3/2 = 706.095{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.370{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}
 
Badness (Sintel): 0.645
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 64/63, 65/63, 99/98
 
{{Mapping|legend=0| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}
 
Optimal tunings:
* WE: ~7/5 = 599.9732{{c}}, ~3/2 = 708.8873{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9227{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}
 
Badness (Sintel): 1.14
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 52/51, 64/63, 65/63, 99/98
 
{{Mapping|legend=0| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.8871{{c}}, ~3/2 = 708.6725{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.8176{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}
 
Badness (Sintel): 1.06
 
==== Pajarina ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 64/63, 78/77, 99/98
 
{{Mapping|legend=0| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}
 
Optimal tunings:
* WE: ~7/5 = 598.7732{{c}}, ~3/2 = 704.6889{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.3950{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}
 
Badness (Sintel): 0.923
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 64/63, 78/77, 85/84, 99/98
 
{{Mapping|legend=0| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.0204{{c}}, ~3/2 = 705.2572{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.5660{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}
 
Badness (Sintel): 0.936
 
==== Pajarita ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 50/49, 64/63, 66/65
 
{{Mapping|legend=0| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}
 
Optimal tunings:
* WE: ~7/5 = 598.3048{{c}}, ~3/2 = 705.4512{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.9238{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12f, 22f, 34dff }}
 
Badness (Sintel): 0.937
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 40/39, 50/49, 64/63, 66/65, 85/84
 
{{Mapping|legend=0| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}
 
Optimal tunings:
* WE: ~7/5 = 598.6103{{c}}, ~3/2 = 706.3076{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.2256{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}
 
Badness (Sintel): 0.968
 
=== Pajarous ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 55/54, 64/63
 
{{Mapping|legend=0| 2 0 11 12 -9 | 0 1 -2 -2 5 }}
 
Optimal tunings:
* WE: ~7/5 = 599.4055{{c}}, ~3/2 = 708.8747{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 709.5508{{c}}
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = 709.091 (13\22)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.803]
 
{{Optimal ET sequence|legend=0| 10, 12e, 22, 120bce, 142bce }}
 
Badness (Sintel): 0.937
 
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63
 
{{Mapping|legend=0| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.408{{c}}, ~3/2 = 708.878{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 709.544{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12e, 22 }}
 
Badness (Sintel): 0.766
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 55/54, 64/63, 65/63
 
{{Mapping|legend=0| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}
 
Optimal tunings:
* WE: ~7/5 = 599.9064{{c}}, ~3/2 = 710.1289{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2325{{c}}
 
{{Optimal ET sequence|legend=0| 10, 22 }}
 
Badness (Sintel): 1.04
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 52/51, 55/54, 64/63, 65/63
 
{{Mapping|legend=0| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.8239{{c}}, ~3/2 = 710.0128{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2067{{c}}
 
{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}
 
Badness (Sintel): 0.930
 
==== Pajaro ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 50/49, 55/54, 64/63
 
{{Mapping|legend=0| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}
 
Optimal tunings:
* WE: ~7/5 = 598.8257{{c}}, ~3/2 = 709.4266{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8414{{c}}
 
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}
 
Badness (Sintel): 1.13
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 40/39, 50/49, 55/54, 64/63, 85/84
 
{{Mapping|legend=0| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}
 
Optimal tunings:
* WE: ~7/5 = 598.8865{{c}}, ~3/2 = 709.5472{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8704{{c}}
 
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}
 
Badness (Sintel): 1.01
 
=== Pajaric ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 50/49, 56/55
 
{{Mapping|legend=0| 2 0 11 12 7 | 0 1 -2 -2 0 }}
 
Optimal tunings:
* WE: ~7/5 = 597.4807{{c}}, ~3/2 = 702.5616{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.0542{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12, 22e }}
 
Badness (Sintel): 0.787
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 45/44, 50/49, 56/55
 
{{Mapping|legend=0| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}
 
Optimal tunings:
* WE: ~7/5 = 597.1952{{c}}, ~3/2 = 704.1350{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.1989{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}
 
Badness (Sintel): 0.845
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 34/33, 40/39, 45/44, 50/49, 56/55
 
{{Mapping|legend=0| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}
 
Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~3/2 = 705.7702{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9719{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}
 
Badness (Sintel): 0.896
 
=== Hemipaj ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 64/63, 121/120
 
{{Mapping|legend=0| 2 1 9 10 8 | 0 2 -4 -4 -1 }}
 
: mapping generators: ~2, ~16/11
 
Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~16/11 = 652.7788{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 653.7119{{c}}
 
{{Optimal ET sequence|legend=0| 2, 20, 22 }}
 
Badness (Sintel): 1.29
 
=== Hemifourths ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 64/63, 243/242
 
{{Mapping|legend=0| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
: mapping generators: ~2, ~55/32
 
Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~55/32 = 950.8475{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~55/32 = 953.1172{{c}}
 
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
 
Badness (Sintel): 1.62
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 64/63, 78/77, 144/143
 
{{Mapping|legend=0| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}
 
Optimal tunings:
* WE: ~7/5 = 598.6748{{c}}, ~26/15 = 950.9691{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.1052{{c}}
 
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
 
Badness (Sintel): 1.19
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 64/63, 78/77, 85/84, 144/143
 
{{Mapping|legend=0| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}
 
Optimal tunings:
* WE: ~7/5 = 598.8411{{c}}, ~26/15 = 951.3687{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.2169{{c}}
 
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
 
Badness (Sintel): 1.11
 
== Srutal ==
{{See also| Srutal vs diaschismic }}
 
Srutal can be described as the {{nowrap| 34d & 46 }} temperament, where 7/4 is located at 15 generator steps, or the double-augmented fifth (C–Gx). As such, it weakly extends [[leapfrog]]. 80edo and [[126edo]] are among the possible tunings. Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 [[julius]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2048/2025, 4375/4374
 
{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4046{{c}}, ~3/2 = 704.1150{{c}}
: [[error map]]: {{val| -1.191 +0.969 +1.289 +0.044 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.7646{{c}}
: error map: {{val| 0.000 +2.810 +4.157 +2.643 }}
 
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
 
{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}
 
[[Badness]] (Sintel): 2.32
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 896/891, 1331/1323
 
{{Mapping|legend=0| 2 0 11 -42 -28 | 0 1 -2 15 11 }}
 
Optimal tunings:
* WE: ~45/32 = 599.4413{{c}}, ~3/2 = 704.1999{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8017{{c}}
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
 
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}
 
Badness (Sintel): 1.17
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 176/175, 325/324, 364/363
 
{{Mapping|legend=0| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}
 
Optimal tunings:
* WE: ~45/32 = 599.5490{{c}}, ~3/2 = 704.3516{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8347{{c}}
 
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
 
{{Optimal ET sequence|legend=0| 34d, 46, 80 }}
 
Badness (Sintel): 1.04
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 169/168, 176/175, 221/220, 256/255
 
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}
 
Optimal tunings:
* WE: ~17/12 = 599.6459{{c}}, ~3/2 = 704.4237{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8083{{c}}
 
Tuning ranges:
* 17-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
 
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126 }}
 
Badness (Sintel): 0.947
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255
 
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}
 
Optimal tunings:
* WE: ~17/12 = 599.6371{{c}}, ~3/2 = 704.4790{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8745{{c}}
 
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}
 
Badness (Sintel): 1.04
 
==== Srutaloo ====
Srutaloo adds 576/575, 736/729 or 208/207, and rhymes with [[skidoo]].
 
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255
 
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}
 
Optimal tunings:
* WE: ~17/12 = 599.6690{{c}}, ~3/2 = 704.5098{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8713{{c}}
 
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}
 
Badness (Sintel): 0.971
 
===== 29-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255
 
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}
 
Optimal tunings:
* WE: ~17/12 = 599.6664{{c}}, ~3/2 = 704.5138{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8807{{c}}
 
{{Optimal ET sequence|legend=0| 34dhj, 46, 80 }}
 
Badness (Sintel): 1.10
 
===== 31-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
 
Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 217/216, 221/220, 232/231, 256/255
 
{{Mapping|legend=0| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}
 
Optimal tunings:
* WE: ~17/12 = 599.8115{{c}}, ~3/2 = 704.5958{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8086{{c}}
 
{{Optimal ET sequence|legend=0| 46, 80, 126 }}
 
Badness (Sintel): 1.44
 
== Keen ==
Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the {{nowrap| 22 & 34 }} temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 875/864, 2048/2025
 
{{Mapping|legend=1| 2 0 11 -23 | 0 1 -2 9 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.6603{{c}}, ~3/2 = 707.1707{{c}}
: [[error map]]: {{val| -0.679 +4.536 -3.033 -2.591 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5294{{c}}
: error map: {{val| 0.000 +5.574 -1.373 -1.061 }}
 
{{Optimal ET sequence|legend=1| 22, 56, 78, 134b }}
 
[[Badness]] (Sintel): 2.13
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 385/384, 1232/1215
 
{{Mapping|legend=0| 2 0 11 -23 26 | 0 1 -2 9 -6 }}
 
Optimal tunings:
* WE: ~45/32 = 599.6286{{c}}, ~3/2 = 707.1712{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5984{{c}}
 
{{Optimal ET sequence|legend=0| 22, 56, 78 }}
 
Badness (Sintel): 1.50
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 105/104, 144/143, 1078/1053
 
{{Mapping|legend=0| 2 0 11 -23 26 -18 | 0 1 -2 9 -6 8 }}
 
Optimal tunings:
* WE: ~45/32 = 599.3498{{c}}, ~3/2 = 706.4009{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.1309{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 34, 56f }}
 
Badness (Sintel): 1.85
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 100/99, 105/104, 119/117, 144/143, 154/153
 
{{Mapping|legend=0| 2 0 11 -23 26 -18 5 | 0 1 -2 9 -6 8 1}}
 
Optimal tunings:
* WE: ~17/12 = 599.4053{{c}}, ~3/2 = 706.4544{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.1243{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 34, 56f }}
 
Badness (Sintel): 1.54
 
==== Keenic ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 100/99, 352/351, 385/384
 
{{Mapping|legend=0| 2 0 11 -23 26 36 | 0 1 -2 9 -6 -9 }}
 
Optimal tunings:
* WE: ~45/32 = 599.8547{{c}}, ~3/2 = 707.0858{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.2596{{c}}
 
{{Optimal ET sequence|legend=0| 22, 34, 56 }}
 
Badness (Sintel): 1.67
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 100/99, 136/135, 154/153, 256/255
 
{{Mapping|legend=0| 2 0 11 -23 26 36 5 | 0 1 -2 9 -6 -9 1 }}
 
Optimal tunings:
* WE: ~17/12 = 599.8338{{c}}, ~3/2 = 707.0558{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.2537{{c}}
 
{{Optimal ET sequence|legend=0| 22, 34, 56 }}
 
Badness (Sintel): 1.37
 
== Bidia ==
Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the {{nowrap| 12 & 68 }} temperament; its ploidacot is tetraploid monocot. Scales of bidia [[cluster temperament|cluster]] around [[12edo]], with a small residue left behind when three semitones exceed the quarter-octave period. This residue represents [[64/63]], and somewhat peculiarly, [[81/80]] is represented by ''two'' of these intervals.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2048/2025, 3136/3125
 
{{Mapping|legend=1| 4 0 22 43 | 0 1 -2 -5 }}
: mapping generators: ~25/21, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 299.6887{{c}}, ~3/2 = 704.6318{{c}}
: [[error map]]: {{val| -1.245 +1.432 +0.064 +0.854 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 705.5070{{c}}
: error map: {{val| 0.000 +3.552 +2.672 +3.639 }}
 
{{Optimal ET sequence|legend=1| 12, …, 56, 68, 80, 148d }}
 
[[Badness]] (Sintel): 1.43
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 896/891, 1375/1372
 
{{Mapping|legend=0| 4 0 22 43 71 | 0 1 -2 -5 -9 }}
 
Optimal tunings:
* WE: ~25/21 = 299.6809{{c}}, ~3/2 = 704.3367{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.2170{{c}}
 
{{Optimal ET sequence|legend=0| 12, 56e, 68, 80 }}
 
Badness (Sintel): 1.33
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 325/324, 640/637, 896/891
 
{{Mapping|legend=0| 4 0 22 43 71 -36 | 0 1 -2 -5 -9 8 }}
 
Optimal tunings:
* WE: ~25/21 = 299.7538{{c}}, ~3/2 = 704.7222{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3241{{c}}
 
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
 
Badness (Sintel): 1.70
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 176/175, 256/255, 325/324, 640/637
 
{{Mapping|legend=0| 4 0 22 43 71 -36 10 | 0 1 -2 -5 -9 8 1 }}
 
Optimal tunings:
* WE: ~25/21 = 299.7883{{c}}, ~3/2 = 704.8365{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3496{{c}}
 
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
 
Badness (Sintel): 1.46
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637
 
{{Mapping|legend=0| 4 0 22 43 71 -36 10 17 | 0 1 -2 -5 -9 8 1 0 }}
 
Optimal tunings:
* WE: ~19/16 = 299.7967{{c}}, ~3/2 = 704.8609{{c}}
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3519{{c}}
 
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
 
Badness (Sintel): 1.25
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 136/135, 176/175, 190/189, 253/252, 256/255, 325/324, 640/637
 
{{Mapping|legend=0| 4 0 22 43 71 -36 10 17 -20 | 0 1 -2 -5 -9 8 1 0 6 }}
 
Optimal tunings:
* WE: ~19/16 = 299.7961{{c}}, ~3/2 = 704.8577{{c}}
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3413{{c}}
 
{{Optimal ET sequence|legend=0| 12, 68, 80, 148di }}
 
Badness (Sintel): 1.24
 
== Shrutar ==
Shrutar adds 245/243 to the commas, and also tempers out [[6144/6125]]. It can also be described as {{nowrap| 22 & 46 }}. Its generator can be taken as either ~36/35 or ~35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. Its ploidacot is diploid alpha-dicot. [[68edo]] makes for a good tuning, but another excellent choice is a generator of 14<sup>(1/7)</sup>, making 7's just.
 
By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.
 
Additionally, shrutar can employ the standard diaschismic mapping of prime 17, and most naturally represents the 2.3.5.7.11.17 subgroup temperament where 15:16:17:18 and 32:33:34:35:36 are equalized. Shrutar canonically maps primes 13, 19, and 23 as the 46 & 68 temperament; these mappings are significantly more complex and need finer tuning, however.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 245/243, 2048/2025
 
{{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }}
: mapping generators: ~45/32, ~35/24
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.5401{{c}}, ~35/24 = 652.3108{{c}}
: [[error map]]: {{val| -0.920 +2.207 +0.304 -1.730 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~35/24 = 652.7736{{c}}
: error map: {{val| 0.000 +3.592 +2.592 +0.589 }}
 
{{Optimal ET sequence|legend=1| 22, 46, 68, 182b, 250bc }}
 
[[Badness]] (Sintel): 1.20
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 245/243
 
{{Mapping|legend=0| 2 1 9 -2 8 | 0 2 -4 7 -1 }}
 
Optimal tunings:
* WE: ~45/32 = 599.7721{{c}}, ~16/11 = 652.4321{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6672{{c}}
 
{{Optimal ET sequence|legend=0| 22, 46, 68, 114 }}
 
Badness (Sintel): 0.876
 
=== 13-limit ===
Note that an alternative, simpler mapping for prime 13 exists at -7 gens, as in [[#Shru]]. This mapping is especially valuable when considering shrutar as a no-7's 17-limit temperament (22 & 24, supported by 46, 68f and 70), as the complexity from mapping 7 at +7 gens is one of the motivations for picking a +16 gen mapping for 13. This temperament isn't as accurate, but it is simple, and makes many [[MOS]]es, specifically 2L 2''n''s for ''n'' up to [[2L 20s|10]], followed by [[22L 2s]] for the 24-note MOS, with a quarter-tone generator equivalent to ~11/8.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 176/175, 196/195, 245/243
 
{{Mapping|legend=0| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}
 
Optimal tunings:
* WE: ~45/32 = 599.7699{{c}}, ~16/11 = 652.4035{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6374{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}
 
Badness (Sintel): 1.16
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195
 
{{Mapping|legend=0| 2 1 9 -2 8 -10 6 | 0 2 -4 7 -1 16 2 }}
 
Optimal tunings:
* WE: ~17/12 = 599.7995{{c}}, ~16/11 = 652.4287{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6334{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}
 
Badness (Sintel): 0.953
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342
 
{{Mapping|legend=0| 2 1 9 -2 8 -10 6 -10 | 0 2 -4 7 -1 16 2 17 }}
 
Optimal tunings:
* WE: ~17/12 = 599.8060{{c}}, ~16/11 = 652.5190{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.7164{{c}}
 
{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114, 182bef }}
 
Badness (Sintel): 1.07
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 253/252, 343/342
 
{{Mapping|legend=0| 2 1 9 -2 8 -10 6 -10 -4 | 0 2 -4 7 -1 16 2 17 12 }}
 
Optimal tunings:
* WE: ~17/12 = 599.7879{{c}}, ~16/11 = 652.4776{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6926{{c}}
 
{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114 }}
 
Badness (Sintel): 1.03
 
== Shru ==
Shru tempers out 392/375 and slices the compound semitone into two generators of ~10/7. Its ploidacot is diploid alpha-dicot, the same as shrutar. It can be seen as one way of merging [[parapyth]] with [[diaschismic]] (and hence as a lower-accuracy rank 2 tuning of [[varda]]).
 
Note that shru is especially efficient as a no-7's 17-limit temperament, as its 7 is the highest damage, and without it, the other primes can be tuned more accurately.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 392/375, 1323/1280
 
{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
: mapping generators: ~45/32, ~10/7
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 600.2519{{c}}, ~10/7 = 650.4083{{c}}
: [[error map]]: {{val| +0.504 -0.887 +14.321 -18.096 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~10/7 = 650.1017{{c}}
: error map: {{val| 0.000 -1.752 +13.279 -19.334 }}
 
{{Optimal ET sequence|legend=1| 2, 22d, 24 }}
 
[[Badness]] (Sintel): 3.99
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 77/75, 1323/1280
 
{{Mapping|legend=0| 2 1 9 11 8 | 0 2 -4 -5 -1 }}
 
Optimal tunings:
* WE: ~17/12 = 600.2356{{c}}, ~10/7 = 650.3856{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}
 
{{Optimal ET sequence|legend=0| 2, 22d, 24 }}
 
Badness (Sintel): 2.10
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 77/75, 105/104, 507/500
 
{{Mapping|legend=0| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}
 
Optimal tunings:
* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}
 
{{Optimal ET sequence|legend=0| 2, 24 }}
 
Badness (Sintel): 2.12
 
== Sruti ==
{{Redirect|Sruti|the tuning concept in Indian music|Shruti}}
 
Sruti tempers out 19683/19600, setting itself up as a [[hemipyth]] temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as {{nowrap| 24 & 34d }}; its ploidacot is diploid dicot. [[58edo]] may be recommended as a tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2048/2025, 19683/19600
 
{{Mapping|legend=1| 2 0 11 -15 | 0 2 -4 13 }}
: mapping generators: ~45/32, ~140/81
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.2764{{c}}, ~140/81 = 950.7284{{c}}
: [[error map]]: {{val| -1.447 -0.498 +2.813 +1.497 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~140/81 = 951.8227{{c}}
: error map: {{val| 0.000 +1.690 +6.395 +4.869 }}
 
{{Optimal ET sequence|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}
 
[[Badness]] (Sintel): 2.97
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 243/242, 896/891
 
{{Mapping|legend=0| 2 0 11 -15 -1 | 0 2 -4 13 5 }}
 
Optimal tunings:
* WE: ~45/32 = 599.1951{{c}}, ~121/70 = 950.5864{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~121/70 = 951.7972{{c}}
 
{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdee, 208ccddee, 266ccddeee }}
 
Badness (Sintel): 1.37
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 176/175, 351/350, 676/675
 
{{Mapping|legend=0| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}
 
Optimal tunings:
* WE: ~45/32 = 599.1479{{c}}, ~26/15 = 950.5337{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~26/15 = 951.8314{{c}}
 
{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdeef, 208ccddeeff, 266ccddeeefff }}
 
Badness (Sintel): 0.983
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 144/143, 170/169, 176/175, 221/220
 
{{Mapping|legend=0| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}
 
Optimal tunings:
* WE: ~17/12 = 599.3003{{c}}, ~26/15 = 950.7465{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~26/15 = 951.8142{{c}}
 
{{Optimal ET sequence|legend=0| 24, 34d, 58 }}
 
Badness (Sintel): 1.05
 
== Anguirus ==
As another hemipyth temperament, anguirus tempers out 49/48. It can be described as the {{nowrap| 10 & 24 }} temperament; its ploidacot is diploid dicot, the same as sruti.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 49/48, 2048/2025
 
{{Mapping|legend=1| 2 0 11 4 | 0 2 -4 1 }}
: mapping generators: ~45/32, ~7/4
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 600.2758{{c}}, ~7/4 = 953.4593{{c}}
: [[error map]]: {{val| +0.552 +4.964 +2.883 -14.264 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~7/4 = 953.0188{{c}}
: error map: {{val| 0.000 +4.083 +1.611 -15.807 }}
 
{{Optimal ET sequence|legend=1| 10, 24, 34 }}
 
[[Badness]] (Sintel): 1.97
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 56/55, 243/242
 
{{Mapping|legend=0| 2 0 11 4 -1 | 0 2 -4 1 5 }}
 
Optimal tunings:
* WE: ~45/32 = 599.9250{{c}}, ~7/4 = 952.0646{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.1784{{c}}
 
{{Optimal ET sequence|legend=0| 10, 24, 34 }}
 
Badness (Sintel): 1.63
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 56/55, 91/90, 243/242
 
{{Mapping|legend=0| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}
 
Optimal tunings:
* WE: ~45/32 = 599.7575{{c}}, ~7/4 = 951.9241{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.2980{{c}}
 
{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}
 
Badness (Sintel): 1.27
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 49/48, 56/55, 91/90, 119/117, 154/153
 
{{Mapping|legend=0| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}
 
Optimal tunings:
* WE: ~17/12 = 599.7925{{c}}, ~7/4 = 952.0004{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~7/4 = 952.3178{{c}}
 
{{Optimal ET sequence|legend=0| 10, 24, 34 }}
 
Badness (Sintel): 1.10
 
== Echidnic ==
Echidnic tempers out 686/675 and [[1029/1024]]. It has the same semi-octave period as diaschismic, but slices the generator of a fifth into three ~8/7's. It can be described as the {{nowrap| 10 & 46 }} temperament; its ploidacot is diploid tricot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 686/675, 1029/1024
 
{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
: mapping generators: ~45/32, ~8/7
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.7208{{c}}, ~8/7 = 234.8330{{c}}
: [[error map]]: {{val| -0.558 +1.986 +2.733 -5.334 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~8/7 = 234.9539{{c}}
: error map: {{val| 0.000 +2.907 +3.963 -3.780 }}
 
{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}
 
[[Badness]] (Sintel): 1.83
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 686/675
 
{{Mapping|legend=0| 2 2 7 6 3 | 0 3 -6 -1 10 }}
 
Optimal tunings:
* WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }}
 
Badness (Sintel): 1.49
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 169/168, 385/384, 441/440
 
{{Mapping|legend=0| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}
 
Optimal tunings:
* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
 
Badness (Sintel): 1.19
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
 
{{Mapping|legend=0| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}
 
Optimal tunings:
* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
 
Badness (Sintel): 0.983
 
; Music
* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]
* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)
 
== Quadrasruta ==
Named by [[Xenllium]] in 2022, quadrasruta tempers out 2401/2400, the breedsma, and extends [[buzzard]]. It may be described as {{nowrap| 58 & 68 }}; its ploidacot is diploid alpha-tetracot. 126edo may be recommended as a tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2048/2025, 2401/2400
 
{{Mapping|legend=1| 2 0 11 8 | 0 4 -8 -3 }}
: mapping generators: ~45/32, ~21/16
 
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4443{{c}}, ~21/16 = 475.7746{{c}}
: [[error map]]: {{val| -1.111 +1.143 +1.377 -0.595 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~21/16 = 476.2394{{c}}
: error map: {{val| 0.000 +3.003 +3.771 +2.456 }}
 
{{Optimal ET sequence|legend=1| 10, …, 58, 68, 126, 446bbccd }}
 
[[Badness]] (Sintel): 1.86
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 896/891, 2401/2400
 
{{Mapping|legend=0| 2 0 11 8 22 | 0 4 -8 -3 -19 }}
 
Optimal tunings:
* WE: ~45/32 = 599.4648{{c}}, ~21/16 = 475.6929{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1507{{c}}
 
{{Optimal ET sequence|legend=0| 10e, …, 58, 126, 184c, 310bccde }}
 
Badness (Sintel): 1.62
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 196/195, 512/507, 676/675
 
{{Mapping|legend=0| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}
 
Optimal tunings:
* WE: ~45/32 = 599.3787{{c}}, ~21/16 = 475.6065{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1345{{c}}
 
{{Optimal ET sequence|legend=0| 10e, …, 58, 126f, 184cff }}
 
Badness (Sintel): 1.18
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 170/169, 176/175, 196/195, 256/255
 
{{Mapping|legend=0| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}
 
Optimal tunings:
* WE: ~17/12 = 599.5077{{c}}, ~21/16 = 475.7713{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.1814{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 58, 126f }}
 
Badness (Sintel): 1.21
 
=== Quadrafourths ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 2048/2025
 
{{Mapping|legend=0| 2 0 11 8 -1 | 0 4 -8 -3 10 }}
 
Optimal tunings:
* WE: ~45/32 = 599.2593{{c}}, ~21/16 = 475.4292{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0088{{c}}
 
{{Optimal ET sequence|legend=0| 10, 48c, 58, 184cee, 242ccdeee }}
 
Badness (Sintel): 1.62
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 196/195, 243/242, 676/675
 
{{Mapping|legend=0| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}
 
Optimal tunings:
* WE: ~45/32 = 599.2147{{c}}, ~21/16 = 475.4052{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0253{{c}}
 
{{Optimal ET sequence|legend=0| 10, 48c, 58, 126eef, 184ceeff, 242ccdeeeff }}
 
Badness (Sintel): 1.11
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 144/143, 170/169, 196/195, 221/220
 
{{Mapping|legend=0| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}
 
Optimal tunings:
* WE: ~17/12 = 599.3353{{c}}, ~21/16 = 475.5495{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.0691{{c}}
 
{{Optimal ET sequence|legend=0| 10, 48c, 58 }}
 
Badness (Sintel): 1.13
 
== Subgroup extensions ==
=== Srutal archagall (2.3.5.17) ===
{{See also | Fiventeen }}
 
This extension of 5-limit diaschismic adds prime 17 and which with respect to [[MVP archagall]] is able to express the harmonics [[75/1|75]] and [[85/1|85]] in their appropriate prime subgroup. It achieves this by equating [[85/64]] with [[4/3]] by tempering out their difference of [[256/255]] (S16). Therefore it also tempers out [[289/288]] (S17) and thus equates [[17/15]] with [[9/8]] due to tempering out [[136/135]] (S16⋅S17). It could be described as the 10 & 12 temperament with strong emphasis on 12edo being the better tuning on the 2.3.5.17 subgroup, implying ideal tunings of 34edo, 46edo or 80edo.
 
Subgroup: 2.3.5.17
 
Comma list: 136/135, 256/255
 
{{Mapping|legend=2| 2 0 11 5 | 0 1 -2 1 }}
: mapping generators: ~17/12, ~3
 
Optimal tunings:
* WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }}
 
Badness (Sintel): 0.212
 
[[Category:Diaschismic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
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