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Dicot family: Difference between revisions

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**Imported revision 147269605 - Original comment: **
Decanonicalize septimal dicot. - 2.3.5.11-subgroup eudicot (no need for explicit documentation if it's canonical)
 
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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 '''dicot family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] [[25/24]], the classical chromatic semitone. Dicot was likely the first named of the temperaments ending in -cot, as it is the only one to correspond with a proper botanical term (referring to plants with two embryonic leaves) and it is the most inaccurate.  
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2010-06-06 17:36:31 UTC</tt>.<br>
: The original revision id was <tt>147269605</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 dicot family is 25/24, the [[chromatic semitone]]. Its [[monzo]] is |-3 -1 2&gt;, and flipping that yields &lt;&lt;2 1 -3|| for the [[wedgie]]. This tells us the generator is a third (major and minor mean the same thing), and that two thirds gives a fifth. In fact, (5/4)^2 = 3/2 * 25/24. Possible tunings for dicot are [[7edo]], [[24edo]] using the val &lt;24 38 55| and [[31edo]] using the val &lt;31 49 71|. In a sense, what dicot is all about is using neutral thirds and pretending that's 5-limit, and like any temperament which seems to involve pretending dicot is at the edge of what can sensibly be called a temperament at all.


==Seven limit children==
== Dicot ==
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Septimal dicot, with wedgie &lt;&lt;2 1 6 -3 4 11|| adds 28/27, retaining the same period and generator, decimal with wedgie &lt;&lt;4 2 2 -6 -8 -1|| adds 49/48, and sidi with wedgie &lt;&lt;4 2 9 -3 6 15|| adds 245/243. Here decimal divides the period to 1/2 octave, and sidi uses 9/7 as a generator, with two of them making up the combined 5/3 and 8/5 neutral sixth.
{{Main| Dicot }}


The head of this family, dicot, is [[generator|generated]] by a classical third (major and minor mean the same thing), and two such thirds give a fifth. In fact, {{nowrap|(5/4)<sup>2</sup> {{=}} (3/2)(25/24)}}. Its [[ploidacot]] is the same as its name, dicot.


</pre></div>
Possible tunings for dicot are [[7edo]], [[10edo]], [[17edo]], [[24edo]] using the val {{val| 24 38 55 }} (24c), and [[31edo]] using the val {{val| 31 49 71 }} (31c). In a sense, what dicot is all about is using neutral thirds and sixths and pretending that these are 5-limit, and like any temperament which seems to involve a lot of "pretending", dicot is close to the edge of what can be sensibly called a temperament at all. In other words, it is an [[exotemperament]].
<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;Dicot family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 5-limit parent comma for the dicot family is 25/24, the &lt;a class="wiki_link" href="/chromatic%20semitone"&gt;chromatic semitone&lt;/a&gt;. Its &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt; is |-3 -1 2&amp;gt;, and flipping that yields &amp;lt;&amp;lt;2 1 -3|| for the &lt;a class="wiki_link" href="/wedgie"&gt;wedgie&lt;/a&gt;. This tells us the generator is a third (major and minor mean the same thing), and that two thirds gives a fifth. In fact, (5/4)^2 = 3/2 * 25/24. Possible tunings for dicot are &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt;, &lt;a class="wiki_link" href="/24edo"&gt;24edo&lt;/a&gt; using the val &amp;lt;24 38 55| and &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; using the val &amp;lt;31 49 71|. In a sense, what dicot is all about is using neutral thirds and pretending that's 5-limit, and like any temperament which seems to involve pretending dicot is at the edge of what can sensibly be called a temperament at all.&lt;br /&gt;
[[Subgroup]]: 2.3.5
&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;
[[Comma list]]: 25/24
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. Septimal dicot, with wedgie &amp;lt;&amp;lt;2 1 6 -3 4 11|| adds 28/27, retaining the same period and generator, decimal with wedgie &amp;lt;&amp;lt;4 2 2 -6 -8 -1|| adds 49/48, and sidi with wedgie &amp;lt;&amp;lt;4 2 9 -3 6 15|| adds 245/243. Here decimal divides the period to 1/2 octave, and sidi uses 9/7 as a generator, with two of them making up the combined 5/3 and 8/5 neutral sixth.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
{{Mapping|legend=1| 1 1 2 | 0 2 1 }}
: mapping generators: ~2, ~5/4
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1206.283{{c}}, ~5/4 = 350.420{{c}}
: [[error map]]: {{val| +6.283 +5.167 -23.328 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 351.086{{c}}
: error map: {{val| 0.000 +0.216 -35.228 }}
 
[[Tuning ranges]]:  
* [[5-odd-limit]] [[diamond monotone]]: ~5/4 = [300.000, 400.000] (1\4 to 1\3)
* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [315.641, 386.314] (full comma to untempered)
 
{{Optimal ET sequence|legend=1| 3, 4, 7, 17, 24c, 31c }}
 
[[Badness]] (Sintel): 0.306
 
=== Overview to extensions ===
==== 7-limit extensions ====
The second comma of the comma list defines which [[7-limit]] family member we are looking at. Mujannabic adds [[36/35]], flattie adds [[21/20]], sharpie adds [[28/27]], and dichotic adds [[64/63]], all retaining the same period and generator.
 
The dicot comma, 25/24, factors into the 7-limit as ([[49/48]])⋅([[50/49]]). Since [[49/48]] is the difference between [[8/7]] and [[7/6]], and [[50/49]] is the difference between [[7/5]] and [[10/7]], it makes sense to extend dicot to temper them all out, leading to decimal, a weak extension where the octave and twelfth are split in halves. Other weak extensions include sidi, which adds [[245/243]], and jamesbond, which adds [[16/15]]. Here sidi uses 14/9 as a generator, with two of them making up the combined [[5/2]][[~]][[12/5]] neutral tenth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.
 
Temperaments discussed elsewhere are:
* ''[[Geryon]]'' → [[Very low accuracy temperaments #Geryon|Very low accuracy temperaments]]
* ''[[Jamesbond]]'' → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]]
 
The rest are considered in each sections below.
 
==== Subgroup extensions ====
In the 11-limit, we have the identity 25/24 = ([[45/44]])⋅([[55/54]]), so it makes sense to temper out all of them. This leads to the very natural subgroup temperament where [[11/9]]~[[27/22]] is mapped to the neutral third. As such, this is also the path that most of the septimal extensions take to get their 11-limit versions.
 
An alternative identity is 25/24 = ([[33/32]])⋅([[100/99]]), and tempering out these commas leads to the 2.3.5.11-subgroup restriction of some of the temperaments below.
 
=== 2.3.5.11 subgroup ===
Subgroup: 2.3.5.11
 
Comma list: 25/24, 45/44
 
Subgroup val mapping: {{mapping| 1 1 2 2 | 0 2 1 5 }}
 
Gencom mapping: {{mapping| 1 1 2 0 2 | 0 2 1 0 5 }}
 
Optimal tunings:
* WE: ~2 = 1206.750{{c}}, ~5/4 = 348.684{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 348.954{{c}}
 
{{Optimal ET sequence|legend=0| 3e, 4e, 7, 24c, 31c }}
 
Badness (Sintel): 0.370
 
==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13
 
Comma list: 25/24, 40/39, 45/44
 
Subgroup val mapping: {{mapping| 1 1 2 2 4 | 0 2 1 5 -1 }}
 
Gencom mapping: {{mapping| 1 1 2 0 2 4 | 0 2 1 0 5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1202.433{{c}}, ~5/4 = 351.237{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 350.978{{c}}
 
{{Optimal ET sequence|legend=0| 3e, 7, 17 }}
 
Badness (Sintel): 0.536
 
== Mujannabic ==
Mujannabic extends dicot such that [[7/6]] and [[9/7]] are also conflated with 5/4~6/5. Although 5/4–6/5 covers a giant block of pitches already, 7/6 and 9/7 are often considered as thirds too. On that account one could argue for the utility of this extension despite the relatively poor accuracy.
 
Mujannabic was known as ''septimal dicot'' in earlier materials such as [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 15/14, 25/24
 
{{Mapping|legend=1| 1 1 2 2 | 0 2 1 3 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1205.532{{c}}, ~6/5 = 337.931{{c}}
: [[error map]]: {{val| +5.532 -20.561 -37.319 +56.032 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.561{{c}}
: error map: {{val| 0.000 -24.834 -47.753 +46.856 }}
 
{{Optimal ET sequence|legend=1| 3d, 4, 7 }}
 
[[Badness]] (Sintel): 0.504
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 15/14, 22/21, 25/24
 
Mapping: {{mapping| 1 1 2 2 2 | 0 2 1 3 5 }}
 
Optimal tunings:
* WE: ~2 = 1203.346{{c}}, ~6/5 = 343.078{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 343.260{{c}}
 
{{Optimal ET sequence|legend=0| 3de, 4e, 7 }}
 
Badness (Sintel): 0.656
 
=== Eudicot ===
Subgroup: 2.3.5.7.11
 
Comma list: 15/14, 25/24, 33/32
 
Mapping: {{mapping| 1 1 2 2 4 | 0 2 1 3 -2 }}
 
Optimal tunings:
* WE: ~2 = 1205.828{{c}}, ~6/5 = 337.683{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 336.909{{c}}
 
{{Optimal ET sequence|legend=0| 3d, 4, 7, 18bc, 25bccd }}
 
Badness (Sintel): 0.896
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 15/14, 25/24, 33/32, 40/39
 
Mapping: {{mapping| 1 1 2 2 4 4 | 0 2 1 3 -2 -1 }}
 
Optimal tunings:
* WE: ~2 = 1202.660{{c}}, ~6/5 = 339.597{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 339.104{{c}}
 
{{Optimal ET sequence|legend=0| 3d, 4, 7 }}
 
Badness (Sintel): 0.985
 
== Flattie ==
This temperament used to be known as ''flat''. Unlike mujannabic where 7/6 is added to the neutral third, here [[8/7]] is added instead.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 21/20, 25/24
 
{{Mapping|legend=1| 1 1 2 3 | 0 2 1 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1220.466{{c}}, ~6/5 = 337.577{{c}}
: [[error map]]: {{val| +20.466 -6.335 -7.804 -45.004 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 335.391{{c}}
: error map: {{val| 0.000 -31.173 -50.922 -104.217 }}
 
{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
 
[[Badness]] (Sintel): 0.642
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 21/20, 25/24, 33/32
 
Mapping: {{mapping| 1 1 2 3 4 | 0 2 1 -1 -2 }}
 
Optimal tunings:
* WE: ~2 = 1216.069{{c}}, ~6/5 = 342.052{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 338.467{{c}}
 
{{Optimal ET sequence|legend=0| 3, 4, 7d }}
 
Badness (Sintel): 0.826
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 14/13, 21/20, 25/24, 33/32
 
Mapping: {{mapping| 1 1 2 3 4 4 | 0 2 1 -1 -2 -1 }}
 
Optimal tunings:
* WE: ~2 = 1211.546{{c}}, ~6/5 = 344.304{{c}}
* CWE: ~2 = 1200.000{{c}}, ~6/5 = 341.373{{c}}
 
{{Optimal ET sequence|legend=0| 3, 4, 7d }}
 
Badness (Sintel): 0.968
 
== Sharpie ==
This temperament used to be known as ''sharp''. This is where you find 7/6 at the major second and [[7/4]] at the major sixth.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 25/24, 28/27
 
{{Mapping|legend=1| 1 1 2 1 | 0 2 1 6 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.488{{c}}, ~5/4 = 358.680{{c}}
: [[error map]]: {{val| +2.488 +17.893 -22.658 -14.258 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 358.495{{c}}
: error map: {{val| 0.000 +15.035 -27.818 -17.854 }}
 
{{Optimal ET sequence|legend=1| 3d, 7d, 10 }}
 
[[Badness]] (Sintel): 0.732
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 25/24, 28/27, 35/33
 
Mapping: {{mapping| 1 1 2 1 2 | 0 2 1 6 5 }}
 
Optimal tunings:
* WE: ~2 = 1201.518{{c}}, ~5/4 = 356.557{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 356.457{{c}}
 
{{Optimal ET sequence|legend=0| 3de, 7d, 10, 17d }}
 
Badness (Sintel): 0.739
 
== Dichotic ==
In dichotic, 7/4 is found at a stack of two perfect fourths.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 25/24, 64/63
 
{{Mapping|legend=1| 1 1 2 4 | 0 2 1 -4 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.802{{c}}, ~5/4 = 356.502{{c}}
: [[error map]]: {{val| +0.802 +11.851 -28.208 +8.374 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 356.275{{c}}
: error map: {{val| 0.000 +10.595 -30.039 +6.074 }}
 
{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c }}
 
[[Badness]] (Sintel): 0.951
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 25/24, 45/44, 64/63
 
Mapping: {{mapping| 1 1 2 4 2 | 0 2 1 -4 5 }}
 
Optimal tunings:
* WE: ~2 = 1199.504{{c}}, ~5/4 = 354.115{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.236{{c}}
 
{{Optimal ET sequence|legend=0| 7, 10, 17 }}
 
Badness (Sintel): 1.01
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 25/24, 40/39, 45/44, 64/63
 
Mapping: {{mapping| 1 1 2 4 2 4 | 0 2 1 -4 5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1199.289{{c}}, ~5/4 = 354.156{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.340{{c}}
 
{{Optimal ET sequence|legend=0| 7, 10, 17, 27ce, 44cce }}
 
Badness (Sintel): 0.896
 
=== Dichotomic ===
Subgroup: 2.3.5.7.11
 
Comma list: 22/21, 25/24, 33/32
 
Mapping: {{mapping| 1 1 2 4 4 | 0 2 1 -4 -2 }}
 
Optimal tunings:
* WE: ~2 = 1203.949{{c}}, ~5/4 = 355.239{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.024{{c}}
 
{{Optimal ET sequence|legend=0| 3, 7, 10e }}
 
Badness (Sintel): 1.05
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 22/21, 25/24, 33/32, 40/39
 
Mapping: {{mapping| 1 1 2 4 4 4 | 0 2 1 -4 -2 -1 }}
 
Optimal tunings:
* WE: ~2 = 1202.979{{c}}, ~5/4 = 355.193{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 354.254{{c}}
 
{{Optimal ET sequence|legend=0| 3, 7, 10e }}
 
Badness (Sintel): 0.940
 
=== Dichosis ===
Subgroup: 2.3.5.7.11
 
Comma list: 25/24, 35/33, 64/63
 
Mapping: {{mapping| 1 1 2 4 5 | 0 2 1 -4 -5 }}
 
Optimal tunings:
* WE: ~2 = 1197.526{{c}}, ~5/4 = 359.915{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.745{{c}}
 
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}
 
Badness (Sintel): 1.37
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 25/24, 35/33, 40/39, 64/63
 
Mapping: {{mapping| 1 1 2 4 5 4 | 0 2 1 -4 -5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1197.922{{c}}, ~5/4 = 360.021{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5/4 = 360.722{{c}}
 
{{Optimal ET sequence|legend=0| 3, 7e, 10 }}
 
Badness (Sintel): 1.15
 
== Decimal ==
{{Main| Decimal }}
{{See also| Jubilismic clan }}
 
Decimal tempers out 49/48 and [[50/49]], and has a semi-octave period for 7/5~10/7 and a hemitwelfth generator for 7/4~12/7. Its ploidacot is diploid dicot. [[10edo]] makes for a good tuning, from which it derives its name. [[14edo]] in the 14c val and [[24edo]] in the 24c val are also among the possibilities.
 
Decimal can be extended to the 11-limit by the usual path of tempering out 45/44 and 55/54. There is an alternative due to the identity 50/49 = ([[99/98]])⋅([[100/99]]), in which case it also tempers out 33/32. The two mappings meet at the 14c val of [[14edo]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 25/24, 49/48
 
{{Mapping|legend=1| 2 0 3 4 | 0 2 1 1 }}
 
: mapping generators: ~7/5, ~7/4
 
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 603.286{{c}}, ~7/4 = 953.637{{c}} (~7/6 = 252.935{{c}})
: [[error map]]: {{val| +6.571 +5.318 -22.821 -2.047 }}
* [[CWE]]: ~7/5 = 600.000{{c}}, ~7/4 = 950.957{{c}} (~7/6 = 249.043{{c}})
: error map: {{val| 0.000 -0.041 -35.357 -17.869 }}
 
{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd }}
 
[[Badness]] (Sintel): 0.717
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 25/24, 45/44, 49/48
 
Mapping: {{mapping| 2 0 3 4 -1 | 0 2 1 1 5 }}
 
Optimal tunings:
* WE: ~7/5 = 603.558{{c}}, ~7/4 = 952.121{{c}} (~7/6 = 254.996{{c}})
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 948.610{{c}} (~7/6 = 251.390{{c}})
 
{{Optimal ET sequence|legend=0| 4e, 10, 14c, 24c }}
 
Badness (Sintel): 0.883
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 25/24, 45/44, 49/48, 91/90
 
Mapping: {{mapping| 2 0 3 4 -1 1| 0 2 1 1 5 4}}
 
Optimal tunings:
* WE: ~7/5 = 603.612{{c}}, ~7/4 = 953.663{{c}} (~7/6 = 253.562{{c}})
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 950.116{{c}} (~7/6 = 249.884{{c}})
 
{{Optimal ET sequence|legend=0| 4ef, 10, 14cf, 24cf }}
 
Badness (Sintel): 0.881
 
=== Decimated ===
Subgroup: 2.3.5.7.11
 
Comma list: 25/24, 33/32, 49/48
 
Mapping: {{mapping| 2 0 3 4 10 | 0 2 1 1 -2 }}
 
Optimal tunings:
* WE: ~7/5 = 604.535{{c}}, ~7/4 = 952.076{{c}} (~7/6 = 256.994{{c}})
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 946.108{{c}} (~7/6 = 253.892{{c}})
 
{{Optimal ET sequence|legend=0| 4, 10e, 14c }}
 
Badness (Sintel): 1.04
 
=== Decibel ===
Subgroup: 2.3.5.7.11
 
Comma list: 25/24, 35/33, 49/48
 
Mapping: {{mapping| 2 0 3 4 7 | 0 2 1 1 0 }}
 
Optimal tunings:
* WE: ~7/5 = 599.404{{c}}, ~7/4 = 955.557{{c}} (~8/7 = 243.251{{c}})
* CWE: ~7/5 = 600.000{{c}}, ~7/4 = 956.169{{c}} (~8/7 = 243.831{{c}})
 
{{Optimal ET sequence|legend=0| 4, 6, 10 }}
 
Badness (Sintel): 1.07
 
== Sidi ==
Sidi tempers out [[245/243]], and splits [[5/2]][[~]][[12/5]] in two. Its [[ploidacot]] is beta-tetracot. This relates it to [[squares]], to which it can be used as a simpler alternative. 14edo in the 14c val can be used as a tuning, in which case it is identical to squares, however.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 25/24, 245/243
 
{{Mapping|legend=1| 1 -1 1 -3 | 0 4 2 9 }}
 
: mapping generators: ~2, ~14/9
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1207.178{{c}}, ~14/9 = 777.414{{c}}
: [[error map]]: {{val| +7.178 +0.523 -24.308 +6.367 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~14/9 = 773.872{{c}}
: error map: {{val| 0.000 -6.464 -38.569 -3.973 }}
 
{{Optimal ET sequence|legend=1| 3d, …, 11cd, 14c }}
 
[[Badness]] (Sintel): 1.43
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 25/24, 45/44, 99/98
 
Mapping: {{mapping| 1 -1 1 -3 -3 | 0 4 2 9 10 }}
 
Optimal tunings:
* WE: ~2 = 1207.200{{c}}, ~11/7 = 777.363{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/7 = 773.777{{c}}
 
{{Optimal ET sequence|legend=0| 3de, …, 11cdee, 14c }}
 
Badness (Sintel): 1.09
 
== Sida ==
Named by [[Xenllium]] in 2026, sida is described as the {{nowrap| 3 & 14c }} temperment, and tempers out [[1323/1280]] and [[4000/3969]]. Its [[ploidacot]] is beta-tetracot, the same as [[#Sidi|sidi]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 25/24, 1323/1280
 
{{Mapping|legend=1| 1 -1 1 6 | 0 4 2 -5 }}
: mapping generators: ~2, ~32/21
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1209.021{{c}}, ~32/21 = 778.298{{c}}
: [[error map]]: {{val| +9.021 +2.216 -20.696 -6.188 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~32/21 = 772.785{{c}}
: error map: {{val| 0.000 -10.816 -40.744 -32.749 }}
 
{{Optimal ET sequence|legend=1| 3, 11c, 14c, 45ccdd }}
 
[[Badness]] (Sintel): 2.12
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 25/24, 33/32, 245/242
 
Mapping: {{mapping| 1 3 3 1 2 | 0 -4 -2 5 4 }}
 
Optimal tunings:
* WE: ~2 = 1209.621{{c}}, ~11/7 = 772.376{{c}}
* CWE: ~2 = 1200.000{{c}}, ~11/7 = 772.247{{c}}
 
{{Optimal ET sequence|legend=0| 3, 11c, 14c }}
 
Badness (Sintel): 1.54
 
[[Category:Temperament families]]
[[Category:Dicot family| ]] <!-- main article -->
[[Category:Rank 2]]
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