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Keemic temperaments: 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>
This is a collection of [[rank-2 temperament|linear]] [[regular temperament|temperaments]] that [[tempering out|temper out]] the [[keema]] ({{monzo|legend=1| -5 -3 3 1 }}, [[ratio]]: 875/864), with [[S-expression]] S5/S6. Its fundamental equivalence entails that [[6/5]] is sharpened so that it stacks three times to reach [[7/4]], and the interval between 6/5 and [[5/4]] is compressed so that [[7/6]]–6/5–5/4–[[9/7]] are set equidistant from each other. As the canonical extension of rank-3 [[keemic]] to the [[11-limit]] tempers out the commas [[100/99]] and [[385/384]] (whereby ([[6/5]])<sup>2</sup> is identified with [[16/11]]), this provides a clean way to extend the various keemic temperaments to the 11-limit as well.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-04-17 17:00:44 UTC</tt>.<br>
: The original revision id was <tt>321733400</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">These temper out the keema, |-5 -3 3 1&gt; = 875/864. Keemic temperaments include magic, keemun, flattone, porcupine, doublewide, superkleismic, sycamore anbd quasitemp.


=Quasitemp=
Full [[7-limit]] keemic temperaments discussed elsewhere are:
Commas: 875/854, 2401/2400
* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
* ''[[Mujannabic]]'' (+25/24) → [[Dicot family #Dicot|Dicot family]]
* [[Porcupine]] (+64/63) → [[Porcupine family #Septimal porcupine|Porcupine family]]
* [[Monkey]] (+5120/5103) → [[Tetracot family #Monkey|Tetracot family]]
* [[Magic]] (+225/224) → [[Magic family #Septimal magic|Magic family]]
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* ''[[Doublewide]]'' (+50/49) → [[Jubilismic clan #Doublewide|Jubilismic clan]]
* [[Superkleismic]] (+1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]
* ''[[Sycamore]]'' (+686/675) → [[Sycamore family #Septimal sycamore|Sycamore family]]
* ''[[Undeka]]'' (+3200/3087) → [[11th-octave temperaments #Undeka|11th-octave temperaments]]


POTE generator ~25/21 = 292.710
Discussed below are quasitemp, chromo, barbad, hyperkleismic, and sevond, in the order of increasing [[TE logflat badness]].


Map: [&lt;1 5 5 5|, &lt;0 -14 -11 -9|]
== Quasitemp ==
Wedgie: &lt;&lt;14 11 9 -15 -25 -10||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasitemp]].''
EDOs: 37, 41
Badness: 0.0603


==11-limit==
Quasitemp tempers out [[2401/2400]] in addition to 875/864 and may be described as the {{nowrap| 37 & 41 }} temperament. It is characterized by equating the interval between the pental and septimal thirds ([[36/35]]) with the classical chromatic semitone ([[25/24]]), and by tempering together the septimal dieses of [[49/48]] and [[50/49]]. In that sense, it is opposed to [[orwellismic temperaments]], in particular [[myna]], where the distance between the pental and septimal thirds is the same as the septimal dieses and different from the classical chromatic semitone.
Commas: 100/99, 385/384, 1375/1372


POTE generator: ~25/21 = 292.547
Quasitemp can also be thought of as a [[strong extension]] of the 2.5/3.7/3-subgroup temperament called [[gariberttet]], which is defined by tempering out [[3125/3087]]. In gariberttet, three generators reach [[5/3]] and five reach [[7/3]], so that the generator itself has the interpretation of [[25/21]]. This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. Quasitemp tempering out 875/864 entails that [[8/7]] is found after 9 generators, from which the mappings of 3 and 5 follow.  


Map: [&lt;1 5 5 5 2|, &lt;0 -14 -11 -9 6|]
Note that the generator is given as 25/21's octave complement, 42/25, in the data that follow, since a stack of 14 such generators octave-reduced is the perfect fifth, whence the temperament's [[ploidacot]] is iota-14-cot. This generator is equated to [[22/13]] for the 13-limit extension, tempering out [[275/273]].
EDOs: 37, 41, 119, 160c, 201ce
 
Badness: 0.0432</pre></div>
[[Subgroup]]: 2.3.5.7
<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;Keemic temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;These temper out the keema, |-5 -3 3 1&amp;gt; = 875/864. Keemic temperaments include magic, keemun, flattone, porcupine, doublewide, superkleismic, sycamore anbd quasitemp.&lt;br /&gt;
[[Comma list]]: 875/864, 2401/2400
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Quasitemp"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Quasitemp&lt;/h1&gt;
{{Mapping|legend=1| 1 -9 -6 -4 | 0 14 11 9 }}
Commas: 875/854, 2401/2400&lt;br /&gt;
: mapping generators: ~2, ~42/25
&lt;br /&gt;
 
POTE generator ~25/21 = 292.710&lt;br /&gt;
[[Optimal tuning]]s:
&lt;br /&gt;
* [[WE]]: ~2 = 1200.9237{{c}}, ~42/25 = 907.9887{{c}}
Map: [&amp;lt;1 5 5 5|, &amp;lt;0 -14 -11 -9|]&lt;br /&gt;
: [[error map]]: {{val| +0.924 +1.573 -3.981 -0.623 }}
Wedgie: &amp;lt;&amp;lt;14 11 9 -15 -25 -10||&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 907.3471{{c}}
EDOs: 37, 41&lt;br /&gt;
: error map: {{val| 0.000 +0.905 -5.495 -2.702 }}
Badness: 0.0603&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 4, …, 37, 41 }}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Quasitemp-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
 
Commas: 100/99, 385/384, 1375/1372&lt;br /&gt;
[[Badness]] (Sintel): 1.53
&lt;br /&gt;
 
POTE generator: ~25/21 = 292.547&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
Map: [&amp;lt;1 5 5 5 2|, &amp;lt;0 -14 -11 -9 6|]&lt;br /&gt;
 
EDOs: 37, 41, 119, 160c, 201ce&lt;br /&gt;
Comma list: 100/99, 385/384, 1375/1372
Badness: 0.0432&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Mapping: {{mapping| 1 -9 -6 -4 8 | 0 14 11 9 -6 }}
 
Optimal tunings:
* WE: ~2 = 1199.9585{{c}}, ~42/25 = 907.4221{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.4521{{c}}
 
{{Optimal ET sequence|legend=0| 4, 37, 41, 119 }}
 
Badness (Sintel): 1.43
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 196/195, 275/273, 385/384
 
Mapping: {{mapping| 1 -9 -6 -4 8 9 | 0 14 11 9 -6 -7 }}
 
Optimal tunings:
* WE: ~2 = 1199.4376{{c}}, ~22/13 = 907.1175{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 907.5314{{c}}
 
{{Optimal ET sequence|legend=0| 4, 37, 41, 78, 119f }}
 
Badness (Sintel): 1.36
 
=== Quato ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 625/616
 
Mapping: {{mapping| 1 -9 -6 -4 -23 | 0 14 11 9 35 }}
 
Optimal tunings:
* WE: ~2 = 1201.2729{{c}}, ~42/25 = 908.1116{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.2109{{c}}
 
{{Optimal ET sequence|legend=0| 41, 127cd, 168cd }}
 
Badness (Sintel): 1.36
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 243/242, 275/273, 325/324
 
Mapping: {{mapping| 1 -9 -6 -4 -23 -22 | 0 14 11 9 35 34 }}
 
Optimal tunings:
* WE: ~2 = 1201.4078{{c}}, ~42/25 = 908.1362{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.1370{{c}}
 
{{Optimal ET sequence|legend=0| 41, 86ce }}
 
Badness (Sintel): 1.24
 
== Chromo ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Chromo]].''
 
Chromo represents the [[13edf]] chain as a rank-2 temperament, with [[6/5]] and [[5/4]] mapped to 6 and 7 steps, respectively. Since the difference of those two intervals is abbreviated considerably from just, keemic provides the most meaningful 7-limit extension (setting [[7/6]], 6/5, 5/4, [[9/7]] equidistant) so that the temperament then approximates the [[4:5:6:7]] tetrad with 0:7:13:18 generator steps.
 
Note that if one allows a more complex mapping for prime 7 and wants a larger prime limit, one may prefer [[escapade]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 875/864, 2430/2401
 
{{Mapping|legend=1| 1 1 2 2 | 0 13 7 18 }}
: mapping generators: ~2, ~36/35
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.4060{{c}}, ~36/35 = 53.8791{{c}}
: [[error map]]: {{val| +1.406 -0.121 -6.348 +3.810 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 53.9055{{c}}
: error map: {{val| 0.000 -1.183 -8.975 +1.474 }}
 
{{Optimal ET sequence|legend=1| 22, 45, 67c }}
 
[[Badness]] (Sintel): 2.30
 
== Barbad ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 875/864, 16875/16807
 
{{Mapping|legend=1| 1 -10 -5 -10 | 0 19 12 21 }}
: mapping generators: ~2, ~98/75
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0462{{c}}, ~75/49 = 732.3071{{c}}
: [[error map]]: {{val| +1.046 +1.418 -3.859 -0.838 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/49 = 731.7183{{c}}
: error map: {{val| 0.000 +0.692 -5.694 -2.742 }}
 
{{Optimal ET sequence|legend=0| 18, 23d, 41 }}
 
[[Badness]] (Sintel): 2.80
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/242, 540/539, 625/616
 
Mapping: {{mapping| 1 -10 -5 -10 -13 | 0 19 12 21 27 }}
 
Optimal tunings:  
* WE: ~2 = 1200.8513{{c}}, ~75/49 = 732.1519{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/49 = 731.6740{{c}}
 
{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}
 
Badness (Sintel): 1.66
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 196/195, 245/242, 275/273
 
Mapping: {{mapping| 1 -10 -5 -10 -13 -3 | 0 19 12 21 27 11 }}
 
Optimal tunings:
* WE: ~2 = 1199.7960{{c}}, ~20/13 = 731.6053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 731.7208{{c}}
 
{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}
 
Badness (Sintel): 1.62
 
== Hyperkleismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 875/864, 51200/50421
 
{{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }}
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0290{{c}}, ~6/5 = 323.7882{{c}}
: [[error map]]: {{val| +0.029 +2.358 -5.759 +2.597 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 323.7816{{c}}
: error map: {{val| 0.000 +2.332 -5.808 +2.519 }}
 
{{Optimal ET sequence|legend=1| 26, 37, 63 }}
 
[[Badness]] (Sintel): 3.99
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 385/384, 2420/2401
 
Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}
 
Optimal tunings:
* WE: ~2 = 1199.9010{{c}}, ~6/5 = 323.7691{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7931{{c}}
 
{{Optimal ET sequence|legend=0| 26, 37, 63 }}
 
Badness (Sintel): 2.16
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 169/168, 275/273, 385/384
 
Mapping: {{mapping| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}
 
Optimal tunings:  
* WE: ~2 = 1200.0524{{c}}, ~6/5 = 323.8039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7912{{c}}
 
{{Optimal ET sequence|legend=0| 26, 37, 63 }}
 
Badness (Sintel): 1.48
 
== Sevond ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sevond (5-limit)]].''
 
10/9 is tempered to be exactly 1\7. Therefore 3/2 is 1 generator sharp of a 7edo step and 5/4 is 2 generators sharp.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 875/864, 327680/321489
 
{{Mapping|legend=1| 7 0 -6 53 | 0 1 2 -3 }}
: mapping generators: ~10/9, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~10/9 = 171.4007{{c}}, ~3/2 = 705.4982{{c}}
: [[error map]]: {{val| -0.195 +3.348 -4.112 -0.499 }}
* [[CWE]]: ~10/9 = 171.4286{{c}}, ~3/2 = 705.6057{{c}}
: error map: {{val| 0.000 +3.651 -3.674 +0.071 }}
 
{{Optimal ET sequence|legend=1| 7, …, 56, 63, 119 }}
 
[[Badness]] (Sintel): 5.23
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 385/384, 6655/6561
 
Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}
 
Optimal tunings:
* WE: ~11/10 = 171.3859{{c}}, ~3/2 = 705.3421{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.4973{{c}}
 
{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}
 
Badness (Sintel): 2.33
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 169/168, 352/351, 385/384
 
Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}
 
Optimal tunings:
* WE: ~11/10 = 171.4163{{c}}, ~3/2 = 705.2930{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.3402{{c}}
 
{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}
 
Badness (Sintel): 1.70
 
[[Category:Temperament collections]]
[[Category:Keemic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Keemic_temperaments"