Keemic temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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These temper out the keema, {{monzo| -5 -3 3 1 }} = [[875/864]] = {{S|5/S6}}, whose 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 [[Keemic family#Undecimal supermagic|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>2012-05-20 16:52:24 UTC</tt>.<br>
: The original revision id was <tt>337626628</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">[[toc]]


These temper out the keema, |-5 -3 3 1&gt; = 875/864. Keemic temperaments include magic, keemun, flattone, porcupine, doublewide, superkleismic, sycamore and quasitemp.
Full [[7-limit]] keemic temperaments discussed elsewhere are:
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* ''[[Doublewide]]'' (+50/49) → [[Jubilismic clan #Doublewide|Jubilismic clan]]
* [[Porcupine]] (+64/63) → [[Porcupine family #Septimal porcupine|Porcupine family]]
* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
* [[Magic]] (+225/224) → [[Magic family #Septimal magic|Magic family]]
* ''[[Sycamore]]'' (+686/675) → [[Sycamore family #Septimal sycamore|Sycamore family]]
* [[Superkleismic]] (+1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]
* ''[[Undeka]]'' (+3200/3087) → [[11th-octave temperaments #Undeka|11th-octave temperaments]]


=Quasitemp=
Discussed below are quasitemp, chromo, barbad, hyperkleismic, and sevond.
Comma: 6103515625/5804752896


POTE generator: ~3125/2592 = 292.702
== Quasitemp ==
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Quasitemp]].''


Map: [&lt;1 5 5|, &lt;0 -14 -11|]
Quasitemp is a full 7-limit strong extension of [[gariberttet]], the 2.5/3.7/3 subgroup temperament 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]] (which is equated to [[13/11]] in the 13-limit extension). This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. In quasitemp, tempering out 875/864 entails that [[8/7]] is found after 9 generators, from which the mappings of 3 and 5 follow.
EDOs: 37, 41
Badness: 0.7678


==7-limit==
[[Subgroup]]: 2.3.5.7
Commas: 875/854, 2401/2400


POTE generator ~25/21 = 292.710
[[Comma list]]: 875/864, 2401/2400


Map: [&lt;1 5 5 5|, &lt;0 -14 -11 -9|]
{{Mapping|legend=1| 1 5 5 5 | 0 -14 -11 -9 }}
Wedgie: &lt;&lt;14 11 9 -15 -25 -10||
EDOs: 37, 41
Badness: 0.0603


==11-limit==
: Mapping generators: ~2, ~25/21
Commas: 100/99, 385/384, 1375/1372


POTE generator: ~25/21 = 292.547
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 292.710


Map: [&lt;1 5 5 5 2|, &lt;0 -14 -11 -9 6|]
{{Optimal ET sequence|legend=1| 4, 37, 41 }}
EDOs: 37, 41, 119, 160c, 201ce
Badness: 0.0432


==Quato==
[[Badness]]: 0.060269
Commas: 243/242, 441/440, 625/616


POTE generator: ~25/21 = 292.851
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 5 5 12|, &lt;0 -14 -11 -9 -35|]
Comma list: 100/99, 385/384, 1375/1372
EDOs: 41, 127cd, 168cd
Badness: 0.0412


===13-limit===
Mapping: {{mapping| 1 5 5 5 2 | 0 -14 -11 -9 6 }}
Commas: 105/104, 243/242, 275/273, 325/324


POTE generator: ~13/11 = 292.928
Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.547


Map: [&lt;1 5 5 5 12 12|, &lt;0 -14 -11 -9 -35 -34|]
{{Optimal ET sequence|legend=1| 4, 37, 41, 119 }}
EDOs: 41, 86ce, 127cd
Badness: 0.0301


=Barbad=
Badness: 0.043209
Commas: 875/864, 16875/16807


POTE generator: ~75/49 = 468.331
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 9 7 11|, &lt;0 -19 -12 -21|]
Comma list: 100/99, 196/195, 275/273, 385/384
Wedgie: &lt;&lt;19 12 21 -25 -20 15||
EDOs: 18, 23d, 41
Badness: 0.1104


==11-limit==
Mapping: {{mapping| 1 5 5 5 2 2 | 0 -14 -11 -9 6 7 }}
Commas: 245/242, 540/539, 625/616


POTE generator: ~98/75 = 468.367
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.457


Map: [&lt;1 9 7 11 14|, &lt;0 -19 -12 -21 -27|]
{{Optimal ET sequence|legend=1| 4, 37, 41, 78, 119f }}
EDOs: 41, 228cd
Badness: 0.0501


==13-limit==
Badness: 0.032913
Commas: 144/143, 196/195, 245/242, 275/273


POTE generator: ~13/10 = 468.270
=== Quato ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 9 7 11 14 8|, &lt;0 -19 -12 -21 -27 -11|]
Comma list: 243/242, 441/440, 625/616
EDOs: 41
Badness: 0.0392


Mapping: {{mapping| 1 5 5 5 12 | 0 -14 -11 -9 -35 }}


</pre></div>
Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.851
<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;&lt;!-- ws:start:WikiTextTocRule:16:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Quasitemp"&gt;Quasitemp&lt;/a&gt;&lt;/div&gt;
{{Optimal ET sequence|legend=1| 41, 127cd, 168cd }}
&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Quasitemp-7-limit"&gt;7-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Quasitemp-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
Badness: 0.041170
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Quasitemp-Quato"&gt;Quato&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Quasitemp-Quato-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
==== 13-limit ====
&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Barbad"&gt;Barbad&lt;/a&gt;&lt;/div&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Barbad-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Barbad-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
Comma list: 105/104, 243/242, 275/273, 325/324
&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;br /&gt;
Mapping: {{mapping| 1 5 5 5 12 12 | 0 -14 -11 -9 -35 -34 }}
These temper out the keema, |-5 -3 3 1&amp;gt; = 875/864. Keemic temperaments include magic, keemun, flattone, porcupine, doublewide, superkleismic, sycamore and quasitemp.&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.928
&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;
 
Comma: 6103515625/5804752896&lt;br /&gt;
{{Optimal ET sequence|legend=1| 41, 86ce, 127cd }}
&lt;br /&gt;
 
POTE generator: ~3125/2592 = 292.702&lt;br /&gt;
Badness: 0.030081
&lt;br /&gt;
 
Map: [&amp;lt;1 5 5|, &amp;lt;0 -14 -11|]&lt;br /&gt;
== Chromo ==
EDOs: 37, 41&lt;br /&gt;
: ''For the 5-limit version of this temperament, see [[Miscellaneous 5-limit temperaments #Chromo]].''
Badness: 0.7678&lt;br /&gt;
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.
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Quasitemp-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;7-limit&lt;/h2&gt;
Note that if one allows a more complex mapping for prime 7 and wants a larger prime limit, one may prefer [[escapade]].
Commas: 875/854, 2401/2400&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
POTE generator ~25/21 = 292.710&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 875/864, 2430/2401
Map: [&amp;lt;1 5 5 5|, &amp;lt;0 -14 -11 -9|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;14 11 9 -15 -25 -10||&lt;br /&gt;
{{Mapping|legend=1| 1 1 2 2 | 0 13 7 18 }}
EDOs: 37, 41&lt;br /&gt;
 
Badness: 0.0603&lt;br /&gt;
: Mapping generators: ~2, ~25/24
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Quasitemp-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 53.816
Commas: 100/99, 385/384, 1375/1372&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 22, 45, 67c }}
POTE generator: ~25/21 = 292.547&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]]: 0.090769
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;
== Barbad ==
Badness: 0.0432&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Quasitemp-Quato"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Quato&lt;/h2&gt;
[[Comma list]]: 875/864, 16875/16807
Commas: 243/242, 441/440, 625/616&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 9 7 11 | 0 -19 -12 -21 }}
POTE generator: ~25/21 = 292.851&lt;br /&gt;
 
&lt;br /&gt;
: Mapping generators: ~2, ~98/75
Map: [&amp;lt;1 5 5 5 12|, &amp;lt;0 -14 -11 -9 -35|]&lt;br /&gt;
 
EDOs: 41, 127cd, 168cd&lt;br /&gt;
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98/75 = 468.331
Badness: 0.0412&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 18, 23d, 41 }}
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="Quasitemp-Quato-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;13-limit&lt;/h3&gt;
 
Commas: 105/104, 243/242, 275/273, 325/324&lt;br /&gt;
[[Badness]]: 0.110448
&lt;br /&gt;
 
POTE generator: ~13/11 = 292.928&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
Map: [&amp;lt;1 5 5 5 12 12|, &amp;lt;0 -14 -11 -9 -35 -34|]&lt;br /&gt;
 
EDOs: 41, 86ce, 127cd&lt;br /&gt;
Comma list: 245/242, 540/539, 625/616
Badness: 0.0301&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 9 7 11 14 | 0 -19 -12 -21 -27 }}
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Barbad"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Barbad&lt;/h1&gt;
 
Commas: 875/864, 16875/16807&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1\1, ~98/75 = 468.367
&lt;br /&gt;
 
POTE generator: ~75/49 = 468.331&lt;br /&gt;
{{Optimal ET sequence|legend=1| 18e, 23de, 41, 228ccdd }}
&lt;br /&gt;
 
Map: [&amp;lt;1 9 7 11|, &amp;lt;0 -19 -12 -21|]&lt;br /&gt;
Badness: 0.050105
Wedgie: &amp;lt;&amp;lt;19 12 21 -25 -20 15||&lt;br /&gt;
 
EDOs: 18, 23d, 41&lt;br /&gt;
=== 13-limit ===
Badness: 0.1104&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Barbad-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11-limit&lt;/h2&gt;
Comma list: 144/143, 196/195, 245/242, 275/273
Commas: 245/242, 540/539, 625/616&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 9 7 11 14 8 | 0 -19 -12 -21 -27 -11 }}
POTE generator: ~98/75 = 468.367&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 468.270
Map: [&amp;lt;1 9 7 11 14|, &amp;lt;0 -19 -12 -21 -27|]&lt;br /&gt;
 
EDOs: 41, 228cd&lt;br /&gt;
{{Optimal ET sequence|legend=1| 18e, 23de, 41 }}
Badness: 0.0501&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.039183
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Barbad-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h2&gt;
 
Commas: 144/143, 196/195, 245/242, 275/273&lt;br /&gt;
== Hyperkleismic ==
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
POTE generator: ~13/10 = 468.270&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 875/864, 51200/50421
Map: [&amp;lt;1 9 7 11 14 8|, &amp;lt;0 -19 -12 -21 -27 -11|]&lt;br /&gt;
 
EDOs: 41&lt;br /&gt;
{{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }}
Badness: 0.0392&lt;/body&gt;&lt;/html&gt;</pre></div>
 
: Mapping generators: ~2, ~6/5
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 323.780
 
{{Optimal ET sequence|legend=1| 26, 37, 63 }}
 
[[Badness]]: 0.157830
 
=== 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 tuning (POTE): ~2 = 1\1, ~6/5 = 323.796
 
{{Optimal ET sequence|legend=1| 26, 37, 63 }}
 
Badness: 0.065356
 
=== 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 tuning (POTE): ~2 = 1\1, ~6/5 = 323.790
 
{{Optimal ET sequence|legend=1| 26, 37, 63 }}
 
Badness: 0.035724
 
== Sevond ==
10/9 is tempered to be exactly 1\7 of an octave. 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]] ([[POTE]]): ~10/9 = 1\7, ~3/2 = 705.613
 
{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
 
[[Badness]]: 0.206592
 
=== 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 tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.518
 
{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
 
Badness: 0.070437
 
=== 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 tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.344
 
{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
 
Badness: 0.041238
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Keemic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]

Latest revision as of 00:36, 24 June 2025

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.

These temper out the keema, [-5 -3 3 1⟩ = 875/864 = S5/S6, whose 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)2 is identified with 16/11), this provides a clean way to extend the various keemic temperaments to the 11-limit as well.

Full 7-limit keemic temperaments discussed elsewhere are:

Discussed below are quasitemp, chromo, barbad, hyperkleismic, and sevond.

Quasitemp

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Quasitemp.

Quasitemp is a full 7-limit strong extension of gariberttet, the 2.5/3.7/3 subgroup temperament 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 (which is equated to 13/11 in the 13-limit extension). This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. In quasitemp, tempering out 875/864 entails that 8/7 is found after 9 generators, from which the mappings of 3 and 5 follow.

Subgroup: 2.3.5.7

Comma list: 875/864, 2401/2400

Mapping: [⟨1 5 5 5], ⟨0 -14 -11 -9]]

Mapping generators: ~2, ~25/21

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.710

Optimal ET sequence: 4, 37, 41

Badness: 0.060269

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 1375/1372

Mapping: [⟨1 5 5 5 2], ⟨0 -14 -11 -9 6]]

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.547

Optimal ET sequence: 4, 37, 41, 119

Badness: 0.043209

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 275/273, 385/384

Mapping: [⟨1 5 5 5 2 2], ⟨0 -14 -11 -9 6 7]]

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.457

Optimal ET sequence: 4, 37, 41, 78, 119f

Badness: 0.032913

Quato

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 625/616

Mapping: [⟨1 5 5 5 12], ⟨0 -14 -11 -9 -35]]

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.851

Optimal ET sequence: 41, 127cd, 168cd

Badness: 0.041170

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 243/242, 275/273, 325/324

Mapping: [⟨1 5 5 5 12 12], ⟨0 -14 -11 -9 -35 -34]]

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.928

Optimal ET sequence: 41, 86ce, 127cd

Badness: 0.030081

Chromo

For the 5-limit version of this temperament, 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: [⟨1 1 2 2], ⟨0 13 7 18]]

Mapping generators: ~2, ~25/24

Optimal tuning (POTE): ~2 = 1\1, ~25/24 = 53.816

Optimal ET sequence: 22, 45, 67c

Badness: 0.090769

Barbad

Subgroup: 2.3.5.7

Comma list: 875/864, 16875/16807

Mapping: [⟨1 9 7 11], ⟨0 -19 -12 -21]]

Mapping generators: ~2, ~98/75

Optimal tuning (POTE): ~2 = 1\1, ~98/75 = 468.331

Optimal ET sequence: 18, 23d, 41

Badness: 0.110448

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 540/539, 625/616

Mapping: [⟨1 9 7 11 14], ⟨0 -19 -12 -21 -27]]

Optimal tuning (POTE): ~2 = 1\1, ~98/75 = 468.367

Optimal ET sequence: 18e, 23de, 41, 228ccdd

Badness: 0.050105

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 245/242, 275/273

Mapping: [⟨1 9 7 11 14 8], ⟨0 -19 -12 -21 -27 -11]]

Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 468.270

Optimal ET sequence: 18e, 23de, 41

Badness: 0.039183

Hyperkleismic

Subgroup: 2.3.5.7

Comma list: 875/864, 51200/50421

Mapping: [⟨1 -3 -2 2], ⟨0 17 16 3]]

Mapping generators: ~2, ~6/5

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.780

Optimal ET sequence: 26, 37, 63

Badness: 0.157830

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 2420/2401

Mapping: [⟨1 -3 -2 2 4], ⟨0 17 16 3 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.796

Optimal ET sequence: 26, 37, 63

Badness: 0.065356

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 169/168, 275/273, 385/384

Mapping: [⟨1 -3 -2 2 4 1], ⟨0 17 16 3 -2 10]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.790

Optimal ET sequence: 26, 37, 63

Badness: 0.035724

Sevond

10/9 is tempered to be exactly 1\7 of an octave. 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: [⟨7 0 -6 53], ⟨0 1 2 -3]]

Mapping generators: ~10/9, ~3

Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.613

Optimal ET sequence: 7, 56, 63, 119

Badness: 0.206592

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 6655/6561

Mapping: [⟨7 0 -6 53 2], ⟨0 1 2 -3 2]]

Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.518

Optimal ET sequence: 7, 56, 63, 119

Badness: 0.070437

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 169/168, 352/351, 385/384

Mapping: [⟨7 0 -6 53 2 37], ⟨0 1 2 -3 2 -1]]

Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.344

Optimal ET sequence: 7, 56, 63, 119

Badness: 0.041238

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