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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 low [[complexity]], high [[error]], [[regular temperament|temperaments]] which [[temper out]] the septimal quarter-tone, [[36/35]]. 36 is [[Wikipedia: Square triangular number|both]] a [http://mathworld.wolfram.com/SquareNumber.html square] and a [[triangular number]], and this helps make 36/35 a septimal interval of considerable significance. These temperaments equate [[6/5]] with [[7/6]], [[5/4]] with [[9/7]], and [[7/4]] with [[9/5]], so minor and major thirds and sixths are intervals of 5 and 7 at the same time.  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-26 14:36:06 UTC</tt>.<br>
: The original revision id was <tt>205266502</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 are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35. 36 is both a square and a triangular number, and this helps make 36/35 a septimal interval of considerable significance. Temperaments tempering it out include beep, dicot, dominant, father, diminished, august, hystrix, penta, hexadecimal, ripple and gorgo.


==Progression==
Temperaments discussed elsewhere include
Commas: 36/35, 392/375
* [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]
* [[Dominant (temperament)|Dominant]] (+64/63) → [[Meantone family #Dominant|Meantone family]]
* ''[[Armodue]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]
* ''[[Dicot]]'' (+25/24) → [[Dicot family|Dicot family]]
* [[Beep]] (+21/20) → [[Bug family #Beep|Bug family]]
* ''[[August]]'' (+128/125) → [[Augmented family #August|Augmented family]]
* ''[[Gorgo]]'' (+1029/1024) → [[Gamelismic clan #Gorgo|Gamelismic clan]]
* ''[[Hystrix]]'' (+160/147) → [[Porcupine family #Hystrix|Porcupine family]]
* [[Diminished (temperament)|Diminished]] (+50/49) → [[Diminished family #Septimal diminished|Diminished family]]
* ''[[Smate]]'' (+2048/1875) → [[Smate family #Septimal smate|Smate family]]
* ''[[Darkstone]]'' (+1875/1792) → [[Magic family #Darkstone|Magic family]]
* ''[[Ripple]]'' (+2560/2401) → [[Ripple family #Septimal ripple|Ripple family]]
* ''[[Whitewood]]'' (+2187/2048) → [[Whitewood family #Septimal whitewood|Whitewood family]]


[[POTE tuning|POTE generator]]: ~15/14 = 140.927
== Progression ==
{{Distinguish| Progress }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lafayette]].''


Map: [&lt;1 1 2 2|, &lt;0 5 3 7|]
Progression, named by [[Gene Ward Smith]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18891.html Yahoo! Tuning Group | ''Progression temperament'']</ref> can be described as the 8d & 9 temperament, and [[17edo]] is an obvious tuning for it.  
EDOs: 8, 9, 17
Badness: 0.0484


===11-limit===
[[Subgroup]]: 2.3.5.7
Commas: 36/35, 56/55, 77/75


[[POTE tuning|POTE generator]]: ~15/14 = 140.747
[[Comma list]]: 36/35, 392/375


Map: [&lt;1 1 2 2 3|, &lt;0 5 3 7 4|]
{{Mapping|legend=1| 1 1 2 2 | 0 5 3 7 }}
EDOs: 8, 9, 17
Badness: 0.0261


===13-limit===
: mapping generators: ~2, ~15/14
Commas: 26/25, 36/35, 56/55, 66/65


[[POTE tuning|POTE generator]]: ~15/14 = 140.751
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~15/14 = 138.611
: [[error map]]: {{val| 0.000 -8.900 +29.519 +1.451 }}
* [[POTE]]: ~2 = 1200.000, ~15/14 = 140.927
: error map: {{val| 0.000 +2.680 +36.467 +17.663 }}


Map: Map: [&lt;1 1 2 2 3 3|, &lt;0 5 3 7 4 6|]
{{Optimal ET sequence|legend=1| 8d, 9, 17c }}
EDOs: 8, 9, 17
Badness: 0.0182


===17-limit===
[[Badness]] (Smith): 0.048356
Commas: 26/25, 36/35, 51/50, 56/55, 66/65


[[POTE tuning|POTE generator]]: ~15/14 = 141.404
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 1 2 2 3 3 4|, &lt;0 5 3 7 4 6 1|]
Comma list: 36/35, 56/55, 77/75
EDOs: 8, 9, 17
Badness: 0.0167


===19-limit===
Mapping: {{mapping| 1 1 2 2 3 | 0 5 3 7 4 }}
Commas: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65


[[POTE tuning|POTE generator]]: ~15/14 = 140.479
Optimal tunings:
* CTE: ~2 = 1200.000, ~12/11 = 138.556
* POTE: ~2 = 1200.000, ~12/11 = 140.747


Map: [&lt;1 1 2 2 3 3 4 4|, &lt;0 5 3 7 4 6 1 2|]
{{Optimal ET sequence|legend=0| 8d, 9, 17c }}
EDOs: 8, 9, 17, 94
Badness: 0.0168


==Ripple==
Badness (Smith): 0.026050
Commas: 36/35, 2560/2401


[[POTE tuning|POTE generator]]:
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 3 3|, &lt;0 -5 -8 -2|]
Comma list: 26/25, 36/35, 56/55, 66/65
Wedgie: &lt;&lt;5 8 2 1 -11 -18||
 
EDOs: 12
Mapping: {{mapping| 1 1 2 2 3 3 | 0 5 3 7 4 6 }}
Badness: 0.0597
 
</pre></div>
Optimal tunings:
<h4>Original HTML content:</h4>
* CTE: ~2 = 1200.000, ~13/12 = 138.741
<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;Mint temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35. 36 is both a square and a triangular number, and this helps make 36/35 a septimal interval of considerable significance. Temperaments tempering it out include beep, dicot, dominant, father, diminished, august, hystrix, penta, hexadecimal, ripple and gorgo.&lt;br /&gt;
* POTE: ~2 = 1200.000, ~13/12 = 140.751
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Progression"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Progression&lt;/h2&gt;
{{Optimal ET sequence|legend=0| 8d, 9, 17c }}
Commas: 36/35, 392/375&lt;br /&gt;
 
&lt;br /&gt;
Badness (Smith): 0.018158
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 140.927&lt;br /&gt;
 
&lt;br /&gt;
=== 17-limit ===
Map: [&amp;lt;1 1 2 2|, &amp;lt;0 5 3 7|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
EDOs: 8, 9, 17&lt;br /&gt;
 
Badness: 0.0484&lt;br /&gt;
Comma list: 26/25, 36/35, 51/50, 56/55, 66/65
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Progression-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h3&gt;
Mapping: {{mapping| 1 1 2 2 3 3 4 | 0 5 3 7 4 6 1 }}
Commas: 36/35, 56/55, 77/75&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 140.747&lt;br /&gt;
* CTE: ~2 = 1200.000, ~13/12 = 138.649
&lt;br /&gt;
* POTE: ~2 = 1200.000, ~13/12 = 141.404
Map: [&amp;lt;1 1 2 2 3|, &amp;lt;0 5 3 7 4|]&lt;br /&gt;
 
EDOs: 8, 9, 17&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}
Badness: 0.0261&lt;br /&gt;
 
&lt;br /&gt;
Badness (Smith): 0.016743
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Progression-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h3&gt;
 
Commas: 26/25, 36/35, 56/55, 66/65&lt;br /&gt;
=== 19-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 140.751&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65
Map: Map: [&amp;lt;1 1 2 2 3 3|, &amp;lt;0 5 3 7 4 6|]&lt;br /&gt;
 
EDOs: 8, 9, 17&lt;br /&gt;
Mapping: {{mapping| 1 1 2 2 3 3 4 4 | 0 5 3 7 4 6 1 2 }}
Badness: 0.0182&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Progression-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;17-limit&lt;/h3&gt;
* CTE: ~2 = 1200.000, ~13/12 = 138.750
Commas: 26/25, 36/35, 51/50, 56/55, 66/65&lt;br /&gt;
* POTE: ~2 = 1200.000, ~13/12 = 140.479
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 141.404&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 2 2 3 3 4|, &amp;lt;0 5 3 7 4 6 1|]&lt;br /&gt;
Badness (Smith): 0.016841
EDOs: 8, 9, 17&lt;br /&gt;
 
Badness: 0.0167&lt;br /&gt;
== Subklei ==
&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Delorean]].''
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-Progression-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;19-limit&lt;/h3&gt;
 
Commas: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 140.479&lt;br /&gt;
[[Comma list]]: 36/35, 1029/1000
&lt;br /&gt;
 
Map: [&amp;lt;1 1 2 2 3 3 4 4|, &amp;lt;0 5 3 7 4 6 1 2|]&lt;br /&gt;
{{Mapping|legend=1| 1 3 4 4 | 0 -6 -7 -5 }}
EDOs: 8, 9, 17, 94&lt;br /&gt;
 
Badness: 0.0168&lt;br /&gt;
: mapping generators: ~2, ~7/6
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x-Ripple"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Ripple&lt;/h2&gt;
[[Optimal tuning]]s:  
Commas: 36/35, 2560/2401&lt;br /&gt;
* [[CTE]]: ~2 = 1200.000, ~7/6 = 284.986
&lt;br /&gt;
: [[error map]]: {{val| 0.000 -11.870 +18.785 +6.245 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;:&lt;br /&gt;
* [[POTE]]: ~2 = 1200.000, ~7/6 = 284.219
&lt;br /&gt;
: error map: {{val| 0.000 -7.268 +24.154 +10.080 }}
Map: [&amp;lt;1 2 3 3|, &amp;lt;0 -5 -8 -2|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;5 8 2 1 -11 -18||&lt;br /&gt;
{{Optimal ET sequence|legend=1| 4, 13cd, 17c, 21, 38c }}
EDOs: 12&lt;br /&gt;
 
Badness: 0.0597&lt;/body&gt;&lt;/html&gt;</pre></div>
[[Badness]] (Smith): 0.061101
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 77/75, 352/343
 
Mapping: {{mapping| 1 3 4 4 7 | 0 -6 -7 -5 -15 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 284.265
* POTE: ~2 = 1200.000, ~7/6 = 283.253
 
{{Optimal ET sequence|legend=0| 4e, …, 13cdee, 17c }}
 
Badness (Smith): 0.044651
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 26/25, 36/35, 66/65, 352/343
 
Mapping: {{mapping| 1 3 4 4 7 7 | 0 -6 -7 -5 -15 -14 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 283.918
* POTE: ~2 = 1200.000, ~7/6 = 282.856
 
{{Optimal ET sequence|legend=0| 4ef, …, 13cdeef, 17c }}
 
Badness (Smith): 0.032324
 
=== Subkla ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 56/55, 1029/1000
 
Mapping: {{mapping| 1 3 4 4 3 | 0 -6 -7 -5 2 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 284.870
* POTE: ~2 = 1200.000, ~7/6 = 283.822
 
{{Optimal ET sequence|legend=0| 4, 17c, 21, 38ce }}
 
Badness (Smith): 0.047247
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 36/35, 56/55, 66/65, 640/637
 
Mapping: {{mapping| 1 3 4 4 3 3 | 0 -6 -7 -5 2 3 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 284.758
* POTE: ~2 = 1200.000, ~7/6 = 283.882
 
{{Optimal ET sequence|legend=0| 4, 17c, 38ce }}
 
Badness (Smith): 0.036701
 
== Slurpee ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Slurpee]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 36/35, 51200/50421
 
{{Mapping|legend=1| 1 2 3 3 | 0 -7 -11 -3 }}
 
: mapping generators: ~2, ~21/20
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~21/20 = 72.776
: [[error map]]: {{val| 0.000 -11.385 +13.153 +12.847 }}
* [[POTE]]: ~2 = 1200.000, ~21/20 = 72.303
: error map: {{val| 0.000 -8.075 +18.355 +14.265 }}
 
{{Optimal ET sequence|legend=1| 16, 17c, 33 }}
 
[[Badness]] (Smith): 0.114869
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 121/120, 352/343
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -7 -11 -3 -9 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~21/20 = 72.681
* POTE: ~2 = 1200.000, ~21/20 = 72.297
 
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}
 
Badness (Smith): 0.050514
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 36/35, 66/65, 143/140, 352/343
 
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -7 -11 -3 -9 -5 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~21/20 = 72.653
* POTE: ~2 = 1200.000, ~21/20 = 72.409
 
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}
 
Badness (Smith): 0.033102
 
== Penta ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 28/25, 36/35
 
{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}
 
: mapping generators: ~2, ~7/6
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~7/6 = 231.845
: [[error map]]: {{val| 0.000 -6.421 +77.376 -41.447 }}
* [[POTE]]: ~2 = 1200.000, ~7/6 = 239.977
: error map: {{val| 0.000 +17.976 +93.640 -8.918 }}
 
{{Optimal ET sequence|legend=1| 1bd, …, 4bcd, 5 }}
 
[[Badness]] (Smith): 0.046882
 
== Shallowtone ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Shallowtone (5-limit)]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 36/35, 295245/262144
 
{{Mapping|legend=1| 1 0 18 -16 | 0 1 -10 12 }}
 
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~3/2 = 681.278
: [[error map]]: {{val| 0.000 -20.677 +0.906 +6.510 }}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 681.245
: error map: {{val| 0.000 -20.710 +1.239 +6.110 }}
 
{{Optimal ET sequence|legend=1| 7, 30b, 37b }}
 
[[Badness]] (Smith): 0.308
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 45/44, 72171/65536
 
Mapping: {{Mapping| 1 0 18 -16 16 | 0 1 -10 12 -8 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 681.254
* CWE: ~2 = 1200.000, ~3/2 = 681.327
 
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}
 
Badness (Smith): 0.130
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 36/35, 45/44, 16731/16384
 
Mapping: {{Mapping| 1 0 18 -16 16 -1 | 0 1 -10 12 -8 3 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 681.237
* CWE: ~2 = 1200.000, ~3/2 = 681.267
 
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}
 
Badness (Smith): 0.0773
 
== Notes ==
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Mint temperaments| ]] <!-- main article -->
[[Category:Mint| ]] <!-- key article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Mint_temperaments"