Mint 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 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>2012-01-07 20:04:05 UTC</tt>.<br>
: The original revision id was <tt>290290885</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|flat]]


These are low complexity, high error temperaments tempering out the septimal quarter-tone, [[36_35|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.
Temperaments discussed elsewhere include
* [[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]]


=Progression=
== Progression ==
Commas: 36/35, 392/375
{{Distinguish| Progress }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lafayette]].''


POTE generator: ~15/14 = 140.927
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.  


Map: [&lt;1 1 2 2|, &lt;0 5 3 7|]
[[Subgroup]]: 2.3.5.7
Wedgie: &lt;&lt;5 3 7 -7 -3 8||
EDOs: 9, 17c
Badness: 0.0484


==11-limit==
[[Comma list]]: 36/35, 392/375
Commas: 36/35, 56/55, 77/75


[[POTE tuning|POTE generator]]: ~15/14 = 140.747
{{Mapping|legend=1| 1 1 2 2 | 0 5 3 7 }}


Map: [&lt;1 1 2 2 3|, &lt;0 5 3 7 4|]
: mapping generators: ~2, ~15/14
EDOs: 8, 9, 17
Badness: 0.0261


==13-limit==
[[Optimal tuning]]s:
Commas: 26/25, 36/35, 56/55, 66/65
* [[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 }}


[[POTE tuning|POTE generator]]: ~15/14 = 140.751
{{Optimal ET sequence|legend=1| 8d, 9, 17c }}


Map: Map: [&lt;1 1 2 2 3 3|, &lt;0 5 3 7 4 6|]
[[Badness]] (Smith): 0.048356
EDOs: 8, 9, 17
Badness: 0.0182


==17-limit==
=== 11-limit ===
Commas: 26/25, 36/35, 51/50, 56/55, 66/65
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE generator]]: ~15/14 = 141.404
Comma list: 36/35, 56/55, 77/75


Map: [&lt;1 1 2 2 3 3 4|, &lt;0 5 3 7 4 6 1|]
Mapping: {{mapping| 1 1 2 2 3 | 0 5 3 7 4 }}
EDOs: 8, 9, 17
Badness: 0.0167


===19-limit===
Optimal tunings:
Commas: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65
* CTE: ~2 = 1200.000, ~12/11 = 138.556
* POTE: ~2 = 1200.000, ~12/11 = 140.747


[[POTE tuning|POTE generator]]: ~15/14 = 140.479
{{Optimal ET sequence|legend=0| 8d, 9, 17c }}


Map: [&lt;1 1 2 2 3 3 4 4|, &lt;0 5 3 7 4 6 1 2|]
Badness (Smith): 0.026050
EDOs: 8, 9, 17, [[94edo|94]]
Badness: 0.0168


=Ripple=
=== 13-limit ===
Commas: 36/35, 2560/2401
Subgroup: 2.3.5.7.11.13


[[POTE tuning|POTE generator]]: ~21/20 =
Comma list: 26/25, 36/35, 56/55, 66/65


Map: [&lt;1 2 3 3|, &lt;0 -5 -8 -2|]
Mapping: {{mapping| 1 1 2 2 3 3 | 0 5 3 7 4 6 }}
Wedgie: &lt;&lt;5 8 2 1 -11 -18||
EDOs: [[12edo|12]], [[23edo|23]]
Badness: 0.0597


Optimal tunings:
* CTE: ~2 = 1200.000, ~13/12 = 138.741
* POTE: ~2 = 1200.000, ~13/12 = 140.751


{{Optimal ET sequence|legend=0| 8d, 9, 17c }}


Badness (Smith): 0.018158


</pre></div>
=== 17-limit ===
<h4>Original HTML content:</h4>
Subgroup: 2.3.5.7.11.13.17
<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;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Progression"&gt;Progression&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Ripple"&gt;Ripple&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
Comma list: 26/25, 36/35, 51/50, 56/55, 66/65
These are low complexity, high error temperaments tempering out the septimal quarter-tone, &lt;a class="wiki_link" href="/36_35"&gt;36/35&lt;/a&gt;. 36 is both a square and a &lt;a class="wiki_link" href="/triangular%20number"&gt;triangular number&lt;/a&gt;, 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;
 
&lt;br /&gt;
Mapping: {{mapping| 1 1 2 2 3 3 4 | 0 5 3 7 4 6 1 }}
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Progression"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Progression&lt;/h1&gt;
 
Commas: 36/35, 392/375&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* CTE: ~2 = 1200.000, ~13/12 = 138.649
POTE generator: ~15/14 = 140.927&lt;br /&gt;
* POTE: ~2 = 1200.000, ~13/12 = 141.404
&lt;br /&gt;
 
Map: [&amp;lt;1 1 2 2|, &amp;lt;0 5 3 7|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}
Wedgie: &amp;lt;&amp;lt;5 3 7 -7 -3 8||&lt;br /&gt;
 
EDOs: 9, 17c&lt;br /&gt;
Badness (Smith): 0.016743
Badness: 0.0484&lt;br /&gt;
 
&lt;br /&gt;
=== 19-limit ===
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Progression-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
Subgroup: 2.3.5.7.11.13.17.19
Commas: 36/35, 56/55, 77/75&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 140.747&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 1 2 2 3 3 4 4 | 0 5 3 7 4 6 1 2 }}
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 tunings:  
Badness: 0.0261&lt;br /&gt;
* CTE: ~2 = 1200.000, ~13/12 = 138.750
&lt;br /&gt;
* POTE: ~2 = 1200.000, ~13/12 = 140.479
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Progression-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
 
Commas: 26/25, 36/35, 56/55, 66/65&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 140.751&lt;br /&gt;
Badness (Smith): 0.016841
&lt;br /&gt;
 
Map: Map: [&amp;lt;1 1 2 2 3 3|, &amp;lt;0 5 3 7 4 6|]&lt;br /&gt;
== Subklei ==
EDOs: 8, 9, 17&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Delorean]].''
Badness: 0.0182&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Progression-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;17-limit&lt;/h2&gt;
 
Commas: 26/25, 36/35, 51/50, 56/55, 66/65&lt;br /&gt;
[[Comma list]]: 36/35, 1029/1000
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 141.404&lt;br /&gt;
{{Mapping|legend=1| 1 3 4 4 | 0 -6 -7 -5 }}
&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;
: mapping generators: ~2, ~7/6
EDOs: 8, 9, 17&lt;br /&gt;
 
Badness: 0.0167&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;br /&gt;
* [[CTE]]: ~2 = 1200.000, ~7/6 = 284.986
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="Progression-17-limit-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;19-limit&lt;/h3&gt;
: [[error map]]: {{val| 0.000 -11.870 +18.785 +6.245 }}
Commas: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65&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 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 140.479&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 4, 13cd, 17c, 21, 38c }}
Map: [&amp;lt;1 1 2 2 3 3 4 4|, &amp;lt;0 5 3 7 4 6 1 2|]&lt;br /&gt;
 
EDOs: 8, 9, 17, &lt;a class="wiki_link" href="/94edo"&gt;94&lt;/a&gt;&lt;br /&gt;
[[Badness]] (Smith): 0.061101
Badness: 0.0168&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Ripple"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Ripple&lt;/h1&gt;
Subgroup: 2.3.5.7.11
Commas: 36/35, 2560/2401&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 36/35, 77/75, 352/343
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~21/20 = &lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 3 4 4 7 | 0 -6 -7 -5 -15 }}
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 tunings:  
EDOs: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/23edo"&gt;23&lt;/a&gt;&lt;br /&gt;
* CTE: ~2 = 1200.000, ~7/6 = 284.265
Badness: 0.0597&lt;/body&gt;&lt;/html&gt;</pre></div>
* 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]]

Latest revision as of 12:39, 21 August 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.

This is a collection of low complexity, high error, temperaments which temper 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. 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.

Temperaments discussed elsewhere include

Progression

Not to be confused with Progress.
For the 5-limit version, see Miscellaneous 5-limit temperaments #Lafayette.

Progression, named by Gene Ward Smith in 2011[1] can be described as the 8d & 9 temperament, and 17edo is an obvious tuning for it.

Subgroup: 2.3.5.7

Comma list: 36/35, 392/375

Mapping: [⟨1 1 2 2], ⟨0 5 3 7]]

mapping generators: ~2, ~15/14

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 138.611
error map: ⟨0.000 -8.900 +29.519 +1.451]
  • POTE: ~2 = 1200.000, ~15/14 = 140.927
error map: ⟨0.000 +2.680 +36.467 +17.663]

Optimal ET sequence: 8d, 9, 17c

Badness (Smith): 0.048356

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 77/75

Mapping: [⟨1 1 2 2 3], ⟨0 5 3 7 4]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~12/11 = 138.556
  • POTE: ~2 = 1200.000, ~12/11 = 140.747

Optimal ET sequence: 8d, 9, 17c

Badness (Smith): 0.026050

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 56/55, 66/65

Mapping: [⟨1 1 2 2 3 3], ⟨0 5 3 7 4 6]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~13/12 = 138.741
  • POTE: ~2 = 1200.000, ~13/12 = 140.751

Optimal ET sequence: 8d, 9, 17c

Badness (Smith): 0.018158

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 26/25, 36/35, 51/50, 56/55, 66/65

Mapping: [⟨1 1 2 2 3 3 4], ⟨0 5 3 7 4 6 1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~13/12 = 138.649
  • POTE: ~2 = 1200.000, ~13/12 = 141.404

Optimal ET sequence: 8d, 9, 17cg

Badness (Smith): 0.016743

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65

Mapping: [⟨1 1 2 2 3 3 4 4], ⟨0 5 3 7 4 6 1 2]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~13/12 = 138.750
  • POTE: ~2 = 1200.000, ~13/12 = 140.479

Optimal ET sequence: 8d, 9, 17cg

Badness (Smith): 0.016841

Subklei

For the 5-limit version, see Miscellaneous 5-limit temperaments #Delorean.

Subgroup: 2.3.5.7

Comma list: 36/35, 1029/1000

Mapping: [⟨1 3 4 4], ⟨0 -6 -7 -5]]

mapping generators: ~2, ~7/6

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 284.986
error map: ⟨0.000 -11.870 +18.785 +6.245]
  • POTE: ~2 = 1200.000, ~7/6 = 284.219
error map: ⟨0.000 -7.268 +24.154 +10.080]

Optimal ET sequence: 4, 13cd, 17c, 21, 38c

Badness (Smith): 0.061101

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 77/75, 352/343

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: 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: [⟨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: 4ef, …, 13cdeef, 17c

Badness (Smith): 0.032324

Subkla

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 1029/1000

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: 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: [⟨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: 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: [⟨1 2 3 3], ⟨0 -7 -11 -3]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • CTE: ~2 = 1200.000, ~21/20 = 72.776
error map: ⟨0.000 -11.385 +13.153 +12.847]
  • POTE: ~2 = 1200.000, ~21/20 = 72.303
error map: ⟨0.000 -8.075 +18.355 +14.265]

Optimal ET sequence: 16, 17c, 33

Badness (Smith): 0.114869

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 121/120, 352/343

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: 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: [⟨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: 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: [⟨1 1 2 2], ⟨0 3 2 4]]

mapping generators: ~2, ~7/6

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 231.845
error map: ⟨0.000 -6.421 +77.376 -41.447]
  • POTE: ~2 = 1200.000, ~7/6 = 239.977
error map: ⟨0.000 +17.976 +93.640 -8.918]

Optimal ET sequence: 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: [⟨1 0 18 -16], ⟨0 1 -10 12]]

mapping generators: ~2, ~3

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 681.278
error map: ⟨0.000 -20.677 +0.906 +6.510]
  • CWE: ~2 = 1200.000, ~3/2 = 681.245
error map: ⟨0.000 -20.710 +1.239 +6.110]

Optimal ET sequence: 7, 30b, 37b

Badness (Smith): 0.308

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 72171/65536

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: 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: [⟨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: 7, 30b, 37b

Badness (Smith): 0.0773

Notes

  1. ↑ Yahoo! Tuning Group | Progression temperament
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