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>2012-05-17 15:46:38 UTC</tt>.<br>
: The original revision id was <tt>336772140</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 [[http://en.wikipedia.org/wiki/Square_triangular_number|both]] a [[http://mathworld.wolfram.com/SquareNumber.html|square]] and a [[http://en.wikipedia.org/wiki/Triangular_number|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 (temperament)|Armodue]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]
* ''[[Mujannabic]]'' (+25/24) → [[Dicot family #Mujannabic|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]]
* ''[[Rip]]'' (+2560/2401) → [[Ripple family #Rip|Ripple family]]
* ''[[Whitewood]]'' (+2187/2048) → [[Whitewood family #Septimal whitewood|Whitewood family]]


=Progression=
== Penta ==
Comma: 3456/3125
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''


POTE generator: ~25/24 = 140.995
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 1 2|, &lt;0 5 3|]
[[Comma list]]: 28/25, 36/35
EDOs: 8, 9, 17c
Badness: 0.2005


==7-limit==
{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}
Commas: 36/35, 392/375
: mapping generators: ~2, ~7/6


POTE generator: ~15/14 = 140.927
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1186.8093{{c}}, ~7/6 = 237.3390{{c}}
: [[error map]]: {{val| -13.191 -3.129 +61.983 -45.851 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 231.845{{c}}
: error map: {{val| 0.000 +6.291 +85.850 -24.498 }}


Map: [&lt;1 1 2 2|, &lt;0 5 3 7|]
{{Optimal ET sequence|legend=1| 1bd, …, 4bcd, 5 }}
Wedgie: &lt;&lt;5 3 7 -7 -3 8||
EDOs: 9, 17c
Badness: 0.0484


==11-limit==
[[Badness]] (Sintel): 1.19
Commas: 36/35, 56/55, 77/75


[[POTE tuning|POTE generator]]: ~15/14 = 140.747
== Progression ==
{{Distinguish| Progress }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lafayette]].''


Map: [&lt;1 1 2 2 3|, &lt;0 5 3 7 4|]
Named by [[Gene Ward Smith]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18891.html Yahoo! Tuning Group | ''Progression temperament'']</ref>, progression can be described as the {{nowrap| 8d & 9 }} temperament. It has a generator that is a somewhat flat neutral second, three make [[5/4]], five make [[3/2]], and seven make [[7/4]], with a [[ploidacot]] signature of pentacot. [[17edo]] is an obvious tuning for it.  
EDOs: 8, 9, 17
Badness: 0.0261


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 26/25, 36/35, 56/55, 66/65


[[POTE tuning|POTE generator]]: ~15/14 = 140.751
[[Comma list]]: 36/35, 392/375


Map: Map: [&lt;1 1 2 2 3 3|, &lt;0 5 3 7 4 6|]
{{Mapping|legend=1| 1 1 2 2 | 0 5 3 7 }}
EDOs: 8, 9, 17
: mapping generators: ~2, ~15/14
Badness: 0.0182


==17-limit==
[[Optimal tuning]]s:
Commas: 26/25, 36/35, 51/50, 56/55, 66/65
* [[WE]]: ~2 = 1193.9544{{c}}, ~15/14 = 140.2169{{c}}
: [[error map]]: {{val| -6.046 -6.916 +22.246 +0.601 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~15/14 = 139.8991{{c}}
: error map: {{val| 0.000 -2.460 +33.384 +10.468 }}


[[POTE tuning|POTE generator]]: ~15/14 = 141.404
{{Optimal ET sequence|legend=1| 8d, 9, 17c }}


Map: [&lt;1 1 2 2 3 3 4|, &lt;0 5 3 7 4 6 1|]
[[Badness]] (Sintel): 1.22
EDOs: 8, 9, 17
Badness: 0.0167


===19-limit===
=== 11-limit ===
Commas: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE generator]]: ~15/14 = 140.479
Comma list: 36/35, 56/55, 77/75


Map: [&lt;1 1 2 2 3 3 4 4|, &lt;0 5 3 7 4 6 1 2|]
Mapping: {{mapping| 1 1 2 2 3 | 0 5 3 7 4 }}
EDOs: 8, 9, 17, [[94edo|94]]
Badness: 0.0168


=Ripple=
Optimal tunings:
Comma: 6561/6250
* WE: ~2 = 1194.7089{{c}}, ~12/11 = 140.1262{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.8776{{c}}


POTE generator: ~27/25 = 100.838
{{Optimal ET sequence|legend=0| 8d, 9, 17c }}


Map: [&lt;1 2 3|, &lt;0 -5 -8|]
Badness (Sintel): 0.861
EDOs: 12, 71b, 83b, 95b, 107bc, 119bc
Badness: 0.1389


==7-limit==
=== 13-limit ===
Commas: 36/35, 2560/2401
Subgroup: 2.3.5.7.11.13


[[POTE tuning|POTE generator]]: ~21/20 = 99.483
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


==11-limit==
Optimal tunings:
Commas: 36/35, 80/77, 126/121
* WE: ~2 = 1195.3694{{c}}, ~13/12 = 140.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.9749{{c}}


POTE generator: ~21/20 = ~12/11 = 99.385
{{Optimal ET sequence|legend=0| 8d, 9, 17c }}


Map: [&lt;1 2 3 3 4|, &lt;0 -5 -8 -2 -6|]
Badness (Sintel): 0.750
EDOs: 12
Badness: 0.0388


==13-limit==
=== 17-limit ===
Commas: 36/35, 40/39, 66/65, 147/143
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~21/20 = ~12/11 = ~14/13 = 98.572
Comma list: 26/25, 36/35, 51/50, 56/55, 66/65


Map: [&lt;1 2 3 3 4 4|, &lt;0 -5 -8 -2 -6 -3|]
Mapping: {{mapping| 1 1 2 2 3 3 4 | 0 5 3 7 4 6 1 }}
EDOs: 12f
Badness: 0.0316


=Smate=
Optimal tunings:
Comma: 2048/1875
* WE: ~2 = 1193.8350{{c}}, ~13/12 = 140.6779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.5885{{c}}


POTE generator: ~5/4 = 420.855
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}


Map: [&lt;1 3 2|, &lt;0 -4 1|]
Badness (Sintel): 0.853
EDOs: 11, 14, 17c, 20c, 37c
Badness: 0.179


==7-limit==
=== 19-limit ===
Commas: 36/35, 2048/1875
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~5/4 = 422.275
Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65


Map: [&lt;1 3 2 6|, &lt;0 -4 1 -9|]
Mapping: {{mapping| 1 1 2 2 3 3 4 4 | 0 5 3 7 4 6 1 2 }}
Wedgie: &lt;&lt;4 -1 9 -11 3 24||
EDOs: 14, 17c, 37cd
Badness: 0.0779


==11-limit==
Optimal tunings:
Commas: 36/35 56/55 243/242
* WE: ~2 = 1196.1446{{c}}, ~13/12 = 140.0276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.0301{{c}}


POTE generator: ~5/4 = 422.217
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}


Map: [&lt;1 3 2 6 7|, &lt;0 -4 1 -9 -10|]
Badness (Sintel): 1.02
EDOs: 14, 17c, 37cde
Badness: 0.0425


==13-limit==
== Subklei ==
Commas: 26/25, 36/35, 56/55, 243/242
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Delorean]].''


POTE generator: ~5/4 = 423.020
Subklei is likely named for its flatter minor third generator than [[kleismic]]. It tempers out [[1029/1000]] as well as [[2401/2400]] and can be described as {{nowrap| 17c & 21 }}. Its [[ploidacot]] is delta-hexacot. Note that in the data below, the generator is the [[5/3]][[~]][[12/7]] major sixth, so that six generators minus four octaves give the [[3/2|perfect fifth]].  


Map: [&lt;1 3 2 6 7 3|, &lt;0 -4 1 -9 -10 2|]
[[Subgroup]]: 2.3.5.7
EDOs: 14, 17c
Badness: 0.0368


=Hemismate=
[[Comma list]]: 36/35, 1029/1000
commas: 256/245, 392/375


POTE generator: ~8/7 = 210.452
{{Mapping|legend=1| 1 -3 -3 -1 | 0 6 7 5 }}
: mapping generators: ~2, ~5/3


Map: [&lt;1 3 2 3|, &lt;0 -8 2 -1|]
[[Optimal tuning]]s:  
Wedgie: &lt;&lt;8 -2 1 -22 -21 8||
* [[WE]]: ~2 = 1197.5285{{c}}, ~5/3 = 915.8950{{c}}
EDOs: 6, 11, 17c, 40bcd
: [[error map]]: {{val| -2.471 -11.171 +18.366 +3.121 }}
Badness: 0.1543
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 915.4228{{c}}
: error map: {{val| 0.000 -9.418 +21.646 +8.288 }}


==11-limit==
{{Optimal ET sequence|legend=1| 4, 13cd, 17c, 21, 38c }}
Commas: 56/55, 77/75, 256/245


POTE generator: ~8/7 =  210.481
[[Badness]] (Sintel): 1.55


Map: [&lt;1 3 2 3 4|, &lt;0 -8 2 -1 -3|]
=== 11-limit ===
EDOs: 6, 11, 17c, 40bcde
Subgroup: 2.3.5.7.11
Badness: 0.0655


==13-limit==
Comma list: 36/35, 77/75, 352/343
Commas: 26/25, 56/55, 77/75, 256/245


POTE generator: ~8/7 = 210.974
Mapping: {{mapping| 1 -3 -3 -1 -8 | 0 6 7 5 15 }}


Map: [&lt;1 3 2 3 4 3|, &lt;0 -8 2 -1 -3 4|]
Optimal tunings:  
EDOs: 6, 11, 17c
* WE: ~2 = 1196.3345{{c}}, ~5/3 = 913.9469{{c}}
Badness: 0.0505
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.2970{{c}}


=Subklei=
{{Optimal ET sequence|legend=0| 4e, …, 13cdee, 17c }}
Comma: 17496/15625


POTE generator: ~6/5 = 284.293
Badness (Sintel): 1.48


Map: [&lt;1 3 4|, &lt;0 -6 -7|]
==== 13-limit ====
EDOs: 4, 17c, 21, 38c
Subgroup: 2.3.5.7.11.13
Badness: 0.3559


==7-limit==
Comma list: 26/25, 36/35, 66/65, 352/343
Commas: 36/35, 1029/1000


POTE generator: ~6/5 = 284.219
Mapping: {{mapping| 1 3 4 4 7 7 | 0 6 7 5 15 14 }}


Map: [&lt;1 3 4 4|, &lt;0 -6 -7 -5|]
Optimal tunings:  
Wedgie: &lt;&lt;6 7 5 -3 -9 -8||
* WE: ~2 = 1196.0784{{c}}, ~5/3 = 914.1466{{c}}
EDOs: 4, 17c, 21, 38c
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.6712{{c}}
Badness: 0.0611


==11-limit==
{{Optimal ET sequence|legend=0| 4ef, …, 13cdeef, 17c }}
Commas: 36/35, 77/75, 352/343


POTE generator: ~6/5 = 283.253
Badness (Sintel): 1.34


Map: [&lt;1 3 4 4 7|, &lt;0 -6 -7 -5 -15|]
=== Subkla ===
EDOs: 17c, 55cd
Subgroup: 2.3.5.7.11
Badness: 0.0447


==Subkla==
Comma list: 36/35, 56/55, 1029/1000
Commas: 36/35, 56/55, 1029/1000


POTE generator: ~6/5 = 283.822
Mapping: {{mapping| 1 -3 -3 -1 5 | 0 6 7 5 -2 }}


Map: [&lt;1 3 4 4 3|, &lt;0 -6 -7 -5 2|]
Optimal tunings:  
EDOs: 4, 17c, 21, 38ce, 55cde
* WE: ~2 = 1196.3809{{c}}, ~5/3 = 913.4153{{c}}
Badness: 0.0472
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.8804{{c}}


===13-limit===
{{Optimal ET sequence|legend=0| 4, 13cd, 17c, 38ce }}
Commas: 36/35, 56/55, 66/65, 640/637


POTE generator: ~6/5 = 283.882
Badness (Sintel): 1.56


Map: [&lt;1 3 4 4 3 3|, &lt;0 -6 -7 -5 2 3|]
==== 13-limit ====
EDOs: 4, 17c, 21, 38ce, 55cde
Subgroup: 2.3.5.7.11.13
Badness: 0.0367


==13-limit==
Comma list: 36/35, 56/55, 66/65, 640/637
Commas: 26/25, 36/35, 66/65, 352/343


POTE generator: ~6/5 = 282.856
Mapping: {{mapping| 1 -3 -3 -1 5 6 | 0 6 7 5 -2 -3 }}


Map: [&lt;1 3 4 4 7 7|, &lt;0 -6 -7 -5 -15 -14|]
Optimal tunings:  
EDOs: 17c
* WE: ~2 = 1196.5477{{c}}, ~5/3 = 913.4822{{c}}
Badness: 0.0323
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.9416{{c}}


=Slurpee=
{{Optimal ET sequence|legend=0| 4, 17c, 38ce }}
Comma: 177147/156250


POTE generator: ~27/25 = 72.742
Badness (Sintel): 1.52


Map: [&lt;1 2 3|, &lt;0 -7 -11|]
== Naian ==
EDOs: 16, 17c, 33
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Naian]].''
Badness: 0.6893


==7-limit==
Named by [[Xenllium]] in 2026, naian may be described as {{nowrap| 8d & 21 }} with a [[ploidacot]] signature of epsilon-enneacot.
Commas: 36/35, 51200/50421


POTE generator: ~21/20 = 72.303
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 3 3|, &lt;0 -7 -11 -3|]
[[Comma list]]: 36/35, 9604/9375
Wedgie: &lt;&lt;7 11 3 1 -15 -24||
EDOs: 16, 17c, 33, 50cd, 83bcd
Badness: 0.1149


==11-limit==
{{Mapping|legend=1| 1 -4 -2 -4 | 0 9 7 11 }}
Commas: 36/35, 121/120, 352/343
: mapping generators: ~2, ~75/49


POTE generator: ~21/20 = 72.297
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1195.8807{{c}}, ~75/49 = 741.8522{{c}}
: [[error map]]: {{val| -4.119 -8.808 +14.890 +8.026 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/49 = 743.9885{{c}}
: error map: {{val| 0.000 -6.059 +21.606 +15.047 }}


Map: [&lt;1 2 3 3 4|, &lt;0 -7 -11 -3 -9|]
{{Optimal ET sequence|legend=1| 8d, 21, 29cd }}
EDOs: 16, 17c, 33, 50cd, 83bcd
Badness: 0.0505


==13-limit==
[[Badness]] (Sintel): 2.89
Commas: 36/35. 66/65, 143/140, 352/343


POTE generator: ~21/20 = 72.409
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 3 3 4 4|, &lt;0 -7 -11 -3 -9 -5|]
Comma list: 36/35, 56/55, 2541/2500
EDOs: 16, 17c, 33, 83bcd
 
Badness: 0.0331</pre></div>
Mapping: {{mapping| 1 -4 -2 -4 1 | 0 9 7 11 4 }}
<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;Mint temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:54:&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:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;a href="#Progression"&gt;Progression&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt;&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt;&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt; | &lt;a href="#Ripple"&gt;Ripple&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;!-- ws:start:WikiTextTocRule:62: --&gt;&lt;!-- ws:end:WikiTextTocRule:62 --&gt;&lt;!-- ws:start:WikiTextTocRule:63: --&gt;&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt;&lt;!-- ws:end:WikiTextTocRule:64 --&gt;&lt;!-- ws:start:WikiTextTocRule:65: --&gt; | &lt;a href="#Smate"&gt;Smate&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:65 --&gt;&lt;!-- ws:start:WikiTextTocRule:66: --&gt;&lt;!-- ws:end:WikiTextTocRule:66 --&gt;&lt;!-- ws:start:WikiTextTocRule:67: --&gt;&lt;!-- ws:end:WikiTextTocRule:67 --&gt;&lt;!-- ws:start:WikiTextTocRule:68: --&gt;&lt;!-- ws:end:WikiTextTocRule:68 --&gt;&lt;!-- ws:start:WikiTextTocRule:69: --&gt; | &lt;a href="#Hemismate"&gt;Hemismate&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:69 --&gt;&lt;!-- ws:start:WikiTextTocRule:70: --&gt;&lt;!-- ws:end:WikiTextTocRule:70 --&gt;&lt;!-- ws:start:WikiTextTocRule:71: --&gt;&lt;!-- ws:end:WikiTextTocRule:71 --&gt;&lt;!-- ws:start:WikiTextTocRule:72: --&gt; | &lt;a href="#Subklei"&gt;Subklei&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:72 --&gt;&lt;!-- ws:start:WikiTextTocRule:73: --&gt;&lt;!-- ws:end:WikiTextTocRule:73 --&gt;&lt;!-- ws:start:WikiTextTocRule:74: --&gt;&lt;!-- ws:end:WikiTextTocRule:74 --&gt;&lt;!-- ws:start:WikiTextTocRule:75: --&gt;&lt;!-- ws:end:WikiTextTocRule:75 --&gt;&lt;!-- ws:start:WikiTextTocRule:76: --&gt;&lt;!-- ws:end:WikiTextTocRule:76 --&gt;&lt;!-- ws:start:WikiTextTocRule:77: --&gt;&lt;!-- ws:end:WikiTextTocRule:77 --&gt;&lt;!-- ws:start:WikiTextTocRule:78: --&gt; | &lt;a href="#Slurpee"&gt;Slurpee&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:78 --&gt;&lt;!-- ws:start:WikiTextTocRule:79: --&gt;&lt;!-- ws:end:WikiTextTocRule:79 --&gt;&lt;!-- ws:start:WikiTextTocRule:80: --&gt;&lt;!-- ws:end:WikiTextTocRule:80 --&gt;&lt;!-- ws:start:WikiTextTocRule:81: --&gt;&lt;!-- ws:end:WikiTextTocRule:81 --&gt;&lt;!-- ws:start:WikiTextTocRule:82: --&gt;
Optimal tunings:
&lt;!-- ws:end:WikiTextTocRule:82 --&gt;&lt;br /&gt;
* WE: ~2 = 1194.3937{{c}}, ~75/49 = 741.2117{{c}}
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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Square_triangular_number" rel="nofollow"&gt;both&lt;/a&gt; a &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/SquareNumber.html" rel="nofollow"&gt;square&lt;/a&gt; and a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Triangular_number" rel="nofollow"&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;
* CWE: ~2 = 1200.0000{{c}}, ~75/49 = 744.2018{{c}}
&lt;br /&gt;
 
&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;
{{Optimal ET sequence|legend=0| 8d, 21, 29cde }}
Comma: 3456/3125&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.95
POTE generator: ~25/24 = 140.995&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
Map: [&amp;lt;1 1 2|, &amp;lt;0 5 3|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
EDOs: 8, 9, 17c&lt;br /&gt;
 
Badness: 0.2005&lt;br /&gt;
Comma list: 36/35, 56/55, 66/65, 507/500
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Progression-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;7-limit&lt;/h2&gt;
Mapping: {{mapping| 1 -4 -2 -4 1 0 | 0 9 7 11 4 6 }}
Commas: 36/35, 392/375&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~15/14 = 140.927&lt;br /&gt;
* WE: ~2 = 1193.8564{{c}}, ~20/13 = 740.9635{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.2703{{c}}
Map: [&amp;lt;1 1 2 2|, &amp;lt;0 5 3 7|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;5 3 7 -7 -3 8||&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 21, 29cdef }}
EDOs: 9, 17c&lt;br /&gt;
 
Badness: 0.0484&lt;br /&gt;
Badness (Sintel): 1.55
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Progression-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
=== 17-limit ===
Commas: 36/35, 56/55, 77/75&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 140.747&lt;br /&gt;
Comma list: 36/35, 51/50, 56/55, 66/65, 170/169
&lt;br /&gt;
 
Map: [&amp;lt;1 1 2 2 3|, &amp;lt;0 5 3 7 4|]&lt;br /&gt;
Mapping: {{mapping| 1 -4 -2 -4 1 0 1 | 0 9 7 11 4 6 5 }}
EDOs: 8, 9, 17&lt;br /&gt;
 
Badness: 0.0261&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1194.2330{{c}}, ~20/13 = 741.1242{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Progression-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.2679{{c}}
Commas: 26/25, 36/35, 56/55, 66/65&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 21, 29cdef }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 140.751&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.43
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;
== Slurpee ==
Badness: 0.0182&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Slurpee]].''
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Progression-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;17-limit&lt;/h2&gt;
Slurpee may be described as the {{nowrap| 16 & 17c }} temperament. It has a generator that is a somewhat flat semitone of [[~]][[21/20]], three make [[8/7]], seven make [[4/3]], and eleven make [[8/5]], with a [[ploidacot]] signature of omega-heptacot.
Commas: 26/25, 36/35, 51/50, 56/55, 66/65&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 141.404&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 36/35, 51200/50421
Map: [&amp;lt;1 1 2 2 3 3 4|, &amp;lt;0 5 3 7 4 6 1|]&lt;br /&gt;
 
EDOs: 8, 9, 17&lt;br /&gt;
{{Mapping|legend=1| 1 2 3 3 | 0 -7 -11 -3 }}
Badness: 0.0167&lt;br /&gt;
: mapping generators: ~2, ~21/20
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Progression-17-limit-19-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;19-limit&lt;/h3&gt;
[[Optimal tuning]]s:  
Commas: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65&lt;br /&gt;
* [[WE]]: ~2 = 1197.9281{{c}}, ~21/20 = 72.1780{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -2.072 -11.345 +13.512 +8.424 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~15/14 = 140.479&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 72.4941{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 -9.414 +16.251 +13.692 }}
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;
{{Optimal ET sequence|legend=1| 16, 17c, 33 }}
Badness: 0.0168&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 2.91
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Ripple"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Ripple&lt;/h1&gt;
 
Comma: 6561/6250&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE generator: ~27/25 = 100.838&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 36/35, 121/120, 352/343
Map: [&amp;lt;1 2 3|, &amp;lt;0 -5 -8|]&lt;br /&gt;
 
EDOs: 12, 71b, 83b, 95b, 107bc, 119bc&lt;br /&gt;
Mapping: {{mapping| 1 2 3 3 4 | 0 -7 -11 -3 -9 }}
Badness: 0.1389&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Ripple-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;7-limit&lt;/h2&gt;
* WE: ~2 = 1198.4220{{c}}, ~21/20 = 72.2015{{c}}
Commas: 36/35, 2560/2401&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4470{{c}}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~21/20 = 99.483&lt;br /&gt;
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 3|, &amp;lt;0 -5 -8 -2|]&lt;br /&gt;
Badness (Sintel): 1.67
Wedgie: &amp;lt;&amp;lt;5 8 2 1 -11 -18||&lt;br /&gt;
 
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;
=== 13-limit ===
Badness: 0.0597&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Ripple-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;11-limit&lt;/h2&gt;
Comma list: 36/35, 66/65, 143/140, 352/343
Commas: 36/35, 80/77, 126/121&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -7 -11 -3 -9 -5 }}
POTE generator: ~21/20 = ~12/11 = 99.385&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Map: [&amp;lt;1 2 3 3 4|, &amp;lt;0 -5 -8 -2 -6|]&lt;br /&gt;
* WE: ~2 = 1199.0238{{c}}, ~21/20 = 72.3506{{c}}
EDOs: 12&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4956{{c}}
Badness: 0.0388&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Ripple-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;13-limit&lt;/h2&gt;
 
Commas: 36/35, 40/39, 66/65, 147/143&lt;br /&gt;
Badness (Sintel): 1.37
&lt;br /&gt;
 
POTE generator: ~21/20 = ~12/11 = ~14/13 = 98.572&lt;br /&gt;
== Shallowtone ==
&lt;br /&gt;
{{Main| Shallowtone }}
Map: [&amp;lt;1 2 3 3 4 4|, &amp;lt;0 -5 -8 -2 -6 -3|]&lt;br /&gt;
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Shallowtone (5-limit)]].''
EDOs: 12f&lt;br /&gt;
 
Badness: 0.0316&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Smate"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Smate&lt;/h1&gt;
[[Comma list]]: 36/35, 295245/262144
Comma: 2048/1875&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 0 18 -16 | 0 1 -10 12 }}
POTE generator: ~5/4 = 420.855&lt;br /&gt;
: mapping generators: ~2, ~3
&lt;br /&gt;
 
Map: [&amp;lt;1 3 2|, &amp;lt;0 -4 1|]&lt;br /&gt;
[[Optimal tuning]]s:
EDOs: 11, 14, 17c, 20c, 37c&lt;br /&gt;
* [[WE]]: ~2 = 1202.4397{{c}}, ~3/2 = 682.6219{{c}}
Badness: 0.179&lt;br /&gt;
: [[error map]]: {{val| +2.440 -16.893 +6.986 +12.877 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 681.2447{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Smate-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;7-limit&lt;/h2&gt;
: error map: {{val| 0.000 -20.710 +1.239 +6.110 }}
Commas: 36/35, 2048/1875&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7, 30b, 37b }}
POTE generator: ~5/4 = 422.275&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 7.79
Map: [&amp;lt;1 3 2 6|, &amp;lt;0 -4 1 -9|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;4 -1 9 -11 3 24||&lt;br /&gt;
=== 11-limit ===
EDOs: 14, 17c, 37cd&lt;br /&gt;
Subgroup: 2.3.5.7.11
Badness: 0.0779&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 36/35, 45/44, 72171/65536
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Smate-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;11-limit&lt;/h2&gt;
 
Commas: 36/35 56/55 243/242&lt;br /&gt;
Mapping: {{Mapping| 1 0 18 -16 16 | 0 1 -10 12 -8 }}
&lt;br /&gt;
 
POTE generator: ~5/4 = 422.217&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* WE: ~2 = 1202.0285{{c}}, ~3/2 = 682.4922{{c}}
Map: [&amp;lt;1 3 2 6 7|, &amp;lt;0 -4 1 -9 -10|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.3267{{c}}
EDOs: 14, 17c, 37cde&lt;br /&gt;
 
Badness: 0.0425&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Smate-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;13-limit&lt;/h2&gt;
Badness (Sintel): 4.29
Commas: 26/25, 36/35, 56/55, 243/242 &lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
POTE generator: ~5/4 = 423.020&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Map: [&amp;lt;1 3 2 6 7 3|, &amp;lt;0 -4 1 -9 -10 2|]&lt;br /&gt;
Comma list: 27/26, 36/35, 45/44, 16731/16384
EDOs: 14, 17c&lt;br /&gt;
 
Badness: 0.0368&lt;br /&gt;
Mapping: {{Mapping| 1 0 18 -16 16 -1 | 0 1 -10 12 -8 3 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Hemismate"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Hemismate&lt;/h1&gt;
Optimal tunings:
commas: 256/245, 392/375 &lt;br /&gt;
* WE: ~2 = 1201.4928{{c}}, ~3/2 = 682.1188{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.2669{{c}}
POTE generator: ~8/7 = 210.452&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}
Map: [&amp;lt;1 3 2 3|, &amp;lt;0 -8 2 -1|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;8 -2 1 -22 -21 8||&lt;br /&gt;
Badness (Sintel): 3.19
EDOs: 6, 11, 17c, 40bcd&lt;br /&gt;
 
Badness: 0.1543&lt;br /&gt;
== References ==
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Hemismate-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;11-limit&lt;/h2&gt;
[[Category:Temperament collections]]
Commas: 56/55, 77/75, 256/245&lt;br /&gt;
[[Category:Mint temperaments| ]] <!-- main article -->
&lt;br /&gt;
[[Category:Rank 2]]
POTE generator: ~8/7 =  210.481&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 3 2 3 4|, &amp;lt;0 -8 2 -1 -3|]&lt;br /&gt;
EDOs: 6, 11, 17c, 40bcde&lt;br /&gt;
Badness: 0.0655&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Hemismate-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;13-limit&lt;/h2&gt;
Commas: 26/25, 56/55, 77/75, 256/245&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~8/7 = 210.974&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 3 2 3 4 3|, &amp;lt;0 -8 2 -1 -3 4|]&lt;br /&gt;
EDOs: 6, 11, 17c&lt;br /&gt;
Badness: 0.0505&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc17"&gt;&lt;a name="Subklei"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Subklei&lt;/h1&gt;
Comma: 17496/15625&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~6/5 = 284.293&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 3 4|, &amp;lt;0 -6 -7|]&lt;br /&gt;
EDOs: 4, 17c, 21, 38c&lt;br /&gt;
Badness: 0.3559&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Subklei-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;7-limit&lt;/h2&gt;
Commas: 36/35, 1029/1000&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~6/5 = 284.219&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 3 4 4|, &amp;lt;0 -6 -7 -5|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;6 7 5 -3 -9 -8||&lt;br /&gt;
EDOs: 4, 17c, 21, 38c&lt;br /&gt;
Badness: 0.0611&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Subklei-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;11-limit&lt;/h2&gt;
Commas: 36/35, 77/75, 352/343&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~6/5 = 283.253&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 3 4 4 7|, &amp;lt;0 -6 -7 -5 -15|]&lt;br /&gt;
EDOs: 17c, 55cd&lt;br /&gt;
Badness: 0.0447&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Subklei-Subkla"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;Subkla&lt;/h2&gt;
Commas: 36/35, 56/55, 1029/1000&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~6/5 = 283.822&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 3 4 4 3|, &amp;lt;0 -6 -7 -5 2|]&lt;br /&gt;
EDOs: 4, 17c, 21, 38ce, 55cde&lt;br /&gt;
Badness: 0.0472&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc21"&gt;&lt;a name="Subklei-Subkla-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;13-limit&lt;/h3&gt;
Commas: 36/35, 56/55, 66/65, 640/637&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~6/5 = 283.882&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 3 4 4 3 3|, &amp;lt;0 -6 -7 -5 2 3|]&lt;br /&gt;
EDOs: 4, 17c, 21, 38ce, 55cde&lt;br /&gt;
Badness: 0.0367&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Subklei-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;13-limit&lt;/h2&gt;
Commas: 26/25, 36/35, 66/65, 352/343 &lt;br /&gt;
&lt;br /&gt;
POTE generator: ~6/5 = 282.856&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 3 4 4 7 7|, &amp;lt;0 -6 -7 -5 -15 -14|]&lt;br /&gt;
EDOs: 17c&lt;br /&gt;
Badness: 0.0323&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc23"&gt;&lt;a name="Slurpee"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;Slurpee&lt;/h1&gt;
Comma: 177147/156250 &lt;br /&gt;
&lt;br /&gt;
POTE generator: ~27/25 = 72.742&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 3|, &amp;lt;0 -7 -11|]&lt;br /&gt;
EDOs: 16, 17c, 33&lt;br /&gt;
Badness: 0.6893&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc24"&gt;&lt;a name="Slurpee-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;7-limit&lt;/h2&gt;
Commas: 36/35, 51200/50421&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~21/20 = 72.303&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 3 3|, &amp;lt;0 -7 -11 -3|]&lt;br /&gt;
Wedgie: &amp;lt;&amp;lt;7 11 3 1 -15 -24||&lt;br /&gt;
EDOs: 16, 17c, 33, 50cd, 83bcd&lt;br /&gt;
Badness: 0.1149&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc25"&gt;&lt;a name="Slurpee-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;11-limit&lt;/h2&gt;
Commas: 36/35, 121/120, 352/343 &lt;br /&gt;
&lt;br /&gt;
POTE generator: ~21/20 = 72.297&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 3 3 4|, &amp;lt;0 -7 -11 -3 -9|]&lt;br /&gt;
EDOs: 16, 17c, 33, 50cd, 83bcd&lt;br /&gt;
Badness: 0.0505&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc26"&gt;&lt;a name="Slurpee-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;13-limit&lt;/h2&gt;
Commas: 36/35. 66/65, 143/140, 352/343&lt;br /&gt;
&lt;br /&gt;
POTE generator: ~21/20 = 72.409&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 2 3 3 4 4|, &amp;lt;0 -7 -11 -3 -9 -5|]&lt;br /&gt;
EDOs: 16, 17c, 33, 83bcd&lt;br /&gt;
Badness: 0.0331&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 10:15, 29 May 2026

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

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:

  • WE: ~2 = 1186.8093 ¢, ~7/6 = 237.3390 ¢
error map: ⟨-13.191 -3.129 +61.983 -45.851]
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 231.845 ¢
error map: ⟨0.000 +6.291 +85.850 -24.498]

Optimal ET sequence: 1bd, …, 4bcd, 5

Badness (Sintel): 1.19

Progression

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

Named by Gene Ward Smith in 2011[1], progression can be described as the 8d & 9 temperament. It has a generator that is a somewhat flat neutral second, three make 5/4, five make 3/2, and seven make 7/4, with a ploidacot signature of pentacot. 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:

  • WE: ~2 = 1193.9544 ¢, ~15/14 = 140.2169 ¢
error map: ⟨-6.046 -6.916 +22.246 +0.601]
  • CWE: ~2 = 1200.000 ¢, ~15/14 = 139.8991 ¢
error map: ⟨0.000 -2.460 +33.384 +10.468]

Optimal ET sequence: 8d, 9, 17c

Badness (Sintel): 1.22

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:

  • WE: ~2 = 1194.7089 ¢, ~12/11 = 140.1262 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 139.8776 ¢

Optimal ET sequence: 8d, 9, 17c

Badness (Sintel): 0.861

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:

  • WE: ~2 = 1195.3694 ¢, ~13/12 = 140.2080 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.9749 ¢

Optimal ET sequence: 8d, 9, 17c

Badness (Sintel): 0.750

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:

  • WE: ~2 = 1193.8350 ¢, ~13/12 = 140.6779 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.5885 ¢

Optimal ET sequence: 8d, 9, 17cg

Badness (Sintel): 0.853

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:

  • WE: ~2 = 1196.1446 ¢, ~13/12 = 140.0276 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.0301 ¢

Optimal ET sequence: 8d, 9, 17cg

Badness (Sintel): 1.02

Subklei

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

Subklei is likely named for its flatter minor third generator than kleismic. It tempers out 1029/1000 as well as 2401/2400 and can be described as 17c & 21. Its ploidacot is delta-hexacot. Note that in the data below, the generator is the 5/3~12/7 major sixth, so that six generators minus four octaves give the perfect fifth.

Subgroup: 2.3.5.7

Comma list: 36/35, 1029/1000

Mapping: [⟨1 -3 -3 -1], ⟨0 6 7 5]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1197.5285 ¢, ~5/3 = 915.8950 ¢
error map: ⟨-2.471 -11.171 +18.366 +3.121]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 915.4228 ¢
error map: ⟨0.000 -9.418 +21.646 +8.288]

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

Badness (Sintel): 1.55

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -3 -3 -1 -8], ⟨0 6 7 5 15]]

Optimal tunings:

  • WE: ~2 = 1196.3345 ¢, ~5/3 = 913.9469 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 916.2970 ¢

Optimal ET sequence: 4e, …, 13cdee, 17c

Badness (Sintel): 1.48

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:

  • WE: ~2 = 1196.0784 ¢, ~5/3 = 914.1466 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 916.6712 ¢

Optimal ET sequence: 4ef, …, 13cdeef, 17c

Badness (Sintel): 1.34

Subkla

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1196.3809 ¢, ~5/3 = 913.4153 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 915.8804 ¢

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

Badness (Sintel): 1.56

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 56/55, 66/65, 640/637

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

Optimal tunings:

  • WE: ~2 = 1196.5477 ¢, ~5/3 = 913.4822 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 915.9416 ¢

Optimal ET sequence: 4, 17c, 38ce

Badness (Sintel): 1.52

Naian

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

Named by Xenllium in 2026, naian may be described as 8d & 21 with a ploidacot signature of epsilon-enneacot.

Subgroup: 2.3.5.7

Comma list: 36/35, 9604/9375

Mapping: [⟨1 -4 -2 -4], ⟨0 9 7 11]]

mapping generators: ~2, ~75/49

Optimal tunings:

  • WE: ~2 = 1195.8807 ¢, ~75/49 = 741.8522 ¢
error map: ⟨-4.119 -8.808 +14.890 +8.026]
  • CWE: ~2 = 1200.0000 ¢, ~75/49 = 743.9885 ¢
error map: ⟨0.000 -6.059 +21.606 +15.047]

Optimal ET sequence: 8d, 21, 29cd

Badness (Sintel): 2.89

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 2541/2500

Mapping: [⟨1 -4 -2 -4 1], ⟨0 9 7 11 4]]

Optimal tunings:

  • WE: ~2 = 1194.3937 ¢, ~75/49 = 741.2117 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/49 = 744.2018 ¢

Optimal ET sequence: 8d, 21, 29cde

Badness (Sintel): 1.95

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 56/55, 66/65, 507/500

Mapping: [⟨1 -4 -2 -4 1 0], ⟨0 9 7 11 4 6]]

Optimal tunings:

  • WE: ~2 = 1193.8564 ¢, ~20/13 = 740.9635 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.2703 ¢

Optimal ET sequence: 8d, 21, 29cdef

Badness (Sintel): 1.55

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 36/35, 51/50, 56/55, 66/65, 170/169

Mapping: [⟨1 -4 -2 -4 1 0 1], ⟨0 9 7 11 4 6 5]]

Optimal tunings:

  • WE: ~2 = 1194.2330 ¢, ~20/13 = 741.1242 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.2679 ¢

Optimal ET sequence: 8d, 21, 29cdef

Badness (Sintel): 1.43

Slurpee

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

Slurpee may be described as the 16 & 17c temperament. It has a generator that is a somewhat flat semitone of ~21/20, three make 8/7, seven make 4/3, and eleven make 8/5, with a ploidacot signature of omega-heptacot.

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:

  • WE: ~2 = 1197.9281 ¢, ~21/20 = 72.1780 ¢
error map: ⟨-2.072 -11.345 +13.512 +8.424]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 72.4941 ¢
error map: ⟨0.000 -9.414 +16.251 +13.692]

Optimal ET sequence: 16, 17c, 33

Badness (Sintel): 2.91

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:

  • WE: ~2 = 1198.4220 ¢, ~21/20 = 72.2015 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 72.4470 ¢

Optimal ET sequence: 16, 17c, 33

Badness (Sintel): 1.67

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:

  • WE: ~2 = 1199.0238 ¢, ~21/20 = 72.3506 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 72.4956 ¢

Optimal ET sequence: 16, 17c, 33

Badness (Sintel): 1.37

Shallowtone

Main article: 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:

  • WE: ~2 = 1202.4397 ¢, ~3/2 = 682.6219 ¢
error map: ⟨+2.440 -16.893 +6.986 +12.877]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 681.2447 ¢
error map: ⟨0.000 -20.710 +1.239 +6.110]

Optimal ET sequence: 7, 30b, 37b

Badness (Sintel): 7.79

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:

  • WE: ~2 = 1202.0285 ¢, ~3/2 = 682.4922 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 681.3267 ¢

Optimal ET sequence: 7, 30b, 37b

Badness (Sintel): 4.29

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:

  • WE: ~2 = 1201.4928 ¢, ~3/2 = 682.1188 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 681.2669 ¢

Optimal ET sequence: 7, 30b, 37b

Badness (Sintel): 3.19

References

  1. ↑ Yahoo! Tuning Group | Progression temperament
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