Xenharmonic Wiki

Hemimage temperaments: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 339363506 - Original comment: **
 
(71 intermediate revisions by 12 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935). These include chromat, degrees, bicommatic, bisupermajor, and squarschmidt, considered below, as well as the following discussed elsewhere:  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-24 23:23:48 UTC</tt>.<br>
* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
: The original revision id was <tt>339363506</tt>.<br>
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
: The revision comment was: <tt></tt><br>
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic|Magic family]]
<h4>Original Wikitext content:</h4>
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
<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]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Pluto]]'' (+4000/3969) → [[Mirkwai clan #Pluto|Mirkwai clan]]
* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth|Buzzardsmic clan]]
* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]


== Chromat ==
The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an [[Amity family|amity extension]] with third-octave period.


Hemimage temperaments temper out the hemimage comma, |5 -7 -1 3&gt; = 10976/10935.
[[Subgroup]]: 2.3.5.7


=Commatic=
[[Comma list]]: 10976/10935, 235298/234375


==7-limit==
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
Commas: 10976/10935, 50421/50000


POTE generator: ~81/80 = 20.377
: mapping generators: ~63/50, ~28/27


Map: [&lt;2 3 4 5|, &lt;0 5 19 18|]
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~28/27 = 60.528
Wedgie: &lt;&lt;10 38 36 37 29 -23||
EDOs: 58, 118, 294, 412d, 530d
Badness: 0.0843


==11-limit==
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}
Commas: 441/440, 3388/3375, 8019/8000


POTE generator: ~81/80 = 20.390
[[Badness]]: 0.057499


Map: [&lt;2 3 4 5 6|, &lt;0 5 19 18 27|]
=== 11-limit ===
EDOs: 58, 118, 294, 412d
Subgroup: 2.3.5.7.11
Badness: 0.0305


=Chromat=
Comma list: 441/440, 4375/4356, 10976/10935
Commas: 10976/10935, 235298/234375


POTE generator: ~28/27 = 60.528
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}


Map: [&lt;3 4 5 6|, &lt;0 5 13 16|]
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.430
Wedgie: &lt;&lt;15 39 48 27 34 2||
EDOs: 60, 99, 258, 357, 456
Badness: 0.0575


=Degrees=
{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}
Commas: 10976/10935, 390625/388962


POTE generator: ~3/2 = 703.015
Badness: 0.050379


Map: [&lt;20 0 -17 -39|, &lt;0 1 2 3|]
==== 13-limit ====
Wedgie: &lt;&lt;20 40 60 17 39 27||
Subgroup: 2.3.5.7.11.13
EDOs: 60, 80, 140, 640b, 780b, 920b
Badness: 0.1065


==11-limit==
Comma list: 364/363, 441/440, 625/624, 10976/10935
Commas: 1331/1323, 1375/1372, 2200/2187


POTE generator: ~3/2 = 703.231
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}


Map: [&lt;20 0 -17 -39 -26|, &lt;0 1 2 3 3|]
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.428
EDOs: 80, 140, 360, 500be, 860bde
Badness: 0.0468


==13-limit===
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417d }}
Commas: 325/324, 352/351, 1001/1000, 1331/1323


POTE generator: ~3/2 = 703.080
Badness: 0.046006


Map: [&lt;20 0 -17 -39 -26 74|, &lt;0 1 2 3 3 0|]
===== 17-limit =====
EDOs: 60e, 80, 140, 500be, 640be, 780be
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0327


Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757


</pre></div>
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}
<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;Hemimage temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:14:&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:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;a href="#Commatic"&gt;Commatic&lt;/a&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="#Chromat"&gt;Chromat&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt; | &lt;a href="#Degrees"&gt;Degrees&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.438
&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417dg }}
Hemimage temperaments temper out the hemimage comma, |5 -7 -1 3&amp;gt; = 10976/10935.&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.031678
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Commatic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Commatic&lt;/h1&gt;
 
&lt;br /&gt;
==== Catachrome ====
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Commatic-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;7-limit&lt;/h2&gt;
Subgroup: 2.3.5.7.11.13
Commas: 10976/10935, 50421/50000&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 325/324, 441/440, 1001/1000, 10976/10935
POTE generator: ~81/80 = 20.377&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}
Map: [&amp;lt;2 3 4 5|, &amp;lt;0 5 19 18|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;10 38 36 37 29 -23||&lt;br /&gt;
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.378
EDOs: 58, 118, 294, 412d, 530d&lt;br /&gt;
 
Badness: 0.0843&lt;br /&gt;
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Commatic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
Badness: 0.043844
Commas: 441/440, 3388/3375, 8019/8000&lt;br /&gt;
 
&lt;br /&gt;
===== 17-limit =====
POTE generator: ~81/80 = 20.390&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&lt;br /&gt;
 
Map: [&amp;lt;2 3 4 5 6|, &amp;lt;0 5 19 18 27|]&lt;br /&gt;
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913
EDOs: 58, 118, 294, 412d&lt;br /&gt;
 
Badness: 0.0305&lt;br /&gt;
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Chromat"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Chromat&lt;/h1&gt;
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.377
Commas: 10976/10935, 235298/234375&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
POTE generator: ~28/27 = 60.528&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.030218
Map: [&amp;lt;3 4 5 6|, &amp;lt;0 5 13 16|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;15 39 48 27 34 2||&lt;br /&gt;
==== Chromic ====
EDOs: 60, 99, 258, 357, 456&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Badness: 0.0575&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 196/195, 352/351, 729/728, 1875/1859
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Degrees"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Degrees&lt;/h1&gt;
 
Commas: 10976/10935, 390625/388962&lt;br /&gt;
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}
&lt;br /&gt;
 
POTE generator: ~3/2 = 703.015&lt;br /&gt;
Optimal tuning (POTE): ~44/35 = 1\3, ~27/26 = 60.456
&lt;br /&gt;
 
Map: [&amp;lt;20 0 -17 -39|, &amp;lt;0 1 2 3|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
Wedgie: &amp;lt;&amp;lt;20 40 60 17 39 27||&lt;br /&gt;
 
EDOs: 60, 80, 140, 640b, 780b, 920b&lt;br /&gt;
Badness: 0.049857
Badness: 0.1065&lt;br /&gt;
 
&lt;br /&gt;
===== 17-limit =====
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Degrees-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;11-limit&lt;/h2&gt;
Subgroup: 2.3.5.7.11.13.17
Commas: 1331/1323, 1375/1372, 2200/2187&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594
POTE generator: ~3/2 = 703.231&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}
Map: [&amp;lt;20 0 -17 -39 -26|, &amp;lt;0 1 2 3 3|]&lt;br /&gt;
 
EDOs: 80, 140, 360, 500be, 860bde&lt;br /&gt;
Optimal tuning (POTE): ~63/50 = 1\3, ~27/26 = 60.459
Badness: 0.0468&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Degrees-13-limit="&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit=&lt;/h2&gt;
 
Commas: 325/324, 352/351, 1001/1000, 1331/1323&lt;br /&gt;
Badness: 0.031043
&lt;br /&gt;
 
POTE generator: ~3/2 = 703.080&lt;br /&gt;
=== Hemichromat ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
Map: [&amp;lt;20 0 -17 -39 -26 74|, &amp;lt;0 1 2 3 3 0|]&lt;br /&gt;
 
EDOs: 60e, 80, 140, 500be, 640be, 780be&lt;br /&gt;
Comma list: 3025/3024, 10976/10935, 102487/102400
Badness: 0.0327&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}
 
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2511
 
{{Optimal ET sequence|legend=1| 39d, 120cd, 159, 198, 357, 912b }}
 
Badness: 0.067173
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}
 
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2527
 
{{Optimal ET sequence|legend=1| 39df, 120cdff, 159, 198, 357, 912b }}
 
Badness: 0.033420
 
== Bisupermajor ==
{{See also| Very high accuracy temperaments #Kwazy }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 65625/65536
 
{{Mapping|legend=1| 2 1 6 1 | 0 8 -5 17 }}
 
: mapping generators: ~1225/864, ~192/175
 
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~192/175 = 162.806
 
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
 
[[Badness]]: 0.065492
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 3388/3375, 9801/9800
 
Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}
 
Optimal tuning (POTE): ~99/70, ~11/10 = 162.773
 
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 258e, 376de }}
 
Badness: 0.032080
 
== Bicommatic ==
Used to be known simply as the ''commatic'' temperament, the bicommatic temperament has a period of half octave and a generator of 20.4 cents, a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 50421/50000
 
{{Mapping|legend=1| 2 3 4 5 | 0 5 19 18 }}
 
: mapping generators: ~567/400, ~81/80
 
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~81/80 = 20.377
 
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d, 530d }}
 
[[Badness]]: 0.084317
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 3388/3375, 8019/8000
 
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~81/80 = 20.390
 
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}
 
Badness: 0.030461
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 729/728, 1001/1000
 
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 20.427
 
{{Optimal ET sequence|legend=1| 58, 118, 176f }}
 
Badness: 0.026336
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560
 
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 20.378
 
{{Optimal ET sequence|legend=1| 58, 118, 294ffg, 412dffgg }}
 
Badness: 0.022396
 
== Degrees ==
{{ See also | 20th-octave temperaments }}
Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.
 
An obvious extension to the 23-limit exists by equating 4\20 = 1\5 with [[23/20]], 6\20 = 3\10 with [[69/56]], 7\20 with [[23/18]], etc. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]], with 29/28 being especially accurate, and by equating [[29/22]] with 2\5 = 240{{cent}}, we get a uniquely elegant extension to the 29-limit which tempers out ([[33/25]])/([[29/22]]) = [[726/725]], [[784/783|S28 = 784/783]] and [[841/840|S29 = 841/840]]. An edo as large as [[220edo|220]] supports it by patent val, though it does not appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.
 
By equating 37/28 with 2\5 and more accurately 85/74 with 1\5 and 44/37 with 1\4 (among many other equivalences) we get an extension to prime 37 agreeing with many (semi)convergents. By equating 60/41~41/28 with 11\20 or equivalently 56/41~41/30 with 9\20 and by equating 44/41 with 1\10 (among many other equivalences) there is a very efficient extension to prime 41.
 
By looking at the mapping, we observe an 80-note [[mos scale]] is ideal, so that [[80edo]] is in some sense both a trivial and maximally efficient tuning of this temperament. We also observe an abundance of JI interpretations of [[20edo]] by combining primes so that all things require 3 generators, yielding: 37:44:54:56:58:60:69:74:82:85. Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above. The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in this temperament. Note that prime 47 can be added but only really makes sense in rooted form in [[140edo]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 390625/388962
 
{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
 
: mapping generators: ~28/27, ~3
 
[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015 (~126/125 = 16.985)
 
{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}
 
[[Badness]]: 0.106471
 
Badness (Sintel): 2.694
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 1331/1323, 1375/1372, 2200/2187
 
Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}
 
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231 (~100/99 = 16.769)
 
{{Optimal ET sequence|legend=1| 20cd, 60e, 80, 140, 360 }}
 
Badness: 0.046770
 
Badness (Sintel): 1.546
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 1001/1000, 1331/1323
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}
 
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.080 (~100/99 = 16.920)
 
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
 
Badness: 0.032718
 
Badness (Sintel): 1.352
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}
 
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)
 
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
 
Badness (Sintel): 1.171
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}
 
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)
 
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
 
Badness (Sintel): 1.273
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}
 
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.169 (~100/99 = 16.831)
 
{{Optimal ET sequence|legend=1| 20cdei, 60e, 80, 140 }}
 
Badness (Sintel): 1.209
 
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}
 
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171 (~100/99 = 16.829)
 
{{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}
 
Badness (Sintel): 1.134
 
=== no-31's 37-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.37
 
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405, 481/480
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 | 0 1 2 3 3 0 1 0 2 3 3 }}
 
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.222 (~100/99 = 16.778)
 
{{Optimal ET sequence|legend=1| 20cdeijl, 60el, 80, 140 }}
 
Badness (Sintel): 1.127
 
=== no-31's 41-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41
 
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 451/450, 476/475, 481/480, 2871/2870
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 12 | 0 1 2 3 3 0 1 0 2 3 3 3 }}
 
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.207
 
{{Optimal ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}
 
Badness (Sintel): 1.100
 
== Squarschmidt ==
A generator for the squarschimidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as 118&amp;239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&amp;239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 61 4 -29 }}
 
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}
 
: mapping generators: ~2, ~98304/78125
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621
 
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}
 
[[Badness]]: 0.218314
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 29360128/29296875
 
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643
 
{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}
 
[[Badness]]: 0.132821
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 5632/5625, 10976/10935
 
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~44/35 = 396.644
 
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}
 
Badness: 0.038186
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage| ]] <!-- key article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Hemimage_temperaments"