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Ragismic microtemperaments: 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 [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the ragisma, [[4375/4374]] ({{monzo| -1 -7 4 1 }}). The ragisma is the smallest [[7-limit]] [[superparticular ratio]].  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-25 14:44:25 UTC</tt>.<br>
: The original revision id was <tt>150551101</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">The ragisma is 4375/4374, the smallest 7-limit superparticular ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low cmplexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.


===Ennealimmal===
Since {{nowrap|(10/9)<sup>4</sup> {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)<sup>2</sup> }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.
Ennealimmal temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the ennealimma comma, |1 -27 18&gt;, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is &lt;&lt;18 27 18 1 -22 -34||.


Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 equal, though its hardly likely anyone could tell the difference.  
Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
* ''[[Modus]]'' (+64/63) → [[Tetracot family #Modus|Tetracot family]]
* ''[[Flattone]]'' (+81/80) → [[Meantone family #Flattone|Meantone family]]
* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]
* [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]
* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]
* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]]
* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortonic family #Mitonic|Minortonic family]]
* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnuzmic family #Septimal vishnu|Vishnuzmic family]]
* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]
* ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]]
* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]
* ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments #Chlorine|17th-octave temperaments]]
* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments#Dzelic|37th-octave temperaments]]


If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example.) In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS.  
== Supermajor ==
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2<sup>15</sup>)/3, 46 give (2<sup>19</sup>)/5, and 75 give (2<sup>30</sup>)/7. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80-note mos is presumably the place to start, and if that is not enough notes for you, there is always the 171-note mos.


===Supermajor===
[[Subgroup]]: 2.3.5.7
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of &lt;&lt;37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.


===Enneadecal===
[[Comma list]]: 4375/4374, 52734375/52706752
Enndedecal temperament tempers out the enneadeca, |-14 -19 19&gt;, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5 or 7 limits, and [[494edo]] shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.


===Mitonic===
{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}
As a 5-limit temperament, mitonic is a super-accurate microtemperament tempering out the minortone comma, |-16 35 -17&gt;. Flipping that gives the 5-limit wedgie &lt;&lt;17 35 16||, which tells us that 10/9 can be taken as the generator, with 17 of them giving a 6, 18 of them a 20/3, and 35 of them giving a 40. The generator should be tuned about 1/16 of a cent flat, with 6^(1/17) being 0.06423 cents flat and 40^(1/35) being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.


However, as noted before, 32/21 is only a ragisma shy of (10/9)^4, and so a 7-limit interpretation, if not quite so super-accurate, is more or less inevitable. While 559 or 730 are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo]]. The wedgie is now &lt;&lt;17 35 -21 16 -81 -147||, with 21 10/9 generators giving a 64/7. MOS of size 20, 33, 46 or 79 notes can be used for mitonic.
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082


==Nearly Micro==
{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 1106, 1277, 3660, 4937, 6214 }}


===Amity===
[[Badness]]: 0.010836
The generator for amity temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&amp;53 temperament, or by its wedgie, &lt;&lt;5 13 -17 9 -41 -76||. [[99edo]] is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.</pre></div>
 
<h4>Original HTML content:</h4>
=== Semisupermajor ===
<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;Ragismic microtemperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The ragisma is 4375/4374, the smallest 7-limit superparticular ratio. Since (10/9)^4 = 4375/4374 * 32/21, the minor tone 10/9 tends to be an interval of relatively low complexity in temperaments tempering out the ragisma, though when looking at microtemperaments the word &amp;quot;relatively&amp;quot; should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have 7/6 = 4375/4374 * (27/25)^2, so 27/25 also tends to relatively low cmplexity, with the same caveat about &amp;quot;relatively&amp;quot;; however 27/25 is the period for ennealimmal.&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Ennealimmal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Ennealimmal&lt;/h3&gt;
Comma list: 3025/3024, 4375/4374, 35156250/35153041
Ennealimmal temperament tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the ennealimma comma, |1 -27 18&amp;gt;, which leads to the identification of (27/25)^9 with the octave, and gives ennealimmal a period of 1/9 octave. While 27/25 is a 5-limit interval, two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit. Its wedgie is &amp;lt;&amp;lt;18 27 18 1 -22 -34||.&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}
Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40 and 60/49, all of which have their own interesting advantages. Possible tunings are 441, 612, or 3600 equal, though its hardly likely anyone could tell the difference. &lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 435.082
If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example.) In particular, people fond of the idea of &amp;quot;tritaves&amp;quot; as analogous to octaves might consider the 28 or 43 note MOS with generator an approximate 5/3 within 3; for instance as given by 451/970 of a &amp;quot;tritave&amp;quot;. Tetrads have a low enough complexity that (for example) there are nine 1-3/2-7/4-5/2 tetrads in the 28 notes to the tritave MOS, which is equivalent in average step size to a 17 2/3 to the octave MOS. &lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x--Supermajor"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Supermajor&lt;/h3&gt;
 
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.0002 cents flat. 37 of these give (2^15)/3, 46 give (2^19)/5, and 75 give (2^30)/7, leading to a wedgie of &amp;lt;&amp;lt;37 46 75 -13 15 45||. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80 note MOS is presumably the place to start, and if that isn't enough notes for you, there's always the 171 note MOS.&lt;br /&gt;
Badness: 0.012773
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x--Enneadecal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Enneadecal&lt;/h3&gt;
== Enneadecal ==
Enndedecal temperament tempers out the enneadeca, |-14 -19 19&amp;gt;, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be 25/24, 27/25, 10/9, 5/4 or 3/2. To this we may add possible 7-limit generators such as 225/224, 15/14 or 9/7. Since enneadecal tempers out 703125/702464, the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^(1/3). This is the interval needed to adjust the 1/3 comma meantone flat fifths and major thirds of &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt; up to just ones. &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt; is a good tuning for either the 5 or 7 limits, and &lt;a class="wiki_link" href="/494edo"&gt;494edo&lt;/a&gt; shows how to extend the temperament to the 11 or 13 limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use &lt;a class="wiki_link" href="/665edo"&gt;665edo&lt;/a&gt; for a tuning.&lt;br /&gt;
Enneadecal temperament tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen just minor thirds fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)<sup>1/3</sup>. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones. [[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x--Mitonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Mitonic&lt;/h3&gt;
''For the 5-limit temperament, see [[19th-octave temperaments#(5-limit) enneadecal]].''
As a 5-limit temperament, mitonic is a super-accurate microtemperament tempering out the minortone comma, |-16 35 -17&amp;gt;. Flipping that gives the 5-limit wedgie &amp;lt;&amp;lt;17 35 16||, which tells us that 10/9 can be taken as the generator, with 17 of them giving a 6, 18 of them a 20/3, and 35 of them giving a 40. The generator should be tuned about 1/16 of a cent flat, with 6^(1/17) being 0.06423 cents flat and 40^(1/35) being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
However, as noted before, 32/21 is only a ragisma shy of (10/9)^4, and so a 7-limit interpretation, if not quite so super-accurate, is more or less inevitable. While 559 or 730 are still fine as tunings, the error of the 7-limit is lower by a whisker in &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt;. The wedgie is now &amp;lt;&amp;lt;17 35 -21 16 -81 -147||, with 21 10/9 generators giving a 64/7. MOS of size 20, 33, 46 or 79 notes can be used for mitonic.&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 4375/4374, 703125/702464
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-Nearly Micro"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Nearly Micro&lt;/h2&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-Nearly Micro-Amity"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Amity&lt;/h3&gt;
 
The generator for amity temperament is the acute minor third, which means an ordinary 6/5 minor third raised by an 81/80 comma to 243/200, and from this it derives its name. Aside from the ragisma it tempers out the 5-limit amity comma, 1600000/1594323, 5120/5103 and 6144/6125. It can also be described as the 46&amp;amp;53 temperament, or by its wedgie, &amp;lt;&amp;lt;5 13 -17 9 -41 -76||. &lt;a class="wiki_link" href="/99edo"&gt;99edo&lt;/a&gt; is a good tuning for amity, with generator 28/99, and MOS of 11, 18, 25, 32, 46 or 53 notes are available. If you are looking for a different kind of neutral third this could be the temperament for you.&lt;/body&gt;&lt;/html&gt;</pre></div>
: mapping generators: ~28/27, ~3
 
[[Optimal tuning]] ([[CTE]]): ~28/27 = 1\19, ~3/2 = 701.9275 (~225/224 = 7.1907)
 
{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}
 
[[Badness]]: 0.010954
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4375/4374, 16384/16335
 
Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}
 
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 702.1483 (~225/224 = 7.4115)
 
{{Optimal ET sequence|legend=1| 19, 133d, 152, 323e, 475de, 627de }}
 
Badness: 0.043734
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 625/624, 729/728, 2205/2197
 
Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}
 
Optimal tuning (CTE): ~28/27 = 1\19, ~3/2 = 701.9258 (~225/224 = 7.1890)
 
{{Optimal ET sequence|legend=1| 19, 133df, 152f, 323ef }}
 
Badness: 0.033545
 
=== Hemienneadecal ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 234375/234256
 
Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}
 
: mapping generators: ~55/54, ~3
 
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9351 (~225/224 = 7.1983)
 
{{Optimal ET sequence|legend=1| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}
 
Badness: 0.009985
 
==== Hemienneadecalis ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256
 
Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}
 
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9955 (~225/224 = 7.2587)
 
{{Optimal ET sequence|legend=1| 152f, 342f, 494 }}
 
Badness: 0.020782
 
==== Hemienneadec ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213
 
Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}
 
Optimal tuning (CTE): ~55/54 = 1\38, ~3/2 = 701.9812 (~225/224 = 7.2444)
 
{{Optimal ET sequence|legend=1| 152, 342, 494, 1330, 1824, 2318d }}
 
Badness: 0.030391
 
==== Semihemienneadecal ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078
 
Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}
 
: mapping generators: ~55/54 = 1\38, ~55/54, ~429/250
 
Optimal tuning (CTE): ~429/250 = 935.1789 (~144/143 = 12.1895)
 
{{Optimal ET sequence|legend=1| 190, 304d, 494, 684, 1178, 2850, 4028ce }}
 
Badness: 0.014694
 
=== Kalium ===
Named after the 19th element, potassium, and after an archaic variant of the element's name to resolve a name conflict. [[19/16]] can be used as a generator. Since it is enfactored in the 17-limit and lower, it makes no sense to name it for the lower subgroups.
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344
 
Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}
 
Optimal tuning (CTE): ~28/27 = 1\19, ~6545/5928 = 171.244
 
{{Optimal ET sequence|legend=1| 855, 988, 1843 }}
 
== Semidimi ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimi]].''
 
The generator of semidimi temperament is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 3955078125/3954653486
 
{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270
 
{{Optimal ET sequence|legend=1| 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}
 
[[Badness]]: 0.015075
 
== Brahmagupta ==
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.
 
Early in the design of the [[Sagittal]] notation system, Secor and Keenan found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4 ¢ many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of Brahmagupta temperament that has pure octaves and pure fifths, which can also be described as a 17-limit extension having 1/7th octave period (171.4286 ¢) and 1/21st apotome generator (5.4136 ¢).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 70368744177664/70338939985125
 
{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}
 
: mapping generators: ~1157625/1048576, ~27/20
 
[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716
 
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 1106, 1547 }}
 
[[Badness]]: 0.029122
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4000/3993, 4375/4374, 131072/130977
 
Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}
 
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.704
 
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771ee }}
 
Badness: 0.052190
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374
 
Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}
 
Optimal tuning (POTE): ~243/220 = 1\7, ~27/20 = 519.706
 
{{Optimal ET sequence|legend=1| 7, 217, 224, 441, 665, 1771eef }}
 
Badness: 0.023132
 
== Abigail ==
Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."</ref>
 
''For the 5-limit temperament, see [[Very high accuracy temperaments#Abigail]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 2147483648/2144153025
 
{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}
 
: mapping generators: ~46305/32768, ~27/20
 
[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899
 
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
 
[[Badness]]: 0.037000
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 131072/130977
 
Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~1155/1024 = 208.901
 
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764 }}
 
Badness: 0.012860
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
 
Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~44/39 = 208.903
 
{{Optimal ET sequence|legend=1| 46, 178, 224, 270, 494, 764, 1258 }}
 
Badness: 0.008856
 
== Gamera ==
''For the 5-limit temperament, see [[High badness temperaments#Gamera]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 589824/588245
 
{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}
 
: mapping generators: ~2, ~8/7
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336
 
{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}
 
[[Badness]]: 0.037648
 
=== Hemigamera ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 589824/588245
 
Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}
 
: mapping generators: ~99/70, ~8/7
 
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3370
 
{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646, 1068d }}
 
Badness: 0.040955
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024
 
Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 230.3373
 
{{Optimal ET sequence|legend=1| 26, 198, 224, 422, 646f, 1068df }}
 
Badness: 0.020416
 
=== Semigamera ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 14641/14580, 15488/15435
 
Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}
 
: mapping generators: ~2, ~77/72
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1642
 
{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}
 
Badness: 0.078
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580
 
Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/72 = 115.1628
 
{{Optimal ET sequence|legend=1| 73f, 125f, 198, 323, 521 }}
 
Badness: 0.044
 
== Crazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
 
Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament. [[1106edo]] is an strong tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}
 
{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}
 
: mapping generators: ~332150625/234881024, ~1125/1024
 
[[Optimal tuning]]s:
* [[CTE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7475
* [[error map]]: {{val| 0.0000 +0.0253 -0.0514 -0.0133 }}
* [[CWE]]: ~332150625/234881024 = 600.0000, ~1125/1024 = 162.7474
* error map: {{val| 0.0000 +0.0244 -0.0508 -0.0218 }}
 
{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}
 
[[Badness]] (Smith): 0.0394
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 2791309312/2790703125
 
Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}
 
Optimal tunings:
* CTE: ~99/70 = 162.7485, ~1125/1024 = 162.7485
* CWE: ~99/70 = 162.7485, ~1125/1024 = 162.7481
 
{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}
 
Badness (Smith): 0.0170
 
== Orga ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 54975581388800/54936068900769
 
{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}
 
: mapping generators: ~7411887/5242880, ~1310720/1058841
 
[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104
 
{{Optimal ET sequence|legend=1| 26, 244, 270, 836, 1106, 1376, 2482 }}
 
[[Badness]]: 0.040236
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 5767168/5764801
 
Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
 
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836, 1106 }}
 
Badness: 0.016188
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360
 
Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~8/7 = 231.103
 
{{Optimal ET sequence|legend=1| 26, 244, 270, 566, 836f, 1106f }}
 
Badness: 0.021762
 
== Seniority ==
{{See also| Very high accuracy temperaments #Senior }}
 
Aside from the ragisma, the seniority temperament (26 &amp; 145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ({{monzo| -17 62 -35 }}, quadla-sepquingu) is tempered out.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 201768035/201326592
 
{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804
 
{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}
 
[[Badness]]: 0.044877
 
=== Senator ===
The senator temperament (26 &amp; 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It can be extended to the 13- and 17-limit immediately, by adding 364/363 and 595/594 to the comma list in this order.
 
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 4375/4374, 65536/65219
 
Mapping: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
 
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316e, 487ee }}
 
Badness: 0.092238
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 2200/2197, 4375/4374
 
Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 322.793
 
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}
 
Badness: 0.044662
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197
 
Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}
 
Optimal tuning (POTE): ~77/64 = 322.793
 
{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 316ef, 487eef }}
 
Badness: 0.026562
 
== Monzismic ==
: ''For the 5-limit version of this temperament, see [[Very high accuracy temperaments #Monzismic]].
 
The monzismic temperament (53 &amp; 612) tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}
 
{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207
 
{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}
 
[[Badness]]: 0.046569
 
=== Monzism ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 41503/41472, 184549376/184528125
 
Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}
 
Optimal tuning (POTE): ~231/200 = 249.0193
 
{{Optimal ET sequence|legend=1| 53, 559, 612 }}
 
Badness: 0.057083
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625
 
Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}
 
Optimal tuning (POTE): ~231/200 = 249.0199
 
{{Optimal ET sequence|legend=1| 53, 559, 612 }}
 
Badness: 0.053780
 
== Semidimfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semidimfourth]].''
 
The semidimfourth temperament is featured by a semi-diminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, 235298/234375.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 235298/234375
 
[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456
 
{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}
 
[[Badness]]: 0.055249
 
=== Neusec ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 235298/234375
 
Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.547
 
{{Optimal ET sequence|legend=1| 8d, 190, 388 }}
 
Badness: 0.059127
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374
 
Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~12/11 = 151.545
 
{{Optimal ET sequence|legend=1| 8d, 190, 198, 388 }}
 
Badness: 0.030941
 
== Acrokleismic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 2202927104/2197265625
 
{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}
 
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557
 
{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}
 
[[Badness]]: 0.056184
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 41503/41472, 172032/171875
 
Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.558
 
{{Optimal ET sequence|legend=1| 19, 251, 270, 829, 1099, 1369, 1639 }}
 
Badness: 0.036878
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976
 
Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.557
 
{{Optimal ET sequence|legend=1| 19, 251, 270 }}
 
Badness: 0.026818
 
=== Counteracro ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 5632/5625, 117649/117612
 
Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.553
 
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}
 
Badness: 0.042572
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374
 
Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.554
 
{{Optimal ET sequence|legend=1| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}
 
Badness: 0.026028
 
== Quasithird ==
The quasithird temperament is featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 55 -64 20 }}
 
{{Mapping|legend=1| 4 0 -11 | 0 5 16 }}
 
: mapping generators: ~51200000/43046721, ~1594323/1280000
 
[[Optimal tuning]] ([[POTE]]): ~51200000/43046721, ~1594323/1280000 = 380.395
 
{{Optimal ET sequence|legend=1| 60, 104c, 164, 224, 388, 612, 1612, 2224, 2836, 6284, 9120, 15404 }}
 
[[Badness]]: 0.099519
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}
 
{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
 
[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388
 
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}
 
[[Badness]]: 0.061813
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 4296700485/4294967296
 
Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}
 
Optimal tuning (POTE): ~5103/4096 = 380.387 (or ~22/21 = 80.387)
 
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448 }}
 
Badness: 0.021125
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374
 
Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}
 
Optimal tuning (POTE): ~81/65 = 380.385 (or ~22/21 = 80.385)
 
{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}
 
Badness: 0.029501
 
== Deca ==
: ''For 5-limit version of this temperament, see [[10th-octave temperaments #Neon]].''
 
Deca temperament has a period of 1/10 octave and tempers out the [[linus comma]], {{monzo| 11 -10 -10 10 }}, neon comma {{monzo| 21 60 -50 }} and {{monzo| 12 -3 -14 9 }} = 165288374272/164794921875 (satritrizo-asepbigu).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 165288374272/164794921875
 
{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}
 
: mapping generators: ~15/14, ~6/5
 
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577
 
{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}
 
[[Badness]]: 0.080637
 
Badness (Sintel): 2.041
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 391314/390625
 
Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}
 
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.582
 
{{Optimal ET sequence|legend=1| 80, 190, 270, 1000, 1270, 1540e, 1810e }}
 
Badness: 0.024329
 
Badness (Sintel): 0.804
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374
 
Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}
 
Optimal tuning (POTE): ~15/14 = 1\10, ~6/5 = 315.602 (~40/39 = 44.398)
 
{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}
 
Badness: 0.016810
 
Badness (Sintel): 0.695
 
=== no-17's 19-limit ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374, 1521/1520
 
Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}
 
Optimal tuning (CTE): ~15/14 = 1\10, ~6/5 = 315.581 (~39/38 = 44.419)
 
{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}
 
Badness (Sintel): 0.556
 
== Keenanose ==
Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}
 
{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
 
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~{{monzo| 21 3 1 -10 }} = 4.4465
 
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}
 
[[Badness]]: 0.0858
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 117649/117612, 67110351/67108864
 
Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4465
 
{{Optimal ET sequence|legend=1| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}
 
Badness: 0.0308
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612
 
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~385/384 = 4.4466
 
{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 4048 }}
 
Badness: 0.0213
 
== Aluminium ==
Aluminium is named after the 13th element, and tempers out the {{monzo| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 92 -39 -13 }}
 
[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}
 
: mapping generators: ~135/128, ~3
 
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897
 
{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}
 
[[Badness]]: 0.123
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}
 
[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
 
[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 702.0024
 
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}
 
[[Badness]]: 0.126
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 234375/234256, 2097152/2096325
 
Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}
 
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0042
 
{{Optimal ET sequence|legend=1| 494, 1053, 1547, 3588e, 5135e }}
 
Badness: 0.0421
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078
 
Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}
 
Optimal tuning (CTE): ~135/128 = 1\13, ~3/2 = 702.0099
 
{{Optimal ET sequence|legend=1| 494, 1547, 2041, 4576def }}
 
Badness: 0.0286
 
== Countritonic ==
: ''For the 5-limit version of this temperament, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
 
Countritonic (''co-un-tritonic'') can be described as the 53 & 422 temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 68719476736/68356598625
 
{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}
 
: mapping generators: ~2, ~45927/32768
 
[[Optimal tuning]] (CTE): ~2 = 1\1, ~45927/32768 = 588.6216
 
{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}
 
[[Badness]]: 0.133
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 5632/5625, 2621440/2614689
 
Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~539/384 = 588.6258
 
{{Optimal ET sequence|legend=1| 53, 316e, 369, 422, 791e, 1213cde }}
 
Badness: 0.0707
 
=== 13-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625
 
Mapping: {{mapping| 1 6 19 -13 79 | 0 -9 -34 73 154 -74 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~128/91 = 588.6277
 
{{Optimal ET sequence|legend=1| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}
 
Badness: 0.0366
 
== Quatracot ==
{{See also| Stratosphere }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}
 
{{Mapping|legend=1| 2 7 7 23 | 0 -13 -8 -59 }}
 
: mapping generators: ~2278125/1605632, ~448/405
 
[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805
 
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c, 1690bcc }}
 
[[Badness]]: 0.175982
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 1265625/1261568
 
Mapping: {{mapping| 2 7 7 23 19 | 0 -13 -8 -59 -41 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 176.806
 
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1052c }}
 
Badness: 0.041043
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 729/728, 1575/1573, 2200/2197
 
Mapping: {{mapping| 2 7 7 23 19 13 | 0 -13 -8 -59 -41 -19 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~195/176 = 176.804
 
{{Optimal ET sequence|legend=1| 190, 224, 414, 638, 1690bcc, 2328bccde }}
 
Badness: 0.022643
 
== Moulin ==
Moulin has a generator of 22/13, and it is named after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". It can be described as the 494 & 1619 temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}
 
{{Mapping|legend=1| 1 57 38 248 | 0 -73 -47 -323 }}
 
: mapping generators: ~2, ~6422528/3796875
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6422528/3796875 = 910.9323
 
{{Optimal ET sequence|legend=1| 494, 1125, 1619 }}
 
[[Badness]]: 0.234
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 759375/758912, 100663296/100656875
 
Mapping: {{mapping| 1 57 38 248 -14 | 0 -73 -47 -323 23 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~1024/605 = 910.9323
 
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}
 
Badness: 0.0678
 
=== 13-limit ===
Since 11/8 is within 23 generators, the 25 tone MOS (4L 21s) of this temperament contains the 8:11:13 triad.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078
 
Mapping: {{mapping| 1 57 38 248 -14 -13 | 0 -73 -47 -323 23 22 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~22/13 = 910.9323
 
{{Optimal ET sequence|legend=1| 494, 1125, 1619, 2113 }}
 
Badness: 0.0271
 
== Palladium ==
: ''For the 5-limit version of this temperament, see [[46th-octave temperaments]]''.
 
The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minortones (10/9) fall short of seven octaves. This temperament can be described as 46 &amp; 414 temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}
 
{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
 
: mapping generators: ~83349/81920, ~3
 
[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074
 
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874d }}
 
[[Badness]]: 0.308505
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 134775333/134217728
 
Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}
 
Optimal tuning (POTE): ~8192/8085 = 1\46, ~3/2 = 701.5951
 
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de }}
 
Badness: 0.073783
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364
 
Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}
 
Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6419
 
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334de }}
 
Badness: 0.040751
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224
 
Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}
 
Optimal tuning (POTE): ~65/64 = 1\46, ~3/2 = 701.6425
 
{{Optimal ET sequence|legend=1| 46, 368, 414, 460, 874de, 1334deg }}
 
Badness: 0.022441
 
== Oviminor ==
{{See also| Syntonic–kleismic equivalence continuum }}
 
Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}
 
{{Mapping|legend=1| 1 50 51 147 | 0 -184 -185 -548 }}
 
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~6/5 = 315.7501
 
{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}
 
[[Badness]]: 0.582
 
== Octoid ==
''For the 5-limit temperament, see [[8th-octave temperaments#Octoid (5-limit)]].''
 
The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai]]). In the 11-limit, it tempers out 540/539, 1375/1372, and 6250/6237. In this temperament, one period gives both 12/11 and 49/45, two gives 25/21, three gives 35/27, and four gives both 99/70 and 140/99.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 16875/16807
 
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
 
: mapping generators: ~49/45, ~7/5
 
[[Optimal tuning]] ([[POTE]]): ~49/45 = 1\8, ~7/5 = 583.940
 
[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
 
{{Optimal ET sequence|legend=1| 8d, 72, 152, 224 }}
 
[[Badness]]: 0.042670
 
Scales: [[octoid72]], [[octoid80]]
 
=== 11-limit ===
The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimaxing the damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, if one wants to use 80edo as the tuning, one must use octopus — not octoid — as 80edo doesn't temper 324/323, 375/374, 495/494, 625/624, 715/714 or 729/728.
 
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 4000/3993
 
Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}
 
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.962
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]
 
{{Optimal ET sequence|legend=1| 72, 152, 224 }}
 
Badness: 0.014097
 
Scales: [[octoid72]], [[octoid80]]
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 625/624, 729/728, 1375/1372
 
Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}
 
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.905
 
{{Optimal ET sequence|legend=1| 72, 152f, 224 }}
 
Badness: 0.015274
 
Scales: [[octoid72]], [[octoid80]]
 
; Music
* ''Dreyfus'' (archived 2010) by [[Gene Ward Smith]] – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – octoid[72] in 224edo tuning
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 375/374, 540/539, 625/624, 715/714, 729/728
 
Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}
 
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.842
 
{{Optimal ET sequence|legend=1| 72, 152fg, 224, 296, 520g }}
 
Badness: 0.014304
 
Scales: [[octoid72]], [[octoid80]]
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714
 
Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}
 
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.932
 
{{Optimal ET sequence|legend=1| 72, 152fg, 224 }}
 
Badness: 0.016036
 
Scales: [[octoid72]], [[octoid80]]
 
==== Octopus ====
A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4 = 300{{cent}}.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 325/324, 364/363, 540/539
 
Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}
 
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.892
 
{{Optimal ET sequence|legend=1| 72, 152, 224f }}
 
Badness: 0.021679
 
Scales: [[octoid72]], [[octoid80]]
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 289/288, 325/324, 540/539
 
Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}
 
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 583.811
 
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 296ffg }}
 
Badness: 0.015614
 
Scales: [[Octoid72]], [[Octoid80]]
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399
 
Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}
 
Optimal tuning (POTE): ~12/11 = 1\8, ~7/5 = 584.064
 
{{Optimal ET sequence|legend=1| 72, 152, 224fg, 376ffgh }}
 
Badness: 0.016321
 
Scales: [[Octoid72]], [[Octoid80]]
 
==== Hexadecoid ====
{{ See also | 16th-octave temperaments }}
 
Hexadecoid (80 &amp; 144) has a period of 1/16 octave and tempers out 4225/4224.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224
 
Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
 
: mapping generators: ~448/429, ~7/5
 
Optimal tuning (POTE): ~448/429 = 1\16, ~13/8 = 841.015
 
{{Optimal ET sequence|legend=1| 80, 144, 224 }}
 
Badness: 0.030818
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224
 
Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}
 
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.932
 
{{Optimal ET sequence|legend=1| 80, 144, 224, 528dg }}
 
Badness: 0.028611
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444
 
Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 -3 -4 -5 -3 1 2 0 }}
 
Optimal tuning (POTE): ~117/112 = 1\16, ~13/8 = 840.896
 
{{Optimal ET sequence|legend=1| 80, 144, 224, 304dh, 528dghh }}
 
Badness: 0.023731
 
== Parakleismic ==
{{Main| Parakleismic }}
 
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, and 11-limit adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 1224440064/1220703125
 
{{Mapping|legend=1| 1 5 6 | 0 -13 -14 }}
 
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.240
 
{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}
 
[[Badness]]: 0.043279
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3136/3125, 4375/4374
 
{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}
 
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.181
 
{{Optimal ET sequence|legend=1| 19, 80, 99, 217, 316, 415 }}
 
[[Badness]]: 0.027431
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 3136/3125, 4375/4374
 
Mapping: {{mapping| 1 5 6 12 -6 | 0 -13 -14 -35 36 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.251
 
{{Optimal ET sequence|legend=1| 19, 99, 118 }}
 
Badness: 0.049711
 
=== Paralytic ===
The ''paralytic'' temperament (118&amp;217) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 &amp; 217 tempers out 1001/1000, 1575/1573, and 3584/3575.
 
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 3136/3125, 4375/4374
 
Mapping: {{mapping| 1 5 6 12 25 | 0 -13 -14 -35 -82 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.220
 
{{Optimal ET sequence|legend=1| 19e, 99e, 118, 217, 335, 552d, 887dd }}
 
Badness: 0.036027
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374
 
Mapping: {{mapping| 1 5 6 12 25 -16 | 0 -13 -14 -35 -82 75 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.214
 
{{Optimal ET sequence|legend=1| 99e, 118, 217, 552d, 769de }}
 
Badness: 0.044710
 
==== Paraklein ====
The ''paraklein'' temperament (19e &amp; 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 625/624, 729/728
 
Mapping: {{mapping| 1 5 6 12 25 15 | 0 -13 -14 -35 -82 -43 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.225
 
{{Optimal ET sequence|legend=1| 19e, 99ef, 118, 217ff, 335ff }}
 
Badness: 0.037618
 
=== Parkleismic ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 1375/1372, 2200/2187
 
Mapping: {{mapping| 1 5 6 12 20 | 0 -13 -14 -35 -63 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.060
 
{{Optimal ET sequence|legend=1| 19e, 80, 179, 259cd }}
 
Badness: 0.055884
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 176/175, 325/324, 1375/1372
 
Mapping: {{mapping| 1 5 6 12 20 10 | 0 -13 -14 -35 -63 -24 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.075
 
{{Optimal ET sequence|legend=1| 19e, 80, 179 }}
 
Badness: 0.036559
 
=== Paradigmic ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 896/891, 3136/3125
 
Mapping: {{mapping| 1 5 6 12 -1 | 0 -13 -14 -35 17 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.096
 
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}
 
Badness: 0.041720
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 325/324, 540/539, 832/825
 
Mapping: {{mapping| 1 5 6 12 -1 10 | 0 -13 -14 -35 17 -24 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.080
 
{{Optimal ET sequence|legend=1| 19, 61d, 80, 99e, 179e }}
 
Badness: 0.035781
 
=== Semiparakleismic ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 3136/3125, 4375/4374
 
Mapping: {{mapping| 2 10 12 24 19 | 0 -13 -14 -35 -23 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.181
 
{{Optimal ET sequence|legend=1| 80, 118, 198, 316, 514c, 830c }}
 
Badness: 0.034208
 
==== Semiparamint ====
This extension was named ''semiparakleismic'' in the earlier materials.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374
 
Mapping: {{mapping| 2 10 12 24 19 -1 | 0 -13 -14 -35 -23 16 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 315.156
 
{{Optimal ET sequence|legend=1| 80, 118, 198 }}
 
Badness: 0.033775
 
==== Semiparawolf ====
This extension was named ''gentsemiparakleismic'' in the earlier materials.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 325/324, 364/363, 3136/3125
 
Mapping: {{mapping| 2 10 12 24 19 20 | 0 -13 -14 -35 -23 -24 }}
 
Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 315.184
 
{{Optimal ET sequence|legend=1| 80, 118f, 198f }}
 
Badness: 0.040467
 
== Counterkleismic ==
{{See also| High badness temperaments #Counterhanson}}
 
In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as 19 &amp; 224 temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 158203125/157351936
 
{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }}
 
: mapping generators: ~2, ~5/3
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.060
 
{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }}
 
[[Badness]]: 0.090553
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4375/4374, 2097152/2096325
 
Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.071
 
{{Optimal ET sequence|legend=1| 19, 205, 224 }}
 
Badness: 0.070952
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 625/624, 729/728, 10985/10976
 
Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.070
 
{{Optimal ET sequence|legend=1| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}
 
Badness: 0.033874
 
=== Counterlytic ===
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 4375/4374, 496125/495616
 
Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065
 
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}
 
Badness: 0.065400
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 729/728, 1375/1372, 10985/10976
 
Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.065
 
{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}
 
Badness: 0.029782
 
== Quincy ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 823543/819200
 
{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1715 = 16.613
 
{{Optimal ET sequence|legend=1| 72, 217, 289 }}
 
[[Badness]]: 0.079657
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 4000/3993, 4375/4374
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.613
 
{{Optimal ET sequence|legend=1| 72, 217, 289 }}
 
Badness: 0.030875
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 676/675, 4375/4374
 
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602
 
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}
 
Badness: 0.023862
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155
 
Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.602
 
{{Optimal ET sequence|legend=1| 72, 145, 217, 289 }}
 
Badness: 0.014741
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675
 
Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~100/99 = 16.594
 
{{Optimal ET sequence|legend=1| 72, 145, 217 }}
 
Badness: 0.015197
 
== Sfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 64827/64000
 
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 26.287
 
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
 
[[Badness]]: 0.123291
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 441/440, 4375/4374
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.286
 
{{Optimal ET sequence|legend=1| 45e, 46, 91e, 137de }}
 
Badness: 0.054098
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 169/168, 325/324, 441/440
 
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.310
 
{{Optimal ET sequence|legend=1| 45ef, 46, 91ef, 137def }}
 
Badness: 0.033067
 
=== Sfour ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 2401/2376, 4375/4374
 
Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.246
 
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
 
Badness: 0.076567
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 364/363, 385/384, 4375/4374
 
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 26.239
 
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
 
Badness: 0.051893
 
== Trideci ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tridecatonic]].''
 
The trideci temperament (26 &amp; 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from "tridecim" (Latin for "[[wikipedia:13|thirteen]]").
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 83349/81920
 
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
 
[[Optimal tuning]] ([[POTE]]): ~256/245 = 1\13, ~3/2 = 699.1410
 
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }}
 
[[Badness]]: 0.184585
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/242, 385/384, 4375/4374
 
Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}
 
Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.6179
 
{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdde }}
 
Badness: 0.084590
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 245/242, 325/324, 385/384
 
Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}
 
Optimal tuning (POTE): ~22/21 = 1\13, ~3/2 = 699.2969
 
{{Optimal ET sequence|legend=1| 26, 65f, 91f, 156dff }}
 
Badness: 0.052366
 
== Counterorson ==
Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]].
 
Subgroup: 2.3.5.7
 
Comma list: 4375/4374, {{monzo| 154 -54 -21 -7 }}
 
Mapping: {{mapping| 1 0 -21 85 | 0 7 103 -363 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~{{monzo| 66 -23 -9 -3 }} = 271.7113
 
{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}
 
Badness: 0.312806
 
== Notes ==
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Ragismic microtemperaments| ]] <!-- main article -->
[[Category:Ragismic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Microtemperaments]]
[[Category:Abigail]]
[[Category:Deca]]
[[Category:Enneadecal]]
[[Category:Ennealimmal]]
[[Category:Gamera]]
[[Category:Mitonic]]
[[Category:Octoid]]
[[Category:Parakleismic]]
[[Category:Quincy]]
[[Category:Supermajor]]
Retrieved from "https://en.xen.wiki/w/Ragismic_microtemperaments"