Breedsmic 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 page discusses miscellaneous [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[breedsma]], {{monzo| -5 -1 -2 4 }} = 2401/2400. This is the amount by which two [[49/40]] intervals exceed [[3/2]], and by which two [[60/49]] intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system ([[12edo]], for example) which does not possess a neutral third cannot be tempering out the breedsma.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-07-04 05:39:02 UTC</tt>.<br>
: The original revision id was <tt>151486757</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">Breedsmic temperaments are rank two temperaments tempering out the breedsma, 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.


It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.
The breedsma is also the amount by which four stacked [[10/7]] intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, [[25/24]]. We might note also that (49/40)(10/7) = 7/4 and (49/40)(10/7)<sup>2</sup> = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.


===Tertiaseptal===
Temperaments discussed elsewhere include:
Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.
* ''[[Decimal]]'' (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]
* ''[[Beatles]]'' (+64/63 or 686/675) → [[Archytas clan #Beatles|Archytas clan]]
* [[Squares]] (+81/80) → [[Meantone family #Squares|Meantone family]]
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Greenwoodmic temperaments #Greenwood|Greenwoodmic temperaments]]
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]]
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* [[Ennealimmal]] (+4375/4374) → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]]
* ''[[Quadritikleismic]]'' (+15625/15552) → [[Kleismic family #Quadritikleismic|Kleismic family]]
* [[Harry]] (+19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Sesquiquartififths]]'' (+32805/32768) → [[Schismatic family #Sesquiquartififths|Schismatic family]]
* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]


===Neptune===
== Hemififths ==
Neptune adds 48828125/48771072 = |-12 -5 11 -2&gt; to the list of commas, and may be described as the 68&amp;171 temperament. It has 10/7 or 7/5 as a generator. The generator chain goes merrily on, stacking one 10/7 over another. until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just.
{{Main| Hemififths }}


Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the 11-limit, where (7/5)^3 equates to 11/4. This may be described as &lt;&lt;40 22 21 -3 ...|| or 68&amp;103, and 171 can still be used as a tuning, with val &lt;171 271 397 480  591|.
Hemififths may be described as the {{nowrap| 41 & 58 }} temperament, tempering out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator; its [[ploidacot]] is dicot. [[99edo]] and [[140edo]] provides good tunings, and [[239edo]] an even better one; and other possible tunings are 160<sup>(1/25)</sup>, giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14<sup>(1/13)</sup>, giving just 7's. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17- and 24-note mos are suited. The full force of this highly accurate temperament can be found using the 41-note mos or even the 34-note 2mos{{clarify}}.


An article on Neptune as an analog of miracle can be found [[http://tech.groups.yahoo.com/group/tuning-math/message/6001|here]].
By adding [[243/242]] (which also means [[441/440]], [[540/539]] and [[896/891]]) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99edo is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.


===Harry===
[[Subgroup]]: 2.3.5.7
Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&amp;72 temperament, with wedgie &lt;&lt;12 34 20 26 -2 -49||. The period is half an octave, and the generator 21/20, with generator tunings of 9/130 or 14/202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.


Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is &lt;&lt;12 34 20 30 ...||.
[[Comma list]]: 2401/2400, 5120/5103


Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with &lt;&lt;12 34 20 30 52 ...|| as the octave wedgie. [[130edo]] is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}


===Quasiorwell===
: mapping generators: ~2, ~49/40
In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&gt;. It has a wedgie &lt;&lt;38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.


Adding 3025/3024 extends to the 11-limit and gives &lt;&lt;38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning.</pre></div>
[[Optimal tuning]]s:
<h4>Original HTML content:</h4>
* [[CTE]]: ~2 = 1200.0000, ~49/40 = 351.4464
<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;Breedsmic temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Breedsmic temperaments are rank two temperaments tempering out the breedsma, 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.&lt;br /&gt;
: [[error map]]: {{val| 0.0000 +0.9379 -0.1531 -0.0224 }}
&lt;br /&gt;
* [[POTE]]: ~2 = 1200.0000, ~49/40 = 351.4774
It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.&lt;br /&gt;
: error map: {{val| 0.0000 +0.9999 +0.6221 +0.0307 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Tertiaseptal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Tertiaseptal&lt;/h3&gt;
[[Minimax tuning]]:
Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt; makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.&lt;br /&gt;
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
&lt;br /&gt;
: {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x--Neptune"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Neptune&lt;/h3&gt;
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
Neptune adds 48828125/48771072 = |-12 -5 11 -2&amp;gt; to the list of commas, and may be described as the 68&amp;amp;171 temperament. It has 10/7 or 7/5 as a generator. The generator chain goes merrily on, stacking one 10/7 over another. until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt; makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just.&lt;br /&gt;
 
&lt;br /&gt;
[[Algebraic generator]]: (2 + sqrt(2))/2
Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the 11-limit, where (7/5)^3 equates to 11/4. This may be described as &amp;lt;&amp;lt;40 22 21 -3 ...|| or 68&amp;amp;103, and 171 can still be used as a tuning, with val &amp;lt;171 271 397 480  591|.&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338 }}
An article on Neptune as an analog of miracle can be found &lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/6001" rel="nofollow"&gt;here&lt;/a&gt;.&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Smith): 0.022243
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x--Harry"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Harry&lt;/h3&gt;
 
Harry adds cataharry, 19683/19600, to the set of commas. It may be described as the 58&amp;amp;72 temperament, with wedgie &amp;lt;&amp;lt;12 34 20 26 -2 -49||. The period is half an octave, and the generator 21/20, with generator tunings of 9/130 or 14/202 being good choices. MOS of size 14, 16, 30, 44 or 58 are among the scale choices.&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
Harry becomes much more interesting as we move to the 11-limit, where we can add 243/242, 441/440 and 540/539 to the set of commas. 130 and especially 202 still make for good tuning choices, and the octave part of the wedgie is &amp;lt;&amp;lt;12 34 20 30 ...||.&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 243/242, 441/440, 896/891
Similar comments apply to the 13-limit, where we can add 351/350 and 364/363 to the commas, with &amp;lt;&amp;lt;12 34 20 30 52 ...|| as the octave wedgie. &lt;a class="wiki_link" href="/130edo"&gt;130edo&lt;/a&gt; is again a good tuning choice, but even better might be tuning 7s justly, which can be done via a generator of 83.1174 cents. 72 notes of harry gives plenty of room even for the 13-limit harmonies.&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x--Quasiorwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Quasiorwell&lt;/h3&gt;
 
In addition to 2401/2400, quasiorwell tempers out 29360128/29296875 = |22 -1 -10 1&amp;gt;. It has a wedgie &amp;lt;&amp;lt;38 -3 8 -93 -94 27||. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31&amp;amp;270 temperament, and as one might expect, 61/270 makes for an excellent tuning choice. Other possibilities are (7/2)^(1/8), giving just 7s, or 384^(1/38), giving pure fifths.&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* CTE: ~2 = 1200.0000, ~11/9 = 351.4289
Adding 3025/3024 extends to the 11-limit and gives &amp;lt;&amp;lt;38 -3 8 64 ...|| for the initial wedgie, and as expected, 270 remains an excellent tuning.&lt;/body&gt;&lt;/html&gt;</pre></div>
* POTE: ~2 = 1200.0000, ~11/9 = 351.5206
 
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}
 
Badness (Smith): 0.023498
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 196/195, 243/242, 364/363
 
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~11/9 = 351.4331
* POTE: ~2 = 1200.0000, ~11/9 = 351.5734
 
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}
 
Badness (Smith): 0.019090
 
=== Semihemi ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3388/3375, 5120/5103
 
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
 
: mapping generators: ~99/70, ~400/231
 
Optimal tunings:
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4722
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5047
 
{{Optimal ET sequence|legend=0| 58, 140, 198 }}
 
Badness (Smith): 0.042487
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 676/675, 847/845, 1716/1715
 
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}
 
Optimal tunings:  
* CTE: ~99/70 = 600.0000, ~49/40 = 351.4674
* POTE: ~99/70 = 600.0000, ~49/40 = 351.5019
 
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}
 
Badness (Smith): 0.021188
 
=== Quadrafifths ===
This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four.
 
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 5120/5103
 
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
 
: Mapping generators: ~2, ~243/220
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~243/220 = 175.7284
* POTE: ~2 = 1200.0000, ~243/220 = 175.7378
 
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}
 
Badness (Smith): 0.040170
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 847/845, 2401/2400, 3025/3024
 
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~72/65 = 175.7412
* POTE: ~2 = 1200.0000, ~72/65 = 175.7470
 
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}
 
Badness (Smith): 0.031144
 
== Tertiaseptal ==
{{Main| Tertiaseptal }}
 
Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the 31 &amp; 171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning, although 171edo - [[31edo]] = [[140edo]] also makes sense, and in very high limits 140edo + 171edo = [[311edo]] is especially notable. The 15- or 16-note mos can be used to explore no-threes harmony, and the 31-note mos gives plenty of room for those as well.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 65625/65536
 
{{Mapping|legend=1| 1 3 2 3 | 0 -22 5 -3 }}
 
: Mapping generators: ~2, ~256/245
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~256/245 = 77.191
 
{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
 
[[Badness]]: 0.012995
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 65625/65536
 
Mapping: {{mapping| 1 3 2 3 7 | 0 -22 5 -3 -55 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171, 202 }}
 
Badness: 0.035576
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 441/440, 625/624, 3584/3575
 
Mapping: {{mapping| 1 3 2 3 7 1 | 0 -22 5 -3 -55 42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109e, 140e, 171 }}
 
Badness: 0.036876
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
 
Mapping: {{mapping| 1 3 2 3 7 1 1 | 0 -22 5 -3 -55 42 48 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109eg, 140e, 171 }}
 
Badness: 0.027398
 
=== Tertia ===
Subgroup:2.3.5.7.11
 
Comma list: 385/384, 1331/1323, 1375/1372
 
Mapping: {{mapping| 1 3 2 3 5 | 0 -22 5 -3 -24 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 171e, 311e }}
 
Badness: 0.030171
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 385/384, 625/624, 1331/1323
 
Mapping: {{mapping| 1 3 2 3 5 1 | 0 -22 5 -3 -24 42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109, 140, 311e, 451ee }}
 
Badness: 0.028384
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
 
Mapping: {{mapping| 1 3 2 3 5 1 1 | 0 -22 5 -3 -24 42 48 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162
 
Optimal ET sequence: {{Optimal ET sequence| 31, 109g, 140, 311e, 451ee }}
 
Badness: 0.022416
 
=== Tertiaseptia ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 6250/6237, 65625/65536
 
Mapping: {{mapping| 1 3 2 3 -4 | 0 -22 5 -3 116 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde }}
 
Badness: 0.056926
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400
 
Mapping: {{mapping| 1 3 2 3 -4 1 | 0 -22 5 -3 116 42 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf }}
 
Badness: 0.027474
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 | 0 -22 5 -3 116 42 48 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.018773
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 | 0 -22 5 -3 116 42 48 -105 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg }}
 
Badness: 0.017653
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 | 0 -22 5 -3 116 42 48 -105 117 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168
 
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfgg }}
 
Badness: 0.015123
 
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 | 0 -22 5 -3 116 42 48 -105 117 60 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167
 
Optimal ET sequence: {{Optimal ET sequence| 140, 311, 762g, 1073g, 1384cfggj }}
 
Badness: 0.012181
 
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
 
Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 | 0 -22 5 -3 116 42 48 -105 117 60 -94 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.012311
 
==== 37-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
 
Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.010949
 
==== 41-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
 
Comma list: 595/594, 625/624, 697/696, 703/702, 714/713, 784/783, 820/819, 833/832, 875/874, 900/899, 931/930
 
Mapping: {{mapping| 1 3 2 3 -4 1 1 11 -3 1 11 0 6 | 0 -22 5 -3 116 42 48 -105 117 60 -94 81 -10 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169
 
Optimal ET sequence: {{Optimal ET sequence| 140, 171, 311 }}
 
Badness: 0.009825
 
=== Hemitert ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 65625/65536
 
Mapping: {{mapping| 1 3 2 3 6 | 0 -44 10 -6 -79 }}
 
: Mapping generators: ~2, ~45/44
 
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596
 
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 342 }}
 
Badness: 0.015633
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095
 
Mapping: {{mapping| 1 3 2 3 6 1 | 0 -44 10 -6 -79 84 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588
 
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 964f, 1275f, 1586cff }}
 
Badness: 0.033573
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095
 
Mapping: {{mapping| 1 3 2 3 6 1 1 | 0 -44 10 -6 -79 84 96 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589
 
Optimal ET sequence: {{Optimal ET sequence| 31, 280, 311, 653f, 964f }}
 
Badness: 0.025298
 
=== Semitert ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 9801/9800, 65625/65536
 
Mapping: {{mapping| 2 6 4 6 1 | 0 -22 5 -3 46 }}
 
: Mapping generators: ~99/70, ~256/245
 
Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193
 
Optimal ET sequence: {{Optimal ET sequence| 62e, 140, 202, 342 }}
 
Badness: 0.025790
 
== Quasiorwell ==
In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = {{monzo| 22 -1 -10 1 }}. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31 &amp; 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths.
 
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 29360128/29296875
 
{{Mapping|legend=1| 1 31 0 9 | 0 -38 3 -8 }}
 
: Mapping generators: ~2, ~875/512
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1024/875 = 271.107
 
{{Optimal ET sequence|legend=1| 31, 177, 208, 239, 270, 571, 841, 1111 }}
 
[[Badness]]: 0.035832
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 5632/5625
 
Mapping: {{mapping| 1 31 0 9 53 | 0 -38 3 -8 -64 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111
 
Optimal ET sequence: {{Optimal ET sequence| 31, 208, 239, 270 }}
 
Badness: 0.017540
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095
 
Mapping: {{mapping| 1 31 0 9 53 -59 | 0 -38 3 -8 -64 81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107
 
Optimal ET sequence: {{Optimal ET sequence| 31, 239, 270, 571, 841, 1111 }}
 
Badness: 0.017921
 
== Neominor ==
The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 177147/175616
 
{{Mapping|legend=1| 1 3 12 8 | 0 -6 -41 -22 }}
 
: Mapping generators: ~2, ~189/160
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~189/160 = 283.280
 
{{Optimal ET sequence|legend=1| 72, 161, 233, 305 }}
 
[[Badness]]: 0.088221
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 35937/35840
 
Mapping: {{mapping| 1 3 12 8 7 | 0 -6 -41 -22 -15 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276
 
Optimal ET sequence: {{Optimal ET sequence| 72, 161, 233, 305 }}
 
Badness: 0.027959
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 243/242, 364/363, 441/440
 
Mapping: {{mapping| 1 3 12 8 7 7 | 0 -6 -41 -22 -15 -14 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294
 
Optimal ET sequence: {{Optimal ET sequence| 72, 161f, 233f }}
 
Badness: 0.026942
 
== Emmthird ==
The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 14348907/14336000
 
{{Mapping|legend=1| 1 11 42 25 | 0 -14 -59 -33 }}
 
: Mapping generators: ~2, ~2187/1372
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2744/2187 = 392.988
 
{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
 
[[Badness]]: 0.016736
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 1792000/1771561
 
Mapping: {{mapping| 1 11 42 25 27 | 0 -14 -59 -33 -35 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991
 
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
 
Badness: 0.052358
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 364/363, 441/440, 2200/2197
 
Mapping: {{mapping| 1 11 42 25 27 38 | 0 -14 -59 -33 -35 -51 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989
 
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
 
Badness: 0.026974
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197
 
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985
 
Optimal ET sequence: {{Optimal ET sequence| 58, 113, 171 }}
 
Badness: 0.023205
 
== Quinmite ==
The generator for quinmite is quasi-tempered minor third [[25/21]], flatter than 6/5 by the starling comma, [[126/125]]. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth", given by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 1959552/1953125
 
{{Mapping|legend=1| 1 27 24 20 | 0 -34 -29 -23 }}
 
: Mapping generators: ~2, ~42/25
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 302.997
 
{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc }}
 
[[Badness]]: 0.037322
 
== Unthirds ==
Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[Temperament merging|join]] of two tuning systems well-known for their high accuracy in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.
 
The commas it tempers out include the [[breedsma]] (2401/2400), the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 68359375/68024448
 
{{Mapping|legend=1| 1 29 33 25 | 0 -42 -47 -34 }}
 
: Mapping generators: ~2, ~6125/3888
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3969/3125 = 416.717
 
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
 
[[Badness]]: 0.075253
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 4000/3993
 
Mapping: {{mapping| 1 29 33 25 25 | 0 -42 -47 -34 -33 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718
 
Optimal ET sequence: {{Optimal ET sequence| 72, 167, 239, 311 }}
 
Badness: 0.022926
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
 
Mapping: {{mapping| 1 29 33 25 25 99 | 0 -42 -47 -34 -33 -146 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716
 
Optimal ET sequence: {{Optimal ET sequence| 72, 239f, 311, 694, 1005c }}
 
Badness: 0.020888
 
== Newt ==
Newt has a generator of a neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]]. It can be described as the 41 & 270 temperament, and extends naturally to the no-17 19-limit, a.k.a. '''neonewt'''. [[270edo]] and [[311edo]] are obvious tuning choices, but [[581edo]] and especially [[851edo]] work much better.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 33554432/33480783
 
{{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}
 
: mapping generators: ~2, ~49/40
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 351.113
 
{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}
 
[[Badness]]: 0.041878
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 19712/19683
 
Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115
 
Optimal ET sequence: {{Optimal ET sequence| 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972 }}
 
Badness: 0.019461
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095
 
Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
 
Optimal ET sequence: {{Optimal ET sequence| 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b }}
 
Badness: 0.013830
 
=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400
 
Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117
 
Optimal ET sequence: {{Optimal ET sequence| 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb }}
 
== Septidiasemi ==
{{Main| Septidiasemi }}
 
Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 2152828125/2147483648
 
{{Mapping|legend=1| 1 25 -31 -8 | 0 -26 37 12 }}
 
: Mapping generators: ~2, ~28/15
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 119.297
 
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd }}
 
[[Badness]]: 0.044115
 
=== Sedia ===
The ''sedia'' temperament (10&amp;161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.
 
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 939524096/935859375
 
Mapping: {{mapping| 1 25 -31 -8 62 | 0 -26 37 12 -65 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279
 
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332 }}
 
Badness: 0.090687
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 441/440, 2200/2197, 3584/3575
 
Mapping: {{mapping| 1 25 -31 -8 62 1 | 0 -26 37 12 -65 3 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281
 
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 835eeff }}
 
Badness: 0.045773
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
 
Mapping: {{mapping| 1 25 -31 -8 62 1 23 | 0 -26 37 12 -65 3 -21 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281
 
Optimal ET sequence: {{Optimal ET sequence| 10, 151, 161, 171, 332, 503ef, 835eeff }}
 
Badness: 0.027322
 
== Maviloid ==
{{See also| Ragismic microtemperaments #Parakleismic }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 1224440064/1220703125
 
{{Mapping|legend=1| 1 31 34 26 | 0 -52 -56 -41 }}
 
: Mapping generators: ~2, ~1296/875
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1296/875 = 678.810
 
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
 
[[Badness]]: 0.057632
 
== Subneutral ==
{{See also| Luna family }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 274877906944/274658203125
 
{{Mapping|legend=1| 1 19 0 6 | 0 -60 8 -11 }}
 
: Mapping generators: ~2, ~57344/46875
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~57344/46875 = 348.301
 
{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}
 
[[Badness]]: 0.045792
 
== Osiris ==
{{See also| Metric microtemperaments #Geb }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 31381059609/31360000000
 
{{Mapping|legend=1| 1 13 33 21 | 0 -32 -86 -51 }}
 
: Mapping generators: ~2, ~2800/2187
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~2800/2187 = 428.066
 
{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}
 
[[Badness]]: 0.028307
 
== Gorgik ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 28672/28125
 
{{Mapping|legend=1| 1 5 1 3 | 0 -18 7 -1 }}
 
: Mapping generators: ~2, ~8/7
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 227.512
 
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
 
[[Badness]]: 0.158384
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 2401/2400, 2560/2541
 
Mapping: {{mapping| 1 5 1 3 1 | 0 -18 7 -1 13 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500
 
Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
 
Badness: 0.059260
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 196/195, 364/363, 512/507
 
Mapping: {{mapping| 1 5 1 3 1 2 | 0 -18 7 -1 13 9 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493
 
Optimal ET sequence: {{Optimal ET sequence| 21, 37, 58, 153bcef, 211bccdeeff }}
 
Badness: 0.032205
 
== Fibo ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 341796875/339738624
 
{{Mapping|legend=1| 1 19 8 10 | 0 -46 -15 -19 }}
 
: Mapping generators: ~2, ~125/96
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 454.310
 
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}
 
Badness: 0.100511
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1375/1372, 43923/43750
 
Mapping: {{mapping| 1 19 8 10 8 | 0 -46 -15 -19 -12 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318
 
Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}
 
Badness: 0.056514
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 385/384, 625/624, 847/845, 1375/1372
 
Mapping: {{mapping| 1 19 8 10 8 9 | 0 -46 -15 -19 -12 -14 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316
 
Optimal ET sequence: {{Optimal ET sequence| 37, 66b, 103, 140, 243e }}
 
Badness: 0.027429
 
== Mintone ==
In addition to 2401/2400, mintone tempers out 177147/175000 = {{monzo| -3 11 -5 -1 }} in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58 &amp; 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 177147/175000
 
{{Mapping|legend=1| 1 5 9 7 | 0 -22 -43 -27 }}
 
: Mapping generators: ~2, ~10/9
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 186.343
 
{{Optimal ET sequence|legend=1| 45, 58, 103, 161, 586b, 747bc, 908bbc }}
 
[[Badness]]: 0.125672
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 43923/43750
 
Mapping: {{mapping| 1 5 9 7 12 | 0 -22 -43 -27 -55 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345
 
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586b, 747bc }}
 
Badness: 0.039962
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 351/350, 441/440, 847/845
 
Mapping: {{mapping| 1 5 9 7 12 11 | 0 -22 -43 -27 -55 -47 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347
 
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}
 
Badness: 0.021849
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
 
Mapping: {{mapping| 1 5 9 7 12 11 3 | 0 -22 -43 -27 -55 -47 7 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348
 
Optimal ET sequence: {{Optimal ET sequence| 58, 103, 161, 425b, 586bf }}
 
Badness: 0.020295
 
== Catafourth ==
{{See also| Sensipent family }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 78732/78125
 
{{Mapping|legend=1| 1 13 17 13 | 0 -28 -36 -25 }}
 
: Mapping generators: ~2, ~250/189
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~250/189 = 489.235
 
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
 
Badness: 0.079579
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 78408/78125
 
Mapping: {{mapping| 1 13 17 13 32 | 0 -28 -36 -25 -70 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252
 
Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363, 493e, 856be }}
 
Badness: 0.036785
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 351/350, 441/440, 10985/10976
 
Mapping: {{mapping| 1 13 17 13 32 9 | 0 -28 -36 -25 -70 -13 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256
 
Optimal ET sequence: {{Optimal ET sequence| 103, 130, 233, 363 }}
 
Badness: 0.021694
 
== Cotritone ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 390625/387072
 
{{Mapping|legend=1| 1 17 9 10 | 0 -30 -13 -14 }}
 
: Mappping generators: ~2, ~10/7
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 583.385
 
{{Optimal ET sequence|legend=1| 35, 37, 72, 109, 181, 253 }}
 
[[Badness]]: 0.098322
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1375/1372, 4000/3993
 
Mapping: {{mapping| 1 17 9 10 5 | 0 -30 -13 -14 -3 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
 
Optimal ET sequence: {{Optimal ET sequence| 35, 37, 72, 109, 181, 253 }}
 
Badness: 0.032225
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 364/363, 385/384, 625/624
 
Mapping: {{mapping| 1 17 9 10 5 15 | 0 -30 -13 -14 -3 -22 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387
 
Optimal ET sequence: {{Optimal ET sequence| 37, 72, 109, 181f }}
 
Badness: 0.028683
 
== Quasimoha ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Quasimoha]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 3645/3584
 
{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}
 
: Mapping generators: ~2, ~49/40
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 348.603
 
{{Optimal ET sequence|legend=1| 31, 117c, 148bc, 179bc }}
 
[[Badness]]: 0.110820
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 1815/1792
 
Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639
 
Optimal ET sequence: {{Optimal ET sequence| 31, 86ce, 117ce, 148bce }}
 
Badness: 0.046181
 
== Lockerbie ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''
Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[120/77]] or [[77/60]]. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.
 
The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament is defined as 103 & 270, hence the name. The name is proposed by Eliora, who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.
 
Lockerbie also has a unique extension that adds the 41st harmonic such that the generator below 600 cents is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}
 
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
 
: Mapping generators: ~2, ~3828125/2985984
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1071
: [[error map]]: {{val| 0.0000 -0.0270 +0.1502 -0.1120 }}
* [[CWE]]: ~2 = 1200.0000, ~3828125/2985984 = 431.1072
: error map: {{val| 0.0000 -0.0205 +0.1547 -0.1081 }}
 
{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913 }}
 
[[Badness]] (Smith): 0.0597
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 766656/765625
 
Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~77/60 = 431.1082
* CWE: ~2 = 1200.0000, ~77/60 = 431.1078
 
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}
 
Badness (Smith): 0.0262
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224
 
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000, ~77/60 = 431.1085
* CWE: ~2 = 1200.0000, ~77/60 = 431.1069
 
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}
 
Badness (Smith): 0.0160
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
 
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~77/60 = 431.107
* CWE: ~2 = 1200.000, ~77/60 = 431.108
 
{{Optimal ET sequence|legend=0| 103, 167, 270 }}
 
Badness (Smith): 0.0210
 
=== 2.3.5.7.11.13.17.41 subgroup ===
Subgroup: 2.3.5.7.11.13.17.41
 
Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
 
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 5 | 0 74 51 44 82 27 42 1 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~41/32 = 431.107
* CWE: ~2 = 1200.000, ~41/32 = 431.111
 
{{Optimal ET sequence|legend=0| 103, 167, 270 }}
 
== Hemigoldis ==
: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''
 
Though fairly complex in the [[7-limit]], hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~[[21/19]] to add prime 19 or perhaps more accurately ~[[31/28]] to add prime 7, or even simply as ~[[32/29]] to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again [[89edo]] is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 549755813888/533935546875
 
{{Mapping|legend=1| 1 21 -9 2 | 0 -24 14 1 }}
 
: mapping generators: ~2, ~7/4
 
[[Optimal tuning]] ([[CWE]]): ~2 = 1200.000, ~7/4 = 970.690
 
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}
 
[[Badness]] (Sintel): 4.40
 
== Surmarvelpyth ==
''Surmarvelpyth'' is named for the generator fifth, 675/448 being 225/224 (marvel comma) sharp of 3/2. It can be described as the 311 & 431 temperament, starting with the 7-limit to the 19-limit.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, {{monzo| 93 -32 -17 -1 }}
 
{{Mapping|legend=1| 1 43 -74 -25 | 0 -70 129 47 }}
 
: Mapping generators: ~2, ~675/448
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~675/448 = 709.9719
 
{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }}
 
[[Badness]]: 0.202249
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 820125/819896, 2097152/2096325
 
Mapping: {{mapping| 1 43 -74 -25 36 | 0 -70 129 47 -55 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9720
 
Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795 }}
 
Badness: 0.052308
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167
 
Mapping: {{mapping| 1 43 -74 -25 36 25 | 0 -70 129 47 -55 -36 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9723
 
Optimal ET sequence: {{Optimal ET sequence| 120, 191, 311, 742, 1053, 1795f }}
 
Badness: 0.032503
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619
 
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 | 0 -70 129 47 -55 -36 181 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722
 
Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}
 
Badness: 0.020995
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984
 
Mapping: {{mapping| 1 43 -74 -25 36 25 -103 -49 | 0 -70 129 47 -55 -36 181 90 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722
 
Optimal ET sequence: {{Optimal ET sequence| 120g, 191g, 311, 431, 742, 1795f }}
 
Badness: 0.013771
 
== Notes ==
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Rank 2]]

Latest revision as of 20:57, 18 August 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This page discusses miscellaneous rank-2 temperaments tempering out the breedsma, [-5 -1 -2 4⟩ = 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.

The breedsma is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, 25/24. We might note also that (49/40)(10/7) = 7/4 and (49/40)(10/7)2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.

Temperaments discussed elsewhere include:

Hemififths

Main article: Hemififths

Hemififths may be described as the 41 & 58 temperament, tempering out 5120/5103, the hemifamity comma, and 10976/10935, hemimage. It has a neutral third as a generator; its ploidacot is dicot. 99edo and 140edo provides good tunings, and 239edo an even better one; and other possible tunings are 160(1/25), giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14(1/13), giving just 7's. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17- and 24-note mos are suited. The full force of this highly accurate temperament can be found using the 41-note mos or even the 34-note 2mos[clarification needed].

By adding 243/242 (which also means 441/440, 540/539 and 896/891) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99edo is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding 144/143 brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be 16/13. 99 remains a good tuning choice.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 5120/5103

Mapping: [⟨1 1 -5 -1], ⟨0 2 25 13]]

mapping generators: ~2, ~49/40

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~49/40 = 351.4464
error map: ⟨0.0000 +0.9379 -0.1531 -0.0224]
  • POTE: ~2 = 1200.0000, ~49/40 = 351.4774
error map: ⟨0.0000 +0.9999 +0.6221 +0.0307]

Minimax tuning:

[[1 0 0 0⟩, [7/5 0 2/25 0⟩, [0 0 1 0⟩, [8/5 0 13/25 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5

Algebraic generator: (2 + sqrt(2))/2

Optimal ET sequence: 41, 58, 99, 239, 338

Badness (Smith): 0.022243

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 896/891

Mapping: [⟨1 1 -5 -1 2], ⟨0 2 25 13 5]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~11/9 = 351.4289
  • POTE: ~2 = 1200.0000, ~11/9 = 351.5206

Optimal ET sequence: 17c, 41, 58, 99e

Badness (Smith): 0.023498

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 243/242, 364/363

Mapping: [⟨1 1 -5 -1 2 4], ⟨0 2 25 13 5 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~11/9 = 351.4331
  • POTE: ~2 = 1200.0000, ~11/9 = 351.5734

Optimal ET sequence: 17c, 41, 58, 99ef, 157eff

Badness (Smith): 0.019090

Semihemi

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3388/3375, 5120/5103

Mapping: [⟨2 0 -35 -15 -47], ⟨0 2 25 13 34]]

mapping generators: ~99/70, ~400/231

Optimal tunings:

  • CTE: ~99/70 = 600.0000, ~49/40 = 351.4722
  • POTE: ~99/70 = 600.0000, ~49/40 = 351.5047

Optimal ET sequence: 58, 140, 198

Badness (Smith): 0.042487

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845, 1716/1715

Mapping: [⟨2 0 -35 -15 -47 -37], ⟨0 2 25 13 34 28]]

Optimal tunings:

  • CTE: ~99/70 = 600.0000, ~49/40 = 351.4674
  • POTE: ~99/70 = 600.0000, ~49/40 = 351.5019

Optimal ET sequence: 58, 140, 198, 536f

Badness (Smith): 0.021188

Quadrafifths

This has been logged as semihemififths in Graham Breed's temperament finder, but quadrafifths arguably makes more sense because it straight-up splits the fifth in four.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 5120/5103

Mapping: [⟨1 1 -5 -1 8], ⟨0 4 50 26 -31]]

Mapping generators: ~2, ~243/220

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~243/220 = 175.7284
  • POTE: ~2 = 1200.0000, ~243/220 = 175.7378

Optimal ET sequence: 41, 157, 198, 239, 676b, 915be

Badness (Smith): 0.040170

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 2401/2400, 3025/3024

Mapping: [⟨1 1 -5 -1 8 10], ⟨0 4 50 26 -31 -43]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~72/65 = 175.7412
  • POTE: ~2 = 1200.0000, ~72/65 = 175.7470

Optimal ET sequence: 41, 157, 198, 437f, 635bcff

Badness (Smith): 0.031144

Tertiaseptal

Main article: Tertiaseptal

Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152, the rainy comma. It can be described as the 31 & 171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. 171edo makes for an excellent tuning, although 171edo - 31edo = 140edo also makes sense, and in very high limits 140edo + 171edo = 311edo is especially notable. The 15- or 16-note mos can be used to explore no-threes harmony, and the 31-note mos gives plenty of room for those as well.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 65625/65536

Mapping: [⟨1 3 2 3], ⟨0 -22 5 -3]]

Mapping generators: ~2, ~256/245

Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.191

Optimal ET sequence: 31, 109, 140, 171

Badness: 0.012995

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 65625/65536

Mapping: [⟨1 3 2 3 7], ⟨0 -22 5 -3 -55]]

Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.227

Optimal ET sequence: 31, 109e, 140e, 171, 202

Badness: 0.035576

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 441/440, 625/624, 3584/3575

Mapping: [⟨1 3 2 3 7 1], ⟨0 -22 5 -3 -55 42]]

Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.203

Optimal ET sequence: 31, 109e, 140e, 171

Badness: 0.036876

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575

Mapping: [⟨1 3 2 3 7 1 1], ⟨0 -22 5 -3 -55 42 48]]

Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.201

Optimal ET sequence: 31, 109eg, 140e, 171

Badness: 0.027398

Tertia

Subgroup:2.3.5.7.11

Comma list: 385/384, 1331/1323, 1375/1372

Mapping: [⟨1 3 2 3 5], ⟨0 -22 5 -3 -24]]

Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.173

Optimal ET sequence: 31, 109, 140, 171e, 311e

Badness: 0.030171

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 625/624, 1331/1323

Mapping: [⟨1 3 2 3 5 1], ⟨0 -22 5 -3 -24 42]]

Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.158

Optimal ET sequence: 31, 109, 140, 311e, 451ee

Badness: 0.028384

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 385/384, 561/560, 625/624, 715/714

Mapping: [⟨1 3 2 3 5 1 1], ⟨0 -22 5 -3 -24 42 48]]

Optimal tuning (POTE): ~2 = 1\1, ~22/21 = 77.162

Optimal ET sequence: 31, 109g, 140, 311e, 451ee

Badness: 0.022416

Tertiaseptia

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 6250/6237, 65625/65536

Mapping: [⟨1 3 2 3 -4], ⟨0 -22 5 -3 116]]

Optimal tuning (POTE): ~2 = 1\1, ~256/245 = 77.169

Optimal ET sequence: 140, 171, 311, 1695c, 2006bcd, 2317bcd, 2628bccde, 2939bccde, 3250bccde

Badness: 0.056926

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400

Mapping: [⟨1 3 2 3 -4 1], ⟨0 -22 5 -3 116 42]]

Optimal tuning (POTE): ~2 = 1\1, ~117/112 = 77.168

Optimal ET sequence: 140, 171, 311, 1073, 1384cf, 1695cf, 2006bcdf

Badness: 0.027474

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197

Mapping: [⟨1 3 2 3 -4 1 1], ⟨0 -22 5 -3 116 42 48]]

Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169

Optimal ET sequence: 140, 171, 311

Badness: 0.018773

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 625/624, 833/832, 1156/1155, 1216/1215, 2200/2197

Mapping: [⟨1 3 2 3 -4 1 1 11], ⟨0 -22 5 -3 116 42 48 -105]]

Optimal tuning (POTE): ~2 = 1\1, ~68/65 = 77.169

Optimal ET sequence: 140, 171, 311, 1384cfgg, 1695cfgg, 2006bcdfgg

Badness: 0.017653

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215

Mapping: [⟨1 3 2 3 -4 1 1 11 -3], ⟨0 -22 5 -3 116 42 48 -105 117]]

Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.168

Optimal ET sequence: 140, 311, 762g, 1073g, 1384cfgg

Badness: 0.015123

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155

Mapping: [⟨1 3 2 3 -4 1 1 11 -3 1], ⟨0 -22 5 -3 116 42 48 -105 117 60]]

Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.167

Optimal ET sequence: 140, 311, 762g, 1073g, 1384cfggj

Badness: 0.012181

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014

Mapping: [⟨1 3 2 3 -4 1 1 11 -3 1 11], ⟨0 -22 5 -3 116 42 48 -105 117 60 -94]]

Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169

Optimal ET sequence: 140, 171, 311

Badness: 0.012311

37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37

Comma list: 595/594, 625/624, 703/702, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014

Mapping: [⟨1 3 2 3 -4 1 1 11 -3 1 11 0], ⟨0 -22 5 -3 116 42 48 -105 117 60 -94 81]]

Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.170

Optimal ET sequence: 140, 171, 311

Badness: 0.010949

41-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41

Comma list: 595/594, 625/624, 697/696, 703/702, 714/713, 784/783, 820/819, 833/832, 875/874, 900/899, 931/930

Mapping: [⟨1 3 2 3 -4 1 1 11 -3 1 11 0 6], ⟨0 -22 5 -3 116 42 48 -105 117 60 -94 81 -10]]

Optimal tuning (POTE): ~2 = 1\1, ~23/22 = 77.169

Optimal ET sequence: 140, 171, 311

Badness: 0.009825

Hemitert

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 65625/65536

Mapping: [⟨1 3 2 3 6], ⟨0 -44 10 -6 -79]]

Mapping generators: ~2, ~45/44

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.596

Optimal ET sequence: 31, 280, 311, 342

Badness: 0.015633

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095

Mapping: [⟨1 3 2 3 6 1], ⟨0 -44 10 -6 -79 84]]

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.588

Optimal ET sequence: 31, 280, 311, 964f, 1275f, 1586cff

Badness: 0.033573

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095

Mapping: [⟨1 3 2 3 6 1 1], ⟨0 -44 10 -6 -79 84 96]]

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.589

Optimal ET sequence: 31, 280, 311, 653f, 964f

Badness: 0.025298

Semitert

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 65625/65536

Mapping: [⟨2 6 4 6 1], ⟨0 -22 5 -3 46]]

Mapping generators: ~99/70, ~256/245

Optimal tuning (POTE): ~99/70 = 1\2, ~256/245 = 77.193

Optimal ET sequence: 62e, 140, 202, 342

Badness: 0.025790

Quasiorwell

In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 = [22 -1 -10 1⟩. It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31 & 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)1/8, giving just 7's, or 3841/38, giving pure fifths.

Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 29360128/29296875

Mapping: [⟨1 31 0 9], ⟨0 -38 3 -8]]

Mapping generators: ~2, ~875/512

Optimal tuning (POTE): ~2 = 1\1, ~1024/875 = 271.107

Optimal ET sequence: 31, 177, 208, 239, 270, 571, 841, 1111

Badness: 0.035832

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 5632/5625

Mapping: [⟨1 31 0 9 53], ⟨0 -38 3 -8 -64]]

Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.111

Optimal ET sequence: 31, 208, 239, 270

Badness: 0.017540

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095

Mapping: [⟨1 31 0 9 53 -59], ⟨0 -38 3 -8 -64 81]]

Optimal tuning (POTE): ~2 = 1\1, ~90/77 = 271.107

Optimal ET sequence: 31, 239, 270, 571, 841, 1111

Badness: 0.017921

Neominor

The generator for neominor temperament is tridecimal minor third 13/11, also known as Neo-gothic minor third.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 177147/175616

Mapping: [⟨1 3 12 8], ⟨0 -6 -41 -22]]

Mapping generators: ~2, ~189/160

Optimal tuning (POTE): ~2 = 1\1, ~189/160 = 283.280

Optimal ET sequence: 72, 161, 233, 305

Badness: 0.088221

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 35937/35840

Mapping: [⟨1 3 12 8 7], ⟨0 -6 -41 -22 -15]]

Optimal tuning (POTE): ~2 = 1\1, ~33/28 = 283.276

Optimal ET sequence: 72, 161, 233, 305

Badness: 0.027959

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 243/242, 364/363, 441/440

Mapping: [⟨1 3 12 8 7 7], ⟨0 -6 -41 -22 -15 -14]]

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 283.294

Optimal ET sequence: 72, 161f, 233f

Badness: 0.026942

Emmthird

The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 14348907/14336000

Mapping: [⟨1 11 42 25], ⟨0 -14 -59 -33]]

Mapping generators: ~2, ~2187/1372

Optimal tuning (POTE): ~2 = 1\1, ~2744/2187 = 392.988

Optimal ET sequence: 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d

Badness: 0.016736

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 1792000/1771561

Mapping: [⟨1 11 42 25 27], ⟨0 -14 -59 -33 -35]]

Optimal tuning (POTE): ~2 = 1\1, ~1372/1089 = 392.991

Optimal ET sequence: 58, 113, 171

Badness: 0.052358

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 2200/2197

Mapping: [⟨1 11 42 25 27 38], ⟨0 -14 -59 -33 -35 -51]]

Optimal tuning (POTE): ~2 = 1\1, ~180/143 = 392.989

Optimal ET sequence: 58, 113, 171

Badness: 0.026974

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197

Mapping: [⟨1 -3 -17 -8 -8 -13 9], ⟨0 14 59 33 35 51 -15]]

Optimal tuning (POTE): ~2 = 1\1, ~64/51 = 392.985

Optimal ET sequence: 58, 113, 171

Badness: 0.023205

Quinmite

The generator for quinmite is quasi-tempered minor third 25/21, flatter than 6/5 by the starling comma, 126/125. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth", given by Petr Pařízek in 2011[1][2].

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1959552/1953125

Mapping: [⟨1 27 24 20], ⟨0 -34 -29 -23]]

Mapping generators: ~2, ~42/25

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 302.997

Optimal ET sequence: 99, 202, 301, 400, 701, 1101c, 1802c, 2903cc

Badness: 0.037322

Unthirds

Despite the complexity of its mapping, unthirds is an important temperament to the structure of the 11-limit; this is hinted at by unthirds' representation as the 72 & 311 temperament, the join of two tuning systems well-known for their high accuracy in the 11-limit and 41-limit respectively. It is generated by the interval of 14/11 (undecimal major third, hence the name) tuned less than a cent flat, and the 23-note MOS this interval generates serves as a well temperament of, of all things, 23edo. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.

The commas it tempers out include the breedsma (2401/2400), the lehmerisma (3025/3024), the pine comma (4000/3993), the unisquary comma (12005/11979), the argyria (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a restriction of the temperament to the 2.5/3.7/3.11/3 fractional subgroup that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with sqrtphi (whose generator is tuned flat of 72edo's).

Subgroup: 2.3.5.7

Comma list: 2401/2400, 68359375/68024448

Mapping: [⟨1 29 33 25], ⟨0 -42 -47 -34]]

Mapping generators: ~2, ~6125/3888

Optimal tuning (POTE): ~2 = 1\1, ~3969/3125 = 416.717

Optimal ET sequence: 72, 167, 239, 311, 694, 1005c

Badness: 0.075253

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 4000/3993

Mapping: [⟨1 29 33 25 25], ⟨0 -42 -47 -34 -33]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.718

Optimal ET sequence: 72, 167, 239, 311

Badness: 0.022926

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400

Mapping: [⟨1 29 33 25 25 99], ⟨0 -42 -47 -34 -33 -146]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.716

Optimal ET sequence: 72, 239f, 311, 694, 1005c

Badness: 0.020888

Newt

Newt has a generator of a neutral third (0.2 cents flat of 49/40) and tempers out the garischisma. It can be described as the 41 & 270 temperament, and extends naturally to the no-17 19-limit, a.k.a. neonewt. 270edo and 311edo are obvious tuning choices, but 581edo and especially 851edo work much better.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 33554432/33480783

Mapping: [⟨1 1 19 11], ⟨0 2 -57 -28]]

mapping generators: ~2, ~49/40

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.113

Optimal ET sequence: 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201

Badness: 0.041878

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 19712/19683

Mapping: [⟨1 1 19 11 -10], ⟨0 2 -57 -28 46]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.115

Optimal ET sequence: 41, 147ce, 188, 229, 270, 581, 851, 1121, 1972

Badness: 0.019461

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095

Mapping: [⟨1 1 19 11 -10 -20], ⟨0 2 -57 -28 46 81]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117

Optimal ET sequence: 41, 147cef, 188f, 229, 270, 581, 851, 2283b, 3134b

Badness: 0.013830

2.3.5.7.11.13.19 subgroup (neonewt)

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400

Mapping: [⟨1 1 19 11 -10 -20 18], ⟨0 2 -57 -28 46 81 -47]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 351.117

Optimal ET sequence: 41, 147cefh, 188f, 229, 270, 581, 851, 3134b, 3985b, 4836bb

Septidiasemi

Main article: Septidiasemi

Aside from 2401/2400, septidiasemi tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of 15/14). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 2152828125/2147483648

Mapping: [⟨1 25 -31 -8], ⟨0 -26 37 12]]

Mapping generators: ~2, ~28/15

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.297

Optimal ET sequence: 10, 151, 161, 171, 3581bcdd, 3752bcdd, 3923bcdd, 4094bcdd, 4265bccdd, 4436bccdd, 4607bccdd

Badness: 0.044115

Sedia

The sedia temperament (10&161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 939524096/935859375

Mapping: [⟨1 25 -31 -8 62], ⟨0 -26 37 12 -65]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.279

Optimal ET sequence: 10, 151, 161, 171, 332

Badness: 0.090687

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 441/440, 2200/2197, 3584/3575

Mapping: [⟨1 25 -31 -8 62 1], ⟨0 -26 37 12 -65 3]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281

Optimal ET sequence: 10, 151, 161, 171, 332, 835eeff

Badness: 0.045773

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575

Mapping: [⟨1 25 -31 -8 62 1 23], ⟨0 -26 37 12 -65 3 -21]]

Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 119.281

Optimal ET sequence: 10, 151, 161, 171, 332, 503ef, 835eeff

Badness: 0.027322

Maviloid

See also: Ragismic microtemperaments § Parakleismic

Subgroup: 2.3.5.7

Comma list: 2401/2400, 1224440064/1220703125

Mapping: [⟨1 31 34 26], ⟨0 -52 -56 -41]]

Mapping generators: ~2, ~1296/875

Optimal tuning (POTE): ~2 = 1\1, ~1296/875 = 678.810

Optimal ET sequence: 76, 99, 274, 373, 472, 571, 1043, 1614

Badness: 0.057632

Subneutral

See also: Luna family

Subgroup: 2.3.5.7

Comma list: 2401/2400, 274877906944/274658203125

Mapping: [⟨1 19 0 6], ⟨0 -60 8 -11]]

Mapping generators: ~2, ~57344/46875

Optimal tuning (POTE): ~2 = 1\1, ~57344/46875 = 348.301

Optimal ET sequence: 31, …, 348, 379, 410, 441, 1354, 1795, 2236

Badness: 0.045792

Osiris

See also: Metric microtemperaments § Geb

Subgroup: 2.3.5.7

Comma list: 2401/2400, 31381059609/31360000000

Mapping: [⟨1 13 33 21], ⟨0 -32 -86 -51]]

Mapping generators: ~2, ~2800/2187

Optimal tuning (POTE): ~2 = 1\1, ~2800/2187 = 428.066

Optimal ET sequence: 157, 171, 1012, 1183, 1354, 1525, 1696

Badness: 0.028307

Gorgik

Subgroup: 2.3.5.7

Comma list: 2401/2400, 28672/28125

Mapping: [⟨1 5 1 3], ⟨0 -18 7 -1]]

Mapping generators: ~2, ~8/7

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.512

Optimal ET sequence: 21, 37, 58, 153bc, 211bccd, 269bccd

Badness: 0.158384

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 2401/2400, 2560/2541

Mapping: [⟨1 5 1 3 1], ⟨0 -18 7 -1 13]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.500

Optimal ET sequence: 21, 37, 58, 153bce, 211bccdee, 269bccdee

Badness: 0.059260

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 196/195, 364/363, 512/507

Mapping: [⟨1 5 1 3 1 2], ⟨0 -18 7 -1 13 9]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 227.493

Optimal ET sequence: 21, 37, 58, 153bcef, 211bccdeeff

Badness: 0.032205

Fibo

Subgroup: 2.3.5.7

Comma list: 2401/2400, 341796875/339738624

Mapping: [⟨1 19 8 10], ⟨0 -46 -15 -19]]

Mapping generators: ~2, ~125/96

Optimal tuning (POTE): ~2 = 1\1, ~125/96 = 454.310

Optimal ET sequence: 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd

Badness: 0.100511

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 43923/43750

Mapping: [⟨1 19 8 10 8], ⟨0 -46 -15 -19 -12]]

Optimal tuning (POTE): ~2 = 1\1, ~100/77 = 454.318

Optimal ET sequence: 37, 66b, 103, 140, 243e

Badness: 0.056514

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624, 847/845, 1375/1372

Mapping: [⟨1 19 8 10 8 9], ⟨0 -46 -15 -19 -12 -14]]

Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 454.316

Optimal ET sequence: 37, 66b, 103, 140, 243e

Badness: 0.027429

Mintone

In addition to 2401/2400, mintone tempers out 177147/175000 = [-3 11 -5 -1⟩ in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58 & 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 177147/175000

Mapping: [⟨1 5 9 7], ⟨0 -22 -43 -27]]

Mapping generators: ~2, ~10/9

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.343

Optimal ET sequence: 45, 58, 103, 161, 586b, 747bc, 908bbc

Badness: 0.125672

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 43923/43750

Mapping: [⟨1 5 9 7 12], ⟨0 -22 -43 -27 -55]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.345

Optimal ET sequence: 58, 103, 161, 425b, 586b, 747bc

Badness: 0.039962

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 441/440, 847/845

Mapping: [⟨1 5 9 7 12 11], ⟨0 -22 -43 -27 -55 -47]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.347

Optimal ET sequence: 58, 103, 161, 425b, 586bf

Badness: 0.021849

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 351/350, 441/440, 561/560, 847/845

Mapping: [⟨1 5 9 7 12 11 3], ⟨0 -22 -43 -27 -55 -47 7]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 186.348

Optimal ET sequence: 58, 103, 161, 425b, 586bf

Badness: 0.020295

Catafourth

See also: Sensipent family

Subgroup: 2.3.5.7

Comma list: 2401/2400, 78732/78125

Mapping: [⟨1 13 17 13], ⟨0 -28 -36 -25]]

Mapping generators: ~2, ~250/189

Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.235

Optimal ET sequence: 27, 76, 103, 130

Badness: 0.079579

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 78408/78125

Mapping: [⟨1 13 17 13 32], ⟨0 -28 -36 -25 -70]]

Optimal tuning (POTE): ~2 = 1\1, ~250/189 = 489.252

Optimal ET sequence: 103, 130, 233, 363, 493e, 856be

Badness: 0.036785

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 441/440, 10985/10976

Mapping: [⟨1 13 17 13 32 9], ⟨0 -28 -36 -25 -70 -13]]

Optimal tuning (POTE): ~2 = 1\1, ~65/49 = 489.256

Optimal ET sequence: 103, 130, 233, 363

Badness: 0.021694

Cotritone

Subgroup: 2.3.5.7

Comma list: 2401/2400, 390625/387072

Mapping: [⟨1 17 9 10], ⟨0 -30 -13 -14]]

Mappping generators: ~2, ~10/7

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.385

Optimal ET sequence: 35, 37, 72, 109, 181, 253

Badness: 0.098322

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 4000/3993

Mapping: [⟨1 17 9 10 5], ⟨0 -30 -13 -14 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387

Optimal ET sequence: 35, 37, 72, 109, 181, 253

Badness: 0.032225

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 364/363, 385/384, 625/624

Mapping: [⟨1 17 9 10 5 15], ⟨0 -30 -13 -14 -3 -22]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 583.387

Optimal ET sequence: 37, 72, 109, 181f

Badness: 0.028683

Quasimoha

For the 5-limit version of this temperament, see High badness temperaments #Quasimoha.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3645/3584

Mapping: [⟨1 1 9 6], ⟨0 2 -23 -11]]

Mapping generators: ~2, ~49/40

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 348.603

Optimal ET sequence: 31, 117c, 148bc, 179bc

Badness: 0.110820

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 1815/1792

Mapping: [⟨1 1 9 6 2], ⟨0 2 -23 -11 5]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.639

Optimal ET sequence: 31, 86ce, 117ce, 148bce

Badness: 0.046181

Lockerbie

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

Lockerbie can be described as the 103 & 270 temperament. Its generator is 120/77 or 77/60. An obvious tuning is given by 270edo, but 373edo and especially 643edo work as well.

The temperament derives its name from the Scottish town, where a flight numbered 103 crashed with 270 casualties, and the temperament is defined as 103 & 270, hence the name. The name is proposed by Eliora, who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.

Lockerbie also has a unique extension that adds the 41st harmonic such that the generator below 600 cents is also on the same step in 103 or 270 as 41/32, which means that 616/615 is tempered out.

Subgroup: 2.3.5.7

Comma list: 2401/2400, [24 13 -18 -1⟩

Mapping: [⟨1 -25 -16 -13], ⟨0 74 51 44]]

Mapping generators: ~2, ~3828125/2985984

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~3828125/2985984 = 431.1071
error map: ⟨0.0000 -0.0270 +0.1502 -0.1120]
  • CWE: ~2 = 1200.0000, ~3828125/2985984 = 431.1072
error map: ⟨0.0000 -0.0205 +0.1547 -0.1081]

Optimal ET sequence: 103, 167, 270, 643, 913

Badness (Smith): 0.0597

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 766656/765625

Mapping: [⟨1 -25 -16 -13 -26], ⟨0 74 51 44 82]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~77/60 = 431.1082
  • CWE: ~2 = 1200.0000, ~77/60 = 431.1078

Optimal ET sequence: 103, 167, 270, 643, 913, 1183e

Badness (Smith): 0.0262

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224

Mapping: [⟨1 -25 -16 -13 -26 -6], ⟨0 74 51 44 82 27]]

Optimal tunings:

  • CTE: ~2 = 1200.0000, ~77/60 = 431.1085
  • CWE: ~2 = 1200.0000, ~77/60 = 431.1069

Optimal ET sequence: 103, 167, 270, 643, 913f

Badness (Smith): 0.0160

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224

Mapping: [⟨1 -25 -16 -13 -26 -6 -11], ⟨0 74 51 44 82 27 42]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~77/60 = 431.107
  • CWE: ~2 = 1200.000, ~77/60 = 431.108

Optimal ET sequence: 103, 167, 270

Badness (Smith): 0.0210

2.3.5.7.11.13.17.41 subgroup

Subgroup: 2.3.5.7.11.13.17.41

Comma list: 616/615, 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224

Mapping: [⟨1 -25 -16 -13 -26 -6 -11 5], ⟨0 74 51 44 82 27 42 1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~41/32 = 431.107
  • CWE: ~2 = 1200.000, ~41/32 = 431.111

Optimal ET sequence: 103, 167, 270

Hemigoldis

For the 5-limit version, see Diaschismic–gothmic equivalence continuum #Goldis.

Though fairly complex in the 7-limit, hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~21/19 to add prime 19 or perhaps more accurately ~31/28 to add prime 7, or even simply as ~32/29 to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again 89edo is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 549755813888/533935546875

Mapping: [⟨1 21 -9 2], ⟨0 -24 14 1]]

mapping generators: ~2, ~7/4

Optimal tuning (CWE): ~2 = 1200.000, ~7/4 = 970.690

Optimal ET sequence: 21, 47b, 68, 157, 382bccd, 529bccd

Badness (Sintel): 4.40

Surmarvelpyth

Surmarvelpyth is named for the generator fifth, 675/448 being 225/224 (marvel comma) sharp of 3/2. It can be described as the 311 & 431 temperament, starting with the 7-limit to the 19-limit.

Subgroup: 2.3.5.7

Comma list: 2401/2400, [93 -32 -17 -1⟩

Mapping: [⟨1 43 -74 -25], ⟨0 -70 129 47]]

Mapping generators: ~2, ~675/448

Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9719

Optimal ET sequence: 120, 191, 311, 742, 1053, 2848, 3901

Badness: 0.202249

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 820125/819896, 2097152/2096325

Mapping: [⟨1 43 -74 -25 36], ⟨0 -70 129 47 -55]]

Optimal tuning (CTE): ~2 = 1\1, ~675/448 = 709.9720

Optimal ET sequence: 120, 191, 311, 742, 1053, 1795

Badness: 0.052308

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167

Mapping: [⟨1 43 -74 -25 36 25], ⟨0 -70 129 47 -55 -36]]

Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9723

Optimal ET sequence: 120, 191, 311, 742, 1053, 1795f

Badness: 0.032503

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619

Mapping: [⟨1 43 -74 -25 36 25 -103], ⟨0 -70 129 47 -55 -36 181]]

Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722

Optimal ET sequence: 120g, 191g, 311, 431, 742, 1795f

Badness: 0.020995

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984

Mapping: [⟨1 43 -74 -25 36 25 -103 -49], ⟨0 -70 129 47 -55 -36 181 90]]

Optimal tuning (CTE): ~2 = 1\1, ~98/65 = 709.9722

Optimal ET sequence: 120g, 191g, 311, 431, 742, 1795f

Badness: 0.013771

Notes

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
  2. ↑ Yahoo! Tuning Group | 2D temperament names, part I -- reclassified temperaments from message #101780
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