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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
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By tempering a JI scale using N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in [[http://www.huygens-fokker.org/scala/|Scala]] using the Quantize command: either type in "Quantize/consistent N" on the bottom, or use the pull-down menu under "Modify", check the box saying "Consistent" and type N (without a decimal point) into "Resolution". | By tempering a JI scale using N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in [[http://www.huygens-fokker.org/scala/|Scala]] using the Quantize command: either type in "Quantize/consistent N" on the bottom, or use the pull-down menu under "Modify", check the box saying "Consistent" and type N (without a decimal point) into "Resolution". | ||
To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q <= p, if d is the absolute value in cents of the difference between the tuning of q given by the [[POTE tuning]] and the POTE tuning rounded to the nearest N-edo value, then d < 600/N, from which it follows that N < 600/d. Likewise, if e is the absolute value of the error of q in the patent val tuning, then e < 600/N and so N < 600/e. If N-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. We have two distances from the patent val, one to the POTE tuning and one to the the JI tuning, both bounded by 600/N, and so by the triangle inequality the distance from the JI tuning to the POTE tuning, which is the error of the prime q in the POTE tuning, is bounded by 1200/N. Hence, N < 1200/error(q). If now we take the minimum value for 1200/error(prime) for all the odd primes up to p, we obtain an upper bound for N. | To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q <= p, if d is the absolute value in cents of the difference between the tuning of q given by the [[POTE tuning]] and the POTE tuning rounded to the nearest N-edo value, then d < 600/N, from which it follows that N < 600/d. Likewise, if e is the absolute value of the error of q in the patent val tuning, then e < 600/N and so N < 600/e. If N-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. We have two distances from the patent val, one to the POTE tuning and one to the the JI tuning, both bounded by 600/N, and so by the triangle inequality the distance from the JI tuning to the POTE tuning, which is the error of the prime q in the POTE tuning, is bounded by 1200/N. Hence, N < 1200/error(q). If now we take the minimum value for 1200/error(prime) for all the odd primes up to p, we obtain an upper bound for N. | ||
Below are tabulated some values. | Below are tabulated some values. | ||
==5-limit rank two== | ==5-limit rank two== | ||
16/15: [[8edo]] | 16/15: [[8edo]] | ||
648/625: [[12edo]] | 648/625: [[12edo]] | ||
| Line 36: | Line 36: | ||
7629394531250/7625597484987: [[3501edo]] | 7629394531250/7625597484987: [[3501edo]] | ||
==7-limit rank two== | ==7-limit rank two== | ||
Name: Optimal patent val: Val name: 1000*badness | Name: Optimal patent val: Val name: 1000*badness | ||
[[Trienstonic clan|Father]]: [[5edo]] [<1 0 4 -2|, <0 1 -1 3|] 21.312 | [[Trienstonic clan|Father]]: [[5edo]] [<1 0 4 -2|, <0 1 -1 3|] 21.312 | ||
| Line 111: | Line 111: | ||
[[Ragismic microtemperaments|Supermajor]]: [[6214edo]] 80&171 10.836 | [[Ragismic microtemperaments|Supermajor]]: [[6214edo]] 80&171 10.836 | ||
==7-limit rank three== | ==7-limit rank three== | ||
Comma: Optimal patent val: 10^6 * badness | Comma: Optimal patent val: 10^6 * badness | ||
36/35: [[12edo]] 63.675 | 36/35: [[12edo]] 63.675 | ||
| Line 126: | Line 126: | ||
65625/65536: [[171edo]] 172.601 | 65625/65536: [[171edo]] 172.601 | ||
126/125: [[185edo]] 69.867 | 126/125: [[185edo]] 69.867 | ||
1029/1024: [[190edo]] | 1029/1024: [[190edo]] 176.105 | ||
225/224: [[197edo]] 36.517 | 225/224: [[197edo]] 36.517 | ||
4000/3969: [[215edo]] 215.550 | 4000/3969: [[215edo]] 215.550 | ||
| Line 142: | Line 142: | ||
78125000/78121827: [[101654edo]] 20.457 | 78125000/78121827: [[101654edo]] 20.457 | ||
==11-limit rank two== | ==11-limit rank two== | ||
Name: Optimal patent val: Val name: 1000*badness | Name: Optimal patent val: Val name: 1000*badness | ||
[[Jubilismic clan|Diminished]]: [[12edo]] 4&12 22.132 | [[Jubilismic clan|Diminished]]: [[12edo]] 4&12 22.132 | ||
| Line 198: | Line 198: | ||
[[Ragismic microtemperaments|Hemiennealimmal]]: [[1566edo]] 72&198 6.283 | [[Ragismic microtemperaments|Hemiennealimmal]]: [[1566edo]] 72&198 6.283 | ||
==11-limit rank three== | ==11-limit rank three== | ||
Name: Optimal patent val: Val name: 10^6 * badness | Name: Optimal patent val: Val name: 10^6 * badness | ||
[[Archytas family|Vulcan]]: [[15edo]] 2&3&10 577.113 | [[Archytas family|Vulcan]]: [[15edo]] 2&3&10 577.113 | ||
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[[Kalismic temperaments|Odin]]: [[7464edo]] 12&42&72 116.151 | [[Kalismic temperaments|Odin]]: [[7464edo]] 12&42&72 116.151 | ||
==13-limit rank two== | ==13-limit rank two== | ||
[[Marvel temperaments|Negri]]: [[19edo]] 9&10 17.833 | [[Marvel temperaments|Negri]]: [[19edo]] 9&10 17.833 | ||
[[Starling temperaments|Sensis]]: [[19edo]] | [[Starling temperaments|Sensis]]: [[19edo]] [<1 6 8 11 6 10|, <0 -8 -9 -13 -4 -10|] 20.017 | ||
[[Augmented family|August]]: [[21edo]] 9&21 18.448 | [[Augmented family|August]]: [[21edo]] 9&21 18.448 | ||
[[Meantone family|Huygens]]: [[31edo]] | [[Meantone family|Huygens]]: [[31edo]] [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|] 18.048 | ||
[[Semicomma family|Winston]]: [[31edo]] | [[Semicomma family|Winston]]: [[31edo]] | ||
[[Gamelismic clan|Miracle]]: [[41edo]] 10&31 18.669 | [[Gamelismic clan|Miracle]]: [[41edo]] 10&31 18.669</pre></div> | ||
</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Optimal patent val</title></head><body>Given any collection of p-limit commas, there is a finite list of p-limit <a class="wiki_link" href="/Patent%20val">patent vals</a> tempering out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the unique patent val which has the lowest <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures">TE error</a>; this is the <em>optimal (TE) patent val</em> for the temperament defined by the commas. Note that other definitions of error, such as maximum p-limit error, or maximum q-limit error where q is the largest odd number less than the prime above p, lead to different results.<br /> | <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"><html><head><title>Optimal patent val</title></head><body>Given any collection of p-limit commas, there is a finite list of p-limit <a class="wiki_link" href="/Patent%20val">patent vals</a> tempering out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the unique patent val which has the lowest <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures">TE error</a>; this is the <em>optimal (TE) patent val</em> for the temperament defined by the commas. Note that other definitions of error, such as maximum p-limit error, or maximum q-limit error where q is the largest odd number less than the prime above p, lead to different results.<br /> | ||
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16/15: <a class="wiki_link" href="/8edo">8edo</a><br /> | 16/15: <a class="wiki_link" href="/8edo">8edo</a><br /> | ||
648/625: <a class="wiki_link" href="/12edo">12edo</a><br /> | 648/625: <a class="wiki_link" href="/12edo">12edo</a><br /> | ||
27/25: <a class="wiki_link" href="/14edo">14edo</a><br /> | 27/25: <a class="wiki_link" href="/14edo">14edo</a><br /> | ||
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Name: Optimal patent val: Val name: 1000*badness<br /> | Name: Optimal patent val: Val name: 1000*badness<br /> | ||
<a class="wiki_link" href="/Trienstonic%20clan">Father</a>: <a class="wiki_link" href="/5edo">5edo</a> [&lt;1 0 4 -2|, &lt;0 1 -1 3|] 21.312<br /> | <a class="wiki_link" href="/Trienstonic%20clan">Father</a>: <a class="wiki_link" href="/5edo">5edo</a> [&lt;1 0 4 -2|, &lt;0 1 -1 3|] 21.312<br /> | ||
<a class="wiki_link" href="/Archytas%20clan">Mother</a>: <a class="wiki_link" href="/5edo">5edo</a> 2&amp;3 24.152<br /> | <a class="wiki_link" href="/Archytas%20clan">Mother</a>: <a class="wiki_link" href="/5edo">5edo</a> 2&amp;3 24.152<br /> | ||
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<br /> | <br /> | ||
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Comma: Optimal patent val: 10^6 * badness<br /> | Comma: Optimal patent val: 10^6 * badness<br /> | ||
36/35: <a class="wiki_link" href="/12edo">12edo</a> 63.675<br /> | 36/35: <a class="wiki_link" href="/12edo">12edo</a> 63.675<br /> | ||
49/48: <a class="wiki_link" href="/19edo">19edo</a> 115.514<br /> | 49/48: <a class="wiki_link" href="/19edo">19edo</a> 115.514<br /> | ||
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65625/65536: <a class="wiki_link" href="/171edo">171edo</a> 172.601<br /> | 65625/65536: <a class="wiki_link" href="/171edo">171edo</a> 172.601<br /> | ||
126/125: <a class="wiki_link" href="/185edo">185edo</a> 69.867<br /> | 126/125: <a class="wiki_link" href="/185edo">185edo</a> 69.867<br /> | ||
1029/1024: <a class="wiki_link" href="/190edo">190edo</a> | 1029/1024: <a class="wiki_link" href="/190edo">190edo</a> 176.105<br /> | ||
225/224: <a class="wiki_link" href="/197edo">197edo</a> 36.517<br /> | 225/224: <a class="wiki_link" href="/197edo">197edo</a> 36.517<br /> | ||
4000/3969: <a class="wiki_link" href="/215edo">215edo</a> 215.550<br /> | 4000/3969: <a class="wiki_link" href="/215edo">215edo</a> 215.550<br /> | ||
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Name: Optimal patent val: Val name: 1000*badness<br /> | Name: Optimal patent val: Val name: 1000*badness<br /> | ||
<a class="wiki_link" href="/Jubilismic%20clan">Diminished</a>: <a class="wiki_link" href="/12edo">12edo</a> 4&amp;12 22.132<br /> | <a class="wiki_link" href="/Jubilismic%20clan">Diminished</a>: <a class="wiki_link" href="/12edo">12edo</a> 4&amp;12 22.132<br /> | ||
<a class="wiki_link" href="/Marvel%20temperaments">Negri</a>: <a class="wiki_link" href="/19edo">19edo</a> 9&amp;10 26.190<br /> | <a class="wiki_link" href="/Marvel%20temperaments">Negri</a>: <a class="wiki_link" href="/19edo">19edo</a> 9&amp;10 26.190<br /> | ||
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Name: Optimal patent val: Val name: 10^6 * badness<br /> | Name: Optimal patent val: Val name: 10^6 * badness<br /> | ||
<a class="wiki_link" href="/Archytas%20family">Vulcan</a>: <a class="wiki_link" href="/15edo">15edo</a> 2&amp;3&amp;10 577.113<br /> | <a class="wiki_link" href="/Archytas%20family">Vulcan</a>: <a class="wiki_link" href="/15edo">15edo</a> 2&amp;3&amp;10 577.113<br /> | ||
<a class="wiki_link" href="/Jubilismic%20family">Festival</a>: <a class="wiki_link" href="/26edo">26edo</a> 2&amp;10&amp;26 688.510<br /> | <a class="wiki_link" href="/Jubilismic%20family">Festival</a>: <a class="wiki_link" href="/26edo">26edo</a> 2&amp;10&amp;26 688.510<br /> | ||
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<a class="wiki_link" href="/Marvel%20temperaments">Negri</a>: <a class="wiki_link" href="/19edo">19edo</a> 9&amp;10 17.833<br /> | <a class="wiki_link" href="/Marvel%20temperaments">Negri</a>: <a class="wiki_link" href="/19edo">19edo</a> 9&amp;10 17.833<br /> | ||
<a class="wiki_link" href="/Starling%20temperaments">Sensis</a>: <a class="wiki_link" href="/19edo">19edo</a> <br /> | <a class="wiki_link" href="/Starling%20temperaments">Sensis</a>: <a class="wiki_link" href="/19edo">19edo</a> [&lt;1 6 8 11 6 10|, &lt;0 -8 -9 -13 -4 -10|] 20.017 <br /> | ||
<a class="wiki_link" href="/Augmented%20family">August</a>: <a class="wiki_link" href="/21edo">21edo</a> 9&amp;21 18.448<br /> | <a class="wiki_link" href="/Augmented%20family">August</a>: <a class="wiki_link" href="/21edo">21edo</a> 9&amp;21 18.448<br /> | ||
<a class="wiki_link" href="/Meantone%20family">Huygens</a>: <a class="wiki_link" href="/31edo">31edo</a><br /> | <a class="wiki_link" href="/Meantone%20family">Huygens</a>: <a class="wiki_link" href="/31edo">31edo</a> [&lt;1 0 -4 -13 -25 -20|, &lt;0 1 4 10 18 15|] 18.048<br /> | ||
<a class="wiki_link" href="/Semicomma%20family">Winston</a>: <a class="wiki_link" href="/31edo">31edo</a><br /> | <a class="wiki_link" href="/Semicomma%20family">Winston</a>: <a class="wiki_link" href="/31edo">31edo</a><br /> | ||
<a class="wiki_link" href="/Gamelismic%20clan">Miracle</a>: <a class="wiki_link" href="/41edo">41edo</a> 10&amp;31 18.669</body></html></pre></div> | <a class="wiki_link" href="/Gamelismic%20clan">Miracle</a>: <a class="wiki_link" href="/41edo">41edo</a> 10&amp;31 18.669</body></html></pre></div> | ||
Revision as of 07:00, 22 February 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author guest and made on 2011-02-22 07:00:03 UTC.
- The original revision id was 203933000.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
Given any collection of p-limit commas, there is a finite list of p-limit [[Patent val|patent vals]] tempering out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the unique patent val which has the lowest [[Tenney-Euclidean temperament measures|TE error]]; this is the //optimal (TE) patent val// for the temperament defined by the commas. Note that other definitions of error, such as maximum p-limit error, or maximum q-limit error where q is the largest odd number less than the prime above p, lead to different results. By tempering a JI scale using N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in [[http://www.huygens-fokker.org/scala/|Scala]] using the Quantize command: either type in "Quantize/consistent N" on the bottom, or use the pull-down menu under "Modify", check the box saying "Consistent" and type N (without a decimal point) into "Resolution". To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q <= p, if d is the absolute value in cents of the difference between the tuning of q given by the [[POTE tuning]] and the POTE tuning rounded to the nearest N-edo value, then d < 600/N, from which it follows that N < 600/d. Likewise, if e is the absolute value of the error of q in the patent val tuning, then e < 600/N and so N < 600/e. If N-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. We have two distances from the patent val, one to the POTE tuning and one to the the JI tuning, both bounded by 600/N, and so by the triangle inequality the distance from the JI tuning to the POTE tuning, which is the error of the prime q in the POTE tuning, is bounded by 1200/N. Hence, N < 1200/error(q). If now we take the minimum value for 1200/error(prime) for all the odd primes up to p, we obtain an upper bound for N. Below are tabulated some values. ==5-limit rank two== 16/15: [[8edo]] 648/625: [[12edo]] 27/25: [[14edo]] 25/24: [[17edo]] 250/243: [[22edo]] 135/128: [[23edo]] 128/125: [[39edo]] 3125/3072: [[60edo]] 2048/2025: [[80edo]] 81/80: [[81edo]] 393216/390625: [[164edo]] 2109375/2097152: [[296edo]] 15625/15552: [[458edo]] 78732/78125: [[539edo]] 32805/32768: [[749edo]] 1600000/1594323: [[873edo]] 6115295232/6103515625: [[1400edo]] 1224440064/1220703125: [[1496edo]] 274877906944/274658203125: [[1559edo]] 7629394531250/7625597484987: [[3501edo]] ==7-limit rank two== Name: Optimal patent val: Val name: 1000*badness [[Trienstonic clan|Father]]: [[5edo]] [<1 0 4 -2|, <0 1 -1 3|] 21.312 [[Archytas clan|Mother]]: [[5edo]] 2&3 24.152 [[Meantone family|Sharptone]]: [[5edo]] [<1 0 -4 -2|, <0 1 4 3|] 24.848 [[Dicot family|Dicot]]: [[7edo]] 4&7 19.935 [[Bug family|Beep]]: [[9edo]] 4&5 18.638 [[Dicot family|Sharp]]: [[10edo]] [<1 1 2 1|, <0 2 1 6|] 28.942 [[Dicot family|Decimal]]: [[10edo]] 4&10 28.334 [[Meantone family|Dominant]]: [[12edo]] 5&7 20.690 [[Jubilismic clan|Diminished]]: [[12edo]] 4&12 22.401 [[Augmented family|August]]: [[12edo]] 9&12 26.459 [[Augmented family|Hexe]]: [[12edo]] 6&12 57.730 [[Archytas clan|Blacksmith]]: [[15edo]] 5&10 25.640 [[Gamelismic clan|Gorgo]]: [[16edo]] 5&16 46.385 [[Trienstonic clan|Octokaidecal]]: [[18edo]] 10&18 36.747 [[Marvel temperaments|Negri]]: [[19edo]] 1&9 26.483 [[Meantone family|Godzilla]]: [[19edo]] 5&19 26.747 [[Kleismic family|Keemun]]: [[19edo]] 4&15 27.408 [[Diaschismic family|Pajara]]: [[22edo]] 2&10 20.033 [[Jubilismic clan|Lemba]]: [[26edo]] 10&16 62.208 [[Augmented family|Augene]]: [[27edo]] 3&12 24.816 [[Archytas clan|Beatles]]: [[27edo]] 10&27 45.872 [[Meantone family|Mothra]]: [[31edo]] 5&26 37.146 [[Meantone family|Injera]]: [[38edo]] 12&26 31.130 [[Magic family|Magic]]: [[41edo]] 19&41 18.918 [[Gamelismic clan|Superkleismic]]: [[41edo]] 15&26 47.932 [[Meantone family|Flattone]]: [[45edo]] 7&19 38.553 [[Starling temperaments|Sensi]]: [[46edo]] 19&27 25.622 [[Jubilismic clan|Doublewide]]: [[48edo]] 4&18 43.462 [[Archytas clan|Superpyth]]: [[49edo]] 5&17 32.318 [[Meantone family|Liese]]: [[55edo]] 19&55 46.706 [[Gamelismic clan|Miracle]]: [[72edo]] 10&21 16.742 [[Marvel temperaments|Wizard]]: [[72edo]] 22&50 40.846 [[Meantone family|Meantone]]: [[81edo]] 12&19 13.707 [[Starling temperaments|Myna]]: [[89edo]] 4&27 27.044 [[Marvel temperaments|Septimin]]: [[91edo]] 9&32 54.502 [[Schismatic family|Garibaldi]]: [[94edo]] 12&29 21.644 [[Hemimean clan|Misty]]: [[99edo]] 12&87 36.802 [[Starling temperaments|Nusecond]]: [[101edo]] 31&101 50.389 [[Tetracot family|Octacot]]: [[109edo]] 27&41 33.845 [[Hemimean clan|Semisept]]: [[111edo]] 18&31 50.472 [[Gamelismic clan|Guiron]]: [[118edo]] 36&41 47.544 [[Marvel temperaments|Tritonic]]: [[122edo]] 2&29 47.578 [[Marvel temperaments|Slender]]: [[125edo]] 1&31 56.934 [[Gamelismic clan|Rodan]]: [[128edo]] 5&41 37.112 [[Semicomma family|Orwell]]: [[137edo]] 9&22 20.375 [[Breedsmic temperaments|Tertiaseptal]]: [[171edo]] 31&109 12.995 [[Schismatic family|Pontiac]]: [[171edo]] 53&118 14.133 [[Ragismic microtemperaments|Mitonic]]: [[171edo]] 46&125 25.184 [[Mutt family|Mutt]]: [[171edo]] 3&84 28.406 [[Starling temperaments|Valentine]]: [[185edo]] 15&31 31.056 [[Gamelismic clan|Unidec]]: [[190edo]] 26&46 38.393 [[Kleismic family|Catakleismic]]: [[197edo]] 19&53 21.501 [[Kleismic family|Quadritikleismic]]: [[212edo]] 4&68 39.231 [[Ragismic microtemperaments|Octoid]]: [224edo]] 72&152 42.670 [[Pythagorean family|Compton]]: [[228edo]] 12&60 35.686 [[Kleismic family|Tritikleismic]]: [[231edo]] 15&57 56.337 [[Wuerschmidt family|Hemiwuerschmidt]]: [[328edo]] 6&31 20.307 [[Breedsmic temperaments|Hemififths]]: [[338edo]] 41&58 22.243 [[Ragismic microtemperaments|Amity]]: [[350edo]] 7&39 23.649 [[Schismatic family|Bischismic]]: [[378edo]] 12&118 54.744 [[Ragismic microtemperaments|Parakleismic]]: [[415edo]] 19&80 27.431 [[Ragismic microtemperaments|Gamera]]: [[422edo]] 26&73 37.648 [[Kleismic family|Countercata]]: [[473edo]] 34&53 52.128 [[Breedsmic temperaments|Harry]]: [[534edo]] 58&72 34.077 [[Ragismic microtemperaments|Ennealimmal]]: [[612edo]] 27&45 3.610 [[Breedsmic temperaments|Quasiorwell]]: [[1111edo]] 31&177 35.832 [[Schismatic family|Sesquiquartififths]]: [[1498edo]] 41&89 11.244 [[Schismatic family|Term]]: [[1722edo]] 12&159 19.950 [[Gammic family|Neptune]]: [[1778edo]] 35&68 23.427 [[Ragismic microtemperaments|Abigail]]: [[1798edo]] 46&132 37.000 [[Ragismic microtemperaments|Enneadecal]]: [[2185edo]] 19&152 10.954 [[Ragismic microtemperaments|Supermajor]]: [[6214edo]] 80&171 10.836 ==7-limit rank three== Comma: Optimal patent val: 10^6 * badness 36/35: [[12edo]] 63.675 49/48: [[19edo]] 115.514 875/864: [[41edo]] 212.413 525/512: [[45edo]] 476.454 686/675: [[46edo]] 320.094 50/49: [[48edo]] 127.208 64/63: [[49edo]] 98.719 1029/1000: [[55edo]] 678.164 3125/3087: [[94edo]] 613.731 1728/1715: [[111edo]] 187.341 2430/2401: [[137edo]] 518.625 65625/65536: [[171edo]] 172.601 126/125: [[185edo]] 69.867 1029/1024: [[190edo]] 176.105 225/224: [[197edo]] 36.517 4000/3969: [[215edo]] 215.550 16875/16807: [[224edo]] 341.931 245/243: [[283edo]] 121.191 6144/6125: [[381edo]] 142.375 3136/3125: [[446edo]] 160.039 19683/19600: [[587edo]] 300.987 10976/10935: [[695edo]] 199.139 703125/702464: [[2185edo]] 149.828 2401/2400: [[2749edo]] 15.288 420175/419904: [[4306edo]] 86.449 4375/4374: [[8419edo]] 11.665 250047/250000: [[12555edo]] 21.301 78125000/78121827: [[101654edo]] 20.457 ==11-limit rank two== Name: Optimal patent val: Val name: 1000*badness [[Jubilismic clan|Diminished]]: [[12edo]] 4&12 22.132 [[Marvel temperaments|Negri]]: [[19edo]] 9&10 26.190 [[Diaschismic family|Pajara]]: [[22edo]] 12&22 20.343 [[Meantone family|Mohajira]]: [[31edo]] 7&24 26.064 [[Meantone family|Squares]]: [[31edo]] 31&45 56.826 [[Starling temperaments|Casablanca]]: [[31edo]] 31&42 67.291 [[Kleismic family|Keemun]]: [[34edo]] 4&15 27.410 [[Schismatic family|Cassandra]]: [[41edo]] 12&29 23.556 [[Gamelismic clan|Superkleismic]]: [[41edo]] 15&26 25.659 [[Starling temperaments|Sensi]]: [[46edo]] 19&27 37.942 [[Jubilismic clan|Doublewide]]: [[48edo]] 4&18 32.058 [[Semicomma family|Orwell]]: [[53edo]] 9&22 15.231 [[Diaschismic family|Diaschismic]]: [[58edo]] 12&46 25.034 [[Breedsmic temperaments|Hemififths]]: [[58edo]] 41&58 23.498 [[Porcupine family|Porcupine]]: [[59edo]] 7&15 21.562 [[Gamelismic clan|Miracle]]: [[72edo]] 10&31 10.684 [[Kleismic family|Catakleismic]]: [[72edo]] 19&53 21.849 [[Pythagorean family|Compton]]: [[72edo]] 12&72 22.235 [[Starling temperaments|Valentine]]: [[77edo]] 15&31 16.687 [[Diaschismic family|Keen]]: [[78edo]] 22&56 45.270 [[Diaschismic family|Echidna]]: [[80edo]] 22&58 25.987 [[Meantone family|Meanpop]]: [[81edo]] 19&31 21.543 [[Gamelismic clan|Rodan]]: [[87edo]] 5&41 23.093 [[Meantone family|Mothra]]: [[88edo]] 5&26 25.642 [[Starling temperaments|Myna]]: [[89edo]] 31&58 16.842 [[Marvel temperaments|Septimin]]: [[91edo]] 9&32 31.309 [[Schismatic family|Garibaldi]]: [[94edo]] 41&53 27.396 [[Starling temperaments|Nusecond]]: [[101edo]] 31&101 25.621 [[Magic family|Magic]]: [[104edo]] 19&22 25.621 [[Meantone family|Huygens]]: [[105edo]] 12&31 17.027 [[Diaschismic family|Shrutar]]: [[114edo]] 22&46 26.489 [[Gamelismic clan|Hemithirds]]: [[118edo]] 31&87 19.003 [[Ragismic microtemperaments|Parakleismic]]: [[118edo]] 19&99 49.711 [[Marvel temperaments|Tritonic]]: [[122edo]] 31&91 45.456 [[Marvel temperaments|Slender]]: [[125edo]] 31&63 25.342 [[Wuerschmidt family|Hemiwuerschmidt]]: [[130edo]] 31&130 21.069 [[Diaschismic family|Echidnic]]: [[148edo]] 10&46 45.127 [[Mirkwai clan|Grendel]]: [[152edo]] 31&121 19.845 [[Mirkwai clan|Kwai]]: [[152edo]] 41&111 26.219 [[Schismatic family|Guiron]]: [[159edo]] 41&77 26.648 [[Marvel temperaments|Wizard]]: [[166edo]] 22&50 18.539 [[Gamelismic clan|Unidec]]: [[190edo]] 26&46 15.479 [[Breedsmic temperaments|Harry]]: [[202edo]] 58&72 15.867 [[Breedsmic temperaments|Tertiaseptal]]: [[202edo]] 31&171 35.576 [[Ragismic microtemperaments|Octoid]]: [[224edo]] 72&152 14.097 [[Kleismic family|Countercata]]: [[227edo]] 34&53 39.770 [[Kleismic family|Tritikleismic]]: [[231edo]] 15&57 19.333 [[Breedsmic temperaments|Quasiorwell]]: [[270edo]] 31&208 17.540 [[Marvel temperaments|Marvo]]: [[281edo]] 72&281 31.685 [[Kleismic family|Quadritikleismic]]: [[284edo]] 4&68 23.406 [[Ragismic microtemperaments|Hemiamity]]: [[350edo]] 46&106 31.307 [[Ragismic microtemperaments|Abigail]]: [[764edo]] 46&132 12.860 [[Ragismic microtemperaments|Hemiennealimmal]]: [[1566edo]] 72&198 6.283 ==11-limit rank three== Name: Optimal patent val: Val name: 10^6 * badness [[Archytas family|Vulcan]]: [[15edo]] 2&3&10 577.113 [[Jubilismic family|Festival]]: [[26edo]] 2&10&26 688.510 [[Starling family|Thrasher]]: [[34edo]] 3&4&12 480.188 [[Didymus rank three family|Calliope]]: [[45edo]] 7&12&19 529.778 [[Jubilismic family|Jubilee]]: [[48edo]] 4&12&22 559.842 [[Archytas family|Ares]]: [[49edo]] 3&7&12 468.874 [[Porcupine rank three family|Sonic]]: [[51edo]] 7&15&51 523.158 [[Nuwell family|Big Brother]]: [[53edo]] 9&22&70 506.556 [[Marvel family|Prodigy]]: [[72edo]] 2&10&29 334.745 [[Didymus rank three family|Erato]]: [[81edo]] 12&19&31 557.521 [[Didymus rank three family|Euterpe]]: [[88edo]] 5&12&26 535.809 [[Starling family|Thrush]]: [[89edo]] 3&12&31 352.591 [[Starling family|Aplonis]]: [[89edo]] 19&31&58 647.511 [[Porwell family|Zeus]]: [[99edo]] 6&7&9 399.882 [[Marvel family|Apollo]]: [[104edo]] 12&19&22 529.766 [[Orwellismic family|Guanyin]]: [[111edo]] 9&22&58 411.252 [[Marvel family|Minerva]]: [[127edo]] 1&9&12 381.064 [[Didymus rank three family|Clio]]: [[129edo]] 7&12&24 737.590 [[Marvel family|Marvel]]: [[166edo]] 9&10&22 255.033 [[Gamelismic family|Portent]]: [[190edo]] 5&10&26 234.092 [[Breed family|Jove]]: [[202edo]] 10&31&58 241.055 [[Marvel family|Spectacle]]: [[281edo]] 10&31&281 498.669 [[Porwell family|Jupiter]]: [[282edo]] 9&22&108 562.368 [[Breed family|Agni]]: [[284edo]] 4&6&27 493.790 [[Werckismic temperaments|Varuna]]: [[320edo]] 12&26&46 418.714 [[Werckismic temperaments|Babar]]: [[491edo]] 15&29&43 603.636 [[Swetismic temperaments|Hades]]: [[578edo]] 22&50&58 361.398 [[Cataharry family|Albus]]: [[640edo]] 19&53&58 621.100 [[Mirkwai family|Indra]]: [[703edo]] 10&31&152 266.020 [[Kalismic temperaments|Loki]]: [[1092edo]] 12&22&86 492.550 [[Lehmerismic temperaments|Ganesha]]: [[1502edo]] [[Breed family|Freya]]: [[1566edo]] 10&31&167 169.702 [[Lehmerismic temperaments|Tyr]]: [[1566edo]] 15&57&183 258.374 [[Breed family|Baldur]]: [[1696edo]] 4&58&72 166.115 [[Lehmerismic temperaments|Skadi]]: [[2014edo]] 31&97&121 348.944 [[Ragisma family|Thor]]: [[2554edo]] 26&46&152 88.834 [[Breed family|Ennealimmic]]: [[2619edo]] 27&45&171 274.685 [[Kalismic temperaments|Odin]]: [[7464edo]] 12&42&72 116.151 ==13-limit rank two== [[Marvel temperaments|Negri]]: [[19edo]] 9&10 17.833 [[Starling temperaments|Sensis]]: [[19edo]] [<1 6 8 11 6 10|, <0 -8 -9 -13 -4 -10|] 20.017 [[Augmented family|August]]: [[21edo]] 9&21 18.448 [[Meantone family|Huygens]]: [[31edo]] [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|] 18.048 [[Semicomma family|Winston]]: [[31edo]] [[Gamelismic clan|Miracle]]: [[41edo]] 10&31 18.669
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<html><head><title>Optimal patent val</title></head><body>Given any collection of p-limit commas, there is a finite list of p-limit <a class="wiki_link" href="/Patent%20val">patent vals</a> tempering out the commas. The list is not guaranteed to contain any members, but in most actual circumstances it will. If the list is not empty, then among these patent vals will be found the unique patent val which has the lowest <a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures">TE error</a>; this is the <em>optimal (TE) patent val</em> for the temperament defined by the commas. Note that other definitions of error, such as maximum p-limit error, or maximum q-limit error where q is the largest odd number less than the prime above p, lead to different results.<br /> <br /> By tempering a JI scale using N-edo found on the list below we automatically temper it to the corresponding temperament. This can be done in <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/" rel="nofollow">Scala</a> using the Quantize command: either type in "Quantize/consistent N" on the bottom, or use the pull-down menu under "Modify", check the box saying "Consistent" and type N (without a decimal point) into "Resolution".<br /> <br /> To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q <= p, if d is the absolute value in cents of the difference between the tuning of q given by the <a class="wiki_link" href="/POTE%20tuning">POTE tuning</a> and the POTE tuning rounded to the nearest N-edo value, then d < 600/N, from which it follows that N < 600/d. Likewise, if e is the absolute value of the error of q in the patent val tuning, then e < 600/N and so N < 600/e. If N-edo defines an optimal patent val, then the patent val will be identical to the val obtained by rounding the POTE tuning to the nearest N-edo value. We have two distances from the patent val, one to the POTE tuning and one to the the JI tuning, both bounded by 600/N, and so by the triangle inequality the distance from the JI tuning to the POTE tuning, which is the error of the prime q in the POTE tuning, is bounded by 1200/N. Hence, N < 1200/error(q). If now we take the minimum value for 1200/error(prime) for all the odd primes up to p, we obtain an upper bound for N.<br /> <br /> Below are tabulated some values.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-5-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:0 -->5-limit rank two</h2> 16/15: <a class="wiki_link" href="/8edo">8edo</a><br /> 648/625: <a class="wiki_link" href="/12edo">12edo</a><br /> 27/25: <a class="wiki_link" href="/14edo">14edo</a><br /> 25/24: <a class="wiki_link" href="/17edo">17edo</a><br /> 250/243: <a class="wiki_link" href="/22edo">22edo</a><br /> 135/128: <a class="wiki_link" href="/23edo">23edo</a><br /> 128/125: <a class="wiki_link" href="/39edo">39edo</a><br /> 3125/3072: <a class="wiki_link" href="/60edo">60edo</a><br /> 2048/2025: <a class="wiki_link" href="/80edo">80edo</a><br /> 81/80: <a class="wiki_link" href="/81edo">81edo</a><br /> 393216/390625: <a class="wiki_link" href="/164edo">164edo</a><br /> 2109375/2097152: <a class="wiki_link" href="/296edo">296edo</a><br /> 15625/15552: <a class="wiki_link" href="/458edo">458edo</a><br /> 78732/78125: <a class="wiki_link" href="/539edo">539edo</a><br /> 32805/32768: <a class="wiki_link" href="/749edo">749edo</a><br /> 1600000/1594323: <a class="wiki_link" href="/873edo">873edo</a><br /> 6115295232/6103515625: <a class="wiki_link" href="/1400edo">1400edo</a><br /> 1224440064/1220703125: <a class="wiki_link" href="/1496edo">1496edo</a><br /> 274877906944/274658203125: <a class="wiki_link" href="/1559edo">1559edo</a><br /> 7629394531250/7625597484987: <a class="wiki_link" href="/3501edo">3501edo</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-7-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:2 -->7-limit rank two</h2> Name: Optimal patent val: Val name: 1000*badness<br /> <a class="wiki_link" href="/Trienstonic%20clan">Father</a>: <a class="wiki_link" href="/5edo">5edo</a> [<1 0 4 -2|, <0 1 -1 3|] 21.312<br /> <a class="wiki_link" href="/Archytas%20clan">Mother</a>: <a class="wiki_link" href="/5edo">5edo</a> 2&3 24.152<br /> <a class="wiki_link" href="/Meantone%20family">Sharptone</a>: <a class="wiki_link" href="/5edo">5edo</a> [<1 0 -4 -2|, <0 1 4 3|] 24.848<br /> <a class="wiki_link" href="/Dicot%20family">Dicot</a>: <a class="wiki_link" href="/7edo">7edo</a> 4&7 19.935<br /> <a class="wiki_link" href="/Bug%20family">Beep</a>: <a class="wiki_link" href="/9edo">9edo</a> 4&5 18.638<br /> <a class="wiki_link" href="/Dicot%20family">Sharp</a>: <a class="wiki_link" href="/10edo">10edo</a> [<1 1 2 1|, <0 2 1 6|] 28.942<br /> <a class="wiki_link" href="/Dicot%20family">Decimal</a>: <a class="wiki_link" href="/10edo">10edo</a> 4&10 28.334<br /> <a class="wiki_link" href="/Meantone%20family">Dominant</a>: <a class="wiki_link" href="/12edo">12edo</a> 5&7 20.690<br /> <a class="wiki_link" href="/Jubilismic%20clan">Diminished</a>: <a class="wiki_link" href="/12edo">12edo</a> 4&12 22.401<br /> <a class="wiki_link" href="/Augmented%20family">August</a>: <a class="wiki_link" href="/12edo">12edo</a> 9&12 26.459<br /> <a class="wiki_link" href="/Augmented%20family">Hexe</a>: <a class="wiki_link" href="/12edo">12edo</a> 6&12 57.730<br /> <a class="wiki_link" href="/Archytas%20clan">Blacksmith</a>: <a class="wiki_link" href="/15edo">15edo</a> 5&10 25.640<br /> <a class="wiki_link" href="/Gamelismic%20clan">Gorgo</a>: <a class="wiki_link" href="/16edo">16edo</a> 5&16 46.385<br /> <a class="wiki_link" href="/Trienstonic%20clan">Octokaidecal</a>: <a class="wiki_link" href="/18edo">18edo</a> 10&18 36.747<br /> <a class="wiki_link" href="/Marvel%20temperaments">Negri</a>: <a class="wiki_link" href="/19edo">19edo</a> 1&9 26.483<br /> <a class="wiki_link" href="/Meantone%20family">Godzilla</a>: <a class="wiki_link" href="/19edo">19edo</a> 5&19 26.747<br /> <a class="wiki_link" href="/Kleismic%20family">Keemun</a>: <a class="wiki_link" href="/19edo">19edo</a> 4&15 27.408<br /> <a class="wiki_link" href="/Diaschismic%20family">Pajara</a>: <a class="wiki_link" href="/22edo">22edo</a> 2&10 20.033<br /> <a class="wiki_link" href="/Jubilismic%20clan">Lemba</a>: <a class="wiki_link" href="/26edo">26edo</a> 10&16 62.208<br /> <a class="wiki_link" href="/Augmented%20family">Augene</a>: <a class="wiki_link" href="/27edo">27edo</a> 3&12 24.816<br /> <a class="wiki_link" href="/Archytas%20clan">Beatles</a>: <a class="wiki_link" href="/27edo">27edo</a> 10&27 45.872<br /> <a class="wiki_link" href="/Meantone%20family">Mothra</a>: <a class="wiki_link" href="/31edo">31edo</a> 5&26 37.146<br /> <a class="wiki_link" href="/Meantone%20family">Injera</a>: <a class="wiki_link" href="/38edo">38edo</a> 12&26 31.130<br /> <a class="wiki_link" href="/Magic%20family">Magic</a>: <a class="wiki_link" href="/41edo">41edo</a> 19&41 18.918<br /> <a class="wiki_link" href="/Gamelismic%20clan">Superkleismic</a>: <a class="wiki_link" href="/41edo">41edo</a> 15&26 47.932<br /> <a class="wiki_link" href="/Meantone%20family">Flattone</a>: <a class="wiki_link" href="/45edo">45edo</a> 7&19 38.553<br /> <a class="wiki_link" href="/Starling%20temperaments">Sensi</a>: <a class="wiki_link" href="/46edo">46edo</a> 19&27 25.622<br /> <a class="wiki_link" href="/Jubilismic%20clan">Doublewide</a>: <a class="wiki_link" href="/48edo">48edo</a> 4&18 43.462<br /> <a class="wiki_link" href="/Archytas%20clan">Superpyth</a>: <a class="wiki_link" href="/49edo">49edo</a> 5&17 32.318<br /> <a class="wiki_link" href="/Meantone%20family">Liese</a>: <a class="wiki_link" href="/55edo">55edo</a> 19&55 46.706<br /> <a class="wiki_link" href="/Gamelismic%20clan">Miracle</a>: <a class="wiki_link" href="/72edo">72edo</a> 10&21 16.742<br /> <a class="wiki_link" href="/Marvel%20temperaments">Wizard</a>: <a class="wiki_link" href="/72edo">72edo</a> 22&50 40.846<br /> <a class="wiki_link" href="/Meantone%20family">Meantone</a>: <a class="wiki_link" href="/81edo">81edo</a> 12&19 13.707<br /> <a class="wiki_link" href="/Starling%20temperaments">Myna</a>: <a class="wiki_link" href="/89edo">89edo</a> 4&27 27.044<br /> <a class="wiki_link" href="/Marvel%20temperaments">Septimin</a>: <a class="wiki_link" href="/91edo">91edo</a> 9&32 54.502<br /> <a class="wiki_link" href="/Schismatic%20family">Garibaldi</a>: <a class="wiki_link" href="/94edo">94edo</a> 12&29 21.644<br /> <a class="wiki_link" href="/Hemimean%20clan">Misty</a>: <a class="wiki_link" href="/99edo">99edo</a> 12&87 36.802<br /> <a class="wiki_link" href="/Starling%20temperaments">Nusecond</a>: <a class="wiki_link" href="/101edo">101edo</a> 31&101 50.389<br /> <a class="wiki_link" href="/Tetracot%20family">Octacot</a>: <a class="wiki_link" href="/109edo">109edo</a> 27&41 33.845<br /> <a class="wiki_link" href="/Hemimean%20clan">Semisept</a>: <a class="wiki_link" href="/111edo">111edo</a> 18&31 50.472<br /> <a class="wiki_link" href="/Gamelismic%20clan">Guiron</a>: <a class="wiki_link" href="/118edo">118edo</a> 36&41 47.544<br /> <a class="wiki_link" href="/Marvel%20temperaments">Tritonic</a>: <a class="wiki_link" href="/122edo">122edo</a> 2&29 47.578<br /> <a class="wiki_link" href="/Marvel%20temperaments">Slender</a>: <a class="wiki_link" href="/125edo">125edo</a> 1&31 56.934<br /> <a class="wiki_link" href="/Gamelismic%20clan">Rodan</a>: <a class="wiki_link" href="/128edo">128edo</a> 5&41 37.112<br /> <a class="wiki_link" href="/Semicomma%20family">Orwell</a>: <a class="wiki_link" href="/137edo">137edo</a> 9&22 20.375<br /> <a class="wiki_link" href="/Breedsmic%20temperaments">Tertiaseptal</a>: <a class="wiki_link" href="/171edo">171edo</a> 31&109 12.995<br /> <a class="wiki_link" href="/Schismatic%20family">Pontiac</a>: <a class="wiki_link" href="/171edo">171edo</a> 53&118 14.133<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Mitonic</a>: <a class="wiki_link" href="/171edo">171edo</a> 46&125 25.184<br /> <a class="wiki_link" href="/Mutt%20family">Mutt</a>: <a class="wiki_link" href="/171edo">171edo</a> 3&84 28.406<br /> <a class="wiki_link" href="/Starling%20temperaments">Valentine</a>: <a class="wiki_link" href="/185edo">185edo</a> 15&31 31.056<br /> <a class="wiki_link" href="/Gamelismic%20clan">Unidec</a>: <a class="wiki_link" href="/190edo">190edo</a> 26&46 38.393<br /> <a class="wiki_link" href="/Kleismic%20family">Catakleismic</a>: <a class="wiki_link" href="/197edo">197edo</a> 19&53 21.501<br /> <a class="wiki_link" href="/Kleismic%20family">Quadritikleismic</a>: <a class="wiki_link" href="/212edo">212edo</a> 4&68 39.231<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Octoid</a>: [224edo]] 72&152 42.670<br /> <a class="wiki_link" href="/Pythagorean%20family">Compton</a>: <a class="wiki_link" href="/228edo">228edo</a> 12&60 35.686<br /> <a class="wiki_link" href="/Kleismic%20family">Tritikleismic</a>: <a class="wiki_link" href="/231edo">231edo</a> 15&57 56.337<br /> <a class="wiki_link" href="/Wuerschmidt%20family">Hemiwuerschmidt</a>: <a class="wiki_link" href="/328edo">328edo</a> 6&31 20.307<br /> <a class="wiki_link" href="/Breedsmic%20temperaments">Hemififths</a>: <a class="wiki_link" href="/338edo">338edo</a> 41&58 22.243<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Amity</a>: <a class="wiki_link" href="/350edo">350edo</a> 7&39 23.649<br /> <a class="wiki_link" href="/Schismatic%20family">Bischismic</a>: <a class="wiki_link" href="/378edo">378edo</a> 12&118 54.744<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Parakleismic</a>: <a class="wiki_link" href="/415edo">415edo</a> 19&80 27.431<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Gamera</a>: <a class="wiki_link" href="/422edo">422edo</a> 26&73 37.648<br /> <a class="wiki_link" href="/Kleismic%20family">Countercata</a>: <a class="wiki_link" href="/473edo">473edo</a> 34&53 52.128<br /> <a class="wiki_link" href="/Breedsmic%20temperaments">Harry</a>: <a class="wiki_link" href="/534edo">534edo</a> 58&72 34.077<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Ennealimmal</a>: <a class="wiki_link" href="/612edo">612edo</a> 27&45 3.610<br /> <a class="wiki_link" href="/Breedsmic%20temperaments">Quasiorwell</a>: <a class="wiki_link" href="/1111edo">1111edo</a> 31&177 35.832<br /> <a class="wiki_link" href="/Schismatic%20family">Sesquiquartififths</a>: <a class="wiki_link" href="/1498edo">1498edo</a> 41&89 11.244<br /> <a class="wiki_link" href="/Schismatic%20family">Term</a>: <a class="wiki_link" href="/1722edo">1722edo</a> 12&159 19.950<br /> <a class="wiki_link" href="/Gammic%20family">Neptune</a>: <a class="wiki_link" href="/1778edo">1778edo</a> 35&68 23.427<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Abigail</a>: <a class="wiki_link" href="/1798edo">1798edo</a> 46&132 37.000<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Enneadecal</a>: <a class="wiki_link" href="/2185edo">2185edo</a> 19&152 10.954<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Supermajor</a>: <a class="wiki_link" href="/6214edo">6214edo</a> 80&171 10.836<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x-7-limit rank three"></a><!-- ws:end:WikiTextHeadingRule:4 -->7-limit rank three</h2> Comma: Optimal patent val: 10^6 * badness<br /> 36/35: <a class="wiki_link" href="/12edo">12edo</a> 63.675<br /> 49/48: <a class="wiki_link" href="/19edo">19edo</a> 115.514<br /> 875/864: <a class="wiki_link" href="/41edo">41edo</a> 212.413<br /> 525/512: <a class="wiki_link" href="/45edo">45edo</a> 476.454<br /> 686/675: <a class="wiki_link" href="/46edo">46edo</a> 320.094<br /> 50/49: <a class="wiki_link" href="/48edo">48edo</a> 127.208<br /> 64/63: <a class="wiki_link" href="/49edo">49edo</a> 98.719<br /> 1029/1000: <a class="wiki_link" href="/55edo">55edo</a> 678.164<br /> 3125/3087: <a class="wiki_link" href="/94edo">94edo</a> 613.731<br /> 1728/1715: <a class="wiki_link" href="/111edo">111edo</a> 187.341<br /> 2430/2401: <a class="wiki_link" href="/137edo">137edo</a> 518.625<br /> 65625/65536: <a class="wiki_link" href="/171edo">171edo</a> 172.601<br /> 126/125: <a class="wiki_link" href="/185edo">185edo</a> 69.867<br /> 1029/1024: <a class="wiki_link" href="/190edo">190edo</a> 176.105<br /> 225/224: <a class="wiki_link" href="/197edo">197edo</a> 36.517<br /> 4000/3969: <a class="wiki_link" href="/215edo">215edo</a> 215.550<br /> 16875/16807: <a class="wiki_link" href="/224edo">224edo</a> 341.931<br /> 245/243: <a class="wiki_link" href="/283edo">283edo</a> 121.191<br /> 6144/6125: <a class="wiki_link" href="/381edo">381edo</a> 142.375<br /> 3136/3125: <a class="wiki_link" href="/446edo">446edo</a> 160.039<br /> 19683/19600: <a class="wiki_link" href="/587edo">587edo</a> 300.987<br /> 10976/10935: <a class="wiki_link" href="/695edo">695edo</a> 199.139<br /> 703125/702464: <a class="wiki_link" href="/2185edo">2185edo</a> 149.828<br /> 2401/2400: <a class="wiki_link" href="/2749edo">2749edo</a> 15.288<br /> 420175/419904: <a class="wiki_link" href="/4306edo">4306edo</a> 86.449<br /> 4375/4374: <a class="wiki_link" href="/8419edo">8419edo</a> 11.665<br /> 250047/250000: <a class="wiki_link" href="/12555edo">12555edo</a> 21.301<br /> 78125000/78121827: <a class="wiki_link" href="/101654edo">101654edo</a> 20.457<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="x-11-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit rank two</h2> Name: Optimal patent val: Val name: 1000*badness<br /> <a class="wiki_link" href="/Jubilismic%20clan">Diminished</a>: <a class="wiki_link" href="/12edo">12edo</a> 4&12 22.132<br /> <a class="wiki_link" href="/Marvel%20temperaments">Negri</a>: <a class="wiki_link" href="/19edo">19edo</a> 9&10 26.190<br /> <a class="wiki_link" href="/Diaschismic%20family">Pajara</a>: <a class="wiki_link" href="/22edo">22edo</a> 12&22 20.343<br /> <a class="wiki_link" href="/Meantone%20family">Mohajira</a>: <a class="wiki_link" href="/31edo">31edo</a> 7&24 26.064<br /> <a class="wiki_link" href="/Meantone%20family">Squares</a>: <a class="wiki_link" href="/31edo">31edo</a> 31&45 56.826<br /> <a class="wiki_link" href="/Starling%20temperaments">Casablanca</a>: <a class="wiki_link" href="/31edo">31edo</a> 31&42 67.291<br /> <a class="wiki_link" href="/Kleismic%20family">Keemun</a>: <a class="wiki_link" href="/34edo">34edo</a> 4&15 27.410<br /> <a class="wiki_link" href="/Schismatic%20family">Cassandra</a>: <a class="wiki_link" href="/41edo">41edo</a> 12&29 23.556<br /> <a class="wiki_link" href="/Gamelismic%20clan">Superkleismic</a>: <a class="wiki_link" href="/41edo">41edo</a> 15&26 25.659<br /> <a class="wiki_link" href="/Starling%20temperaments">Sensi</a>: <a class="wiki_link" href="/46edo">46edo</a> 19&27 37.942<br /> <a class="wiki_link" href="/Jubilismic%20clan">Doublewide</a>: <a class="wiki_link" href="/48edo">48edo</a> 4&18 32.058<br /> <a class="wiki_link" href="/Semicomma%20family">Orwell</a>: <a class="wiki_link" href="/53edo">53edo</a> 9&22 15.231<br /> <a class="wiki_link" href="/Diaschismic%20family">Diaschismic</a>: <a class="wiki_link" href="/58edo">58edo</a> 12&46 25.034<br /> <a class="wiki_link" href="/Breedsmic%20temperaments">Hemififths</a>: <a class="wiki_link" href="/58edo">58edo</a> 41&58 23.498<br /> <a class="wiki_link" href="/Porcupine%20family">Porcupine</a>: <a class="wiki_link" href="/59edo">59edo</a> 7&15 21.562<br /> <a class="wiki_link" href="/Gamelismic%20clan">Miracle</a>: <a class="wiki_link" href="/72edo">72edo</a> 10&31 10.684<br /> <a class="wiki_link" href="/Kleismic%20family">Catakleismic</a>: <a class="wiki_link" href="/72edo">72edo</a> 19&53 21.849<br /> <a class="wiki_link" href="/Pythagorean%20family">Compton</a>: <a class="wiki_link" href="/72edo">72edo</a> 12&72 22.235<br /> <a class="wiki_link" href="/Starling%20temperaments">Valentine</a>: <a class="wiki_link" href="/77edo">77edo</a> 15&31 16.687<br /> <a class="wiki_link" href="/Diaschismic%20family">Keen</a>: <a class="wiki_link" href="/78edo">78edo</a> 22&56 45.270<br /> <a class="wiki_link" href="/Diaschismic%20family">Echidna</a>: <a class="wiki_link" href="/80edo">80edo</a> 22&58 25.987<br /> <a class="wiki_link" href="/Meantone%20family">Meanpop</a>: <a class="wiki_link" href="/81edo">81edo</a> 19&31 21.543<br /> <a class="wiki_link" href="/Gamelismic%20clan">Rodan</a>: <a class="wiki_link" href="/87edo">87edo</a> 5&41 23.093<br /> <a class="wiki_link" href="/Meantone%20family">Mothra</a>: <a class="wiki_link" href="/88edo">88edo</a> 5&26 25.642<br /> <a class="wiki_link" href="/Starling%20temperaments">Myna</a>: <a class="wiki_link" href="/89edo">89edo</a> 31&58 16.842<br /> <a class="wiki_link" href="/Marvel%20temperaments">Septimin</a>: <a class="wiki_link" href="/91edo">91edo</a> 9&32 31.309<br /> <a class="wiki_link" href="/Schismatic%20family">Garibaldi</a>: <a class="wiki_link" href="/94edo">94edo</a> 41&53 27.396<br /> <a class="wiki_link" href="/Starling%20temperaments">Nusecond</a>: <a class="wiki_link" href="/101edo">101edo</a> 31&101 25.621<br /> <a class="wiki_link" href="/Magic%20family">Magic</a>: <a class="wiki_link" href="/104edo">104edo</a> 19&22 25.621<br /> <a class="wiki_link" href="/Meantone%20family">Huygens</a>: <a class="wiki_link" href="/105edo">105edo</a> 12&31 17.027<br /> <a class="wiki_link" href="/Diaschismic%20family">Shrutar</a>: <a class="wiki_link" href="/114edo">114edo</a> 22&46 26.489<br /> <a class="wiki_link" href="/Gamelismic%20clan">Hemithirds</a>: <a class="wiki_link" href="/118edo">118edo</a> 31&87 19.003<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Parakleismic</a>: <a class="wiki_link" href="/118edo">118edo</a> 19&99 49.711<br /> <a class="wiki_link" href="/Marvel%20temperaments">Tritonic</a>: <a class="wiki_link" href="/122edo">122edo</a> 31&91 45.456<br /> <a class="wiki_link" href="/Marvel%20temperaments">Slender</a>: <a class="wiki_link" href="/125edo">125edo</a> 31&63 25.342<br /> <a class="wiki_link" href="/Wuerschmidt%20family">Hemiwuerschmidt</a>: <a class="wiki_link" href="/130edo">130edo</a> 31&130 21.069<br /> <a class="wiki_link" href="/Diaschismic%20family">Echidnic</a>: <a class="wiki_link" href="/148edo">148edo</a> 10&46 45.127<br /> <a class="wiki_link" href="/Mirkwai%20clan">Grendel</a>: <a class="wiki_link" href="/152edo">152edo</a> 31&121 19.845<br /> <a class="wiki_link" href="/Mirkwai%20clan">Kwai</a>: <a class="wiki_link" href="/152edo">152edo</a> 41&111 26.219<br /> <a class="wiki_link" href="/Schismatic%20family">Guiron</a>: <a class="wiki_link" href="/159edo">159edo</a> 41&77 26.648<br /> <a class="wiki_link" href="/Marvel%20temperaments">Wizard</a>: <a class="wiki_link" href="/166edo">166edo</a> 22&50 18.539<br /> <a class="wiki_link" href="/Gamelismic%20clan">Unidec</a>: <a class="wiki_link" href="/190edo">190edo</a> 26&46 15.479<br /> <a class="wiki_link" href="/Breedsmic%20temperaments">Harry</a>: <a class="wiki_link" href="/202edo">202edo</a> 58&72 15.867<br /> <a class="wiki_link" href="/Breedsmic%20temperaments">Tertiaseptal</a>: <a class="wiki_link" href="/202edo">202edo</a> 31&171 35.576<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Octoid</a>: <a class="wiki_link" href="/224edo">224edo</a> 72&152 14.097<br /> <a class="wiki_link" href="/Kleismic%20family">Countercata</a>: <a class="wiki_link" href="/227edo">227edo</a> 34&53 39.770<br /> <a class="wiki_link" href="/Kleismic%20family">Tritikleismic</a>: <a class="wiki_link" href="/231edo">231edo</a> 15&57 19.333<br /> <a class="wiki_link" href="/Breedsmic%20temperaments">Quasiorwell</a>: <a class="wiki_link" href="/270edo">270edo</a> 31&208 17.540<br /> <a class="wiki_link" href="/Marvel%20temperaments">Marvo</a>: <a class="wiki_link" href="/281edo">281edo</a> 72&281 31.685<br /> <a class="wiki_link" href="/Kleismic%20family">Quadritikleismic</a>: <a class="wiki_link" href="/284edo">284edo</a> 4&68 23.406<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Hemiamity</a>: <a class="wiki_link" href="/350edo">350edo</a> 46&106 31.307<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Abigail</a>: <a class="wiki_link" href="/764edo">764edo</a> 46&132 12.860<br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Hemiennealimmal</a>: <a class="wiki_link" href="/1566edo">1566edo</a> 72&198 6.283<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="x-11-limit rank three"></a><!-- ws:end:WikiTextHeadingRule:8 -->11-limit rank three</h2> Name: Optimal patent val: Val name: 10^6 * badness<br /> <a class="wiki_link" href="/Archytas%20family">Vulcan</a>: <a class="wiki_link" href="/15edo">15edo</a> 2&3&10 577.113<br /> <a class="wiki_link" href="/Jubilismic%20family">Festival</a>: <a class="wiki_link" href="/26edo">26edo</a> 2&10&26 688.510<br /> <a class="wiki_link" href="/Starling%20family">Thrasher</a>: <a class="wiki_link" href="/34edo">34edo</a> 3&4&12 480.188<br /> <a class="wiki_link" href="/Didymus%20rank%20three%20family">Calliope</a>: <a class="wiki_link" href="/45edo">45edo</a> 7&12&19 529.778<br /> <a class="wiki_link" href="/Jubilismic%20family">Jubilee</a>: <a class="wiki_link" href="/48edo">48edo</a> 4&12&22 559.842<br /> <a class="wiki_link" href="/Archytas%20family">Ares</a>: <a class="wiki_link" href="/49edo">49edo</a> 3&7&12 468.874<br /> <a class="wiki_link" href="/Porcupine%20rank%20three%20family">Sonic</a>: <a class="wiki_link" href="/51edo">51edo</a> 7&15&51 523.158<br /> <a class="wiki_link" href="/Nuwell%20family">Big Brother</a>: <a class="wiki_link" href="/53edo">53edo</a> 9&22&70 506.556<br /> <a class="wiki_link" href="/Marvel%20family">Prodigy</a>: <a class="wiki_link" href="/72edo">72edo</a> 2&10&29 334.745 <br /> <a class="wiki_link" href="/Didymus%20rank%20three%20family">Erato</a>: <a class="wiki_link" href="/81edo">81edo</a> 12&19&31 557.521<br /> <a class="wiki_link" href="/Didymus%20rank%20three%20family">Euterpe</a>: <a class="wiki_link" href="/88edo">88edo</a> 5&12&26 535.809<br /> <a class="wiki_link" href="/Starling%20family">Thrush</a>: <a class="wiki_link" href="/89edo">89edo</a> 3&12&31 352.591<br /> <a class="wiki_link" href="/Starling%20family">Aplonis</a>: <a class="wiki_link" href="/89edo">89edo</a> 19&31&58 647.511<br /> <a class="wiki_link" href="/Porwell%20family">Zeus</a>: <a class="wiki_link" href="/99edo">99edo</a> 6&7&9 399.882<br /> <a class="wiki_link" href="/Marvel%20family">Apollo</a>: <a class="wiki_link" href="/104edo">104edo</a> 12&19&22 529.766<br /> <a class="wiki_link" href="/Orwellismic%20family">Guanyin</a>: <a class="wiki_link" href="/111edo">111edo</a> 9&22&58 411.252<br /> <a class="wiki_link" href="/Marvel%20family">Minerva</a>: <a class="wiki_link" href="/127edo">127edo</a> 1&9&12 381.064<br /> <a class="wiki_link" href="/Didymus%20rank%20three%20family">Clio</a>: <a class="wiki_link" href="/129edo">129edo</a> 7&12&24 737.590<br /> <a class="wiki_link" href="/Marvel%20family">Marvel</a>: <a class="wiki_link" href="/166edo">166edo</a> 9&10&22 255.033<br /> <a class="wiki_link" href="/Gamelismic%20family">Portent</a>: <a class="wiki_link" href="/190edo">190edo</a> 5&10&26 234.092<br /> <a class="wiki_link" href="/Breed%20family">Jove</a>: <a class="wiki_link" href="/202edo">202edo</a> 10&31&58 241.055<br /> <a class="wiki_link" href="/Marvel%20family">Spectacle</a>: <a class="wiki_link" href="/281edo">281edo</a> 10&31&281 498.669<br /> <a class="wiki_link" href="/Porwell%20family">Jupiter</a>: <a class="wiki_link" href="/282edo">282edo</a> 9&22&108 562.368<br /> <a class="wiki_link" href="/Breed%20family">Agni</a>: <a class="wiki_link" href="/284edo">284edo</a> 4&6&27 493.790<br /> <a class="wiki_link" href="/Werckismic%20temperaments">Varuna</a>: <a class="wiki_link" href="/320edo">320edo</a> 12&26&46 418.714<br /> <a class="wiki_link" href="/Werckismic%20temperaments">Babar</a>: <a class="wiki_link" href="/491edo">491edo</a> 15&29&43 603.636<br /> <a class="wiki_link" href="/Swetismic%20temperaments">Hades</a>: <a class="wiki_link" href="/578edo">578edo</a> 22&50&58 361.398<br /> <a class="wiki_link" href="/Cataharry%20family">Albus</a>: <a class="wiki_link" href="/640edo">640edo</a> 19&53&58 621.100<br /> <a class="wiki_link" href="/Mirkwai%20family">Indra</a>: <a class="wiki_link" href="/703edo">703edo</a> 10&31&152 266.020<br /> <a class="wiki_link" href="/Kalismic%20temperaments">Loki</a>: <a class="wiki_link" href="/1092edo">1092edo</a> 12&22&86 492.550<br /> <a class="wiki_link" href="/Lehmerismic%20temperaments">Ganesha</a>: <a class="wiki_link" href="/1502edo">1502edo</a><br /> <a class="wiki_link" href="/Breed%20family">Freya</a>: <a class="wiki_link" href="/1566edo">1566edo</a> 10&31&167 169.702<br /> <a class="wiki_link" href="/Lehmerismic%20temperaments">Tyr</a>: <a class="wiki_link" href="/1566edo">1566edo</a> 15&57&183 258.374<br /> <a class="wiki_link" href="/Breed%20family">Baldur</a>: <a class="wiki_link" href="/1696edo">1696edo</a> 4&58&72 166.115<br /> <a class="wiki_link" href="/Lehmerismic%20temperaments">Skadi</a>: <a class="wiki_link" href="/2014edo">2014edo</a> 31&97&121 348.944<br /> <a class="wiki_link" href="/Ragisma%20family">Thor</a>: <a class="wiki_link" href="/2554edo">2554edo</a> 26&46&152 88.834<br /> <a class="wiki_link" href="/Breed%20family">Ennealimmic</a>: <a class="wiki_link" href="/2619edo">2619edo</a> 27&45&171 274.685<br /> <a class="wiki_link" href="/Kalismic%20temperaments">Odin</a>: <a class="wiki_link" href="/7464edo">7464edo</a> 12&42&72 116.151<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="x-13-limit rank two"></a><!-- ws:end:WikiTextHeadingRule:10 -->13-limit rank two</h2> <a class="wiki_link" href="/Marvel%20temperaments">Negri</a>: <a class="wiki_link" href="/19edo">19edo</a> 9&10 17.833<br /> <a class="wiki_link" href="/Starling%20temperaments">Sensis</a>: <a class="wiki_link" href="/19edo">19edo</a> [<1 6 8 11 6 10|, <0 -8 -9 -13 -4 -10|] 20.017 <br /> <a class="wiki_link" href="/Augmented%20family">August</a>: <a class="wiki_link" href="/21edo">21edo</a> 9&21 18.448<br /> <a class="wiki_link" href="/Meantone%20family">Huygens</a>: <a class="wiki_link" href="/31edo">31edo</a> [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|] 18.048<br /> <a class="wiki_link" href="/Semicomma%20family">Winston</a>: <a class="wiki_link" href="/31edo">31edo</a><br /> <a class="wiki_link" href="/Gamelismic%20clan">Miracle</a>: <a class="wiki_link" href="/41edo">41edo</a> 10&31 18.669</body></html>