Optimal patent val
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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== [[Trienstonic clan|Father]]: [[5edo]] [[Archytas clan|Mother]]: [[5edo]] [[Meantone family|Sharptone]]: [[5edo]] [[Dicot family|Dicot]]: [[7edo]] [[Bug family|Beep]]: [[9edo]] [[Dicot family|Sharp]]: [[10edo]] [[Dicot family|Decimal]]: [[10edo]] [[Meantone family|Dominant]]: [[12edo]] [[Jubilismic clan|Diminished]]: [[12edo]] [[Augmented family|August]]: [[12edo]] [[Archytas clan|Blacksmith]]: [[15edo]] [[Trienstonic clan|Octokaidecal]]: [[18edo]] [[Marvel temperaments|Negri]]: [[19edo]] [[Meantone family|Godzilla]]: [[19edo]] [[Kleismic family|Keemun]]: [[19edo]] [[Diaschismic family|Pajara]]: [[22edo]] [[Augmented family|Augene]]: [[27edo]] [[Meantone family|Mothra]]: [[31edo]] [[Meantone family|Injera]]: [[38edo]] [[Magic family|Magic]]: [[41edo]] [[Starling temperaments|Sensi]]: [[46edo]] [[Archytas clan|Superpyth]]: [[49edo]] [[Gamelismic clan|Miracle]]: [[72edo]] [[Meantone family|Meantone]]: [[81edo]] [[Starling temperaments|Myna]]: [[89edo]] [[Schismatic family|Garibaldi]]: [[94edo]] [[Hemimean clan|Misty]]: [[99edo]] [[Tetracot family|Octacot]]: [[109edo]] [[Gamelismic clan|Rodan]]: [[128edo]] [[Semicomma family|Orwell]]: [[137edo]] [[Breedsmic temperaments|Tertiaseptal]]: [[171edo]] [[Schismatic family|Pontiac]]: [[171edo]] [[Ragismic microtemperaments|Mitonic]]: [[171edo]] [[Mutt family|Mutt]]: [[171edo]] [[Starling temperaments|Valentine]]: [[185edo]] [[Kleismic family|Catakleismic]]: [[197edo]] [[Pythagorean family|Compton]]: [[228edo]] [[Wuerschmidt family|Hemiwuerschmidt]]: [[328edo]] [[Breedsmic temperaments|Hemififths]]: [[338edo]] [[Ragismic microtemperaments|Amity]]: [[350edo]] [[Ragismic microtemperaments|Parakleismic]]: [[415edo]] [[Ragismic microtemperaments|Gamera]]: [[422edo]] [[Breedsmic temperaments|Harry]]: [[534edo]] [[Ragismic microtemperaments|Ennealimmal]]: [[612edo]] [[Breedsmic temperaments|Quasiorwell]]: [[1111edo]] [[Schismatic family|Sesquiquartififths]]: [[1498edo]] [[Schismatic family|Term]]: [[1722edo]] [[Gammic family|Neptune]]: [[1778edo]] [[Ragismic microtemperaments|Enneadecal]]: [[2185edo]] [[Ragismic microtemperaments|Supermajor]]: [[6214edo]] ==7-limit rank three== 36/35: [[12edo]] 49/48: [[19edo]] 875/864: [[41edo]] 525/512: [[45edo]] 686/675: [[46edo]] 50/49: [[48edo]] 64/63: [[49edo]] 1029/1000: [[55edo]] 3125/3087: [[94edo]] 1728/1715: [[111edo]] 2430/2401: [[137edo]] 65625/65536: [[171edo]] 126/125: [[185edo]] 1029/1024: [[190edo]] 225/224: [[197edo]] 4000/3969: [[215edo]] 16875/16807: [[224edo]] 245/243: [[283edo]] 6144/6125: [[381edo]] 3136/3125: [[446edo]] 19683/19600: [[587edo]] 10976/10935: [[695edo]] 703125/702464: [[2185edo]] 2401/2400: [[2749edo]] 420175/419904: [[4306edo]] 4375/4374: [[8419edo]] 250047/250000: [[12555edo]] 78125000/78121827: [[101654edo]] ==11-limit rank two== [[Diaschismic family|Pajara]]: [[22edo]] [[Meantone family|Mohajira]]: [[31edo]] [[Marvel temperaments|Tritonic]]: [[31edo]] [[Schismatic family|Schismic]]: [[41edo]] [[Gamelismic clan|Superkleismic]]: [[41edo]] [[Jubilismic clan|Doublewide]]: [[48edo]] [[Semicomma family|Orwell]]: [[53edo]] [[Diaschismic family|Diaschismic]]: [[58edo]] [[Porcupine family|Porcupine]]: [[59edo]] [[Marvel temperaments|Miracle]]: [[72edo]] [[Kleismic family|Catakleismic]]: [[72edo]] [[Pythagorean family|Compton]]: [[72edo]] [[Starling temperaments|Valentine]]: [[77edo]] [[Meantone family|Meanpop]]: [[81edo]] [[Gamelismic clan|Rodan]]: [[87edo]] [[Meantone family|Mothra]]: [[88edo]] [[Starling temperaments|Myna]]: [[89edo]] [[Schismatic family|Garibaldi]]: [[94edo]] [[Magic family|Magic]]: [[104edo]] [[Meantone family|Huyghens]]: [[105edo]] [[Gamelismic clan|Hemithirds]]: [[118edo]] [[Marvel temperaments|Slender]]: [[125edo]] [[Wuerschmidt family|Hemiwuerschmidt]]: [[130edo]] [[Mirkwai clan|Grendel]]: [[152edo]] [[Marvel temperaments|Wizard]]: [[166edo]] [[Gamelismic clan|Unidec]]: [[190edo]] [[Breedsmic temperaments|Harry]]: [[202edo]] [[Ragismic microtemperaments|Octoid]]: [[224edo]] [[Kleismic family|Tritikleismic]]: [[231edo]] [[Ragismic microtemperaments|Hemiamity]]: [[350edo]] [[Ragismic microtemperaments|Hemiennealimmal]]: [[1566edo]] ==11-limit rank three== [[Starling family|Thrasher]]: [[34edo]] [[Didymus rank three family|Calliope]]: [[45edo]] [[Archytas family|Ares]]: [[49edo]] [[Porcupine rank three family|Sonic]]: [[51edo]] [[Nuwell family|Big Brother]]: [[53edo]] [[Marvel family|Prodigy]]: [[72edo]] [[Didymus rank three family|Euterpe]]: [[88edo]] [[Starling family|Thrush]]: [[89edo]] [[Starling family|Aplonis]]: [[89edo]] [[Porwell family|Zeus]]: [[99edo]] [[Marvel family|Apollo]]: [[104edo]] [[Orwellismic family|Guanyin]]: [[111edo]] [[Marvel family|Minerva]]: [[127edo]] [[Didymus rank three family|Clio]]: [[129edo]] [[Marvel family|Unimarv]]: [[166edo]] [[Gamelismic family|Portent]]: [[190edo]] [[Breed family|Jove]]: [[202edo]] [[Marvel family|Spectacle]]: [[281edo]] [[Breed family|Agni]]: [[284edo]] [[Werckismic temperaments|Varuna]]: [[320edo]] [[Swetismic temperaments|Hades]]: [[578edo]] [[Cataharry family|Albus]]: [[640edo]] [[Mirkwai family|Indra]]: [[703edo]] [[Breed family|Freya]]: [[1566edo]] [[Breed family|Vili]]: [[1583edo]] [[Breed family|Baldur]]: [[1696edo]] [[Ragisma family|Thor]]: [[2554edo]] [[Odin]]: [[7464edo]] ==13-limit rank two== [[Marvel temperaments|Negri]]: [[19edo]] [[Starling temperaments|Sensis]]: [[19edo]] [[Meantone family|Huyghens]]: [[31edo]] [[Semicomma family|Winston]]: [[31edo]]
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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> <a class="wiki_link" href="/Trienstonic%20clan">Father</a>: <a class="wiki_link" href="/5edo">5edo</a><br /> <a class="wiki_link" href="/Archytas%20clan">Mother</a>: <a class="wiki_link" href="/5edo">5edo</a><br /> <a class="wiki_link" href="/Meantone%20family">Sharptone</a>: <a class="wiki_link" href="/5edo">5edo</a><br /> <a class="wiki_link" href="/Dicot%20family">Dicot</a>: <a class="wiki_link" href="/7edo">7edo</a><br /> <a class="wiki_link" href="/Bug%20family">Beep</a>: <a class="wiki_link" href="/9edo">9edo</a><br /> <a class="wiki_link" href="/Dicot%20family">Sharp</a>: <a class="wiki_link" href="/10edo">10edo</a><br /> <a class="wiki_link" href="/Dicot%20family">Decimal</a>: <a class="wiki_link" href="/10edo">10edo</a><br /> <a class="wiki_link" href="/Meantone%20family">Dominant</a>: <a class="wiki_link" href="/12edo">12edo</a><br /> <a class="wiki_link" href="/Jubilismic%20clan">Diminished</a>: <a class="wiki_link" href="/12edo">12edo</a><br /> <a class="wiki_link" href="/Augmented%20family">August</a>: <a class="wiki_link" href="/12edo">12edo</a><br /> <a class="wiki_link" href="/Archytas%20clan">Blacksmith</a>: <a class="wiki_link" href="/15edo">15edo</a><br /> <a class="wiki_link" href="/Trienstonic%20clan">Octokaidecal</a>: <a class="wiki_link" href="/18edo">18edo</a><br /> <a class="wiki_link" href="/Marvel%20temperaments">Negri</a>: <a class="wiki_link" href="/19edo">19edo</a><br /> <a class="wiki_link" href="/Meantone%20family">Godzilla</a>: <a class="wiki_link" href="/19edo">19edo</a><br /> <a class="wiki_link" href="/Kleismic%20family">Keemun</a>: <a class="wiki_link" href="/19edo">19edo</a><br /> <a class="wiki_link" href="/Diaschismic%20family">Pajara</a>: <a class="wiki_link" href="/22edo">22edo</a><br /> <a class="wiki_link" href="/Augmented%20family">Augene</a>: <a class="wiki_link" href="/27edo">27edo</a><br /> <a class="wiki_link" href="/Meantone%20family">Mothra</a>: <a class="wiki_link" href="/31edo">31edo</a><br /> <a class="wiki_link" href="/Meantone%20family">Injera</a>: <a class="wiki_link" href="/38edo">38edo</a><br /> <a class="wiki_link" href="/Magic%20family">Magic</a>: <a class="wiki_link" href="/41edo">41edo</a><br /> <a class="wiki_link" href="/Starling%20temperaments">Sensi</a>: <a class="wiki_link" href="/46edo">46edo</a><br /> <a class="wiki_link" href="/Archytas%20clan">Superpyth</a>: <a class="wiki_link" href="/49edo">49edo</a><br /> <a class="wiki_link" href="/Gamelismic%20clan">Miracle</a>: <a class="wiki_link" href="/72edo">72edo</a><br /> <a class="wiki_link" href="/Meantone%20family">Meantone</a>: <a class="wiki_link" href="/81edo">81edo</a><br /> <a class="wiki_link" href="/Starling%20temperaments">Myna</a>: <a class="wiki_link" href="/89edo">89edo</a><br /> <a class="wiki_link" href="/Schismatic%20family">Garibaldi</a>: <a class="wiki_link" href="/94edo">94edo</a><br /> <a class="wiki_link" href="/Hemimean%20clan">Misty</a>: <a class="wiki_link" href="/99edo">99edo</a><br /> <a class="wiki_link" href="/Tetracot%20family">Octacot</a>: <a class="wiki_link" href="/109edo">109edo</a><br /> <a class="wiki_link" href="/Gamelismic%20clan">Rodan</a>: <a class="wiki_link" href="/128edo">128edo</a><br /> <a class="wiki_link" href="/Semicomma%20family">Orwell</a>: <a class="wiki_link" href="/137edo">137edo</a><br /> <a class="wiki_link" href="/Breedsmic%20temperaments">Tertiaseptal</a>: <a class="wiki_link" href="/171edo">171edo</a><br /> <a class="wiki_link" href="/Schismatic%20family">Pontiac</a>: <a class="wiki_link" href="/171edo">171edo</a><br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Mitonic</a>: <a class="wiki_link" href="/171edo">171edo</a><br /> <a class="wiki_link" href="/Mutt%20family">Mutt</a>: <a class="wiki_link" href="/171edo">171edo</a><br /> <a class="wiki_link" href="/Starling%20temperaments">Valentine</a>: <a class="wiki_link" href="/185edo">185edo</a><br /> <a class="wiki_link" href="/Kleismic%20family">Catakleismic</a>: <a class="wiki_link" href="/197edo">197edo</a><br /> <a class="wiki_link" href="/Pythagorean%20family">Compton</a>: <a class="wiki_link" href="/228edo">228edo</a><br /> <a class="wiki_link" href="/Wuerschmidt%20family">Hemiwuerschmidt</a>: <a class="wiki_link" href="/328edo">328edo</a><br /> <a class="wiki_link" href="/Breedsmic%20temperaments">Hemififths</a>: <a class="wiki_link" href="/338edo">338edo</a><br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Amity</a>: <a class="wiki_link" href="/350edo">350edo</a><br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Parakleismic</a>: <a class="wiki_link" href="/415edo">415edo</a><br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Gamera</a>: <a class="wiki_link" href="/422edo">422edo</a><br /> <a class="wiki_link" href="/Breedsmic%20temperaments">Harry</a>: <a class="wiki_link" href="/534edo">534edo</a><br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Ennealimmal</a>: <a class="wiki_link" href="/612edo">612edo</a><br /> <a class="wiki_link" href="/Breedsmic%20temperaments">Quasiorwell</a>: <a class="wiki_link" href="/1111edo">1111edo</a><br /> <a class="wiki_link" href="/Schismatic%20family">Sesquiquartififths</a>: <a class="wiki_link" href="/1498edo">1498edo</a><br /> <a class="wiki_link" href="/Schismatic%20family">Term</a>: <a class="wiki_link" href="/1722edo">1722edo</a><br /> <a class="wiki_link" href="/Gammic%20family">Neptune</a>: <a class="wiki_link" href="/1778edo">1778edo</a><br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Enneadecal</a>: <a class="wiki_link" href="/2185edo">2185edo</a><br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Supermajor</a>: <a class="wiki_link" href="/6214edo">6214edo</a><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> 36/35: <a class="wiki_link" href="/12edo">12edo</a><br /> 49/48: <a class="wiki_link" href="/19edo">19edo</a><br /> 875/864: <a class="wiki_link" href="/41edo">41edo</a><br /> 525/512: <a class="wiki_link" href="/45edo">45edo</a><br /> 686/675: <a class="wiki_link" href="/46edo">46edo</a><br /> 50/49: <a class="wiki_link" href="/48edo">48edo</a><br /> 64/63: <a class="wiki_link" href="/49edo">49edo</a><br /> 1029/1000: <a class="wiki_link" href="/55edo">55edo</a><br /> 3125/3087: <a class="wiki_link" href="/94edo">94edo</a><br /> 1728/1715: <a class="wiki_link" href="/111edo">111edo</a><br /> 2430/2401: <a class="wiki_link" href="/137edo">137edo</a><br /> 65625/65536: <a class="wiki_link" href="/171edo">171edo</a><br /> 126/125: <a class="wiki_link" href="/185edo">185edo</a><br /> 1029/1024: <a class="wiki_link" href="/190edo">190edo</a><br /> 225/224: <a class="wiki_link" href="/197edo">197edo</a><br /> 4000/3969: <a class="wiki_link" href="/215edo">215edo</a><br /> 16875/16807: <a class="wiki_link" href="/224edo">224edo</a><br /> 245/243: <a class="wiki_link" href="/283edo">283edo</a><br /> 6144/6125: <a class="wiki_link" href="/381edo">381edo</a><br /> 3136/3125: <a class="wiki_link" href="/446edo">446edo</a><br /> 19683/19600: <a class="wiki_link" href="/587edo">587edo</a><br /> 10976/10935: <a class="wiki_link" href="/695edo">695edo</a><br /> 703125/702464: <a class="wiki_link" href="/2185edo">2185edo</a><br /> 2401/2400: <a class="wiki_link" href="/2749edo">2749edo</a><br /> 420175/419904: <a class="wiki_link" href="/4306edo">4306edo</a><br /> 4375/4374: <a class="wiki_link" href="/8419edo">8419edo</a><br /> 250047/250000: <a class="wiki_link" href="/12555edo">12555edo</a><br /> 78125000/78121827: <a class="wiki_link" href="/101654edo">101654edo</a><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> <a class="wiki_link" href="/Diaschismic%20family">Pajara</a>: <a class="wiki_link" href="/22edo">22edo</a><br /> <a class="wiki_link" href="/Meantone%20family">Mohajira</a>: <a class="wiki_link" href="/31edo">31edo</a><br /> <a class="wiki_link" href="/Marvel%20temperaments">Tritonic</a>: <a class="wiki_link" href="/31edo">31edo</a><br /> <a class="wiki_link" href="/Schismatic%20family">Schismic</a>: <a class="wiki_link" href="/41edo">41edo</a><br /> <a class="wiki_link" href="/Gamelismic%20clan">Superkleismic</a>: <a class="wiki_link" href="/41edo">41edo</a><br /> <a class="wiki_link" href="/Jubilismic%20clan">Doublewide</a>: <a class="wiki_link" href="/48edo">48edo</a><br /> <a class="wiki_link" href="/Semicomma%20family">Orwell</a>: <a class="wiki_link" href="/53edo">53edo</a><br /> <a class="wiki_link" href="/Diaschismic%20family">Diaschismic</a>: <a class="wiki_link" href="/58edo">58edo</a><br /> <a class="wiki_link" href="/Porcupine%20family">Porcupine</a>: <a class="wiki_link" href="/59edo">59edo</a><br /> <a class="wiki_link" href="/Marvel%20temperaments">Miracle</a>: <a class="wiki_link" href="/72edo">72edo</a><br /> <a class="wiki_link" href="/Kleismic%20family">Catakleismic</a>: <a class="wiki_link" href="/72edo">72edo</a><br /> <a class="wiki_link" href="/Pythagorean%20family">Compton</a>: <a class="wiki_link" href="/72edo">72edo</a><br /> <a class="wiki_link" href="/Starling%20temperaments">Valentine</a>: <a class="wiki_link" href="/77edo">77edo</a><br /> <a class="wiki_link" href="/Meantone%20family">Meanpop</a>: <a class="wiki_link" href="/81edo">81edo</a><br /> <a class="wiki_link" href="/Gamelismic%20clan">Rodan</a>: <a class="wiki_link" href="/87edo">87edo</a><br /> <a class="wiki_link" href="/Meantone%20family">Mothra</a>: <a class="wiki_link" href="/88edo">88edo</a><br /> <a class="wiki_link" href="/Starling%20temperaments">Myna</a>: <a class="wiki_link" href="/89edo">89edo</a><br /> <a class="wiki_link" href="/Schismatic%20family">Garibaldi</a>: <a class="wiki_link" href="/94edo">94edo</a><br /> <a class="wiki_link" href="/Magic%20family">Magic</a>: <a class="wiki_link" href="/104edo">104edo</a><br /> <a class="wiki_link" href="/Meantone%20family">Huyghens</a>: <a class="wiki_link" href="/105edo">105edo</a><br /> <a class="wiki_link" href="/Gamelismic%20clan">Hemithirds</a>: <a class="wiki_link" href="/118edo">118edo</a><br /> <a class="wiki_link" href="/Marvel%20temperaments">Slender</a>: <a class="wiki_link" href="/125edo">125edo</a><br /> <a class="wiki_link" href="/Wuerschmidt%20family">Hemiwuerschmidt</a>: <a class="wiki_link" href="/130edo">130edo</a><br /> <a class="wiki_link" href="/Mirkwai%20clan">Grendel</a>: <a class="wiki_link" href="/152edo">152edo</a><br /> <a class="wiki_link" href="/Marvel%20temperaments">Wizard</a>: <a class="wiki_link" href="/166edo">166edo</a><br /> <a class="wiki_link" href="/Gamelismic%20clan">Unidec</a>: <a class="wiki_link" href="/190edo">190edo</a><br /> <a class="wiki_link" href="/Breedsmic%20temperaments">Harry</a>: <a class="wiki_link" href="/202edo">202edo</a><br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Octoid</a>: <a class="wiki_link" href="/224edo">224edo</a><br /> <a class="wiki_link" href="/Kleismic%20family">Tritikleismic</a>: <a class="wiki_link" href="/231edo">231edo</a><br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Hemiamity</a>: <a class="wiki_link" href="/350edo">350edo</a><br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Hemiennealimmal</a>: <a class="wiki_link" href="/1566edo">1566edo</a><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> <a class="wiki_link" href="/Starling%20family">Thrasher</a>: <a class="wiki_link" href="/34edo">34edo</a><br /> <a class="wiki_link" href="/Didymus%20rank%20three%20family">Calliope</a>: <a class="wiki_link" href="/45edo">45edo</a><br /> <a class="wiki_link" href="/Archytas%20family">Ares</a>: <a class="wiki_link" href="/49edo">49edo</a><br /> <a class="wiki_link" href="/Porcupine%20rank%20three%20family">Sonic</a>: <a class="wiki_link" href="/51edo">51edo</a><br /> <a class="wiki_link" href="/Nuwell%20family">Big Brother</a>: <a class="wiki_link" href="/53edo">53edo</a><br /> <a class="wiki_link" href="/Marvel%20family">Prodigy</a>: <a class="wiki_link" href="/72edo">72edo</a><br /> <a class="wiki_link" href="/Didymus%20rank%20three%20family">Euterpe</a>: <a class="wiki_link" href="/88edo">88edo</a><br /> <a class="wiki_link" href="/Starling%20family">Thrush</a>: <a class="wiki_link" href="/89edo">89edo</a><br /> <a class="wiki_link" href="/Starling%20family">Aplonis</a>: <a class="wiki_link" href="/89edo">89edo</a><br /> <a class="wiki_link" href="/Porwell%20family">Zeus</a>: <a class="wiki_link" href="/99edo">99edo</a><br /> <a class="wiki_link" href="/Marvel%20family">Apollo</a>: <a class="wiki_link" href="/104edo">104edo</a><br /> <a class="wiki_link" href="/Orwellismic%20family">Guanyin</a>: <a class="wiki_link" href="/111edo">111edo</a><br /> <a class="wiki_link" href="/Marvel%20family">Minerva</a>: <a class="wiki_link" href="/127edo">127edo</a><br /> <a class="wiki_link" href="/Didymus%20rank%20three%20family">Clio</a>: <a class="wiki_link" href="/129edo">129edo</a><br /> <a class="wiki_link" href="/Marvel%20family">Unimarv</a>: <a class="wiki_link" href="/166edo">166edo</a><br /> <a class="wiki_link" href="/Gamelismic%20family">Portent</a>: <a class="wiki_link" href="/190edo">190edo</a><br /> <a class="wiki_link" href="/Breed%20family">Jove</a>: <a class="wiki_link" href="/202edo">202edo</a><br /> <a class="wiki_link" href="/Marvel%20family">Spectacle</a>: <a class="wiki_link" href="/281edo">281edo</a><br /> <a class="wiki_link" href="/Breed%20family">Agni</a>: <a class="wiki_link" href="/284edo">284edo</a><br /> <a class="wiki_link" href="/Werckismic%20temperaments">Varuna</a>: <a class="wiki_link" href="/320edo">320edo</a><br /> <a class="wiki_link" href="/Swetismic%20temperaments">Hades</a>: <a class="wiki_link" href="/578edo">578edo</a><br /> <a class="wiki_link" href="/Cataharry%20family">Albus</a>: <a class="wiki_link" href="/640edo">640edo</a><br /> <a class="wiki_link" href="/Mirkwai%20family">Indra</a>: <a class="wiki_link" href="/703edo">703edo</a><br /> <a class="wiki_link" href="/Breed%20family">Freya</a>: <a class="wiki_link" href="/1566edo">1566edo</a><br /> <a class="wiki_link" href="/Breed%20family">Vili</a>: <a class="wiki_link" href="/1583edo">1583edo</a><br /> <a class="wiki_link" href="/Breed%20family">Baldur</a>: <a class="wiki_link" href="/1696edo">1696edo</a><br /> <a class="wiki_link" href="/Ragisma%20family">Thor</a>: <a class="wiki_link" href="/2554edo">2554edo</a><br /> <a class="wiki_link" href="/Odin">Odin</a>: <a class="wiki_link" href="/7464edo">7464edo</a><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><br /> <a class="wiki_link" href="/Starling%20temperaments">Sensis</a>: <a class="wiki_link" href="/19edo">19edo</a><br /> <a class="wiki_link" href="/Meantone%20family">Huyghens</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></body></html>