Optimal patent val
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-02-16 17:25:38 UTC.
- The original revision id was 202522754.
- 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== 27/25: [[14edo]] 16/15: [[8edo]] 135/128: [[23edo]] 25/24: [[17edo]] 648/625: [[12edo]] 250/243: [[22edo]] 128/125: [[39edo]] 3125/3072: [[60edo]] 81/80: [[81edo]] 2048/2025: [[80edo]] 78732/78125: [[539edo]] 393216/390625: [[164edo]] 2109375/2097152: [[296edo]] 15625/15552: [[458edo]] 1600000/1594323: [[873edo]] 1224440064/1220703125: [[1496edo]] 6115295232/6103515625: [[1400edo]] 32805/32768: [[749edo]] 274877906944/274658203125: [[1559edo]] 7629394531250/7625597484987: [[3501edo]] ==7-limit rank two== [[Ragismic microtemperaments|Ennealimmal]]: [[612edo]] [[Ragismic microtemperaments|Supermajor]]: [[6214edo]] [[Ragismic microtemperaments|Enneadecal]]: [[2185edo]] [[Schismatic family|Sesquiquartififths]]: [[1498edo]] [[Breedsmic temperaments|Tertiaseptal]]: [[171edo]] [[Meantone family|Meantone]]: [[81edo]] [[Schismatic family|Pontiac]]: [[171edo]] [[Gamelismic clan|Miracle]]: [[72edo]] [[Bug family|Beep]]: [[9edo]] [[Magic family|Magic]]: [[41edo]] [[Dicot family|Dicot]]: [[7edo]] [[Schismatic family|Term]]: [[1722edo]] [[Diaschismic family|Pajara]]: [[22edo]] [[Wuerschmidt family|Hemiwuerschmidt]]: [[328edo]] [[Meantone family|Dominant]]: [[12edo]] [[Semicomma family|Orwell]]: [[137edo]] [[Trienstonic clan|Father]]: [[5edo]] [[Kleismic family|Catakleismic]]: [[197edo]] [[Schismatic family|Garibaldi]]: [[94edo]] [[Breedsmic temperaments|Hemififths]]: [[338edo]] [[Diminished]]: [[12edo]] [[Gammic family|Neptune]]: [[1778edo]] [[Ragismic microtemperaments|Amity]]: [[350edo]] [[Archytas clan|Mother]]: [[5edo]] [[Augmented family|Augene]]: [[27edo]] [[Meantone family|Sharptone]]: [[5edo]] [[Ragismic microtemperaments|Mitonic]]: [[171edo]] [[Starling temperaments|Sensi]]: [[46edo]] [[Archytas clan|Blacksmith]]: [[15edo]] [[Augmented family|August]]: [[12edo]] [[Marvel temperaments|Negri]]: [[19edo]] [[Meantone family|Godzilla]]: [[19edo]] [[Starling temperaments|Myna]]: [[89edo]] [[Kleismic family|Keemun]]: [[19edo]] [[Ragismic microtemperaments|Parakleismic]]: [[415edo]] [[Dicot family|Decimal]]: [[10edo]] [[Mutt family|Mutt]]: [[171edo]] [[Dicot family|Sharp]]: [[10edo]] [[Starling temperaments|Valentine]]: [[185edo]] [[Meantone family|Injera]]: [[38edo]] [[Archytas clan|Superpyth]]: [[49edo]] [[Tetracot family|Octacot]]: [[109edo]] [[Breedsmic temperaments|Harry]]: [[534edo]] [[Pythagorean family|Compton]]: [[228edo]] [[Breedsmic temperaments|Quasiorwell]]: [[1111edo]] [[Trienstonic clan|Octokaidecal]]: [[18edo]] [[Hemimean clan|Misty]]: [[99edo]] [[Gamelismic clan|Rodan]]: [[128edo]] [[Meantone family|Mothra]]: [[31edo]] [[Gamera]]: [[422edo]] ==7-limit rank three== 1029/1000: [[55edo]] 36/35: [[12edo]] 525/512: [[45edo]] 49/48: [[19edo]] 50/49: [[48edo]] 686/675: [[46edo]] 64/63: [[49edo]] 875/864: [[41edo]] 3125/3087: [[94edo]] 2430/2401: [[137edo]] 245/243: [[283edo]] 126/125: [[185edo]] 4000/3969: [[215edo]] 1728/1715: [[111edo]] 1029/1024: [[190edo]] 225/224: [[197edo]] 19683/19600: [[587edo]] 16875/16807: [[224edo]] 10976/10935: [[695edo]] 3136/3125: [[446edo]] 6144/6125: [[381edo]] 65625/65536: [[171edo]] 703125/702464: [[2185edo]] 420175/419904: [[4306edo]] 2401/2400: [[2749edo]] 4375/4374: [[8419edo]] 250047/250000: [[12555edo]] 78125000/78121827: [[101654edo]] ==11-limit rank two== [[Ragismic microtemperaments|Hemiennealimmal]]: [[1566edo]] [[Octoid]]: [[224edo]] [[Hemiamity]]: [[350edo]] [[Mirkwai clan|Grendel]]: [[152edo]] [[Gamelismic clan|Unidec]]: [[190edo]] [[Minorsemi]]: [[231edo]] [[Breedsmic temperaments|Harry]]: [[202edo]] [[Marvel temperaments|Wizard]]: [[166edo]] [[Kleismic family|Catakleismic]]: [[72edo]] [[Wuerschmidt family|Hemiwuerschmidt]]: [[130edo]] [[Gamelismic clan|Hemithirds]]: [[118edo]] [[Pythagorean family|Compton]]: [[72edo]] [[Marvel temperaments|Miracle]]: [[72edo]] [[Marvel temperaments|Slender]]: [[125edo]] [[Schismatic family|Garibaldi]]: [[94edo]] [[Gamelismic clan|Rodan]]: [[87edo]] [[Diaschismic family|Diaschismic]]: [[58edo]] [[Starling temperaments|Myna]]: [[89edo]] [[Starling temperaments|Valentine]]: [[77edo]] [[Magic family|Magic]]: [[104edo]] [[Marvel temperaments|Tritonic]]: [[31edo]] [[Schismatic family|Schismic]]: [[41edo]] [[Superkleismic]]: [[41edo]] [[Semicomma family|Orwell]]: [[53edo]] [[Meantone family|Meanpop]]: [[81edo]] [[Meantone family|Mothra]]: [[88edo]] [[Meantone family|Huyghens]]: [[105edo]] [[Meantone family|Mohajira]]: [[31edo]] [[Jubilismic clan|Doublewide]]: [[48edo]] [[Diaschismic family|Pajara]]: [[22edo]] [[Porcupine family|Porcupine]]: [[59edo]] ==11-limit rank three== [[Breed family|Jove]]: [[202edo]] [[Gamelismic family|Portent]]: [[190edo]] [[Didymus rank three family|Euterpe]]: [[88edo]] [[Didymus rank three family|Calliope]]: [[45edo]] [[Didymus rank three family|Clio]]: [[129edo]] [[Breed family|Freya]]: [[1566edo]] [[Marvel family|Unimarv]]: [[166edo]] [[Porwell family|Zeus]]: [[99edo]] [[Mirkwai family|Indra]]: [[703edo]] [[Starling family|Thrush]]: [[89edo]] [[Marvel family|Prodigy]]: [[72edo]] [[Marvel family|Minerva]]: [[127edo]] [[Odin]]: [[7464edo]] [[Ragisma family|Thor]]: [[2554edo]] [[Breed family|Baldur]]: [[1696edo]] [[Hades]]: [[578edo]] ==13-limit rank two== [[Marvel temperaments|Negri]]: [[19edo]] [[Sensis]]: [[19edo]] [[Meantone family|Huyghens]]: [[31edo]]
Original HTML content:
<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> 27/25: <a class="wiki_link" href="/14edo">14edo</a><br /> 16/15: <a class="wiki_link" href="/8edo">8edo</a><br /> 135/128: <a class="wiki_link" href="/23edo">23edo</a><br /> 25/24: <a class="wiki_link" href="/17edo">17edo</a><br /> 648/625: <a class="wiki_link" href="/12edo">12edo</a><br /> 250/243: <a class="wiki_link" href="/22edo">22edo</a><br /> 128/125: <a class="wiki_link" href="/39edo">39edo</a><br /> 3125/3072: <a class="wiki_link" href="/60edo">60edo</a><br /> 81/80: <a class="wiki_link" href="/81edo">81edo</a><br /> 2048/2025: <a class="wiki_link" href="/80edo">80edo</a><br /> 78732/78125: <a class="wiki_link" href="/539edo">539edo</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 /> 1600000/1594323: <a class="wiki_link" href="/873edo">873edo</a><br /> 1224440064/1220703125: <a class="wiki_link" href="/1496edo">1496edo</a><br /> 6115295232/6103515625: <a class="wiki_link" href="/1400edo">1400edo</a><br /> 32805/32768: <a class="wiki_link" href="/749edo">749edo</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="/Ragismic%20microtemperaments">Ennealimmal</a>: <a class="wiki_link" href="/612edo">612edo</a><br /> <a class="wiki_link" href="/Ragismic%20microtemperaments">Supermajor</a>: <a class="wiki_link" href="/6214edo">6214edo</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="/Schismatic%20family">Sesquiquartififths</a>: <a class="wiki_link" href="/1498edo">1498edo</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="/Meantone%20family">Meantone</a>: <a class="wiki_link" href="/81edo">81edo</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="/Gamelismic%20clan">Miracle</a>: <a class="wiki_link" href="/72edo">72edo</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="/Magic%20family">Magic</a>: <a class="wiki_link" href="/41edo">41edo</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="/Schismatic%20family">Term</a>: <a class="wiki_link" href="/1722edo">1722edo</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="/Wuerschmidt%20family">Hemiwuerschmidt</a>: <a class="wiki_link" href="/328edo">328edo</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="/Semicomma%20family">Orwell</a>: <a class="wiki_link" href="/137edo">137edo</a><br /> <a class="wiki_link" href="/Trienstonic%20clan">Father</a>: <a class="wiki_link" href="/5edo">5edo</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="/Schismatic%20family">Garibaldi</a>: <a class="wiki_link" href="/94edo">94edo</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="/Diminished">Diminished</a>: <a class="wiki_link" href="/12edo">12edo</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">Amity</a>: <a class="wiki_link" href="/350edo">350edo</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="/Augmented%20family">Augene</a>: <a class="wiki_link" href="/27edo">27edo</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="/Ragismic%20microtemperaments">Mitonic</a>: <a class="wiki_link" href="/171edo">171edo</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">Blacksmith</a>: <a class="wiki_link" href="/15edo">15edo</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="/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="/Starling%20temperaments">Myna</a>: <a class="wiki_link" href="/89edo">89edo</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="/Ragismic%20microtemperaments">Parakleismic</a>: <a class="wiki_link" href="/415edo">415edo</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="/Mutt%20family">Mutt</a>: <a class="wiki_link" href="/171edo">171edo</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="/Starling%20temperaments">Valentine</a>: <a class="wiki_link" href="/185edo">185edo</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="/Archytas%20clan">Superpyth</a>: <a class="wiki_link" href="/49edo">49edo</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="/Breedsmic%20temperaments">Harry</a>: <a class="wiki_link" href="/534edo">534edo</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="/Breedsmic%20temperaments">Quasiorwell</a>: <a class="wiki_link" href="/1111edo">1111edo</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="/Hemimean%20clan">Misty</a>: <a class="wiki_link" href="/99edo">99edo</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="/Meantone%20family">Mothra</a>: <a class="wiki_link" href="/31edo">31edo</a><br /> <a class="wiki_link" href="/Gamera">Gamera</a>: <a class="wiki_link" href="/422edo">422edo</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> 1029/1000: <a class="wiki_link" href="/55edo">55edo</a><br /> 36/35: <a class="wiki_link" href="/12edo">12edo</a><br /> 525/512: <a class="wiki_link" href="/45edo">45edo</a><br /> 49/48: <a class="wiki_link" href="/19edo">19edo</a><br /> 50/49: <a class="wiki_link" href="/48edo">48edo</a><br /> 686/675: <a class="wiki_link" href="/46edo">46edo</a><br /> 64/63: <a class="wiki_link" href="/49edo">49edo</a><br /> 875/864: <a class="wiki_link" href="/41edo">41edo</a><br /> 3125/3087: <a class="wiki_link" href="/94edo">94edo</a><br /> 2430/2401: <a class="wiki_link" href="/137edo">137edo</a><br /> 245/243: <a class="wiki_link" href="/283edo">283edo</a><br /> 126/125: <a class="wiki_link" href="/185edo">185edo</a><br /> 4000/3969: <a class="wiki_link" href="/215edo">215edo</a><br /> 1728/1715: <a class="wiki_link" href="/111edo">111edo</a><br /> 1029/1024: <a class="wiki_link" href="/190edo">190edo</a><br /> 225/224: <a class="wiki_link" href="/197edo">197edo</a><br /> 19683/19600: <a class="wiki_link" href="/587edo">587edo</a><br /> 16875/16807: <a class="wiki_link" href="/224edo">224edo</a><br /> 10976/10935: <a class="wiki_link" href="/695edo">695edo</a><br /> 3136/3125: <a class="wiki_link" href="/446edo">446edo</a><br /> 6144/6125: <a class="wiki_link" href="/381edo">381edo</a><br /> 65625/65536: <a class="wiki_link" href="/171edo">171edo</a><br /> 703125/702464: <a class="wiki_link" href="/2185edo">2185edo</a><br /> 420175/419904: <a class="wiki_link" href="/4306edo">4306edo</a><br /> 2401/2400: <a class="wiki_link" href="/2749edo">2749edo</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="/Ragismic%20microtemperaments">Hemiennealimmal</a>: <a class="wiki_link" href="/1566edo">1566edo</a><br /> <a class="wiki_link" href="/Octoid">Octoid</a>: <a class="wiki_link" href="/224edo">224edo</a><br /> <a class="wiki_link" href="/Hemiamity">Hemiamity</a>: <a class="wiki_link" href="/350edo">350edo</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="/Gamelismic%20clan">Unidec</a>: <a class="wiki_link" href="/190edo">190edo</a><br /> <a class="wiki_link" href="/Minorsemi">Minorsemi</a>: <a class="wiki_link" href="/231edo">231edo</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="/Marvel%20temperaments">Wizard</a>: <a class="wiki_link" href="/166edo">166edo</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="/Wuerschmidt%20family">Hemiwuerschmidt</a>: <a class="wiki_link" href="/130edo">130edo</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="/Pythagorean%20family">Compton</a>: <a class="wiki_link" href="/72edo">72edo</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="/Marvel%20temperaments">Slender</a>: <a class="wiki_link" href="/125edo">125edo</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="/Gamelismic%20clan">Rodan</a>: <a class="wiki_link" href="/87edo">87edo</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="/Starling%20temperaments">Myna</a>: <a class="wiki_link" href="/89edo">89edo</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="/Magic%20family">Magic</a>: <a class="wiki_link" href="/104edo">104edo</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="/Superkleismic">Superkleismic</a>: <a class="wiki_link" href="/41edo">41edo</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="/Meantone%20family">Meanpop</a>: <a class="wiki_link" href="/81edo">81edo</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="/Meantone%20family">Huyghens</a>: <a class="wiki_link" href="/105edo">105edo</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="/Jubilismic%20clan">Doublewide</a>: <a class="wiki_link" href="/48edo">48edo</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="/Porcupine%20family">Porcupine</a>: <a class="wiki_link" href="/59edo">59edo</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="/Breed%20family">Jove</a>: <a class="wiki_link" href="/202edo">202edo</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="/Didymus%20rank%20three%20family">Euterpe</a>: <a class="wiki_link" href="/88edo">88edo</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="/Didymus%20rank%20three%20family">Clio</a>: <a class="wiki_link" href="/129edo">129edo</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="/Marvel%20family">Unimarv</a>: <a class="wiki_link" href="/166edo">166edo</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="/Mirkwai%20family">Indra</a>: <a class="wiki_link" href="/703edo">703edo</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="/Marvel%20family">Prodigy</a>: <a class="wiki_link" href="/72edo">72edo</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="/Odin">Odin</a>: <a class="wiki_link" href="/7464edo">7464edo</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="/Breed%20family">Baldur</a>: <a class="wiki_link" href="/1696edo">1696edo</a><br /> <a class="wiki_link" href="/Hades">Hades</a>: <a class="wiki_link" href="/578edo">578edo</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="/Sensis">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></body></html>