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 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== [[Ennealimmal]]: [[612edo]] [[Supermajor]]: [[6214edo]] [[Enneadecal]]: [[2185edo]] [[Sesquiquartififths]]: [[1498edo]] [[Tertiaseptal]]: [[171edo]] [[Meantone]]: [[81edo]] [[Pontiac]]: [[171edo]] [[Miracle]]: [[72edo]] [[Beep]]: [[9edo]] [[Magic]]: [[41edo]] [[Dicot]]: [[7edo]] [[Term]]: [[1722edo]] [[Pajara]]: [[22edo]] [[Hemiwuerschmidt]]: [[328edo]] [[Dominant]]: [[12edo]] [[Orwell]]: [[137edo]] [[Father]]: [[5edo]] [[Catakleismic]]: [[197edo]] [[Garibaldi]]: [[94edo]] [[Hemififths]]: [[338edo]] [[Diminished]]: [[12edo]] [[Neptune]]: [[1778edo]] [[Amity]]: [[350edo]] [[Mother]]: [[5edo]] [[Augene]]: [[27edo]] [[Sharptone]]: [[5edo]] [[Mitonic]]: [[171edo]] [[Sensi]]: [[46edo]] [[Blacksmith]]: [[15edo]] [[August]]: [[12edo]] [[Negri]]: [[19edo]] [[Godzilla]]: [[19edo]] [[Myna]]: [[89edo]] [[Keemun]]: [[19edo]] [[Parakleismic]]: [[415edo]] [[Decimal]]: [[10edo]] [[Mutt]]: [[171edo]] [[Sharp]]: [[10edo]] [[Valentine]]: [[185edo]] [[Injera]]: [[38edo]] [[Superpyth]]: [[49edo]] [[Octacot]]: [[109edo]] [[Harry]]: [[534edo]] [[Compton]]: [[228edo]] [[Quasiorwell]]: [[1111edo]] [[Octokaidecal]]: [[18edo]] [[Misty]]: [[99edo]] [[Rodan]]: [[128edo]] [[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== [[Hemiennealimmal]]: [[1566edo]] [[Octoid]]: [[224edo]] [[Hemiamity]]: [[350edo]] [[Grendel]]: [[152edo]] [[Unidec]]: [[190edo]] [[Minorsemi]]: [[231edo]] [[Harry]]: [[202edo]] [[Wizard]]: [[166edo]] [[Catakleismic]]: [[72edo]] [[Hemiwuerschmidt]]: [[130edo]] [[Hemithirds]]: [[118edo]] [[Compton]]: [[72edo]] [[Miracle]]: [[72edo]] [[Slender]]: [[125edo]]
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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 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".<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="/Ennealimmal">Ennealimmal</a>: <a class="wiki_link" href="/612edo">612edo</a><br /> <a class="wiki_link" href="/Supermajor">Supermajor</a>: <a class="wiki_link" href="/6214edo">6214edo</a><br /> <a class="wiki_link" href="/Enneadecal">Enneadecal</a>: <a class="wiki_link" href="/2185edo">2185edo</a><br /> <a class="wiki_link" href="/Sesquiquartififths">Sesquiquartififths</a>: <a class="wiki_link" href="/1498edo">1498edo</a><br /> <a class="wiki_link" href="/Tertiaseptal">Tertiaseptal</a>: <a class="wiki_link" href="/171edo">171edo</a><br /> <a class="wiki_link" href="/Meantone">Meantone</a>: <a class="wiki_link" href="/81edo">81edo</a><br /> <a class="wiki_link" href="/Pontiac">Pontiac</a>: <a class="wiki_link" href="/171edo">171edo</a><br /> <a class="wiki_link" href="/Miracle">Miracle</a>: <a class="wiki_link" href="/72edo">72edo</a><br /> <a class="wiki_link" href="/Beep">Beep</a>: <a class="wiki_link" href="/9edo">9edo</a><br /> <a class="wiki_link" href="/Magic">Magic</a>: <a class="wiki_link" href="/41edo">41edo</a><br /> <a class="wiki_link" href="/Dicot">Dicot</a>: <a class="wiki_link" href="/7edo">7edo</a><br /> <a class="wiki_link" href="/Term">Term</a>: <a class="wiki_link" href="/1722edo">1722edo</a><br /> <a class="wiki_link" href="/Pajara">Pajara</a>: <a class="wiki_link" href="/22edo">22edo</a><br /> <a class="wiki_link" href="/Hemiwuerschmidt">Hemiwuerschmidt</a>: <a class="wiki_link" href="/328edo">328edo</a><br /> <a class="wiki_link" href="/Dominant">Dominant</a>: <a class="wiki_link" href="/12edo">12edo</a><br /> <a class="wiki_link" href="/Orwell">Orwell</a>: <a class="wiki_link" href="/137edo">137edo</a><br /> <a class="wiki_link" href="/Father">Father</a>: <a class="wiki_link" href="/5edo">5edo</a><br /> <a class="wiki_link" href="/Catakleismic">Catakleismic</a>: <a class="wiki_link" href="/197edo">197edo</a><br /> <a class="wiki_link" href="/Garibaldi">Garibaldi</a>: <a class="wiki_link" href="/94edo">94edo</a><br /> <a class="wiki_link" href="/Hemififths">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="/Neptune">Neptune</a>: <a class="wiki_link" href="/1778edo">1778edo</a><br /> <a class="wiki_link" href="/Amity">Amity</a>: <a class="wiki_link" href="/350edo">350edo</a><br /> <a class="wiki_link" href="/Mother">Mother</a>: <a class="wiki_link" href="/5edo">5edo</a><br /> <a class="wiki_link" href="/Augene">Augene</a>: <a class="wiki_link" href="/27edo">27edo</a><br /> <a class="wiki_link" href="/Sharptone">Sharptone</a>: <a class="wiki_link" href="/5edo">5edo</a><br /> <a class="wiki_link" href="/Mitonic">Mitonic</a>: <a class="wiki_link" href="/171edo">171edo</a><br /> <a class="wiki_link" href="/Sensi">Sensi</a>: <a class="wiki_link" href="/46edo">46edo</a><br /> <a class="wiki_link" href="/Blacksmith">Blacksmith</a>: <a class="wiki_link" href="/15edo">15edo</a><br /> <a class="wiki_link" href="/August">August</a>: <a class="wiki_link" href="/12edo">12edo</a><br /> <a class="wiki_link" href="/Negri">Negri</a>: <a class="wiki_link" href="/19edo">19edo</a><br /> <a class="wiki_link" href="/Godzilla">Godzilla</a>: <a class="wiki_link" href="/19edo">19edo</a><br /> <a class="wiki_link" href="/Myna">Myna</a>: <a class="wiki_link" href="/89edo">89edo</a><br /> <a class="wiki_link" href="/Keemun">Keemun</a>: <a class="wiki_link" href="/19edo">19edo</a><br /> <a class="wiki_link" href="/Parakleismic">Parakleismic</a>: <a class="wiki_link" href="/415edo">415edo</a><br /> <a class="wiki_link" href="/Decimal">Decimal</a>: <a class="wiki_link" href="/10edo">10edo</a><br /> <a class="wiki_link" href="/Mutt">Mutt</a>: <a class="wiki_link" href="/171edo">171edo</a><br /> <a class="wiki_link" href="/Sharp">Sharp</a>: <a class="wiki_link" href="/10edo">10edo</a><br /> <a class="wiki_link" href="/Valentine">Valentine</a>: <a class="wiki_link" href="/185edo">185edo</a><br /> <a class="wiki_link" href="/Injera">Injera</a>: <a class="wiki_link" href="/38edo">38edo</a><br /> <a class="wiki_link" href="/Superpyth">Superpyth</a>: <a class="wiki_link" href="/49edo">49edo</a><br /> <a class="wiki_link" href="/Octacot">Octacot</a>: <a class="wiki_link" href="/109edo">109edo</a><br /> <a class="wiki_link" href="/Harry">Harry</a>: <a class="wiki_link" href="/534edo">534edo</a><br /> <a class="wiki_link" href="/Compton">Compton</a>: <a class="wiki_link" href="/228edo">228edo</a><br /> <a class="wiki_link" href="/Quasiorwell">Quasiorwell</a>: <a class="wiki_link" href="/1111edo">1111edo</a><br /> <a class="wiki_link" href="/Octokaidecal">Octokaidecal</a>: <a class="wiki_link" href="/18edo">18edo</a><br /> <a class="wiki_link" href="/Misty">Misty</a>: <a class="wiki_link" href="/99edo">99edo</a><br /> <a class="wiki_link" href="/Rodan">Rodan</a>: <a class="wiki_link" href="/128edo">128edo</a><br /> <a class="wiki_link" href="/Mothra">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="/Hemiennealimmal">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="/Grendel">Grendel</a>: <a class="wiki_link" href="/152edo">152edo</a><br /> <a class="wiki_link" href="/Unidec">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="/Harry">Harry</a>: <a class="wiki_link" href="/202edo">202edo</a><br /> <a class="wiki_link" href="/Wizard">Wizard</a>: <a class="wiki_link" href="/166edo">166edo</a><br /> <a class="wiki_link" href="/Catakleismic">Catakleismic</a>: <a class="wiki_link" href="/72edo">72edo</a><br /> <a class="wiki_link" href="/Hemiwuerschmidt">Hemiwuerschmidt</a>: <a class="wiki_link" href="/130edo">130edo</a><br /> <a class="wiki_link" href="/Hemithirds">Hemithirds</a>: <a class="wiki_link" href="/118edo">118edo</a><br /> <a class="wiki_link" href="/Compton">Compton</a>: <a class="wiki_link" href="/72edo">72edo</a><br /> <a class="wiki_link" href="/Miracle">Miracle</a>: <a class="wiki_link" href="/72edo">72edo</a><br /> <a class="wiki_link" href="/Slender">Slender</a>: <a class="wiki_link" href="/125edo">125edo</a></body></html>