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 defintions 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. Given N-edo, and the odd primes up to p, let us set e = max_q |round(Nlog2(q))/N - log2(q)|; that is, e is the maximum absolute value of the error of the patent val for N-edo, measured in octaves, for all of the odd primes q up to p. Then e < 1/2N, since the error cannot be more than half the size of a step of N-edo. From this it follows that N < 1/2e, or if we measure e in cents, we can also say N < 600/e. If e is the maximum absolute error in cents for the [[POTE tuning]] of the temperament defined by the collection of commas, then we need only search as far as 600/e for the optimal patent val. 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]]
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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 defintions 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 /> Given N-edo, and the odd primes up to p, let us set e = max_q |round(Nlog2(q))/N - log2(q)|; that is, e is the maximum absolute value of the error of the patent val for N-edo, measured in octaves, for all of the odd primes q up to p. Then e < 1/2N, since the error cannot be more than half the size of a step of N-edo. From this it follows that N < 1/2e, or if we measure e in cents, we can also say N < 600/e. If e is the maximum absolute error in cents for the <a class="wiki_link" href="/POTE%20tuning">POTE tuning</a> of the temperament defined by the collection of commas, then we need only search as far as 600/e for the optimal patent val.<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></body></html>