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Optimal patent val

Revision as of 00:28, 15 February 2011 by Wikispaces>genewardsmith (**Imported revision 201829688 - Original comment: **)

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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.

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 eq 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 eq < 600/N, from which it follows that N < 600/eq. 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. If we take the minimum value for 1200/error(q) for the odd primes up to p, where error(q) is the POTE tuning error of q, we obtain by application of the triangle inequality 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]]

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 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 />
To limit the search range when finding the optimal patent val a useful observation is this: given N-edo, and an odd prime q &lt;= p, if eq 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 eq &lt; 600/N, from which it follows that N &lt; 600/eq. 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. If we take the minimum value for 1200/error(q) for the odd primes up to p, where error(q) is the POTE tuning error of q, we obtain by application of the triangle inequality an upper bound for N.<br />
<br />
Below are tabulated some values.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><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:&lt;h2&gt; --><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:&lt;h2&gt; --><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: [224edo]]<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></body></html>
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