Optimal patent val: Difference between revisions
Jump to navigation
Jump to search
Wikispaces>genewardsmith **Imported revision 201827910 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 201829688 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-15 00: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-15 00:28:12 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>201829688</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 8: | Line 8: | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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. | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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 | 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. | Below are tabulated some values. | ||
| Line 118: | Line 118: | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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 /> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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 /> | <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 | 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 /> | <br /> | ||
Below are tabulated some values.<br /> | Below are tabulated some values.<br /> | ||
Revision as of 00:28, 15 February 2011
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 2011-02-15 00:28:12 UTC.
- The original revision id was 201829688.
- 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 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 <= 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 < 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.<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: [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>