Nearest just interval: Difference between revisions
Wikispaces>genewardsmith **Imported revision 236965526 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 237019800 - Original comment: page reordered - the reverse approximations is the less interesting (in my opinion)** |
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<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: | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-16 02:40:58 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>237019800</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>page reordered - the reverse approximations is the less interesting (in my opinion)</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
| Line 12: | Line 12: | ||
The [[http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents|semiconvergents]] of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely [[http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations|best relative approximation]]. Here it is required that |qr - p| is less than |nr - m| for any n < q. | The [[http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents|semiconvergents]] of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely [[http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations|best relative approximation]]. Here it is required that |qr - p| is less than |nr - m| for any n < q. | ||
==Examples== | == Examples == | ||
=== Approximations for Ratios (of Pure Intervals) === | |||
The best rational approximations to log2(3/2) define edos which have especially good approximations to the fifth (701.955000865... [[cent|cents]]): | |||
|| **Step\EDO** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents || | |||
|| ... || ... || ... || ... || | |||
||= 1 \ 1 || 0.0 ||= 1200.0 ||= 498.04 || | |||
||= 1 \ 2 || 1.0 ||= 600.00 ||= -101.96 || | |||
||= 2 \ 3 || 2.585 ||= 800.00 ||= 98.045 || | |||
||= 3 \ [[5edo|5]] || 3.907 ||= 720.00 ||= 18.045 || | |||
||= 4 \ [[7edo|7]] || 4.807 ||= 685.7143 ||= -16.2407 || | |||
||= 7 \ [[12edo|12]] || 6.392 ||= 700.00 ||= -1.955 || | |||
||= 17 \ [[29edo|29]] || 8.945 ||= 703.4483 ||= 1.4933 || | |||
||= 24 \ [[41edo|41]] || 9.943 ||= 702.43902 ||= 0.48402 || | |||
||= 31 \ [[53edo|53]] || 10.682 ||= 701.88679 ||= -0.06821 || | |||
=== Approximation for Logarihmic Measures === | |||
The 600-cent interval sqrt(2) (6 steps of [[12edo]], "Tritone") approximates following ratios: | The 600-cent interval sqrt(2) (6 steps of [[12edo]], "Tritone") approximates following ratios: | ||
|| **freq. ratio** || **log2([[Tenney Height]])** || **size** in cents || **"error"** in cents || | || **freq. ratio** || **log2([[Tenney Height]])** || **size** in cents || **"error"** in cents || | ||
| Line 36: | Line 52: | ||
|| ... || ... || ... || ... || | || ... || ... || ... || ... || | ||
</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<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>Nearest just interval</title></head><body>An irrational interval or ratio of frequencies given by a real number r has an infinite list of <em>nearest just intervals</em>; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call <em>best rational approximations</em>. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from sqrt(2) = 600 cents, but |4/3 - sqrt(2)| = .08088 whereas |3/2 - sqrt(2)| = 0.08479, which is larger.<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>Nearest just interval</title></head><body>An irrational interval or ratio of frequencies given by a real number r has an infinite list of <em>nearest just intervals</em>; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call <em>best rational approximations</em>. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from sqrt(2) = 600 cents, but |4/3 - sqrt(2)| = .08088 whereas |3/2 - sqrt(2)| = 0.08479, which is larger.<br /> | ||
| Line 55: | Line 60: | ||
The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents" rel="nofollow">semiconvergents</a> of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations" rel="nofollow">best relative approximation</a>. Here it is required that |qr - p| is less than |nr - m| for any n &lt; q.<br /> | The <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents" rel="nofollow">semiconvergents</a> of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations" rel="nofollow">best relative approximation</a>. Here it is required that |qr - p| is less than |nr - m| for any n &lt; q.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 -->Examples</h2> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:0 --> Examples </h2> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Examples-Approximations for Ratios (of Pure Intervals)"></a><!-- ws:end:WikiTextHeadingRule:2 --> Approximations for Ratios (of Pure Intervals) </h3> | |||
The best rational approximations to log2(3/2) define edos which have especially good approximations to the fifth (701.955000865... <a class="wiki_link" href="/cent">cents</a>):<br /> | |||
<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
<td><strong> | <td><strong>Step\EDO</strong><br /> | ||
</td> | </td> | ||
<td><strong> | <td><strong>log(<a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)</strong><br /> | ||
</td> | </td> | ||
<td><strong>size</strong> in cents<br /> | <td><strong>size</strong> in cents<br /> | ||
| Line 81: | Line 88: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">1 | <td style="text-align: center;">1 \ 1<br /> | ||
</td> | </td> | ||
<td | <td>0.0<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">1200.0<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">498.04<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">1 \ 2<br /> | ||
</td> | </td> | ||
<td | <td>1.0<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">600.00<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">101.96<br /> | <td style="text-align: center;">-101.96<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">2 \ 3<br /> | ||
</td> | </td> | ||
<td | <td>2.585<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">800.00<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">98.045<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">3 \ <a class="wiki_link" href="/5edo">5</a><br /> | ||
</td> | </td> | ||
<td | <td>3.907<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">720.00<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">18.045<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">4 \ <a class="wiki_link" href="/7edo">7</a><br /> | ||
</td> | </td> | ||
<td | <td>4.807<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">685.7143<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">-16.2407<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">7 \ <a class="wiki_link" href="/12edo">12</a><br /> | ||
</td> | </td> | ||
<td><br /> | <td>6.392<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">700.00<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-1.955<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">17 \ <a class="wiki_link" href="/29edo">29</a><br /> | ||
</td> | </td> | ||
<td><br /> | <td>8.945<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">703.4483<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">1.4933<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">24 \ <a class="wiki_link" href="/41edo">41</a><br /> | ||
</td> | </td> | ||
<td><br /> | <td>9.943<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">702.43902<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">0.48402<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td style="text-align: center;">31 \ <a class="wiki_link" href="/53edo">53</a><br /> | ||
</td> | </td> | ||
<td>. | <td>10.682<br /> | ||
</td> | </td> | ||
<td>. | <td style="text-align: center;">701.88679<br /> | ||
</td> | </td> | ||
<td>. | <td style="text-align: center;">-0.06821<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 173: | Line 180: | ||
<br /> | <br /> | ||
The | <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-Examples-Approximation for Logarihmic Measures"></a><!-- ws:end:WikiTextHeadingRule:4 --> Approximation for Logarihmic Measures </h3> | ||
The 600-cent interval sqrt(2) (6 steps of <a class="wiki_link" href="/12edo">12edo</a>, &quot;Tritone&quot;) approximates following ratios:<br /> | |||
| Line 180: | Line 188: | ||
<td><strong>freq. ratio</strong><br /> | <td><strong>freq. ratio</strong><br /> | ||
</td> | </td> | ||
<td><strong> | <td><strong>log2(<a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)</strong><br /> | ||
</td> | </td> | ||
<td><strong>size</strong> in cents<br /> | <td><strong>size</strong> in cents<br /> | ||
| Line 204: | Line 212: | ||
<td style="text-align: center;">0.0<br /> | <td style="text-align: center;">0.0<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">600.0<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">3 / 2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">2.585<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">701.96<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">101.96<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">4 / 3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">3.585<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">498.04<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-101.96<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">7 / 5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">5.129<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">582.51<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-17.49<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">17 / 12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">7.672<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">603.000<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">3.000<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td style="text-align: center;">24 / 17<br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td>. | <td style="text-align: center;">597.000<br /> | ||
</td> | </td> | ||
<td>. | <td style="text-align: center;">-3.000<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td | <td style="text-align: center;">99 / 70<br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td style="text-align: center;">600.0883<br /> | ||
</td> | </td> | ||
<td | <td style="text-align: center;">0.0883<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td style="text-align: center;">140 / 99<br /> | ||
</td> | </td> | ||
<td> | <td><br /> | ||
</td> | </td> | ||
<td>. | <td style="text-align: center;">599.9117<br /> | ||
</td> | </td> | ||
<td>. | <td style="text-align: center;">-0.0883<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td | <td>...<br /> | ||
</td> | </td> | ||
<td> | <td>...<br /> | ||
</td> | </td> | ||
<td | <td>...<br /> | ||
</td> | </td> | ||
<td | <td>...<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | |||
<br /> | |||
The 300-cent interval 2^(1/4) (3 steps of <a class="wiki_link" href="/12edo">12edo</a>, &quot;minor third&quot;) approximates following ratios:<br /> | |||
<table class="wiki_table"> | |||
<tr> | <tr> | ||
<td | <td><strong>freq. ratio</strong><br /> | ||
</td> | </td> | ||
<td> | <td><strong>log(<a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)</strong><br /> | ||
</td> | </td> | ||
<td | <td><strong>size</strong> in cents<br /> | ||
</td> | </td> | ||
<td | <td><strong>&quot;error&quot;</strong> in cents<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td | <td>...<br /> | ||
</td> | </td> | ||
<td> | <td>...<br /> | ||
</td> | </td> | ||
<td | <td>...<br /> | ||
</td> | </td> | ||
<td | <td>...<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">1 / 1<br /> | ||
</td> | </td> | ||
<td> | <td style="text-align: center;">0.0<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">0.0<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">300.0<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">6 / 5<br /> | ||
</td> | </td> | ||
<td>4. | <td style="text-align: center;">4.907<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">315.64<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">15.64<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">13 / 11<br /> | ||
</td> | </td> | ||
<td> | <td style="text-align: center;">7.160<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">289.21<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-10.79<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">19 / 16<br /> | ||
</td> | </td> | ||
<td>8. | <td style="text-align: center;">8.248<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">297.51<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">-2.49<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">25 / 21<br /> | ||
</td> | </td> | ||
<td>9. | <td style="text-align: center;">9.036<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">301.84<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">1.84<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td | <td>...<br /> | ||
</td> | </td> | ||
<td> | <td>...<br /> | ||
</td> | </td> | ||
<td | <td>...<br /> | ||
</td> | </td> | ||
<td | <td>...<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||