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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | An irrational interval or ratio of frequencies given by a real number r has an infinite list of ''nearest just intervals''; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call ''best rational approximations''. A ratio of integers p/q with q > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < 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 √2 = 600 cents, but |4/3 - √2| = .08088 whereas |3/2 - √2| = 0.08479, which is larger. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| |
| : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2012-12-12 17:25:20 UTC</tt>.<br>
| |
| : The original revision id was <tt>391770730</tt>.<br>
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| : The revision comment was: <tt>added some links to intervals</tt><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>
| |
| <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">An irrational interval or ratio of frequencies given by a real number r has an infinite list of //nearest just intervals//; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call //best rational approximations//. A ratio of integers p/q with q > 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n < 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 √2 = 600 cents, but |4/3 - √2| = .08088 whereas |3/2 - √2| = 0.08479, which is larger.
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|
| |
|
| Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number such as 3/2 or ∜5 is often of interest. | | Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number such as 3/2 or ∜5 is often of interest. |
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| |
|
| 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. |
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| == Examples == | | == Examples == |
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| === Approximations for Ratios (of Pure Intervals) === | | === 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]]): | | 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 ||
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|
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|
| * //for approximations of the harmonic seventh see [[7_4#Approximations]]//
| | {| class="wikitable" |
| | |- |
| | | | '''Step\EDO''' |
| | | | '''log([[Tenney_Height|Tenney Height]])''' |
| | | | '''size''' in cents |
| | | | '''"error"''' in cents |
| | |- |
| | | | ... |
| | | | ... |
| | | | ... |
| | | | ... |
| | |- |
| | | style="text-align:center;" | 1 \ 1 |
| | | | 0.0 |
| | | style="text-align:center;" | 1200.0 |
| | | style="text-align:center;" | 498.04 |
| | |- |
| | | style="text-align:center;" | 1 \ 2 |
| | | | 1.0 |
| | | style="text-align:center;" | 600.00 |
| | | style="text-align:center;" | -101.96 |
| | |- |
| | | style="text-align:center;" | 2 \ 3 |
| | | | 2.585 |
| | | style="text-align:center;" | 800.00 |
| | | style="text-align:center;" | 98.045 |
| | |- |
| | | style="text-align:center;" | 3 \ [[5edo|5]] |
| | | | 3.907 |
| | | style="text-align:center;" | 720.00 |
| | | style="text-align:center;" | 18.045 |
| | |- |
| | | style="text-align:center;" | 4 \ [[7edo|7]] |
| | | | 4.807 |
| | | style="text-align:center;" | 685.7143 |
| | | style="text-align:center;" | -16.2407 |
| | |- |
| | | style="text-align:center;" | 7 \ [[12edo|12]] |
| | | | 6.392 |
| | | style="text-align:center;" | 700.00 |
| | | style="text-align:center;" | -1.955 |
| | |- |
| | | style="text-align:center;" | 17 \ [[29edo|29]] |
| | | | 8.945 |
| | | style="text-align:center;" | 703.4483 |
| | | style="text-align:center;" | 1.4933 |
| | |- |
| | | style="text-align:center;" | 24 \ [[41edo|41]] |
| | | | 9.943 |
| | | style="text-align:center;" | 702.43902 |
| | | style="text-align:center;" | 0.48402 |
| | |- |
| | | style="text-align:center;" | 31 \ [[53edo|53]] |
| | | | 10.682 |
| | | style="text-align:center;" | 701.88679 |
| | | style="text-align:center;" | -0.06821 |
| | |} |
| | |
| | <ul><li>''for approximations of the harmonic seventh see [[7/4#Approximations|7_4]]''</li></ul> |
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| === Approximation for Logarihmic Measures === | | === 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|12edo]], "Tritone") approximates following ratios: |
| || **freq. ratio** || **log2([[Tenney Height]])** || **size** in cents || **"error"** in cents || | |
| || ... || ... || ... || ... ||
| |
| ||= 1 / 1 ||= 0.0 ||= 0.0 ||= 600.0 ||
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| ||= [[3_2|3 / 2]] ||= 2.585 ||= 701.96 ||= 101.96 ||
| |
| ||= [[4_3|4 / 3]] ||= 3.585 ||= 498.04 ||= -101.96 ||
| |
| ||= [[7_5|7 / 5]] ||= 5.129 ||= 582.51 ||= -17.49 ||
| |
| ||= [[17_12|17 / 12]] ||= 7.672 ||= 603.000 ||= 3.000 ||
| |
| ||= 24 / 17 || ||= 597.000 ||= -3.000 ||
| |
| ||= 99 / 70 || ||= 600.0883 ||= 0.0883 ||
| |
| ||= 140 / 99 || ||= 599.9117 ||= -0.0883 ||
| |
| || ... || ... || ... || ... ||
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| | |
| The 300-cent interval 2^(1/4) (3 steps of [[12edo]], "minor third") approximates following ratios:
| |
| || **freq. ratio** || **log([[Tenney Height]])** || **size** in cents || **"error"** in cents ||
| |
| || ... || ... || ... || ... ||
| |
| ||= 1 / 1 ||= 0.0 ||= 0.0 ||= 300.0 ||
| |
| ||= [[6_5|6 / 5]] ||= 4.907 ||= 315.64 ||= 15.64 ||
| |
| ||= [[13_11|13 / 11]] ||= 7.160 ||= 289.21 ||= -10.79 ||
| |
| ||= [[19_16|19 / 16]] ||= 8.248 ||= 297.51 ||= -2.49 ||
| |
| ||= [[25_21|25 / 21]] ||= 9.036 ||= 301.84 ||= 1.84 ||
| |
| || ... || ... || ... || ... ||
| |
| | |
| </pre></div>
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| <h4>Original HTML content:</h4>
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| <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 √2 = 600 cents, but |4/3 - √2| = .08088 whereas |3/2 - √2| = 0.08479, which is larger.<br />
| |
| <br />
| |
| Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number such as 3/2 or ∜5 is often of interest.<br />
| |
| <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 />
| |
| <!-- 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">
| |
| <tr>
| |
| <td><strong>Step\EDO</strong><br />
| |
| </td>
| |
| <td><strong>log(<a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)</strong><br />
| |
| </td>
| |
| <td><strong>size</strong> in cents<br />
| |
| </td>
| |
| <td><strong>&quot;error&quot;</strong> in cents<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">1 \ 1<br />
| |
| </td>
| |
| <td>0.0<br />
| |
| </td>
| |
| <td style="text-align: center;">1200.0<br />
| |
| </td>
| |
| <td style="text-align: center;">498.04<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">1 \ 2<br />
| |
| </td>
| |
| <td>1.0<br />
| |
| </td>
| |
| <td style="text-align: center;">600.00<br />
| |
| </td>
| |
| <td style="text-align: center;">-101.96<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">2 \ 3<br />
| |
| </td>
| |
| <td>2.585<br />
| |
| </td>
| |
| <td style="text-align: center;">800.00<br />
| |
| </td>
| |
| <td style="text-align: center;">98.045<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">3 \ <a class="wiki_link" href="/5edo">5</a><br />
| |
| </td>
| |
| <td>3.907<br />
| |
| </td>
| |
| <td style="text-align: center;">720.00<br />
| |
| </td>
| |
| <td style="text-align: center;">18.045<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">4 \ <a class="wiki_link" href="/7edo">7</a><br />
| |
| </td>
| |
| <td>4.807<br />
| |
| </td>
| |
| <td style="text-align: center;">685.7143<br />
| |
| </td>
| |
| <td style="text-align: center;">-16.2407<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">7 \ <a class="wiki_link" href="/12edo">12</a><br />
| |
| </td>
| |
| <td>6.392<br />
| |
| </td>
| |
| <td style="text-align: center;">700.00<br />
| |
| </td>
| |
| <td style="text-align: center;">-1.955<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">17 \ <a class="wiki_link" href="/29edo">29</a><br />
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| </td>
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| <td>8.945<br />
| |
| </td>
| |
| <td style="text-align: center;">703.4483<br />
| |
| </td>
| |
| <td style="text-align: center;">1.4933<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">24 \ <a class="wiki_link" href="/41edo">41</a><br />
| |
| </td>
| |
| <td>9.943<br />
| |
| </td>
| |
| <td style="text-align: center;">702.43902<br />
| |
| </td>
| |
| <td style="text-align: center;">0.48402<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">31 \ <a class="wiki_link" href="/53edo">53</a><br />
| |
| </td>
| |
| <td>10.682<br />
| |
| </td>
| |
| <td style="text-align: center;">701.88679<br />
| |
| </td>
| |
| <td style="text-align: center;">-0.06821<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
| | |
| <br />
| |
| <ul><li><em>for approximations of the harmonic seventh see <a class="wiki_link" href="/7_4#Approximations">7_4</a></em></li></ul><br />
| |
| <!-- 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 />
| |
| | |
| | |
| <table class="wiki_table">
| |
| <tr>
| |
| <td><strong>freq. ratio</strong><br />
| |
| </td>
| |
| <td><strong>log2(<a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)</strong><br />
| |
| </td>
| |
| <td><strong>size</strong> in cents<br />
| |
| </td>
| |
| <td><strong>&quot;error&quot;</strong> in cents<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
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| </tr>
| |
| <tr>
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| <td style="text-align: center;">1 / 1<br />
| |
| </td>
| |
| <td style="text-align: center;">0.0<br />
| |
| </td>
| |
| <td style="text-align: center;">0.0<br />
| |
| </td>
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| <td style="text-align: center;">600.0<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
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| <td style="text-align: center;"><a class="wiki_link" href="/3_2">3 / 2</a><br />
| |
| </td>
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| <td style="text-align: center;">2.585<br />
| |
| </td>
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| <td style="text-align: center;">701.96<br />
| |
| </td>
| |
| <td style="text-align: center;">101.96<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;"><a class="wiki_link" href="/4_3">4 / 3</a><br />
| |
| </td>
| |
| <td style="text-align: center;">3.585<br />
| |
| </td>
| |
| <td style="text-align: center;">498.04<br />
| |
| </td>
| |
| <td style="text-align: center;">-101.96<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;"><a class="wiki_link" href="/7_5">7 / 5</a><br />
| |
| </td>
| |
| <td style="text-align: center;">5.129<br />
| |
| </td>
| |
| <td style="text-align: center;">582.51<br />
| |
| </td>
| |
| <td style="text-align: center;">-17.49<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;"><a class="wiki_link" href="/17_12">17 / 12</a><br />
| |
| </td>
| |
| <td style="text-align: center;">7.672<br />
| |
| </td>
| |
| <td style="text-align: center;">603.000<br />
| |
| </td>
| |
| <td style="text-align: center;">3.000<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">24 / 17<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td style="text-align: center;">597.000<br />
| |
| </td>
| |
| <td style="text-align: center;">-3.000<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">99 / 70<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td style="text-align: center;">600.0883<br />
| |
| </td>
| |
| <td style="text-align: center;">0.0883<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">140 / 99<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td style="text-align: center;">599.9117<br />
| |
| </td>
| |
| <td style="text-align: center;">-0.0883<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| </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 />
| |
|
| |
|
| | {| class="wikitable" |
| | |- |
| | | | '''freq. ratio''' |
| | | | '''log2([[Tenney_Height|Tenney Height]])''' |
| | | | '''size''' in cents |
| | | | '''"error"''' in cents |
| | |- |
| | | | ... |
| | | | ... |
| | | | ... |
| | | | ... |
| | |- |
| | | style="text-align:center;" | 1 / 1 |
| | | style="text-align:center;" | 0.0 |
| | | style="text-align:center;" | 0.0 |
| | | style="text-align:center;" | 600.0 |
| | |- |
| | | style="text-align:center;" | [[3/2|3 / 2]] |
| | | style="text-align:center;" | 2.585 |
| | | style="text-align:center;" | 701.96 |
| | | style="text-align:center;" | 101.96 |
| | |- |
| | | style="text-align:center;" | [[4/3|4 / 3]] |
| | | style="text-align:center;" | 3.585 |
| | | style="text-align:center;" | 498.04 |
| | | style="text-align:center;" | -101.96 |
| | |- |
| | | style="text-align:center;" | [[7/5|7 / 5]] |
| | | style="text-align:center;" | 5.129 |
| | | style="text-align:center;" | 582.51 |
| | | style="text-align:center;" | -17.49 |
| | |- |
| | | style="text-align:center;" | [[17/12|17 / 12]] |
| | | style="text-align:center;" | 7.672 |
| | | style="text-align:center;" | 603.000 |
| | | style="text-align:center;" | 3.000 |
| | |- |
| | | style="text-align:center;" | 24 / 17 |
| | | | |
| | | style="text-align:center;" | 597.000 |
| | | style="text-align:center;" | -3.000 |
| | |- |
| | | style="text-align:center;" | 99 / 70 |
| | | | |
| | | style="text-align:center;" | 600.0883 |
| | | style="text-align:center;" | 0.0883 |
| | |- |
| | | style="text-align:center;" | 140 / 99 |
| | | | |
| | | style="text-align:center;" | 599.9117 |
| | | style="text-align:center;" | -0.0883 |
| | |- |
| | | | ... |
| | | | ... |
| | | | ... |
| | | | ... |
| | |} |
|
| |
|
| <table class="wiki_table">
| | The 300-cent interval 2^(1/4) (3 steps of [[12edo|12edo]], "minor third") approximates following ratios: |
| <tr>
| |
| <td><strong>freq. ratio</strong><br />
| |
| </td>
| |
| <td><strong>log(<a class="wiki_link" href="/Tenney%20Height">Tenney Height</a>)</strong><br />
| |
| </td>
| |
| <td><strong>size</strong> in cents<br />
| |
| </td>
| |
| <td><strong>&quot;error&quot;</strong> in cents<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;">1 / 1<br />
| |
| </td>
| |
| <td style="text-align: center;">0.0<br />
| |
| </td>
| |
| <td style="text-align: center;">0.0<br />
| |
| </td>
| |
| <td style="text-align: center;">300.0<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;"><a class="wiki_link" href="/6_5">6 / 5</a><br />
| |
| </td>
| |
| <td style="text-align: center;">4.907<br />
| |
| </td>
| |
| <td style="text-align: center;">315.64<br />
| |
| </td>
| |
| <td style="text-align: center;">15.64<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;"><a class="wiki_link" href="/13_11">13 / 11</a><br />
| |
| </td>
| |
| <td style="text-align: center;">7.160<br />
| |
| </td>
| |
| <td style="text-align: center;">289.21<br />
| |
| </td>
| |
| <td style="text-align: center;">-10.79<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;"><a class="wiki_link" href="/19_16">19 / 16</a><br />
| |
| </td>
| |
| <td style="text-align: center;">8.248<br />
| |
| </td>
| |
| <td style="text-align: center;">297.51<br />
| |
| </td>
| |
| <td style="text-align: center;">-2.49<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td style="text-align: center;"><a class="wiki_link" href="/25_21">25 / 21</a><br />
| |
| </td>
| |
| <td style="text-align: center;">9.036<br />
| |
| </td>
| |
| <td style="text-align: center;">301.84<br />
| |
| </td>
| |
| <td style="text-align: center;">1.84<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| <td>...<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| </body></html></pre></div>
| | {| class="wikitable" |
| | |- |
| | | | '''freq. ratio''' |
| | | | '''log([[Tenney_Height|Tenney Height]])''' |
| | | | '''size''' in cents |
| | | | '''"error"''' in cents |
| | |- |
| | | | ... |
| | | | ... |
| | | | ... |
| | | | ... |
| | |- |
| | | style="text-align:center;" | 1 / 1 |
| | | style="text-align:center;" | 0.0 |
| | | style="text-align:center;" | 0.0 |
| | | style="text-align:center;" | 300.0 |
| | |- |
| | | style="text-align:center;" | [[6/5|6 / 5]] |
| | | style="text-align:center;" | 4.907 |
| | | style="text-align:center;" | 315.64 |
| | | style="text-align:center;" | 15.64 |
| | |- |
| | | style="text-align:center;" | [[13/11|13 / 11]] |
| | | style="text-align:center;" | 7.160 |
| | | style="text-align:center;" | 289.21 |
| | | style="text-align:center;" | -10.79 |
| | |- |
| | | style="text-align:center;" | [[19/16|19 / 16]] |
| | | style="text-align:center;" | 8.248 |
| | | style="text-align:center;" | 297.51 |
| | | style="text-align:center;" | -2.49 |
| | |- |
| | | style="text-align:center;" | [[25/21|25 / 21]] |
| | | style="text-align:center;" | 9.036 |
| | | style="text-align:center;" | 301.84 |
| | | style="text-align:center;" | 1.84 |
| | |- |
| | | | ... |
| | | | ... |
| | | | ... |
| | | | ... |
| | |} |
| | [[Category:approximation]] |
| | [[Category:hack]] |
| | [[Category:just]] |
| | [[Category:math]] |
| | [[Category:theory]] |
| | [[Category:todo:add_examples]] |