List of superparticular intervals: Difference between revisions

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=<span style="color: #800080;">List of Superparticular Intervals</span>= 

[[Superparticular]] numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number other than 1. They appear frequently in [[Just Intonation]] and [[OverToneSeries|Harmonic Series]] music. Adjacent tones in the harmonic series are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio [[20_21|20/21]]. As the overtones get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common [[comma]]s are superparticular ratios.

In addition to names and cents values, the list below includes the factorization of each superparticular ratio as well as the largest prime involved. This is relevant when considering which intervals are characteristic of which [[harmonic limit]]s. [[36_35|36/35]], for instance, is an interval of the [[7-limit]], as it factors to (2<span style="vertical-align: super;">2</span>*3<span style="vertical-align: super;">2</span>)/(5*7), while 37/36 would belong to the 37-limit.

||~ Ratio ||~ Cents Value ||~ Factorization ||~ Prime Limit ||~ Name(s) ||
|| [[2_1|2/1]] || 1200.000 || 2/1 || 2 || (perfect) unison, unity, perfect prime, tonic, duple ||
|| [[3_2|3/2]] || 701.995 || 3/2 || 3 || [[perfect fifth]], 3rd harmonic (octave reduced), diapente ||
|| [[4_3|4/3]] || 498.045 || 2<span style="vertical-align: super;">2</span>/3 || 3 || perfect fourth, 3rd subharmonic (octave reduced), diatessaron ||
|| [[5_4|5/4]] || 386.314 || 5/2<span style="vertical-align: super;">2</span> || 5 || (classic) (5-limit) major third, 5th harmonic (octave reduced) ||
|| [[6_5|6/5]] || 315.641 || (2*3)/5 || 5 || (classic) (5-limit) minor third ||
|| [[7_6|7/6]] || 266.871 || 7/(2*3) || 7 || (septimal) subminor third, septimal minor third, augmented second ||
|| [[8_7|8/7]] || 231.174 || 2<span style="vertical-align: super;">3</span>/7 || 7 || (septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic ||
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Original HTML content:

<html><head><title>List of Superparticular Intervals</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="List of Superparticular Intervals"></a><!-- ws:end:WikiTextHeadingRule:0 --><span style="color: #800080;">List of Superparticular Intervals</span></h1>
 <br />
<a class="wiki_link" href="/Superparticular">Superparticular</a> numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number other than 1. They appear frequently in <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a> and <a class="wiki_link" href="/OverToneSeries">Harmonic Series</a> music. Adjacent tones in the harmonic series are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio <a class="wiki_link" href="/20_21">20/21</a>. As the overtones get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common <a class="wiki_link" href="/comma">comma</a>s are superparticular ratios.<br />
<br />
In addition to names and cents values, the list below includes the factorization of each superparticular ratio as well as the largest prime involved. This is relevant when considering which intervals are characteristic of which <a class="wiki_link" href="/harmonic%20limit">harmonic limit</a>s. <a class="wiki_link" href="/36_35">36/35</a>, for instance, is an interval of the <a class="wiki_link" href="/7-limit">7-limit</a>, as it factors to (2<span style="vertical-align: super;">2</span>*3<span style="vertical-align: super;">2</span>)/(5*7), while 37/36 would belong to the 37-limit.<br />
<br />


<table class="wiki_table">
    <tr>
        <th>Ratio<br />
</th>
        <th>Cents Value<br />
</th>
        <th>Factorization<br />
</th>
        <th>Prime Limit<br />
</th>
        <th>Name(s)<br />
</th>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/2_1">2/1</a><br />
</td>
        <td>1200.000<br />
</td>
        <td>2/1<br />
</td>
        <td>2<br />
</td>
        <td>(perfect) unison, unity, perfect prime, tonic, duple<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/3_2">3/2</a><br />
</td>
        <td>701.995<br />
</td>
        <td>3/2<br />
</td>
        <td>3<br />
</td>
        <td><a class="wiki_link" href="/perfect%20fifth">perfect fifth</a>, 3rd harmonic (octave reduced), diapente<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/4_3">4/3</a><br />
</td>
        <td>498.045<br />
</td>
        <td>2<span style="vertical-align: super;">2</span>/3<br />
</td>
        <td>3<br />
</td>
        <td>perfect fourth, 3rd subharmonic (octave reduced), diatessaron<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/5_4">5/4</a><br />
</td>
        <td>386.314<br />
</td>
        <td>5/2<span style="vertical-align: super;">2</span><br />
</td>
        <td>5<br />
</td>
        <td>(classic) (5-limit) major third, 5th harmonic (octave reduced)<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/6_5">6/5</a><br />
</td>
        <td>315.641<br />
</td>
        <td>(2*3)/5<br />
</td>
        <td>5<br />
</td>
        <td>(classic) (5-limit) minor third<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/7_6">7/6</a><br />
</td>
        <td>266.871<br />
</td>
        <td>7/(2*3)<br />
</td>
        <td>7<br />
</td>
        <td>(septimal) subminor third, septimal minor third, augmented second<br />
</td>
    </tr>
    <tr>
        <td><a class="wiki_link" href="/8_7">8/7</a><br />
</td>
        <td>231.174<br />
</td>
        <td>2<span style="vertical-align: super;">3</span>/7<br />
</td>
        <td>7<br />
</td>
        <td>(septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic<br />
</td>
    </tr>
    <tr>
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@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-09-17 10:54:38 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-09-17 10:55:33 UTC</tt>.<br>
-: The original revision id was <tt>255045006</tt>.<br>
+: The original revision id was <tt>255045152</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>
@@ Line 17: / Line 17: @@
 || [[4_3|4/3]] || 498.045 || 2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;/3 || 3 || perfect fourth, 3rd subharmonic (octave reduced), diatessaron ||
 || [[5_4|5/4]] || 386.314 || 5/2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; || 5 || (classic) (5-limit) major third, 5th harmonic (octave reduced) ||
-|| [[6_5|6/5]] || 315.641 || 3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;/5 || 5 || (classic) (5-limit) minor third ||
+|| [[6_5|6/5]] || 315.641 || (2*3)/5 || 5 || (classic) (5-limit) minor third ||
-|| [[7_6|7/6]] || 266.871 || 7/3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; || 7 || (septimal) subminor third, septimal minor third, augmented second ||
+|| [[7_6|7/6]] || 266.871 || 7/(2*3) || 7 || (septimal) subminor third, septimal minor third, augmented second ||
 || [[8_7|8/7]] || 231.174 || 2&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;/7 || 7 || (septimal) supermajor second, septimal whole tone, diminished third, 7th subharmonic ||
 ||   ||   ||   ||   ||   ||
@@ Line 188: / Line 188: @@
          &lt;td&gt;315.641&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;/5&lt;br /&gt;
+         &lt;td&gt;(2*3)/5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;5&lt;br /&gt;
@@ Line 200: / Line 200: @@
          &lt;td&gt;266.871&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;7/3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;7/(2*3)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;7&lt;br /&gt;

List of superparticular intervals: Difference between revisions

Revision as of 10:55, 17 September 2011

IMPORTED REVISION FROM WIKISPACES

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