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List of superparticular intervals: Difference between revisions

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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;white-space: pre-wrap ! important" class="old-revision-html">=&lt;span style="color: #800080;"&gt;List of Superparticular Intervals&lt;/span&gt;=  
<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">=&lt;span style="color: #800080;"&gt;List of Superparticular Intervals&lt;/span&gt;=  


[[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.
[[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 [[21_20|21/20]]. 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&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;)/(5*7), while 37/36 would belong to the 37-limit.
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&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;)/(5*7), while 37/36 would belong to the 37-limit.
See also: [[Gallery of Just Intervals]]. Many of the names below come from [[http://www.huygens-fokker.org/docs/intervals.html|here]].


||~ Ratio ||~ Cents Value ||~ Factorization ||~ Prime Limit ||~ Name(s) ||
||~ Ratio ||~ Cents Value ||~ Factorization ||~ Prime Limit ||~ Name(s) ||
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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">&lt;html&gt;&lt;head&gt;&lt;title&gt;List of Superparticular Intervals&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="List of Superparticular Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="color: #800080;"&gt;List of Superparticular Intervals&lt;/span&gt;&lt;/h1&gt;
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;List of Superparticular Intervals&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="List of Superparticular Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="color: #800080;"&gt;List of Superparticular Intervals&lt;/span&gt;&lt;/h1&gt;
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&lt;a class="wiki_link" href="/Superparticular"&gt;Superparticular&lt;/a&gt; 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 &lt;a class="wiki_link" href="/Just%20Intonation"&gt;Just Intonation&lt;/a&gt; and &lt;a class="wiki_link" href="/OverToneSeries"&gt;Harmonic Series&lt;/a&gt; music. Adjacent tones in the harmonic series are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio &lt;a class="wiki_link" href="/20_21"&gt;20/21&lt;/a&gt;. 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 &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s are superparticular ratios.&lt;br /&gt;
&lt;a class="wiki_link" href="/Superparticular"&gt;Superparticular&lt;/a&gt; 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 &lt;a class="wiki_link" href="/Just%20Intonation"&gt;Just Intonation&lt;/a&gt; and &lt;a class="wiki_link" href="/OverToneSeries"&gt;Harmonic Series&lt;/a&gt; music. Adjacent tones in the harmonic series are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio &lt;a class="wiki_link" href="/21_20"&gt;21/20&lt;/a&gt;. 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 &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s are superparticular ratios.&lt;br /&gt;
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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 &lt;a class="wiki_link" href="/harmonic%20limit"&gt;harmonic limit&lt;/a&gt;s. &lt;a class="wiki_link" href="/36_35"&gt;36/35&lt;/a&gt;, for instance, is an interval of the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;, as it factors to (2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;)/(5*7), while 37/36 would belong to the 37-limit.&lt;br /&gt;
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 &lt;a class="wiki_link" href="/harmonic%20limit"&gt;harmonic limit&lt;/a&gt;s. &lt;a class="wiki_link" href="/36_35"&gt;36/35&lt;/a&gt;, for instance, is an interval of the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;, as it factors to (2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;)/(5*7), while 37/36 would belong to the 37-limit.&lt;br /&gt;
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See also: &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;Gallery of Just Intervals&lt;/a&gt;. Many of the names below come from &lt;a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/intervals.html" rel="nofollow"&gt;here&lt;/a&gt;.&lt;br /&gt;
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