List of superparticular intervals: Difference between revisions

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-: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-10-28 04:28:07 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-10-28 04:40:39 UTC</tt>.<br>
-: The original revision id was <tt>269489636</tt>.<br>
+: The original revision id was <tt>269491218</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>
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 [[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.
+The list below is ordered by [[harmonic limit]], or the largest prime involved in the prime factorization. [[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.
+[[http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem|Størmer's theorem]] guarantees that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than 2/1, 3/2, 4/3, and 9/8.
 See also: [[Gallery of Just Intervals]]. Many of the names below come from [[http://www.huygens-fokker.org/docs/intervals.html|here]].
@@ Line 50: / Line 52: @@
 || [[55_54|55/54]] || 31.767 || (5*11)/(2*3&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;) ||   ||
 || [[56_55|56/55]] || 31.194 || (2&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;*7)/(5*11) ||   ||
-|| [[99_98|99/98]] || 17.576 || (3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*11)/(2*7&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;) ||   ||
+|| [[99_98|99/98]] || 17.576 || (3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*11)/(2*7&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;) || small undecimal comma ||
 || [[100_99|100/99]] || 17.399 || (2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*5&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;)/(3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*11) || Ptolemy's comma ||
-|| [[121_120|121/120]] || 14.376 || 11^2/(2^3*3*5) ||   ||
+|| [[121_120|121/120]] || 14.376 || 11^2/(2^3*3*5) || undecimal seconds comma ||
 || [[176_175|176/175]] || 9.8646 || (2^4*11)/(5^2*7) ||   ||
-|| [[243_242|243/242]] || 7.1391 || 2^5/(2*11^2) ||   ||
+|| [[243_242|243/242]] || 7.1391 || 2^5/(2*11^2) || neutral third comma ||
 || [[385_384|385/384]] || 4.5026 || (5*7*11)/(2^7*3) || keenanisma ||
-|| [[441_440|441/440]] || 3.9302 || (3^2*7^2)/(2^3*5*11) ||   ||
+|| [[441_440|441/440]] || 3.9302 || (3^2*7^2)/(2^3*5*11) || Werckmeister's undecimal septenarian schisma ||
-|| [[540_539|540/539]] || 3.2090 || (2^2*3^3*5)/(7^2*11) ||   ||
+|| [[540_539|540/539]] || 3.2090 || (2^2*3^3*5)/(7^2*11) || Swets' comma ||
-|| [[3025_3024|3025/3024]] || 0.57240 || (5^2*11^2)/(2^4*3^3*7) ||   ||
+|| [[3025_3024|3025/3024]] || 0.57240 || (5^2*11^2)/(2^4*3^3*7) || Lehmerisma ||
-|| [[9801_9800|9801/9800]] || 0.17665 || (3^4*11^2)/(2^3*5^2*7^2) || Gauss comma ||
+|| [[9801_9800|9801/9800]] || 0.17665 || (3^4*11^2)/(2^3*5^2*7^2) || Gauss comma, kalisma ||
 ||||||||~ 13-limit ||
 || [[13_12|13/12]] || 138.573 || 13/(2&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*3) || tridecimal 2/3-tone ||
@@ Line 69: / Line 71: @@
 || [[66_65|66/65]] || 26.432 || (2*3*11)/(5*13) ||   ||
 || [[78_77|78/77]] || 22.339 || (2*3*13)/(7*11) ||   ||
-|| [[91_90|91/90]] || 19.130 || (7*13)/(2*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*5) ||   ||
+|| [[91_90|91/90]] || 19.130 || (7*13)/(2*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*5) || [[The Biosphere|Biome]] comma ||
-|| [[105_104|105/104]] || 16.567 ||   ||   ||
+|| [[105_104|105/104]] || 16.567 ||   || small tridecimal comma ||
 || [[144_143|144/143]] || 12.064 ||   ||   ||
 || [[169_168|169/168]] || 10.274 ||   ||   ||
@@ Line 84: / Line 86: @@
 || [[1716_1715|1716/1715]] || 1.0092 ||   ||   ||
 || [[2080_2079|2080/2079]] || 0.83252 ||   ||   ||
-|| [[4096_4095|4096/4095]] || 0.42272 ||   ||   ||
+|| [[4096_4095|4096/4095]] || 0.42272 || tridecimal schisma, Sagittal schismina ||   ||
 || [[4225_4224|4225/4224]] || 0.40981 ||   ||   ||
 || [[6656_6655|6656/6655]] || 0.26012 ||   ||   ||
-|| [[10648_10647|10648/10647]] || 0.16260 ||   ||   ||
+|| [[10648_10647|10648/10647]] || 0.16260 || harmonisma ||   ||
 || [[123201_123200|123201/123200]] || 0.014052 ||   ||   ||
 ||||||||~ 17-limit (incomplete) ||
@@ Line 177: / Line 179: @@
 &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;
 &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;
+The list below is ordered by &lt;a class="wiki_link" href="/harmonic%20limit"&gt;harmonic limit&lt;/a&gt;, or the largest prime involved in the prime factorization. &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;
+&lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem" rel="nofollow"&gt;Størmer's theorem&lt;/a&gt; guarantees that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than 2/1, 3/2, 4/3, and 9/8.&lt;br /&gt;
 &lt;br /&gt;
 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;
@@ Line 521: / Line 525: @@
          &lt;td&gt;(3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*11)/(2*7&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;small undecimal comma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 541: / Line 545: @@
          &lt;td&gt;11^2/(2^3*3*5)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;undecimal seconds comma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 561: / Line 565: @@
          &lt;td&gt;2^5/(2*11^2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;neutral third comma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 581: / Line 585: @@
          &lt;td&gt;(3^2*7^2)/(2^3*5*11)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Werckmeister's undecimal septenarian schisma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 591: / Line 595: @@
          &lt;td&gt;(2^2*3^3*5)/(7^2*11)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Swets' comma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 601: / Line 605: @@
          &lt;td&gt;(5^2*11^2)/(2^4*3^3*7)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Lehmerisma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 611: / Line 615: @@
          &lt;td&gt;(3^4*11^2)/(2^3*5^2*7^2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Gauss comma&lt;br /&gt;
+         &lt;td&gt;Gauss comma, kalisma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 705: / Line 709: @@
          &lt;td&gt;(7*13)/(2*3&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;*5)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link" href="/The%20Biosphere"&gt;Biome&lt;/a&gt; comma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 715: / Line 719: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;small tridecimal comma&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 853: / Line 857: @@
          &lt;td&gt;0.42272&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;tridecimal schisma, Sagittal schismina&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 883: / Line 887: @@
          &lt;td&gt;0.16260&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;harmonisma&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;