Prime number: Difference between revisions

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**Imported revision 240196903 - Original comment: **
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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:xenwolf|xenwolf]] and made on <tt>2011-06-29 09:21:47 UTC</tt>.<br>
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-07-06 10:19:56 UTC</tt>.<br>
: The original revision id was <tt>239309681</tt>.<br>
: The original revision id was <tt>240196903</tt>.<br>
: The revision comment was: <tt></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>
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>
<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">todo: add something useful about prime numbers for musicians, composers, microtonalists, xenharmonicians, theorists here.
<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">=Some thoughts about prime numbers in [[EDO]]s=


== The first "Prime edos" ==
Whether a number n is prime or not has quite vital consequences for the properties of the corresponding n-[[edo]], especially for lower numbers.
 
If the octave is divided into a prime number of equal parts, there is no fully symmetric chord, such as the diminished seventh chord in [[12edo]].
 
There is also (besides the full scale of all notes of the edo) no absolutely uniform scale, like the wholetone scale in 12edo.
 
Nor is there a thing like [[http://en.wikipedia.org/wiki/Modes_of_limited_transposition|modes of limited transpostion]], as used by the composer Olivier Messiaen.
 
For these or similar reasons, some musicians seem not to like prime EDOs (e.g. the makers of [[http://www.armodue.com/risorse.htm|Armodue]]).
 
OTOH, primeness may be a desirable feature if you happen to want, e.g., a wholetone scae that is **not** absolutely uniform. (In this case you might like [[19edo]], for example.)
 
The larger the number n is, the less these points matter, since the difference between an **absolutely** uniform scale and an approximated, **nearly** uniform scale eventually become inaudible.
 
 
todo: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, theorists here. XXX
 
==The first "Prime edos"==  
Prime [[edo]]s inherit most of its properties to its multiple edos. The children can often more, but they lose in handiness compared to their parents.
Prime [[edo]]s inherit most of its properties to its multiple edos. The children can often more, but they lose in handiness compared to their parents.


[[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[11edo|11]], [[13edo|13]], [[17edo|17]],  
[[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[11edo|11]], [[13edo|13]], [[17edo|17]],
[[19edo|19]], [[23edo|23]], [[29edo|29]], [[31edo|31]], [[37edo|37]], [[41edo|41]], [[43edo|43]],  
[[19edo|19]], [[23edo|23]], [[29edo|29]], [[31edo|31]], [[37edo|37]], [[41edo|41]], [[43edo|43]],
[[47edo|47]], [[53edo|53]], [[59edo|59]], [[61edo|61]], [[67edo|67]], [[71edo|71]], [[73edo|73]],  
[[47edo|47]], [[53edo|53]], [[59edo|59]], [[61edo|61]], [[67edo|67]], [[71edo|71]], [[73edo|73]],
[[79edo|79]], [[83edo|83]], [[89edo|89]], [[97edo|97]], [[101edo|101]], [[103edo|103]], [[107edo|107]],
[[79edo|79]], [[83edo|83]], [[89edo|89]], [[97edo|97]], [[101edo|101]], [[103edo|103]], [[107edo|107]],
[[109edo|109]], [[113edo|113]], [[127edo|127]], [[131edo|131]], [[137edo|137]], [[139edo|139]], [[149edo|149]],
[[109edo|109]], [[113edo|113]], [[127edo|127]], [[131edo|131]], [[137edo|137]], [[139edo|139]], [[149edo|149]],
Line 19: Line 36:
[[191edo|191]], [[193edo|193]], [[197edo|197]], [[199edo|199]]
[[191edo|191]], [[193edo|193]], [[197edo|197]], [[199edo|199]]


== See also ==
==See also==  
* [[The Prime Harmonic Series]]
* [[The Prime Harmonic Series]]
* [[Monzo]] - an alternative notation for interval ratios
* [[Monzo]] - an alternative notation for interval ratios
* [[prime limit]] or [[Harmonic Limit]]
* [[prime limit]] or [[Harmonic Limit]]


== Links ==
==Links==  
* [[http://www.arndt-bruenner.de/mathe/scripts/primzahlen.htm|Die Primzahlseite]] (German) by Arndt Brünner (helpful tools for prime factorization and ~test)
* [[http://www.arndt-bruenner.de/mathe/scripts/primzahlen.htm|Die Primzahlseite]] (German) by Arndt Brünner (helpful tools for prime factorization and ~test)
* [[http://en.wikipedia.org/wiki/Prime_number|Prime number]] the Wikipedia article</pre></div>
* [[http://en.wikipedia.org/wiki/Prime_number|Prime number]] the Wikipedia article</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">&lt;html&gt;&lt;head&gt;&lt;title&gt;prime numbers&lt;/title&gt;&lt;/head&gt;&lt;body&gt;todo: add something useful about prime numbers for musicians, composers, microtonalists, xenharmonicians, theorists here.&lt;br /&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;prime numbers&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="Some thoughts about prime numbers in EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Some thoughts about prime numbers in &lt;a class="wiki_link" href="/EDO"&gt;EDO&lt;/a&gt;s&lt;/h1&gt;
&lt;br /&gt;
Whether a number n is prime or not has quite vital consequences for the properties of the corresponding n-&lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt;, especially for lower numbers.&lt;br /&gt;
&lt;br /&gt;
If the octave is divided into a prime number of equal parts, there is no fully symmetric chord, such as the diminished seventh chord in &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;. &lt;br /&gt;
&lt;br /&gt;
There is also (besides the full scale of all notes of the edo) no absolutely uniform scale, like the wholetone scale in 12edo.&lt;br /&gt;
&lt;br /&gt;
Nor is there a thing like &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Modes_of_limited_transposition" rel="nofollow"&gt;modes of limited transpostion&lt;/a&gt;, as used by the composer Olivier Messiaen.&lt;br /&gt;
&lt;br /&gt;
For these or similar reasons, some musicians seem not to like prime EDOs (e.g. the makers of &lt;a class="wiki_link_ext" href="http://www.armodue.com/risorse.htm" rel="nofollow"&gt;Armodue&lt;/a&gt;).&lt;br /&gt;
&lt;br /&gt;
OTOH, primeness may be a desirable feature if you happen to want, e.g., a wholetone scae that is &lt;strong&gt;not&lt;/strong&gt; absolutely uniform. (In this case you might like &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, for example.)&lt;br /&gt;
&lt;br /&gt;
The larger the number n is, the less these points matter, since the difference between an &lt;strong&gt;absolutely&lt;/strong&gt; uniform scale and an approximated, &lt;strong&gt;nearly&lt;/strong&gt; uniform scale eventually become inaudible.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
todo: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, theorists here. XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-The first &amp;quot;Prime edos&amp;quot;"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt; The first &amp;quot;Prime edos&amp;quot; &lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Some thoughts about prime numbers in EDOs-The first &amp;quot;Prime edos&amp;quot;"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;The first &amp;quot;Prime edos&amp;quot;&lt;/h2&gt;
Prime &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt;s inherit most of its properties to its multiple edos. The children can often more, but they lose in handiness compared to their parents.&lt;br /&gt;
Prime &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt;s inherit most of its properties to its multiple edos. The children can often more, but they lose in handiness compared to their parents.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/2edo"&gt;2&lt;/a&gt;, &lt;a class="wiki_link" href="/3edo"&gt;3&lt;/a&gt;, &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/11edo"&gt;11&lt;/a&gt;, &lt;a class="wiki_link" href="/13edo"&gt;13&lt;/a&gt;, &lt;a class="wiki_link" href="/17edo"&gt;17&lt;/a&gt;, &lt;br /&gt;
&lt;a class="wiki_link" href="/2edo"&gt;2&lt;/a&gt;, &lt;a class="wiki_link" href="/3edo"&gt;3&lt;/a&gt;, &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/11edo"&gt;11&lt;/a&gt;, &lt;a class="wiki_link" href="/13edo"&gt;13&lt;/a&gt;, &lt;a class="wiki_link" href="/17edo"&gt;17&lt;/a&gt;,&lt;br /&gt;
&lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/23edo"&gt;23&lt;/a&gt;, &lt;a class="wiki_link" href="/29edo"&gt;29&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/37edo"&gt;37&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/43edo"&gt;43&lt;/a&gt;, &lt;br /&gt;
&lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/23edo"&gt;23&lt;/a&gt;, &lt;a class="wiki_link" href="/29edo"&gt;29&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/37edo"&gt;37&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/43edo"&gt;43&lt;/a&gt;,&lt;br /&gt;
&lt;a class="wiki_link" href="/47edo"&gt;47&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/59edo"&gt;59&lt;/a&gt;, &lt;a class="wiki_link" href="/61edo"&gt;61&lt;/a&gt;, &lt;a class="wiki_link" href="/67edo"&gt;67&lt;/a&gt;, &lt;a class="wiki_link" href="/71edo"&gt;71&lt;/a&gt;, &lt;a class="wiki_link" href="/73edo"&gt;73&lt;/a&gt;, &lt;br /&gt;
&lt;a class="wiki_link" href="/47edo"&gt;47&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/59edo"&gt;59&lt;/a&gt;, &lt;a class="wiki_link" href="/61edo"&gt;61&lt;/a&gt;, &lt;a class="wiki_link" href="/67edo"&gt;67&lt;/a&gt;, &lt;a class="wiki_link" href="/71edo"&gt;71&lt;/a&gt;, &lt;a class="wiki_link" href="/73edo"&gt;73&lt;/a&gt;,&lt;br /&gt;
&lt;a class="wiki_link" href="/79edo"&gt;79&lt;/a&gt;, &lt;a class="wiki_link" href="/83edo"&gt;83&lt;/a&gt;, &lt;a class="wiki_link" href="/89edo"&gt;89&lt;/a&gt;, &lt;a class="wiki_link" href="/97edo"&gt;97&lt;/a&gt;, &lt;a class="wiki_link" href="/101edo"&gt;101&lt;/a&gt;, &lt;a class="wiki_link" href="/103edo"&gt;103&lt;/a&gt;, &lt;a class="wiki_link" href="/107edo"&gt;107&lt;/a&gt;,&lt;br /&gt;
&lt;a class="wiki_link" href="/79edo"&gt;79&lt;/a&gt;, &lt;a class="wiki_link" href="/83edo"&gt;83&lt;/a&gt;, &lt;a class="wiki_link" href="/89edo"&gt;89&lt;/a&gt;, &lt;a class="wiki_link" href="/97edo"&gt;97&lt;/a&gt;, &lt;a class="wiki_link" href="/101edo"&gt;101&lt;/a&gt;, &lt;a class="wiki_link" href="/103edo"&gt;103&lt;/a&gt;, &lt;a class="wiki_link" href="/107edo"&gt;107&lt;/a&gt;,&lt;br /&gt;
&lt;a class="wiki_link" href="/109edo"&gt;109&lt;/a&gt;, &lt;a class="wiki_link" href="/113edo"&gt;113&lt;/a&gt;, &lt;a class="wiki_link" href="/127edo"&gt;127&lt;/a&gt;, &lt;a class="wiki_link" href="/131edo"&gt;131&lt;/a&gt;, &lt;a class="wiki_link" href="/137edo"&gt;137&lt;/a&gt;, &lt;a class="wiki_link" href="/139edo"&gt;139&lt;/a&gt;, &lt;a class="wiki_link" href="/149edo"&gt;149&lt;/a&gt;,&lt;br /&gt;
&lt;a class="wiki_link" href="/109edo"&gt;109&lt;/a&gt;, &lt;a class="wiki_link" href="/113edo"&gt;113&lt;/a&gt;, &lt;a class="wiki_link" href="/127edo"&gt;127&lt;/a&gt;, &lt;a class="wiki_link" href="/131edo"&gt;131&lt;/a&gt;, &lt;a class="wiki_link" href="/137edo"&gt;137&lt;/a&gt;, &lt;a class="wiki_link" href="/139edo"&gt;139&lt;/a&gt;, &lt;a class="wiki_link" href="/149edo"&gt;149&lt;/a&gt;,&lt;br /&gt;
Line 41: Line 75:
&lt;a class="wiki_link" href="/191edo"&gt;191&lt;/a&gt;, &lt;a class="wiki_link" href="/193edo"&gt;193&lt;/a&gt;, &lt;a class="wiki_link" href="/197edo"&gt;197&lt;/a&gt;, &lt;a class="wiki_link" href="/199edo"&gt;199&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/191edo"&gt;191&lt;/a&gt;, &lt;a class="wiki_link" href="/193edo"&gt;193&lt;/a&gt;, &lt;a class="wiki_link" href="/197edo"&gt;197&lt;/a&gt;, &lt;a class="wiki_link" href="/199edo"&gt;199&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-See also"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt; See also &lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Some thoughts about prime numbers in EDOs-See also"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;See also&lt;/h2&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/The%20Prime%20Harmonic%20Series"&gt;The Prime Harmonic Series&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Monzo"&gt;Monzo&lt;/a&gt; - an alternative notation for interval ratios&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/prime%20limit"&gt;prime limit&lt;/a&gt; or &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;Harmonic Limit&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/The%20Prime%20Harmonic%20Series"&gt;The Prime Harmonic Series&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Monzo"&gt;Monzo&lt;/a&gt; - an alternative notation for interval ratios&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/prime%20limit"&gt;prime limit&lt;/a&gt; or &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;Harmonic Limit&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt; Links &lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Some thoughts about prime numbers in EDOs-Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Links&lt;/h2&gt;
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.arndt-bruenner.de/mathe/scripts/primzahlen.htm" rel="nofollow"&gt;Die Primzahlseite&lt;/a&gt; (German) by Arndt Brünner (helpful tools for prime factorization and ~test)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Prime_number" rel="nofollow"&gt;Prime number&lt;/a&gt; the Wikipedia article&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.arndt-bruenner.de/mathe/scripts/primzahlen.htm" rel="nofollow"&gt;Die Primzahlseite&lt;/a&gt; (German) by Arndt Brünner (helpful tools for prime factorization and ~test)&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Prime_number" rel="nofollow"&gt;Prime number&lt;/a&gt; the Wikipedia article&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 10:19, 6 July 2011

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author hstraub and made on 2011-07-06 10:19:56 UTC.
The original revision id was 240196903.
The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

=Some thoughts about prime numbers in [[EDO]]s= 

Whether a number n is prime or not has quite vital consequences for the properties of the corresponding n-[[edo]], especially for lower numbers.

If the octave is divided into a prime number of equal parts, there is no fully symmetric chord, such as the diminished seventh chord in [[12edo]]. 

There is also (besides the full scale of all notes of the edo) no absolutely uniform scale, like the wholetone scale in 12edo.

Nor is there a thing like [[http://en.wikipedia.org/wiki/Modes_of_limited_transposition|modes of limited transpostion]], as used by the composer Olivier Messiaen.

For these or similar reasons, some musicians seem not to like prime EDOs (e.g. the makers of [[http://www.armodue.com/risorse.htm|Armodue]]).

OTOH, primeness may be a desirable feature if you happen to want, e.g., a wholetone scae that is **not** absolutely uniform. (In this case you might like [[19edo]], for example.)

The larger the number n is, the less these points matter, since the difference between an **absolutely** uniform scale and an approximated, **nearly** uniform scale eventually become inaudible.


todo: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, theorists here. XXX

==The first "Prime edos"== 
Prime [[edo]]s inherit most of its properties to its multiple edos. The children can often more, but they lose in handiness compared to their parents.

[[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[11edo|11]], [[13edo|13]], [[17edo|17]],
[[19edo|19]], [[23edo|23]], [[29edo|29]], [[31edo|31]], [[37edo|37]], [[41edo|41]], [[43edo|43]],
[[47edo|47]], [[53edo|53]], [[59edo|59]], [[61edo|61]], [[67edo|67]], [[71edo|71]], [[73edo|73]],
[[79edo|79]], [[83edo|83]], [[89edo|89]], [[97edo|97]], [[101edo|101]], [[103edo|103]], [[107edo|107]],
[[109edo|109]], [[113edo|113]], [[127edo|127]], [[131edo|131]], [[137edo|137]], [[139edo|139]], [[149edo|149]],
[[151edo|151]], [[157edo|157]], [[163edo|163]], [[167edo|167]], [[173edo|173]], [[179edo|179]], [[181edo|181]],
[[191edo|191]], [[193edo|193]], [[197edo|197]], [[199edo|199]]

==See also== 
* [[The Prime Harmonic Series]]
* [[Monzo]] - an alternative notation for interval ratios
* [[prime limit]] or [[Harmonic Limit]]

==Links== 
* [[http://www.arndt-bruenner.de/mathe/scripts/primzahlen.htm|Die Primzahlseite]] (German) by Arndt Brünner (helpful tools for prime factorization and ~test)
* [[http://en.wikipedia.org/wiki/Prime_number|Prime number]] the Wikipedia article

Original HTML content:

<html><head><title>prime numbers</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Some thoughts about prime numbers in EDOs"></a><!-- ws:end:WikiTextHeadingRule:0 -->Some thoughts about prime numbers in <a class="wiki_link" href="/EDO">EDO</a>s</h1>
 <br />
Whether a number n is prime or not has quite vital consequences for the properties of the corresponding n-<a class="wiki_link" href="/edo">edo</a>, especially for lower numbers.<br />
<br />
If the octave is divided into a prime number of equal parts, there is no fully symmetric chord, such as the diminished seventh chord in <a class="wiki_link" href="/12edo">12edo</a>. <br />
<br />
There is also (besides the full scale of all notes of the edo) no absolutely uniform scale, like the wholetone scale in 12edo.<br />
<br />
Nor is there a thing like <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Modes_of_limited_transposition" rel="nofollow">modes of limited transpostion</a>, as used by the composer Olivier Messiaen.<br />
<br />
For these or similar reasons, some musicians seem not to like prime EDOs (e.g. the makers of <a class="wiki_link_ext" href="http://www.armodue.com/risorse.htm" rel="nofollow">Armodue</a>).<br />
<br />
OTOH, primeness may be a desirable feature if you happen to want, e.g., a wholetone scae that is <strong>not</strong> absolutely uniform. (In this case you might like <a class="wiki_link" href="/19edo">19edo</a>, for example.)<br />
<br />
The larger the number n is, the less these points matter, since the difference between an <strong>absolutely</strong> uniform scale and an approximated, <strong>nearly</strong> uniform scale eventually become inaudible.<br />
<br />
<br />
todo: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, theorists here. XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Some thoughts about prime numbers in EDOs-The first &quot;Prime edos&quot;"></a><!-- ws:end:WikiTextHeadingRule:2 -->The first &quot;Prime edos&quot;</h2>
 Prime <a class="wiki_link" href="/edo">edo</a>s inherit most of its properties to its multiple edos. The children can often more, but they lose in handiness compared to their parents.<br />
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<a class="wiki_link" href="/2edo">2</a>, <a class="wiki_link" href="/3edo">3</a>, <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/11edo">11</a>, <a class="wiki_link" href="/13edo">13</a>, <a class="wiki_link" href="/17edo">17</a>,<br />
<a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/23edo">23</a>, <a class="wiki_link" href="/29edo">29</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/37edo">37</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/43edo">43</a>,<br />
<a class="wiki_link" href="/47edo">47</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/59edo">59</a>, <a class="wiki_link" href="/61edo">61</a>, <a class="wiki_link" href="/67edo">67</a>, <a class="wiki_link" href="/71edo">71</a>, <a class="wiki_link" href="/73edo">73</a>,<br />
<a class="wiki_link" href="/79edo">79</a>, <a class="wiki_link" href="/83edo">83</a>, <a class="wiki_link" href="/89edo">89</a>, <a class="wiki_link" href="/97edo">97</a>, <a class="wiki_link" href="/101edo">101</a>, <a class="wiki_link" href="/103edo">103</a>, <a class="wiki_link" href="/107edo">107</a>,<br />
<a class="wiki_link" href="/109edo">109</a>, <a class="wiki_link" href="/113edo">113</a>, <a class="wiki_link" href="/127edo">127</a>, <a class="wiki_link" href="/131edo">131</a>, <a class="wiki_link" href="/137edo">137</a>, <a class="wiki_link" href="/139edo">139</a>, <a class="wiki_link" href="/149edo">149</a>,<br />
<a class="wiki_link" href="/151edo">151</a>, <a class="wiki_link" href="/157edo">157</a>, <a class="wiki_link" href="/163edo">163</a>, <a class="wiki_link" href="/167edo">167</a>, <a class="wiki_link" href="/173edo">173</a>, <a class="wiki_link" href="/179edo">179</a>, <a class="wiki_link" href="/181edo">181</a>,<br />
<a class="wiki_link" href="/191edo">191</a>, <a class="wiki_link" href="/193edo">193</a>, <a class="wiki_link" href="/197edo">197</a>, <a class="wiki_link" href="/199edo">199</a><br />
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Some thoughts about prime numbers in EDOs-See also"></a><!-- ws:end:WikiTextHeadingRule:4 -->See also</h2>
 <ul><li><a class="wiki_link" href="/The%20Prime%20Harmonic%20Series">The Prime Harmonic Series</a></li><li><a class="wiki_link" href="/Monzo">Monzo</a> - an alternative notation for interval ratios</li><li><a class="wiki_link" href="/prime%20limit">prime limit</a> or <a class="wiki_link" href="/Harmonic%20Limit">Harmonic Limit</a></li></ul><br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Some thoughts about prime numbers in EDOs-Links"></a><!-- ws:end:WikiTextHeadingRule:6 -->Links</h2>
 <ul><li><a class="wiki_link_ext" href="http://www.arndt-bruenner.de/mathe/scripts/primzahlen.htm" rel="nofollow">Die Primzahlseite</a> (German) by Arndt Brünner (helpful tools for prime factorization and ~test)</li><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Prime_number" rel="nofollow">Prime number</a> the Wikipedia article</li></ul></body></html>
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