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Add a section on coprime numbers
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| en = Prime number
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-08-15 15:55:52 UTC</tt>.<br>
| de = Primzahlen
: The original revision id was <tt>246084097</tt>.<br>
| es =
: The revision comment was: <tt></tt><br>
| ja = 素数
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| ro = numere prime
<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">=Prime numbers in [[EDO]]s=
{{Wikipedia}}


A //prime number// is an integer (whole number) greater than one which is divisible only by itself and one. There are an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13 ... . Whether or not a number //n// is prime has important consequences for the properties of the corresponding //n//-[[edo|EDO]], especially for lower values of //n//.
A '''prime number''' is an integer (whole number) greater than one that is divisible only by itself and one. There are an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13, ….
* 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 scale comprising all notes of the EDO) //no absolutely uniform, octave-repeating 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.
* //n//-EDO does not support any rank two temperament with period a fraction of an octave; all such temperaments are //linear// temperaments.
* Making a chain of any interval of the //n//-EDO, one can reach every tone in //n// steps. (For composite EDOs, this will work with intervals that are co-prime to the EDO, for example 5 degrees of [[12EDO]].)


For these or similar reasons, some musicians do not like the prime EDOs (e.g. the makers of [[http://www.armodue.com/risorse.htm|Armodue]]).
== Prime factorization ==
{{Wikipedia|Integer factorization}}


On the other hand, primality may be desirable if you want, for example, a wholetone scale that is //not// absolutely uniform. In this case you might like [[19edo]] (with whole tone scale 3 3 3 3 3 4, MOS scale of type [[1L 5s|1L+5s]]) or [[17edo]] (with whole tone scale 3 3 3 3 3 2, MOS Scale of type [[5L 1s|5L+1s]]). In general, making a chain of any interval of a prime //n//-EDO, thus treating the interval as the generator of a [[MOSScales|Moment of Symmetry]] scale, one can reach every tone in //n// steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of [[12EDO]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo]]).
By the {{w|fundamental theorem of arithmetic}}, any [[ratio]] can be uniquely represented by a product of prime numbers through prime factorization. It enables the notation of ratios as [[monzo]]s.  


The larger //n// is, the less these points matter, since the difference between an //absolutely// uniform scale and an approximated, //nearly// uniform scale eventually become inaudible.
== Prime equal division ==
{{Main| Prime equal division }}


[TODO: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, ekmelicians and theorists here.]
A prime equal division is an [[equal-step tuning]] that divides a given [[interval]] into a prime number of pitches. They are notable because of many interesting properties.  


==The first 46 Prime EDOs==  
== Coprime numbers ==
Multiples of an EDO, including multiples of a prime EDO, can inherit properties from that EDO, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime EDO is always the least complex EDO divisible by that prime, and these are listed below:
{{Wikipedia|Coprime integers}}
Two integers are '''coprime''' if they have no divisor in common except 1.


[[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[11edo|11]], [[13edo|13]], [[17edo|17]],
== See also ==
[[19edo|19]], [[23edo|23]], [[29edo|29]], [[31edo|31]], [[37edo|37]], [[41edo|41]], [[43edo|43]],
* [[Prime harmonic series]]
[[47edo|47]], [[53edo|53]], [[59edo|59]], [[61edo|61]], [[67edo|67]], [[71edo|71]], [[73edo|73]],
* [[Harmonic limit]]
[[79edo|79]], [[83edo|83]], [[89edo|89]], [[97edo|97]], [[101edo|101]], [[103edo|103]], [[107edo|107]],
* [[List of integer factorizations]]
[[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==  
== External links ==
* [[The Prime Harmonic Series]]
* [http://www.arndt-bruenner.de/mathe/scripts/primzahlen.htm Die Primzahlseite] (German) by Arndt Brünner (helpful tools for prime factorization and ~test)
* [[Monzo]] - an alternative notation for interval ratios
* [[prime limit]] or [[Harmonic Limit]]


==Links==
[[Category:Prime| ]] <!-- main article -->
* [[http://www.arndt-bruenner.de/mathe/scripts/primzahlen.htm|Die Primzahlseite]] (German) by Arndt Brünner (helpful tools for prime factorization and ~test)
[[Category:Elementary math]]
* [[http://en.wikipedia.org/wiki/Prime_number|Prime number]] the Wikipedia article</pre></div>
[[Category:Terms]]
<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;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Prime numbers in EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Prime numbers in &lt;a class="wiki_link" href="/EDO"&gt;EDO&lt;/a&gt;s&lt;/h1&gt;
&lt;br /&gt;
A &lt;em&gt;prime number&lt;/em&gt; is an integer (whole number) greater than one which is divisible only by itself and one. There are an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13 ... . Whether or not a number &lt;em&gt;n&lt;/em&gt; is prime has important consequences for the properties of the corresponding &lt;em&gt;n&lt;/em&gt;-&lt;a class="wiki_link" href="/edo"&gt;EDO&lt;/a&gt;, especially for lower values of &lt;em&gt;n&lt;/em&gt;.&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;If the octave is divided into a prime number of equal parts, there is &lt;em&gt;no fully symmetric chord&lt;/em&gt;, such as the diminished seventh chord in &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;.&lt;/li&gt;&lt;li&gt;There is also (besides the scale comprising all notes of the EDO) &lt;em&gt;no absolutely uniform, octave-repeating scale&lt;/em&gt;, like the wholetone scale in 12edo.&lt;/li&gt;&lt;li&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;/li&gt;&lt;li&gt;&lt;em&gt;n&lt;/em&gt;-EDO does not support any rank two temperament with period a fraction of an octave; all such temperaments are &lt;em&gt;linear&lt;/em&gt; temperaments.&lt;/li&gt;&lt;li&gt;Making a chain of any interval of the &lt;em&gt;n&lt;/em&gt;-EDO, one can reach every tone in &lt;em&gt;n&lt;/em&gt; steps. (For composite EDOs, this will work with intervals that are co-prime to the EDO, for example 5 degrees of &lt;a class="wiki_link" href="/12EDO"&gt;12EDO&lt;/a&gt;.)&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
For these or similar reasons, some musicians do not like the 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;
On the other hand, primality may be desirable if you want, for example, a wholetone scale that is &lt;em&gt;not&lt;/em&gt; absolutely uniform. In this case you might like &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt; (with whole tone scale 3 3 3 3 3 4, MOS scale of type &lt;a class="wiki_link" href="/1L%205s"&gt;1L+5s&lt;/a&gt;) or &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt; (with whole tone scale 3 3 3 3 3 2, MOS Scale of type &lt;a class="wiki_link" href="/5L%201s"&gt;5L+1s&lt;/a&gt;). In general, making a chain of any interval of a prime &lt;em&gt;n&lt;/em&gt;-EDO, thus treating the interval as the generator of a &lt;a class="wiki_link" href="/MOSScales"&gt;Moment of Symmetry&lt;/a&gt; scale, one can reach every tone in &lt;em&gt;n&lt;/em&gt; steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of &lt;a class="wiki_link" href="/12EDO"&gt;12EDO&lt;/a&gt; (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt;).&lt;br /&gt;
&lt;br /&gt;
The larger &lt;em&gt;n&lt;/em&gt; is, the less these points matter, since the difference between an &lt;em&gt;absolutely&lt;/em&gt; uniform scale and an approximated, &lt;em&gt;nearly&lt;/em&gt; uniform scale eventually become inaudible.&lt;br /&gt;
&lt;br /&gt;
[TODO: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, ekmelicians and theorists here.]&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="Prime numbers in EDOs-The first 46 Prime EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;The first 46 Prime EDOs&lt;/h2&gt;
Multiples of an EDO, including multiples of a prime EDO, can inherit properties from that EDO, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime EDO is always the least complex EDO divisible by that prime, and these are listed below:&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="/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="/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="/151edo"&gt;151&lt;/a&gt;, &lt;a class="wiki_link" href="/157edo"&gt;157&lt;/a&gt;, &lt;a class="wiki_link" href="/163edo"&gt;163&lt;/a&gt;, &lt;a class="wiki_link" href="/167edo"&gt;167&lt;/a&gt;, &lt;a class="wiki_link" href="/173edo"&gt;173&lt;/a&gt;, &lt;a class="wiki_link" href="/179edo"&gt;179&lt;/a&gt;, &lt;a class="wiki_link" href="/181edo"&gt;181&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;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="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;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="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>

Latest revision as of 21:03, 20 August 2025

English Wikipedia has an article on:

A prime number is an integer (whole number) greater than one that is divisible only by itself and one. There are an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13, ….

Prime factorization

English Wikipedia has an article on:

By the fundamental theorem of arithmetic, any ratio can be uniquely represented by a product of prime numbers through prime factorization. It enables the notation of ratios as monzos.

Prime equal division

Main article: Prime equal division

A prime equal division is an equal-step tuning that divides a given interval into a prime number of pitches. They are notable because of many interesting properties.

Coprime numbers

English Wikipedia has an article on:

Two integers are coprime if they have no divisor in common except 1.

See also

External links

  • Die Primzahlseite (German) by Arndt Brünner (helpful tools for prime factorization and ~test)
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