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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:genewardsmith|genewardsmith]] and made on <tt>2013-03-31 11:11:22 UTC</tt>.<br>
| de = Primzahlen
: The original revision id was <tt>419045892</tt>.<br>
| es =
: The revision comment was: <tt>Reverted to May 23, 2012 11:39 am: spam</tt><br>
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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 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 ... . 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//. In these instances:
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, ….
* There is //no fully symmetric chord// (such as the diminished seventh chord in [[12edo|12EDO]])
* Excepting the scale comprising all notes of the EDO, there is //no absolutely uniform, octave-repeating scale// (such as the wholetone scale in 12EDO)
* There are no [[http://en.wikipedia.org/wiki/Modes_of_limited_transposition|modes of limited transpostion]], such as as used by the composer Olivier Messiaen
* There is no support for rank-two temperaments whose period is a fraction of the 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 works with intervals that are co-prime to //n//, 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]]) and others love them.
== Prime factorization ==
{{Wikipedia|Integer factorization}}


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|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|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.  


A prime edo is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12edo. Since 12 is 2*2*3, it contains [[2edo|2EDO]], [[3edo|3EDO]], [[4edo|4EDO]] and [[6edo|6EDO]]. All edos with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12edo, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as [[35edo|35EDO]], will work just as well for this purpose.
== Prime equal division ==
{{Main| Prime equal division }}


If you like a certain EDO for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the [[edt|tritave (3/1)]] instead of the octave, can be an option. For example, [[27edt]] is a non-prime system very similar to [[17edo]], while [[19edt|19edt (Stopper tuning)]] is a prime system very similar to the ubiquitous [[12edo]]. (See [[edt#EDO-EDT%20correspondence|EDO-EDT correspondence]] for more of these.) Anyway, for every prime EDO system there is a non-prime [[ED4]] system with identical step sizes.
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 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.
== Coprime numbers ==
{{Wikipedia|Coprime integers}}
Two integers are '''coprime''' if they have no divisor in common except 1.


[TODO: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, ekmelicians and theorists here.]
== See also ==
* [[Prime harmonic series]]
* [[Harmonic limit]]
* [[List of integer factorizations]]


==The first 46 Prime EDOs==  
== External links ==
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:
* [http://www.arndt-bruenner.de/mathe/scripts/primzahlen.htm Die Primzahlseite] (German) by Arndt Brünner (helpful tools for prime factorization and ~test)


[[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[11edo|11]], [[13edo|13]], [[17edo|17]],
[[Category:Prime| ]] <!-- main article -->
[[19edo|19]], [[23edo|23]], [[29edo|29]], [[31edo|31]], [[37edo|37]], [[41edo|41]], [[43edo|43]],
[[Category:Elementary math]]
[[47edo|47]], [[53edo|53]], [[59edo|59]], [[61edo|61]], [[67edo|67]], [[71edo|71]], [[73edo|73]],
[[Category:Terms]]
[[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</pre></div>
<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 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 ... . 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;. In these instances:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;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;Excepting the scale comprising all notes of the EDO, there is &lt;em&gt;no absolutely uniform, octave-repeating scale&lt;/em&gt; (such as the wholetone scale in 12EDO)&lt;/li&gt;&lt;li&gt;There are no &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;, such as as used by the composer Olivier Messiaen&lt;/li&gt;&lt;li&gt;There is no support for rank-two temperaments whose period is a fraction of the 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 works with intervals that are co-prime to &lt;em&gt;n&lt;/em&gt;, for example, 5 degrees of 12EDO)&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;) and others love them.&lt;br /&gt;
&lt;br /&gt;
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;
A prime edo is useful for avoiding intervals and patterns that are familiar-sounding due to their occurrence in 12edo. Since 12 is 2*2*3, it contains &lt;a class="wiki_link" href="/2edo"&gt;2EDO&lt;/a&gt;, &lt;a class="wiki_link" href="/3edo"&gt;3EDO&lt;/a&gt;, &lt;a class="wiki_link" href="/4edo"&gt;4EDO&lt;/a&gt; and &lt;a class="wiki_link" href="/6edo"&gt;6EDO&lt;/a&gt;. All edos with a 2, 3, 4, or 6 in their factorization will share at least one interval with 12edo, if not a whole chord or subset scale. Of course, if the goal is simply to avoid intervals of 12, then non-prime edos which don't have a 2, 3, 4, or 6 in their factorization, such as &lt;a class="wiki_link" href="/35edo"&gt;35EDO&lt;/a&gt;, will work just as well for this purpose.&lt;br /&gt;
&lt;br /&gt;
If you like a certain EDO for its intervals or other reasons, but do not like its primality or non-primality, choosing another equivalence interval, such as the &lt;a class="wiki_link" href="/edt"&gt;tritave (3/1)&lt;/a&gt; instead of the octave, can be an option. For example, &lt;a class="wiki_link" href="/27edt"&gt;27edt&lt;/a&gt; is a non-prime system very similar to &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt;, while &lt;a class="wiki_link" href="/19edt"&gt;19edt (Stopper tuning)&lt;/a&gt; is a prime system very similar to the ubiquitous &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;. (See &lt;a class="wiki_link" href="/edt#EDO-EDT%20correspondence"&gt;EDO-EDT correspondence&lt;/a&gt; for more of these.) Anyway, for every prime EDO system there is a non-prime &lt;a class="wiki_link" href="/ED4"&gt;ED4&lt;/a&gt; system with identical step sizes.&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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