Prime number: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 241852539 - Original comment: **
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
Add a section on coprime numbers
 
(41 intermediate revisions by 20 users not shown)
Line 1: Line 1:
<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>2011-07-18 19:38:16 UTC</tt>.<br>
| de = Primzahlen
: The original revision id was <tt>241852539</tt>.<br>
| es =
: The revision comment was: <tt></tt><br>
| ja = 素数
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| 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 an an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13 ... . Whether a number n is prime or not 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 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.
* N-EDO does not support any rank two temperament with period a fraction of an octave; all such temperaments are //linear// temperaments.


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]]).
== Prime factorization ==
{{Wikipedia|Integer factorization}}


On the other hand, primality may be a desirable feature if you happen to want, e.g., a wholetone scale that is //not// absolutely uniform. (In this case you might like [[19edo]], for example.)
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 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.
== Prime equal division ==
{{Main| 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.


todo: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, theorists here. XXX
== Coprime numbers ==
{{Wikipedia|Coprime integers}}
Two integers are '''coprime''' if they have no divisor in common except 1.


==The first "Prime EDOs"==  
== See also ==
Multiples of an EDO can inherit properties from the EDO, in particular a tuning for certain intervals. The multiple, on the other hand, is necessarily more complex.
* [[Prime harmonic series]]
* [[Harmonic limit]]
* [[List of integer factorizations]]


[[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[11edo|11]], [[13edo|13]], [[17edo|17]],
== External links ==
[[19edo|19]], [[23edo|23]], [[29edo|29]], [[31edo|31]], [[37edo|37]], [[41edo|41]], [[43edo|43]],
* [http://www.arndt-bruenner.de/mathe/scripts/primzahlen.htm Die Primzahlseite] (German) by Arndt Brünner (helpful tools for prime factorization and ~test)
[[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==
[[Category:Prime| ]] <!-- main article -->
* [[The Prime Harmonic Series]]
[[Category:Elementary math]]
* [[Monzo]] - an alternative notation for interval ratios
[[Category:Terms]]
* [[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 which is divisible only by itself and one. There an an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13 ... . Whether a number n is prime or not has important consequences for the properties of the corresponding n-&lt;a class="wiki_link" href="/edo"&gt;EDO&lt;/a&gt;, especially for lower values of n.&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;If the octave is divided into a prime number of equal parts, there is &lt;strong&gt;no fully symmetric chord&lt;/strong&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 full scale of all notes of the edo) &lt;strong&gt;no absolutely uniform scale&lt;/strong&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;N-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;/ul&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;
On the other hand, primality may be a desirable feature if you happen to want, e.g., 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;, 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;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="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;
Multiples of an EDO can inherit properties from the EDO, in particular a tuning for certain intervals. The multiple, on the other hand, is necessarily more complex.&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)
Retrieved from "https://en.xen.wiki/index.php?title=Prime_number&oldid=207815"