Prime number
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=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
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<html><head><title>prime numbers</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><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:<h2> --><h2 id="toc1"><a name="Some thoughts about prime numbers in EDOs-The first "Prime edos""></a><!-- ws:end:WikiTextHeadingRule:2 -->The first "Prime edos"</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 /> <br /> <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 /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><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:<h2> --><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>