Prime number: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-07-06 10:19:56 UTC</tt>.<br>

: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-07-06 10:31:40 UTC</tt>.<br>

: The original revision id was <tt>~~240196903~~</tt>.<br>

: The original revision id was <tt>240198529</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>

Line 8:

<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=

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

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

* 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]].

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.

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.)

OTOH, primeness 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.)

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.

Line 25:

Line 22:

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

==The first "Prime ~~edos~~"==

==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 EDOs 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]],

Line 47:

Line 44:

<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"><html><head><title>prime numbers</title></head><body><h1 id="toc0"><a name="Some thoughts about prime numbers in EDOs"></a>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 />

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 values of n.<br />

<~~br /~~>

<ul><li>If the octave is divided into a prime number of equal parts, there is <strong>no fully symmetric chord</strong>, such as the diminished seventh chord in <a class="wiki_link" href="/12edo">12edo</a>.</li><li>There is also (besides the full scale of all notes of the edo) <strong>no absolutely uniform scale</strong>, like the wholetone scale in 12edo.</li><li>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.</li></ul><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 />

OTOH, primeness may be a desirable feature if you happen to want, e.g., a wholetone scale 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 />

Line 64:

Line 55:

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

<br />

<h2 id="toc1"><a name="Some thoughts about prime numbers in EDOs-The first &quot;Prime ~~edos~~&quot;"></a>The first &quot;Prime ~~edos~~&quot;</h2>

<h2 id="toc1"><a name="Some thoughts about prime numbers in EDOs-The first &quot;Prime EDOs&quot;"></a>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 />

Prime EDOs 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 />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-07-06 10:19:56 UTC</tt>.<br>
+: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-07-06 10:31:40 UTC</tt>.<br>
-: The original revision id was <tt>240196903</tt>.<br>
+: The original revision id was <tt>240198529</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>
@@ Line 8: / Line 8: @@
 <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=
-Whether a number n is prime or not has quite vital consequences for the properties of the corresponding n-[[edo]], especially for lower numbers.
+Whether a number n is prime or not has quite vital consequences for the properties of the corresponding n-[[edo|EDO]], especially for lower values of n.
+* 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]].
-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.
-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.)
+OTOH, primeness 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.)
 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.
@@ Line 25: / Line 22: @@
 todo: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, theorists here. XXX
-==The first "Prime edos"==
+==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 EDOs 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]],
@@ Line 47: / Line 44: @@
 <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;
+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 values of n.&lt;br /&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;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;/ul&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;
+OTOH, primeness may be a desirable feature if you happen to want, e.g., a wholetone scale 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;
@@ Line 64: / Line 55: @@
 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="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;
+&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 EDOs 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;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;