Maximal evenness: Difference between revisions

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-11-05 00:14:29 UTC</tt>.<br>

: The original revision id was <tt>100335277</tt>.<br>

<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">Within every [[edo]] one can specify a "quasi-equal scale" for every smaller edo. The quasi-equal scale would be the closest the parent edo can get to representing the smaller edo.

The quasi-equal scale will be one:

(a) and (b) above imply that the quasi-equal scale will be a [[MOSScales|moment of symmetry scale]].

For instance, here are all the quasi-equal scales available in [[31edo]]:

And here are all the quasi-equal scales available in [[13edo]]:

&lt;span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"&gt;2 .. 6 7

3 .. 4 4 5

4 .. 3 3 3 4

5 .. 2 3 2 3 3

6 .. 2 2 2 2 2 3

7 .. 2 2 2 2 2 2 1

8 .. 2 2 1 2 2 1 2 1

9 .. 2 1 2 1 2 1 2 1 1

10 . 2 1 1 2 1 1 2 1 1 1

11 . 2 1 1 1 1 2 1 1 1 1 1

12 . 1 1 1 1 1 1 1 1 1 1 1 2 &lt;/span&gt;

The quasi-equal scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31th of an octave instead of one 13th).

The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo. </pre></div>

<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;Maximal evenness&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Within every &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt; one can specify a &amp;quot;quasi-equal scale&amp;quot; for every smaller edo. The quasi-equal scale would be the closest the parent edo can get to representing the smaller edo.&lt;br /&gt;

The quasi-equal scale will be one:&lt;br /&gt;

(a) and (b) above imply that the quasi-equal scale will be a &lt;a class="wiki_link" href="/MOSScales"&gt;moment of symmetry scale&lt;/a&gt;.&lt;br /&gt;

And here are all the quasi-equal scales available in &lt;a class="wiki_link" href="/13edo"&gt;13edo&lt;/a&gt;:&lt;br /&gt;

&lt;span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"&gt;2 .. 6 7&lt;br /&gt;

3 .. 4 4 5&lt;br /&gt;

4 .. 3 3 3 4&lt;br /&gt;

5 .. 2 3 2 3 3&lt;br /&gt;

6 .. 2 2 2 2 2 3&lt;br /&gt;

7 .. 2 2 2 2 2 2 1&lt;br /&gt;

8 .. 2 2 1 2 2 1 2 1&lt;br /&gt;

9 .. 2 1 2 1 2 1 2 1 1&lt;br /&gt;

10 . 2 1 1 2 1 1 2 1 1 1&lt;br /&gt;

11 . 2 1 1 1 1 2 1 1 1 1 1&lt;br /&gt;

12 . 1 1 1 1 1 1 1 1 1 1 1 2 &lt;/span&gt;&lt;br /&gt;

The quasi-equal scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31th of an octave instead of one 13th).&lt;br /&gt;