Ed7/3: Difference between revisions

<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:diagonalia|diagonalia]] and made on <tt>2017-01-03 00:20:02 UTC</tt>.<br>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>

<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">&lt;span style="font-size: 19.5px;"&gt;Division of a tenth (e. g. 7/3) into n equal parts&lt;/span&gt;

Division of e. g. the [[7_3|7:3]] into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of [[equivalence]] has not even been posed yet. The utility of 7:3 or another tenth as a base though, is apparent by being the absolute widest range most generally used in popular songs (and even the range of a [[https://en.wikipedia.org/wiki/Dastg%C4%81h-e_M%C4%81hur|dastgah]]) as well as a fairly trivial point to split the difference between the octave and the tritave (which is why I have named the region of intervals between 6 and 7 degrees of 5edo the "Middletown valley", the proper Middletown temperament family being based on an enneatonic scale generated by a third or a fifth optionally with a period of a [wolf] fourth at most 560 cents wide). Incidentally [[Pseudo-traditional harmonic functions of enneatonic scale degrees|enneatonic scales]], especially those equivalent at e. g. 7:3, can sensibly take tetrads as the fundamental complete sonorities of a pseudo-traditional functional harmony due to their seventh degree being as structrally important as it is. Many, though not all, of these scales have a perceptually important pseudo (false) octave, with various degrees of accuracy.

Incidentally, one way to treat 7/3 as an equivalence is the use of the 3:4:5:6:(7) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes two 28/15 to get to 7/2 (tempering out the comma 225/224). So, doing this yields 2 15, 19, and 34 note MOS 2/1 apart. While the notes are rather farther apart, the scheme is uncannily similar to meantone. "Macrobichromatic" might be a practically perfect term for it if it hasn't been named yet.

4&amp;5: Montrose (between 5/4edo and 4/3edo in particular, MOS generated by [pseudo] octaves belong to this branch)

The family of interlaced octatonic scale based temperaments in the "Middletown valley" is called Vesuvius (i. e. the volcano east of Naples).

5&amp;6: Rosablanca

5&amp;7: 8bittone

<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;edX&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="font-size: 19.5px;"&gt;Division of a tenth (e. g. 7/3) into n equal parts&lt;/span&gt;&lt;br /&gt;

Division of e. g. the &lt;a class="wiki_link" href="/7_3"&gt;7:3&lt;/a&gt; into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of &lt;a class="wiki_link" href="/equivalence"&gt;equivalence&lt;/a&gt; has not even been posed yet. The utility of 7:3 or another tenth as a base though, is apparent by being the absolute widest range most generally used in popular songs (and even the range of a &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Dastg%C4%81h-e_M%C4%81hur" rel="nofollow"&gt;dastgah&lt;/a&gt;) as well as a fairly trivial point to split the difference between the octave and the tritave (which is why I have named the region of intervals between 6 and 7 degrees of 5edo the &amp;quot;Middletown valley&amp;quot;, the proper Middletown temperament family being based on an enneatonic scale generated by a third or a fifth optionally with a period of a [wolf] fourth at most 560 cents wide). Incidentally &lt;a class="wiki_link" href="/Pseudo-traditional%20harmonic%20functions%20of%20enneatonic%20scale%20degrees"&gt;enneatonic scales&lt;/a&gt;, especially those equivalent at e. g. 7:3, can sensibly take tetrads as the fundamental complete sonorities of a pseudo-traditional functional harmony due to their seventh degree being as structrally important as it is. Many, though not all, of these scales have a perceptually important pseudo (false) octave, with various degrees of accuracy.&lt;br /&gt;

Incidentally, one way to treat 7/3 as an equivalence is the use of the 3:4:5:6:(7) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes two 28/15 to get to 7/2 (tempering out the comma 225/224). So, doing this yields 2 15, 19, and 34 note MOS 2/1 apart. While the notes are rather farther apart, the scheme is uncannily similar to meantone. &amp;quot;Macrobichromatic&amp;quot; might be a practically perfect term for it if it hasn't been named yet.&lt;br /&gt;

4&amp;amp;5: Montrose (between 5/4edo and 4/3edo in particular, MOS generated by [pseudo] octaves belong to this branch)&lt;br /&gt;

The family of interlaced octatonic scale based temperaments in the &amp;quot;Middletown valley&amp;quot; is called Vesuvius (i. e. the volcano east of Naples).&lt;br /&gt;

The temperaments neighboring Middletown proper are named thus:&lt;br /&gt;

&lt;a class="wiki_link" href="/8edX"&gt;8edX&lt;/a&gt;&lt;br /&gt;

Sort of unsurprisingly, though not so evidently, the golden tuning of edXs will turn out to divide a barely mistuned 5:2 of alomst exactly 45/34edo.&lt;/body&gt;&lt;/html&gt;</pre></div>