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| <h2>IMPORTED REVISION FROM WIKISPACES</h2> | | <span style="font-size: 19.5px;">Division of a tenth (e. g. 7/3) into n equal parts</span> |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:diagonalia|diagonalia]] and made on <tt>2017-01-03 00:20:02 UTC</tt>.<br>
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| : The original revision id was <tt>602985154</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <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"><span style="font-size: 19.5px;">Division of a tenth (e. g. 7/3) into n equal parts</span>
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| | | 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|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. |
| 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. | |
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| 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. | | 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. |
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| 3&6: Tritetrachordal | | 3&6: Tritetrachordal |
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| 4&5: Montrose (between 5/4edo and 4/3edo in particular, MOS generated by [pseudo] octaves belong to this branch) | | 4&5: Montrose (between 5/4edo and 4/3edo in particular, MOS generated by [pseudo] octaves belong to this branch) |
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| 2&7: Terra Rubra | | 2&7: Terra Rubra |
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| 5&6: Rosablanca | | 5&6: Rosablanca |
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| 4&7: Saptimpun (10 1/2) | | 4&7: Saptimpun (10 1/2) |
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| 5&7: 8bittone | | 5&7: 8bittone |
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| [[8edX]] | | [[8edX|8edX]] |
| [[9edX]] | | |
| [[15edX]] | | [[9edX|9edX]] |
| [[16edX]] | | |
| [[17edX]] | | [[15edX|15edX]] |
| [[19edX]] | | |
| | [[16edX|16edX]] |
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| | [[17edX|17edX]] |
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| | [[19edX|19edX]] |
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| 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.</pre></div> | | 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. |
| <h4>Original HTML content:</h4>
| | [[Category:ed7/3]] |
| <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>edX</title></head><body><span style="font-size: 19.5px;">Division of a tenth (e. g. 7/3) into n equal parts</span><br />
| | [[Category:edX]] |
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| Division of e. g. the <a class="wiki_link" href="/7_3">7:3</a> into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of <a class="wiki_link" href="/equivalence">equivalence</a> 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 <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Dastg%C4%81h-e_M%C4%81hur" rel="nofollow">dastgah</a>) 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 &quot;Middletown valley&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 <a class="wiki_link" href="/Pseudo-traditional%20harmonic%20functions%20of%20enneatonic%20scale%20degrees">enneatonic scales</a>, 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.<br />
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| 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. &quot;Macrobichromatic&quot; might be a practically perfect term for it if it hasn't been named yet.<br />
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| The branches of the Middletown family are named thus:<br />
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| 3&amp;6: Tritetrachordal<br />
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| 4&amp;5: Montrose (between 5/4edo and 4/3edo in particular, MOS generated by [pseudo] octaves belong to this branch)<br />
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| 2&amp;7: Terra Rubra<br />
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| The family of interlaced octatonic scale based temperaments in the &quot;Middletown valley&quot; is called Vesuvius (i. e. the volcano east of Naples).<br />
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| The temperaments neighboring Middletown proper are named thus:<br />
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| 5&amp;6: Rosablanca<br />
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| 4&amp;7: Saptimpun (10 1/2)<br />
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| 5&amp;7: 8bittone<br />
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| <a class="wiki_link" href="/8edX">8edX</a><br />
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| <a class="wiki_link" href="/9edX">9edX</a><br />
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| <a class="wiki_link" href="/15edX">15edX</a><br />
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| <a class="wiki_link" href="/16edX">16edX</a><br />
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| <a class="wiki_link" href="/17edX">17edX</a><br />
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| <a class="wiki_link" href="/19edX">19edX</a><br />
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| 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.</body></html></pre></div>
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