33edo: Difference between revisions

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<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:~~Kosmorsky~~|~~Kosmorsky~~]] and made on <tt>2011-07-~~20 10~~:27:12 UTC</tt>.<br>

: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2011-07-21 03:41:06 UTC</tt>.<br>

: The original revision id was <tt>~~242103859~~</tt>.<br>

: The original revision id was <tt>242231351</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>

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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">The //33 equal division// divides the [[octave]] into 33 equal parts of 36.3636 [[cent]]s each. It is not especially good at representing all rational intervals in the [[7-limit]], but it does very well on the 7-limit [[k*N subgroups|3*33 subgroup]] 2.27.15.21. On this subgroup it tunes things to the same tuning as [[99edo]], and as a subgroup patent val it tempers out the same commas. The 99 equal temperaments hemififths, amity, parakleismic, hemiwuerschmidt, ennealimmal and hendecatonic can be reduced to this subgroup and give various possibilities for MOS scales, etc.

While relatively uncommon, ~~33 edo~~ is actually quite an interesting system. As a multiple of ~~11 edo~~, it approximates the 7th and 11th harmonics via Andrew Heathwaite's 4L+3s Orgone modes (see ~~26 edo~~). ~~33 edo~~ also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having an 3L+7s of L=4 s=3. It tunes the perfect fifth about 11 cents flat, allowing a peculiar form of 'commatic meantone' where two fifths are tempered to 10/9 now (leaving the scale be would result in 5L+3s L=4 s=3 buuuut it's more interesting) if you call the ~~11-edo~~ 218-cent interval a sharp 9/8 (in fact so that 9/8 ~ 8/7 if there's a name for such a temperament, if not I call it Camelot), it takes you to the 400-cent major third (1/3 octave, just like ~~12-edo~~), and similarly lowering the 327-cent minor third to 290 c, which if you like could also be called a flat 19th harmonic. So while it might not be the most harmonically accurate temperament, it's structurally interesting, and it touches the 19-limit in it's way.</pre></div>

While relatively uncommon, 33edo is actually quite an interesting system. As a multiple of [[11edo]], it approximates the 7th and 11th harmonics via Andrew Heathwaite's 4L+3s Orgone modes (see [[26edo]]). 33edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having an 3L+7s of L=4 s=3. It tunes the perfect fifth about 11 cents flat, allowing a peculiar form of 'commatic meantone' where two fifths are tempered to 10/9 now (leaving the scale be would result in 5L+3s L=4 s=3 buuuut it's more interesting) if you call the 11edo 218-cent interval a sharp 9/8 (in fact so that 9/8 ~ 8/7 if there's a name for such a temperament, if not I call it Camelot), it takes you to the 400-cent major third (1/3 octave, just like 12edo), and similarly lowering the 327-cent minor third to 290 c, which if you like could also be called a flat 19th harmonic. So while it might not be the most harmonically accurate temperament, it's structurally interesting, and it touches the 19-limit in it's way.</pre></div>

<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>33edo</title></head><body>The <em>33 equal division</em> divides the <a class="wiki_link" href="/octave">octave</a> into 33 equal parts of 36.3636 <a class="wiki_link" href="/cent">cent</a>s each. It is not especially good at representing all rational intervals in the <a class="wiki_link" href="/7-limit">7-limit</a>, but it does very well on the 7-limit <a class="wiki_link" href="/k%2AN%20subgroups">3*33 subgroup</a> 2.27.15.21. On this subgroup it tunes things to the same tuning as <a class="wiki_link" href="/99edo">99edo</a>, and as a subgroup patent val it tempers out the same commas. The 99 equal temperaments hemififths, amity, parakleismic, hemiwuerschmidt, ennealimmal and hendecatonic can be reduced to this subgroup and give various possibilities for MOS scales, etc.<br />

<br />

While relatively uncommon, ~~33 edo~~ is actually quite an interesting system. As a multiple of ~~11 edo~~, it approximates the 7th and 11th harmonics via Andrew Heathwaite's 4L+3s Orgone modes (see ~~26 edo~~). ~~33 edo~~ also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having an 3L+7s of L=4 s=3. It tunes the perfect fifth about 11 cents flat, allowing a peculiar form of 'commatic meantone' where two fifths are tempered to 10/9 now (leaving the scale be would result in 5L+3s L=4 s=3 buuuut it's more interesting) if you call the ~~11-edo~~ 218-cent interval a sharp 9/8 (in fact so that 9/8 ~ 8/7 if there's a name for such a temperament, if not I call it Camelot), it takes you to the 400-cent major third (1/3 octave, just like ~~12-edo~~), and similarly lowering the 327-cent minor third to 290 c, which if you like could also be called a flat 19th harmonic. So while it might not be the most harmonically accurate temperament, it's structurally interesting, and it touches the 19-limit in it's way.</body></html></pre></div>

While relatively uncommon, 33edo is actually quite an interesting system. As a multiple of <a class="wiki_link" href="/11edo">11edo</a>, it approximates the 7th and 11th harmonics via Andrew Heathwaite's 4L+3s Orgone modes (see <a class="wiki_link" href="/26edo">26edo</a>). 33edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having an 3L+7s of L=4 s=3. It tunes the perfect fifth about 11 cents flat, allowing a peculiar form of 'commatic meantone' where two fifths are tempered to 10/9 now (leaving the scale be would result in 5L+3s L=4 s=3 buuuut it's more interesting) if you call the 11edo 218-cent interval a sharp 9/8 (in fact so that 9/8 ~ 8/7 if there's a name for such a temperament, if not I call it Camelot), it takes you to the 400-cent major third (1/3 octave, just like 12edo), and similarly lowering the 327-cent minor third to 290 c, which if you like could also be called a flat 19th harmonic. So while it might not be the most harmonically accurate temperament, it's structurally interesting, and it touches the 19-limit in it's way.</body></html></pre></div>

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 <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:Kosmorsky|Kosmorsky]] and made on <tt>2011-07-20 10:27:12 UTC</tt>.<br>
+: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2011-07-21 03:41:06 UTC</tt>.<br>
-: The original revision id was <tt>242103859</tt>.<br>
+: The original revision id was <tt>242231351</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">The //33 equal division// divides the [[octave]] into 33 equal parts of 36.3636 [[cent]]s each. It is not especially good at representing all rational intervals in the [[7-limit]], but it does very well on the 7-limit [[k*N subgroups|3*33 subgroup]] 2.27.15.21. On this subgroup it tunes things to the same tuning as [[99edo]], and as a subgroup patent val it tempers out the same commas. The 99 equal temperaments hemififths, amity, parakleismic, hemiwuerschmidt, ennealimmal and hendecatonic can be reduced to this subgroup and give various possibilities for MOS scales, etc.
-While relatively uncommon, 33 edo is actually quite an interesting system. As a multiple of 11 edo, it approximates the 7th and 11th harmonics via Andrew Heathwaite's 4L+3s Orgone modes (see 26 edo). 33 edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having an 3L+7s of L=4 s=3. It tunes the perfect fifth about 11 cents flat, allowing a peculiar form of 'commatic meantone' where two fifths are tempered to 10/9 now (leaving the scale be would result in 5L+3s L=4 s=3 buuuut it's more interesting) if you call the 11-edo 218-cent interval a sharp 9/8 (in fact so that 9/8 ~ 8/7 if there's a name for such a temperament, if not I call it Camelot), it takes you to the 400-cent major third (1/3 octave, just like 12-edo), and similarly lowering the 327-cent minor third to 290 c, which if you like could also be called a flat 19th harmonic. So while it might not be the most harmonically accurate temperament, it's structurally interesting, and it touches the 19-limit in it's way.</pre></div>
+While relatively uncommon, 33edo is actually quite an interesting system. As a multiple of [[11edo]], it approximates the 7th and 11th harmonics via Andrew Heathwaite's 4L+3s Orgone modes (see [[26edo]]). 33edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having an 3L+7s of L=4 s=3. It tunes the perfect fifth about 11 cents flat, allowing a peculiar form of 'commatic meantone' where two fifths are tempered to 10/9 now (leaving the scale be would result in 5L+3s L=4 s=3 buuuut it's more interesting) if you call the 11edo 218-cent interval a sharp 9/8 (in fact so that 9/8 ~ 8/7 if there's a name for such a temperament, if not I call it Camelot), it takes you to the 400-cent major third (1/3 octave, just like 12edo), and similarly lowering the 327-cent minor third to 290 c, which if you like could also be called a flat 19th harmonic. So while it might not be the most harmonically accurate temperament, it's structurally interesting, and it touches the 19-limit in it's way.</pre></div>
 <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;33edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;33 equal division&lt;/em&gt; divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 33 equal parts of 36.3636 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. It is not especially good at representing all rational intervals in the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt;, but it does very well on the 7-limit &lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;3*33 subgroup&lt;/a&gt; 2.27.15.21. On this subgroup it tunes things to the same tuning as &lt;a class="wiki_link" href="/99edo"&gt;99edo&lt;/a&gt;, and as a subgroup patent val it tempers out the same commas. The 99 equal temperaments hemififths, amity, parakleismic, hemiwuerschmidt, ennealimmal and hendecatonic can be reduced to this subgroup and give various possibilities for MOS scales, etc.&lt;br /&gt;
 &lt;br /&gt;
-While relatively uncommon, 33 edo is actually quite an interesting system. As a multiple of 11 edo, it approximates the 7th and 11th harmonics via Andrew Heathwaite's 4L+3s Orgone modes (see 26 edo). 33 edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having an 3L+7s of L=4 s=3. It tunes the perfect fifth about 11 cents flat, allowing a peculiar form of 'commatic meantone' where two fifths are tempered to 10/9 now (leaving the scale be would result in 5L+3s L=4 s=3 buuuut it's more interesting) if you call the 11-edo 218-cent interval a sharp 9/8 (in fact so that 9/8 ~ 8/7 if there's a name for such a temperament, if not I call it Camelot), it takes you to the 400-cent major third (1/3 octave, just like 12-edo), and similarly lowering the 327-cent minor third to 290 c, which if you like could also be called a flat 19th harmonic. So while it might not be the most harmonically accurate temperament, it's structurally interesting, and it touches the 19-limit in it's way.&lt;/body&gt;&lt;/html&gt;</pre></div>
+While relatively uncommon, 33edo is actually quite an interesting system. As a multiple of &lt;a class="wiki_link" href="/11edo"&gt;11edo&lt;/a&gt;, it approximates the 7th and 11th harmonics via Andrew Heathwaite's 4L+3s Orgone modes (see &lt;a class="wiki_link" href="/26edo"&gt;26edo&lt;/a&gt;). 33edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having an 3L+7s of L=4 s=3. It tunes the perfect fifth about 11 cents flat, allowing a peculiar form of 'commatic meantone' where two fifths are tempered to 10/9 now (leaving the scale be would result in 5L+3s L=4 s=3 buuuut it's more interesting) if you call the 11edo 218-cent interval a sharp 9/8 (in fact so that 9/8 ~ 8/7 if there's a name for such a temperament, if not I call it Camelot), it takes you to the 400-cent major third (1/3 octave, just like 12edo), and similarly lowering the 327-cent minor third to 290 c, which if you like could also be called a flat 19th harmonic. So while it might not be the most harmonically accurate temperament, it's structurally interesting, and it touches the 19-limit in it's way.&lt;/body&gt;&lt;/html&gt;</pre></div>