33edo: 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:~~Osmiorisbendi~~|~~Osmiorisbendi~~]] and made on <tt>2011-07-~~21 03~~:41:06 UTC</tt>.<br>

: This revision was by author [[User:guest|guest]] and made on <tt>2011-07-28 15:04:39 UTC</tt>.<br>

: The original revision id was <tt>~~242231351~~</tt>.<br>

: The original revision id was <tt>243282611</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">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, 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>

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|3L+7s]] of L=4 s=3. It tunes the perfect fifth about 11 cents flat, such that two fifths are tempered to a good 10/9. Leaving the scale be would result in a flat-tone [[5L 2s|5L+2s]] of L=5 s=4

Now, if you call the 11edo 218-cent interval a sharp 9/8 (in fact so that 9/8 ~ 8/7), 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 quite interesting, and it approximates the full 19-limit consort in it's way. You could even say it tunes the 23rd and 29th harmonics ten cents flat if you were so inclined; as well as getting within two cents of the 37th.

0: 00.000 1/1

1: 36.364 cents

2: 72.727 cents

3: 109.091 17/16

4: 145.455 cents

5: 181.818 10/9

6: 218.182 8/7 9/8

7: 254.545 37/32

8: 290.909 19/16

9: 327.273 6/5

10: 363.636 16/13

11: 400.000 4/3

12: 436.364 9/7

13: 472.727 21/16

14: 509.091 4/3

15: 545.455 11/8

16: 581.818 7/5

17: 618.182 23/16

18: 654.545 cents

19: 690.909 3/2

20: 727.273 cents

21: 763.636 cents

22: 800.000 cents

23: 836.364 13/8

24: 872.727 cents

25: 909.091 cents

26: 945.455 7/4

27: 981.818 7/4

28: 1018.182 9/5

29: 1054.545 cents

30: 1090.909 15/8

31: 1127.273 cents

32: 1163.636 cents

33: 1200.000 cents</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, 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>

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 <a class="wiki_link" href="/3L%207s">3L+7s</a> of L=4 s=3. It tunes the perfect fifth about 11 cents flat, such that two fifths are tempered to a good 10/9. Leaving the scale be would result in a flat-tone <a class="wiki_link" href="/5L%202s">5L+2s</a> of L=5 s=4<br />

<br />

Now, if you call the 11edo 218-cent interval a sharp 9/8 (in fact so that 9/8 ~ 8/7), 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 quite interesting, and it approximates the full 19-limit consort in it's way. You could even say it tunes the 23rd and 29th harmonics ten cents flat if you were so inclined; as well as getting within two cents of the 37th.<br />

<br />

0: 00.000 1/1<br />

1: 36.364 cents<br />

2: 72.727 cents<br />

3: 109.091 17/16<br />

4: 145.455 cents<br />

5: 181.818 10/9<br />

6: 218.182 8/7 9/8<br />

7: 254.545 37/32<br />

8: 290.909 19/16<br />

9: 327.273 6/5<br />

10: 363.636 16/13<br />

11: 400.000 4/3<br />

12: 436.364 9/7<br />

13: 472.727 21/16<br />

14: 509.091 4/3<br />

15: 545.455 11/8<br />

16: 581.818 7/5<br />

17: 618.182 23/16<br />

18: 654.545 cents<br />

19: 690.909 3/2<br />

20: 727.273 cents<br />

21: 763.636 cents<br />

22: 800.000 cents<br />

23: 836.364 13/8<br />

24: 872.727 cents<br />

25: 909.091 cents<br />

26: 945.455 7/4<br />

27: 981.818 7/4<br />

28: 1018.182 9/5<br />

29: 1054.545 cents<br />

30: 1090.909 15/8<br />

31: 1127.273 cents<br />

32: 1163.636 cents<br />

33: 1200.000 cents</body></html></pre></div>

@@ 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:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2011-07-21 03:41:06 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2011-07-28 15:04:39 UTC</tt>.<br>
-: The original revision id was <tt>242231351</tt>.<br>
+: The original revision id was <tt>243282611</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, 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>
+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|3L+7s]] of L=4 s=3. It tunes the perfect fifth about 11 cents flat, such that two fifths are tempered to a good 10/9. Leaving the scale be would result in a flat-tone [[5L 2s|5L+2s]] of L=5 s=4
+Now, if you call the 11edo 218-cent interval a sharp 9/8 (in fact so that 9/8 ~ 8/7), 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 quite interesting, and it approximates the full 19-limit consort in it's way. You could even say it tunes the 23rd and 29th harmonics ten cents flat if you were so inclined; as well as getting within two cents of the 37th.
+: 00.000 1/1
+: 36.364 cents
+: 72.727 cents
+: 109.091 17/16
+: 145.455 cents
+: 181.818 10/9
+: 218.182 8/7 9/8
+: 254.545 37/32
+: 290.909 19/16
+: 327.273 6/5
+: 363.636 16/13
+: 400.000 4/3
+: 436.364 9/7
+: 472.727 21/16
+: 509.091 4/3
+: 545.455 11/8
+: 581.818 7/5
+: 618.182 23/16
+: 654.545 cents
+: 690.909 3/2
+: 727.273 cents
+: 763.636 cents
+: 800.000 cents
+: 836.364 13/8
+: 872.727 cents
+: 909.091 cents
+: 945.455 7/4
+: 981.818 7/4
+: 1018.182 9/5
+: 1054.545 cents
+: 1090.909 15/8
+: 1127.273 cents
+: 1163.636 cents
+: 1200.000 cents</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, 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>
+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 &lt;a class="wiki_link" href="/3L%207s"&gt;3L+7s&lt;/a&gt; of L=4 s=3. It tunes the perfect fifth about 11 cents flat, such that two fifths are tempered to a good 10/9. Leaving the scale be would result in a flat-tone &lt;a class="wiki_link" href="/5L%202s"&gt;5L+2s&lt;/a&gt; of L=5 s=4&lt;br /&gt;
+&lt;br /&gt;
+Now, if you call the 11edo 218-cent interval a sharp 9/8 (in fact so that 9/8 ~ 8/7), 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 quite interesting, and it approximates the full 19-limit consort in it's way. You could even say it tunes the 23rd and 29th harmonics ten cents flat if you were so inclined; as well as getting within two cents of the 37th.&lt;br /&gt;
+&lt;br /&gt;
+: 00.000 1/1&lt;br /&gt;
+: 36.364 cents&lt;br /&gt;
+: 72.727 cents&lt;br /&gt;
+: 109.091 17/16&lt;br /&gt;
+: 145.455 cents&lt;br /&gt;
+: 181.818 10/9&lt;br /&gt;
+: 218.182 8/7 9/8&lt;br /&gt;
+: 254.545 37/32&lt;br /&gt;
+: 290.909 19/16&lt;br /&gt;
+: 327.273 6/5&lt;br /&gt;
+: 363.636 16/13&lt;br /&gt;
+: 400.000 4/3&lt;br /&gt;
+: 436.364 9/7&lt;br /&gt;
+: 472.727 21/16&lt;br /&gt;
+: 509.091 4/3&lt;br /&gt;
+: 545.455 11/8&lt;br /&gt;
+: 581.818 7/5&lt;br /&gt;
+: 618.182 23/16&lt;br /&gt;
+: 654.545 cents&lt;br /&gt;
+: 690.909 3/2&lt;br /&gt;
+: 727.273 cents&lt;br /&gt;
+: 763.636 cents&lt;br /&gt;
+: 800.000 cents&lt;br /&gt;
+: 836.364 13/8&lt;br /&gt;
+: 872.727 cents&lt;br /&gt;
+: 909.091 cents&lt;br /&gt;
+: 945.455 7/4&lt;br /&gt;
+: 981.818 7/4&lt;br /&gt;
+: 1018.182 9/5&lt;br /&gt;
+: 1054.545 cents&lt;br /&gt;
+: 1090.909 15/8&lt;br /&gt;
+: 1127.273 cents&lt;br /&gt;
+: 1163.636 cents&lt;br /&gt;
+: 1200.000 cents&lt;/body&gt;&lt;/html&gt;</pre></div>