33edo: Difference between revisions

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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]] 2.27.15.21.11.13. 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. In particular, the [[Chromatic pairs#Terrain|terrain]] subgroup temperament can be tuned via the 5\33 generator. The full system of harmony provides the optimal patent val for [[Mint temperaments#Slurpee|slurpee temperament]] in the 5, 7, 11 and 13 limits.

Instead of the flat 19\33 fifth you may use the sharp fifth of 20\33, over 25 cents sharp. Two of these lead to a 9/8 of 7\33, which is about 22/19 in size and may be counted as a small third. Between the flat 5\33 version of 9/8 and the sharp 7\33 version there is, of course, a 6\33 = 2\11 [[11edo]] interval of 218 cents. Now 6\33 + 5\33 = 11\33 = 1\3 of an octave, or 400 cents, the same major third as 12edo. Also, we have both a 327 minor third from 9\33 = 3\11, the same as the [[22edo]] minor third, and a flatter 8\33 third of 290 cents, which if you like could also be called a flat 19th harmonic, but much more accurately a 13/11 sharp by 1.7 cents. Another talent it has is that 7/5 is tuned quite accurately by 16\33, and we may put two 8\33 versions of 13/11 together to produce the [[cuthbert triad]]. The 8\33 generator, with MOS of size 5, 9 and 13, gives plenty of scope for these, as well as the 11, 13 and 19 harmonics (taking the generator as a 19/16) which are relatively well in tune.

1: 36.364 48/47

2: 72.727 24/23

4: 145.455 12/11

5: 181.818 10/9

6: 218.182 8/7 9/8 17/15

9: 327.273 6/5

11: 400.000 5/4

12: 436.364 9/7

15: 545.455 11/8

16: 581.818 7/5

19: 690.909 3/2

20: 727.273 32/21

24: 872.727 5/3

25: 909.091 22/13

27: 981.818 7/4

28: 1018.182 9/5

<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; 2.27.15.21.11.13. 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. In particular, the &lt;a class="wiki_link" href="/Chromatic%20pairs#Terrain"&gt;terrain&lt;/a&gt; subgroup temperament can be tuned via the 5\33 generator. The full system of harmony provides the optimal patent val for &lt;a class="wiki_link" href="/Mint%20temperaments#Slurpee"&gt;slurpee temperament&lt;/a&gt; in the 5, 7, 11 and 13 limits.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

Instead of the flat 19\33 fifth you may use the sharp fifth of 20\33, over 25 cents sharp. Two of these lead to a 9/8 of 7\33, which is about 22/19 in size and may be counted as a small third. Between the flat 5\33 version of 9/8 and the sharp 7\33 version there is, of course, a 6\33 = 2\11 &lt;a class="wiki_link" href="/11edo"&gt;11edo&lt;/a&gt; interval of 218 cents. Now 6\33 + 5\33 = 11\33 = 1\3 of an octave, or 400 cents, the same major third as 12edo. Also, we have both a 327 minor third from 9\33 = 3\11, the same as the &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; minor third, and a flatter 8\33 third of 290 cents, which if you like could also be called a flat 19th harmonic, but much more accurately a 13/11 sharp by 1.7 cents. Another talent it has is that 7/5 is tuned quite accurately by 16\33, and we may put two 8\33 versions of 13/11 together to produce the &lt;a class="wiki_link" href="/cuthbert%20triad"&gt;cuthbert triad&lt;/a&gt;. The 8\33 generator, with MOS of size 5, 9 and 13, gives plenty of scope for these, as well as the 11, 13 and 19 harmonics (taking the generator as a 19/16) which are relatively well in tune.&lt;br /&gt;

1: 36.364 48/47&lt;br /&gt;

2: 72.727 24/23&lt;br /&gt;

4: 145.455 12/11&lt;br /&gt;

5: 181.818 10/9&lt;br /&gt;

6: 218.182 8/7 9/8 17/15&lt;br /&gt;

9: 327.273 6/5&lt;br /&gt;

11: 400.000 5/4&lt;br /&gt;

12: 436.364 9/7&lt;br /&gt;

15: 545.455 11/8&lt;br /&gt;

16: 581.818 7/5&lt;br /&gt;

19: 690.909 3/2&lt;br /&gt;

20: 727.273 32/21&lt;br /&gt;

24: 872.727 5/3&lt;br /&gt;

25: 909.091 22/13&lt;br /&gt;

27: 981.818 7/4&lt;br /&gt;

28: 1018.182 9/5&lt;br /&gt;