33edo
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- This revision was by author genewardsmith and made on 2011-07-28 17:05:48 UTC.
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Original Wikitext content:
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|3L+7s]] of L=4 s=3. It tunes the perfect fifth about 11 cents flat, to 19\33, so that two fifths down an octave, 5\33, the approximation of 9/8, is actually half a cent flat from 10/9. The <33 52 76| or 33c val tempers out 81/80 and so leads to a very flat meantone tuning where the major tone is approximately 10/9 in size. Leaving the scale be would result in a flat-tone [[5L 2s|5L+2s]] of L=5 s=4 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]]. 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
Original HTML content:
<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 <a class="wiki_link" href="/3L%207s">3L+7s</a> of L=4 s=3. It tunes the perfect fifth about 11 cents flat, to 19\33, so that two fifths down an octave, 5\33, the approximation of 9/8, is actually half a cent flat from 10/9. The <33 52 76| or 33c val tempers out 81/80 and so leads to a very flat meantone tuning where the major tone is approximately 10/9 in size. 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 /> 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 <a class="wiki_link" href="/11edo">11edo</a> 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 <a class="wiki_link" href="/22edo">22edo</a> 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 <a class="wiki_link" href="/cuthbert%20triad">cuthbert triad</a>. 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>