33edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-11-28 11:46:08 UTC</tt>.

: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-11-28 11:57:26 UTC</tt>.

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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, leading to a near perfect 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 flattone [[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]]. 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.

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 291 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 (if you use the patent val it is an extremely inaccurate 6/5). 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.

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.

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1: 36.364 48/47

2: 72.727 24/23

3: 109.091 ~~[[tel:~~16/15 17/16~~|16/15 17/16]]~~

3: 109.091 16/15 17/16

4: 145.455 12/11

5: 181.818 10/9

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

7: 254.545 7/6 ~~[[tel:22/19 37/32|~~22/19 37/32]]

7: 254.545 7/6 22/19 37/32

8: 290.909 ~~[[tel:13/11 19/16|~~13/11 19/16]]

8: 290.909 13/11 19/16

9: 327.273 6/5

10: 363.636 ~~[[tel:~~16/13 21/17~~|16/13 21/17]]~~

10: 363.636 16/13 21/17

11: 400.000 5/4

12: 436.364 9/7

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15: 545.455 11/8

16: 581.818 7/5

17: 618.182 ~~[[tel:10/7 23/16|~~10/7 23/16]]

17: 618.182 10/7 23/16

18: 654.545 ~~[[tel:19/13 16/11|~~19/13 16/11]]

18: 654.545 19/13 16/11

19: 690.909 3/2

20: 727.273 32/21

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24: 872.727 5/3

25: 909.091 22/13

26: 945.455 ~~[[tel:~~19/11 12/7~~|19/11 12/7]]~~

26: 945.455 19/11 12/7

27: 981.818 7/4

28: 1018.182 9/5

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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, leading to a near perfect 10/9. The &lt;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 flattone <a class="wiki_link" href="/5L%202s">5L+2s</a> 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 <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>. 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.

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 291 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 (if you use the patent val it is an extremely inaccurate 6/5). 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>. 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.