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33edo

Revision as of 01:17, 1 September 2022 by Inthar (talk | contribs) (→Intervals: sign corrections to error column, unify to 3 decimal places + a few other changes)
← 32edo 33edo 34edo →
Prime factorization 3 x 11
Step size 36.3636 ¢ 
Fifth 19\33 (690.909 ¢)
Semitones (A1:m2) 1:4 (36.36 ¢ : 145.5 ¢)
Consistency limit 3
Distinct consistency limit 3

The 33 equal division divides the octave into 33 equal parts of 36.3636 cents each. It is not especially good at representing all rational intervals in the 7-limit, but it does very well on the 7-limit 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 terrain subgroup temperament can be tuned via the 5\33 generator. The full system of harmony provides the optimal patent val for slurpee temperament in the 5, 7, 11 and 13 limits.

While relatively uncommon, 33edo is actually quite an interesting system. As a multiple of 11edo, it approximates the 7th and 11th harmonics via orgone temperament (see 26edo). 33edo also tunes the 13th harmonic slightly flat, allowing it to approximate the 21st and 17th harmonics as well, having a 3L 7s with 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 with 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 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.

Other notable 33edo scales are diasem with L:m:s = 5:3:1 and 5L 4s with L:s = 5:2. This step ratio for 5L 4s is great for its semitone size of 72.7¢.

Approximation of odd harmonics in 33 EDO
Odd harmonic 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
Error absolute (¢) -11.0 +13.7 +13.0 +14.3 -5.9 -4.2 +2.6 +4.1 -6.6 +1.9 -10.1 -9.0 +3.2 -11.4 -17.8
relative (%) -30 +38 +36 +39 -16 -11 +7 +11 -18 +5 -28 -25 +9 -31 -49
Steps (reduced) 52 (19) 77 (11) 93 (27) 105 (6) 114 (15) 122 (23) 129 (30) 135 (3) 140 (8) 145 (13) 149 (17) 153 (21) 157 (25) 160 (28) 163 (31)

Intervals

Step # ET Just Difference
(ET minus Just) 		Extended Pythagorean Notation
Cents Interval Cents
0 0¢ 1/1 0 0 Perfect Unison P1 D
1 36.364 48/47 36.448 +0.085 Augmented Unison A1 D#
2 72.727 24/23 73.681 −0.953 Double-aug 1sn AA1 Dx
3 109.091 16/15 111.731 −2.640 Diminished 2nd d2 Ebb
4 145.455 12/11 150.637 −5.183 Minor 2nd m2 Eb
5 181.818 10/9 182.404 −0.586 Major 2nd M2 E
6 218.182 17/15 216.687 +1.495 Augmented 2nd A2 E#
7 254.545 15/13 247.741 +6.804 Double-aug 2nd/Double-dim 3rd AA2/dd3 Ex/Fbb
8 290.909 13/11 289.210 +1.699 Diminished 3rd d3 Fb
9 327.273 6/5 315.641 +11.631 Minor 3rd m3 F
10 363.636 16/13 359.472 +4.164 Major 3rd M3 F#
11 400.000 5/4 386.314 +13.686 Augmented 3rd A3 Fx
12 436.364 9/7 435.084 +1.280 Double-dim 4th dd4 Gbb
13 472.727 21/16 470.781 +1.946 Diminished 4th d4 Gb
14 509.091 4/3 498.045 +11.046 Perfect 4th P4 G
15 545.455 11/8 551.318 −5.863 Augmented 4th A4 G#
16 581.818 7/5 582.513 −0.694 Double-aug 4th AA4 Gx
17 618.182 10/7 617.488 +0.694 Double-dim 5th dd5 Abb
18 654.545 16/11 648.682 +5.863 Diminished 5th d5 Ab
19 690.909 3/2 701.9550 −11.046 Perfect 5th P5 A
20 727.273 32/21 729.219 -1.946 Augmented 5th A5 A#
21 763.636 14/9 764.9159 −1.280 Double-aug 5th AA5 Ax
22 800.000 8/5 813.686 −13.686 Double-dim 6th d6 Bbb
23 836.364 13/8 840.5276 −4.164 Minor 6th m6 Bb
24 872.727 5/3 884.359 −11.631 Major 6th M6 B
25 909.091 22/13 910.7903 −1.699 Augmented 6th A6 B#
26 945.455 12/7 933.129 +12.325 Double-aug 6th/Double-dim 7th AA6/dd7 Bx/Cbb
27 981.818 30/17 983.313 −1.495 Diminished 7th d7 Cb
28 1018.182 9/5 1017.596 +0.586 Minor 7th m7 C
29 1054.545 11/6 1049.363 +5.183 Major 7th M7 C#
30 1090.909 15/8 1088.268 +2.640 Augmented 7th A7 Cx
31 1127.273 23/12 1126.319 −0.953 Double-dim 8ve dd8 Dbb
32 1163.636 47/24 1163.551 0.085 Diminished 8ve d8 Db
33 1200 2/1 1200 0 Perfect Octave P8 D

Nearby Equal Temperaments:

 

Notable scales

Music

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