33ed4

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33ed4 is the Equal Divisions of the Double Octave into 33 narrow chromatic semitones each of 72.727 cents. It takes out every second step of 33edo and falls between 16edo and 17edo. So even degree 16 or degree 17 can play the role of the octave, depending on the actual melodic or harmonic situation in a given composition. So it can be seen as a kind of Equivocal Tuning.

It has a 9/5 which is 0.6 cents sharp, a 7/5 which is 0.7 cents flat, and a 9/7 which is 1.3 cents sharp. Therefore it is closely related to 13edt, the Bohlen-Pierce scale, although it has no pure 3/1, which is 11.1 cents flat. The lack of a pure fifth makes it also interesting.

Furthermore it has some 11-limit, 13-limit, 17-limit and even 23-limit which are very close (most of them under or nearby 1 cent).

Intervals

degree in cents nearest JI

interval

in cents difference

in cents

1 72,7 24/23 73,7 -1,0
2 145,5 25/23 144,4 1,1
3 218,2 17/15 216,6 1,6
4 290,9 13/11 289,2 1,7
5 363,6 16/13 359,5 4,1
6 436,4 9/7 435,1 1,3
7 509,1 51/38 509,4 -0,3
8 581,8 7/5 582,5 -0,7
9 654,5 19/13 657,0 -2,5
10 727,3 35/23 726,9 0,4
11 800,0 27/17 800,9 -0,9
12 872,7 53/32 873,5 -0,8
13 945,5 19/11 946,2 -0,7
14 1018,2 9/5 1017,6 0,6
15 1090,9 15/8 1088,3 2,6
16 1163,6 45/23 1161,9 1,7
17 1236,4 49/24 1235,7 0,7
18 1309,1 32/15 1311,7 -2,6
19 1381,8 20/9 1382,4 -0,6
20 1454,5 44/19 1453,8 0,7
21 1527,3 29/12 1527,6 -0,3
22 1600,0 68/27 1599,1 0,9
23 1672,7 21/8 1670,8 1,9
24 1745,5 52/19 1743,0 2,5
25 1818,2 20/7 1817,5 0,7
26 1890,9 116/39 1887,1 3,8
27 1963,6 28/9 1964,9 -1,3
28 2036,4 13/4 2040,5 -4,1
29 2109,1 44/13 2110,8 -1,7
30 2181,8 60/17 2183,3 -1,5
31 2254,5 114/31 2254,4 0,1
32 2327,3 23/6 2326,3 1,0
33 2400,0 4/1 2400,0 0,0

Music

Equivocal Tuning by Ahornberg