34ed7

From Xenharmonic Wiki
Jump to: navigation, search

Division of the 7th harmonic into 34 equal parts (34ed7) is related to 12 edo, but with the 7/1 rather than the 2/1 being just. The octave is about 11.0026 cents compressed and the step size is about 99.0831 cents. It is consistent to the 11-integer-limit, but not to the 12-integer-limit. In comparison, 12edo is only consistent up to the 10-integer-limit.

degree cents value corresponding
JI intervals
comments
0 0.0000 exact 1/1
1 99.0831 18/17
2 198.1662 28/25
3 297.2493 19/16
4 396.3325 49/39, 34/27 pseudo-5/4
5 495.4156 4/3
6 594.4987 24/17
7 693.5818 136/91 pseudo-3/2
8 792.6649 30/19, 128/81
9 891.7480 77/46 pseudo-5/3
10 990.8311 85/48, 39/22
11 1089.9143 15/8
12 1188.9974 143/72, 175/88 pseudo-octave
13 1288.0805 21/10, 40/19
14 1387.1636 49/22
15 1486.2467 33/14
16 1585.3298 5/2
17 1684.4130 119/45, 45/17 pseudo-8/3
18 1783.4961 14/5
19 1882.5792 95/32, 98/33 pseudo-3/1
20 1981.6623 22/7
21 2080.7454 133/40, 10/3
22 2179.8285 88/25
23 2278.9116 56/15
24 2377.9948 154/39, 320/81, 336/85 pseudo-4/1
25 2477.0779 46/11
26 2576.1610 133/30
27 2675.2441 169/36
28 2774.3272 119/24 pseudo-5/1
29 2873.4103 21/4 pseudo-16/3
30 2972.4934 39/7
31 3071.5766 112/19 pseudo-6/1
32 3170.6597 25/4
33 3269.7428 119/18
34 3368.8259 exact 7/1 harmonic seventh plus two octaves

34ed7 as a generator

34ed7 can also be thought of as a generator of the 11-limit temperament which tempers out 896/891, 1375/1372, and 4375/4356, which is a cluster temperament with 12 clusters of notes in an octave. This temperament is supported by 12edo, 109edo, and 121edo among others.

5-limit 12&121 (trisa-quingu)

Comma: |37 -16 -5>

POTE generator: ~135/128 = 99.267

Map: [<1 2 1|, <0 -5 16|]

EDOs: 12, 85, 97, 109, 121, 133, 145, 157, 206, 230, 254

Badness: 0.444506

7-limit 12&121

Commas: 4000/3969, 458752/455625

POTE generator: ~135/128 = 99.175

Map: [<1 2 1 0|, <0 -5 16 34|]

EDOs: 12, 97, 109, 121, 206, 230

Badness: 0.111620

11-limit 12&121

Commas: 896/891, 1375/1372, 4375/4356

POTE generator: ~132/125 = 99.156

Map: [<1 2 1 0 -1|, <0 -5 16 34 54|]

EDOs: 12, 109, 121, 230

Badness: 0.056501

13-limit 12f&121

Commas: 352/351, 364/363, 625/624, 2704/2695

POTE generator: ~55/52 = 99.165

Map: [<1 2 1 0 -1 -2|, <0 -5 16 34 54 69|]

EDOs: 12f, 109, 121, 230

Badness: 0.038431

See also