34ed7

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 (quintupole temperament). This temperament is supported by 12EDO, 109EDO, and 121EDO among others.

Trisa-quingu (12&121)
5-limit
Comma: [37 -16 -5⟩
Mapping: [⟨1 2 1], ⟨0 -5 16]]
POTE generator: ~135/128 = 99.267
Vals: 12, 85, 97, 109, 121, 133, 278c, 411bc, 544bc
Badness: 0.444506

Quintupole (12&121)
7-limit
Comma list: 4000/3969, 458752/455625
Mapping: [⟨1 2 1 0], ⟨0 -5 16 34]]
POTE generator: ~135/128 = 99.175
Vals: 12, 97, 109, 121
Badness: 0.111620

11-limit
Comma list: 896/891, 1375/1372, 4375/4356
Mapping: [⟨1 2 1 0 -1], ⟨0 -5 16 34 54]]
POTE generator: ~132/125 = 99.156
Vals: 12, 109, 121, 351bde, 472bdee
Badness: 0.056501

13-limit
Comma list: 352/351, 364/363, 625/624, 2704/2695
Mapping: [⟨1 2 1 0 -1 -2], ⟨0 -5 16 34 54 69]]
POTE generator: ~55/52 = 99.165
Vals: 12f, 109, 121
Badness: 0.038431

17-limit
Comma list: 256/255, 352/351, 364/363, 375/374, 442/441
Mapping: [⟨1 2 1 0 -1 -2 5], ⟨0 -5 16 34 54 69 -11]]
POTE generator: ~18/17 = 99.172
Vals: 12f, 109, 121
Badness: 0.028721

See also

• 12edo: relative EDO
• 19ed3: relative ED3
• 28ed5: relative ED5
• 31ed6: relative ED6
• 40ed10: relative ED10
• 42ed11: relative ED11