67ed5

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67ed5

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Prime factorization	67 (prime)
Step size	41.5868 ¢
Octave	29\67ed5 (1206.02 ¢)
Twelfth	46\67ed5 (1912.99 ¢)
Consistency limit	8
Distinct consistency limit	7

Division of the 5th harmonic into 67 equal parts (67ed5) is related to 29edo, but with the 5/1 rather than the 2/1 being just. The octave is about 6.0164 cents stretched and the step size is about 41.5868 cents.

Theory

67ed5 has a generally sharp tendency for harmonics up to 28. Unlike 29edo, it is only consistent up to the 8-integer-limit, with discrepancy for the 9th harmonic. As an equal temperament, it tempers out 49/48 in the 7-limit; 55/54 in the 11-limit; 65/64 and 91/90 in the 13-limit; 85/84 in the 17-limit; 77/76 in the 19-limit; 70/69 in the 23-limit; 58/57 in the 29-limit; and 93/92 in the 31-limit.

Prime harmonics

Compared to 29edo, 67ed5 has a much better 5/1, 7/1, 11/1, 13/1, and 17/1, at the expense of a much worse 3/1.

The biggest argument in favor of this trade-off is that 29edo’s 7/1 is so inaccurate as to be unusable for many. So, the fact that 67ed5 makes the 3/1 not as good, but still definitely useable, and in return, replaces that unusable 7/1 with almost perfectly in-tune one, could be seen as a worthwhile trade-off.

Approximation of prime harmonics in 67ed5

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+6.0	+11.0	+0.0	-0.3	+7.4	+9.3	+2.3	+17.7	+19.6	-7.4	+1.9
	Relative (%)	+14.5	+26.5	+0.0	-0.7	+17.7	+22.3	+5.5	+42.5	+47.1	-17.9	+4.5
Steps (reduced)		29 (29)	46 (46)	67 (0)	81 (14)	100 (33)	107 (40)	118 (51)	123 (56)	131 (64)	140 (6)	143 (9)

29edo for comparison:

Approximation of prime harmonics in 29edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0	+1.5	-13.9	-17.1	-13.4	-12.9	+19.2	-7.9	-7.6	+4.9	+13.6
	Relative (%)	+0.0	+3.6	-33.6	-41.3	-32.4	-31.3	+46.4	-19.0	-18.3	+11.9	+32.8
Steps (reduced)		29 (0)	46 (17)	67 (9)	81 (23)	100 (13)	107 (20)	119 (3)	123 (7)	131 (15)	141 (25)	144 (28)

67ed5 as a generator

67ed5 can also be thought of as a generator of the 2.3.5.7.11.19 subgroup temperament which tempers out 441/440, 513/512, 4000/3993, and 10125/10108, which is a cluster temperament with 29 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 205821/204800 ~ 210/209 ~ 225/224 ~ 7448/7425 ~ 361/360 ~ 400/399 ~ 1375/1372 ~ 200704/200475 all tempered together. This temperament is supported by 29edo, 202edo, and 231edo.

Intervals

Intervals of 67ed5

degree	cents value	corresponding JI intervals	comments
0	0.0000	exact 1/1
1	41.5868
2	83.1735	22/21, 21/20
3	124.7603	3375/3136
4	166.3471	11/10
5	207.9339	150/133
6	249.5206	800/693, 231/200
7	291.1074	45/38
8	332.6942	40/33
9	374.2809	4455/3584
10	415.8677	80/63, 14/11
11	457.4545
12	499.0413	4/3
13	540.6280
14	582.2148	7/5
15	623.8016	1125/784
16	665.3883	22/15, 147/100
17	706.9751	200/133	pseudo-3/2
18	748.5619	77/50
19	790.1487	30/19
20	831.7354	160/99
21	873.3222	63/38
22	914.9090	95/56, 56/33
23	956.4958
24	998.0825	16/9, 57/32
25	1039.6693
26	1081.2561	28/15
27	1122.8428	375/196
28	1164.4296	49/25
29	1206.0164	800/399, 225/112	pseudo-octave
30	1247.6032	154/75
31	1289.1899	40/19
32	1330.7767	640/297
33	1372.3635	495/224, 42/19
34	1413.9502	95/42, 224/99
35	1455.5370	297/128
36	1497.1238	19/8
37	1538.7106	375/154
38	1580.2973	112/45, 399/160	pseudo-5/2
39	1621.8841	125/49
40	1663.4709	196/75
41	1705.0576	75/28
42	1746.6444
43	1788.2312	160/57, 45/16
44	1829.8180
45	1871.4047	165/56, 56/19
46	1912.9915	190/63
47	1954.5783	99/32
48	1996.1650	19/6
49	2037.7518	250/77
50	2079.3386	133/40	pseudo-10/3
51	2120.9254	500/147, 75/22
52	2162.5121	784/225
53	2204.0989	25/7
54	2245.6857
55	2287.2725	15/4
56	2328.8592
57	2370.4460	55/14, 63/16
58	2412.0328	3584/891
59	2453.6195	33/8
60	2495.2063	38/9
61	2536.7931	1000/231, 693/160
62	2578.3799	133/30
63	2619.9666	50/11
64	2661.5534	3136/675
65	2703.1402	100/21
66	2744.7269
67	2786.3137	exact 5/1	just major third plus two octaves