60edf

← 59edf 60edf 61edf →
Prime factorization	22 × 3 × 5 (highly composite)
Step size	11.6993 ¢
Octave	103\60edf (1205.02 ¢)
Twelfth	163\60edf (1906.98 ¢)
Consistency limit	3
Distinct consistency limit	3

60 equal divisions of the perfect fifth (abbreviated 60edf or 60ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 60 equal parts of about 11.7 ¢ each. Each step represents a frequency ratio of (3/2)^1/60, or the 60th root of 3/2.

Theory

60edf can be thought of as a very octave stretched version of 103edo, or a very compressed version of 102edo, but it actually inherits few properties from either.

It makes available dual versions of primes 2 and 3 from both systems. Yet its mappings of primes 5, 7, 11, 13 and up are actually all different from either of those edos. For example mapping prime 5 to the 238th step (not 237 as in 102edo, nor 239 as in 103edo).

60edf is very similar to 205ed4.

Harmonics

60edf's approximations of primes are strange. Because of its small step size, it's difficult not to hear primes 2, 3, or even 13, even though they have a lot of relative error.

60edf is much more accurate on higher primes than on smaller primes. It approximates all primes from 17 through 31 with less than 29% relative error, but has over 43% rel. err. on 2, 3 and 13.

So perhaps a reasonable - if clunky - way to interpret 60edf, is as a dual-2, dual-3, dual-13 31-limit tuning. Extending it to the 37-limit could also be an option.

Approximation of primes in 60edf
Harmonic		2	3	5	7	11	13	17	19	23	29	31	37	41
Error	Absolute (¢)	+5.02	+5.02	-1.89	+0.56	+1.92	+5.19	-2.97	+3.36	+0.18	-3.35	-1.82	-3.94	+5.53
Error	Relative (%)	+42.9	+42.9	-16.2	+4.8	+16.4	+44.3	-25.4	+28.7	+1.5	-28.6	-15.5	-33.7	+47.2
Steps (reduced)		103 (43)	163 (43)	238 (58)	288 (48)	355 (55)	380 (20)	419 (59)	436 (16)	464 (44)	498 (18)	508 (28)	534 (54)	550 (10)

Approximation of integers in 60edf
Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+5.02	+5.02	-1.65	-1.89	-1.65	+0.56	+3.37	-1.65	+3.13	+1.92	+3.37
Error	Relative (%)	+42.9	+42.9	-14.1	-16.2	-14.1	+4.8	+28.8	-14.1	+26.8	+16.4	+28.8
Steps (reduced)		103 (43)	163 (43)	205 (25)	238 (58)	265 (25)	288 (48)	308 (8)	325 (25)	341 (41)	355 (55)	368 (8)

Approximation of integers in 60edf (continued)
Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+5.19	+5.58	+3.13	-3.31	-2.97	+3.37	+3.36	-3.55	+5.58	-4.76	+0.18	-3.31
Error	Relative (%)	+44.3	+47.7	+26.8	-28.3	-25.4	+28.8	+28.7	-30.3	+47.7	-40.7	+1.5	-28.3
Steps (reduced)		380 (20)	391 (31)	401 (41)	410 (50)	419 (59)	428 (8)	436 (16)	443 (23)	451 (31)	457 (37)	464 (44)	470 (50)

Subsets and supersets

As a highly composite edf, 60edf includes as subsets: 2edf, 3edf, 5edf, 6edf, 9edf, 10edf, 12edf, 15edf, 20edf and 30edf.

This makes it potentially a good polymicrotonal system for using multiple edfs (or their stretched/compressed relative edos/ed4s) simultaneously.

The relative edos/ed4s of its subsets are: 7ed4, 5edo, 9edo/17ed4, 21ed4, 31ed4, 17edo, 21edo/41ed4, 26edo, 34edo and 51edo/103ed4.

The simplest supersets of 60edf are 120edf and 180edf.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	11.7
2	23.4
3	35.1
4	46.8	35/34
5	58.5	28/27
6	70.2
7	81.9	21/20, 22/21
8	93.6
9	105.3
10	117	15/14, 31/29
11	128.7	14/13
12	140.4	25/23
13	152.1
14	163.8	11/10, 34/31
15	175.5	21/19
16	187.2
17	198.9
18	210.6	35/31
19	222.3
20	234
21	245.7	15/13
22	257.4	22/19, 29/25
23	269.1
24	280.8	33/28
25	292.5	13/11
26	304.2
27	315.9
28	327.6	23/19, 35/29
29	339.3
30	351
31	362.7
32	374.4	31/25
33	386.1
34	397.8	29/23
35	409.5	19/15, 33/26
36	421.2	14/11
37	432.9
38	444.6
39	456.3	13/10
40	468
41	479.7
42	491.4
43	503.1
44	514.8	31/23
45	526.5	19/14, 23/17
46	538.2	15/11
47	549.9
48	561.6
49	573.3
50	585	7/5
51	596.7
52	608.4
53	620.1	10/7
54	631.8
55	643.5
56	655.2	19/13
57	666.9	22/15, 25/17
58	678.6	34/23
59	690.3
60	702	3/2

Music

Bryan Deister

60ed(3/2) improv (2025)