68ed12

← 67ed12

68ed12

69ed12 →

Prime factorization	22 × 17
Step size	63.264 ¢
Octave	19\68ed12 (1202.02 ¢)
Twelfth	30\68ed12 (1897.92 ¢) (→ 15\34ed12)
Consistency limit	10
Distinct consistency limit	7

68 equal divisions of the 12th harmonic (abbreviated 68ed12) is a nonoctave tuning system that divides the interval of 12/1 into 68 equal parts of about 63.3 ¢ each. Each step represents a frequency ratio of 12^1/68, or the 68th root of 12.

Theory

68ed12 is very nearly identical to 19edo, but with the 12/1 rather than the 2/1 being just. This results in octaves being stretched by about 2.02 cents. Like 19edo, 68ed12 is consistent to the 10-integer-limit.

Harmonics

Approximation of harmonics in 68ed12

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+2.0	-4.0	+4.0	-2.7	-2.0	-15.8	+6.1	-8.1	-0.7	+24.1	+0.0
	Relative (%)	+3.2	-6.4	+6.4	-4.3	-3.2	-25.0	+9.6	-12.8	-1.1	+38.1	+0.0
Steps (reduced)		19 (19)	30 (30)	38 (38)	44 (44)	49 (49)	53 (53)	57 (57)	60 (60)	63 (63)	66 (66)	68 (0)

Approximation of harmonics in 68ed12 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	-12.0	-13.8	-6.7	+8.1	+29.6	-6.1	+26.9	+1.3	-19.9	+26.1	+12.4	+2.0
	Relative (%)	-19.0	-21.8	-10.6	+12.8	+46.9	-9.6	+42.5	+2.1	-31.4	+41.3	+19.7	+3.2
Steps (reduced)		70 (2)	72 (4)	74 (6)	76 (8)	78 (10)	79 (11)	81 (13)	82 (14)	83 (15)	85 (17)	86 (18)	87 (19)

Subsets and supersets

Since 68 factors into primes as 2² × 17, 68ed12 has subset ed12's 2, 4, 17, and 34.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	63.3	27/26, 28/27, 29/28
2	126.5	14/13, 29/27
3	189.8	19/17, 29/26
4	253.1	22/19, 29/25, 37/32
5	316.3	6/5
6	379.6
7	442.8	22/17, 31/24
8	506.1
9	569.4	25/18, 32/23
10	632.6	36/25
11	695.9
12	759.2	31/20
13	822.4	29/18, 37/23
14	885.7	5/3
15	949	19/11, 26/15
16	1012.2
17	1075.5	13/7
18	1138.8	27/14, 29/15
19	1202	2/1
20	1265.3	27/13
21	1328.5	28/13
22	1391.8	29/13
23	1455.1	37/16
24	1518.3	12/5
25	1581.6
26	1644.9	31/12
27	1708.1
28	1771.4	25/9
29	1834.7	26/9
30	1897.9
31	1961.2	28/9, 31/10
32	2024.4	29/9
33	2087.7	10/3
34	2151
35	2214.2	18/5
36	2277.5
37	2340.8	27/7
38	2404
39	2467.3	25/6
40	2530.6
41	2593.8
42	2657.1
43	2720.4
44	2783.6	5/1
45	2846.9	31/6
46	2910.1
47	2973.4
48	3036.7
49	3099.9	6/1
50	3163.2
51	3226.5
52	3289.7
53	3353
54	3416.3	36/5
55	3479.5
56	3542.8	31/4
57	3606.1
58	3669.3	25/3
59	3732.6
60	3795.8
61	3859.1
62	3922.4
63	3985.6	10/1
64	4048.9
65	4112.2
66	4175.4
67	4238.7
68	4302	12/1

See also

• 11edf – relative edf
• 19edo – relative edo
• 30edt – relative edt
• 49ed6 – relative ed6
• 53ed7 – relative ed7
• 93ed30 – relative ed30