68edo

The 68 equal temperament, often abbreviated 68-tET, 68-EDO, or 68-ET, is the scale derived by dividing the octave into 68 equally-sized steps. Each step represents a frequency ratio of 17.65 cents; this is half of the step size of 34edo, which does well in the 5-limit but not so well in the 7-limit, and one quarter the size of 17edo, which does well in the 3-limit, but not so well in the 5-limit. The luck continues; 68 is a strong 7-limit system, but does not do as well for in 11-limit; though it's certainly usable for that purpose, it does not represent the 11-limit diamond consistently, yet 136edo is contorted in 11-limit anyway.

As a 7-limit system it tempers out 2048/2025, 245/243, 4000/3969, 15625/15552, 3136/3125, 6144/6125 and 2401/2400. It supports octacot, shrutar, hemiwuerschmidt, hemikleismic, clyde and neptune temperaments, and supplies the optimal patent val for 11-limit hemikleismic. It is a sharp-tending system, with the third, fifth and seventh harmonics all sharp.

Diatonic scales:

Negative semitone: 14 14 -1 14 14 14 -1 (E is sharper than F, and B is sharper than C5)

Superpyth: 12 12 4 12 12 12 4

Flattone: 10 10 9 10 10 10 9

Inverse: 8 8 14 8 8 8 14

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 68edo (ordered by absolute error).

Interval, complement	Error (abs., in cents)
7/5, 10/7	0.159
15/13, 26/15	0.682
18/13, 13/9	1.324
8/7, 7/4	1.762
5/4, 8/5	1.922
6/5, 5/3	2.006
7/6, 12/7	2.165
13/12, 24/13	2.604
4/3, 3/2	3.927
15/14, 28/15	4.087
11/8, 16/11	4.259
13/10, 20/13	4.610
14/13, 13/7	4.769
16/15, 15/8	5.849
10/9, 9/5	5.933
14/11, 11/7	6.021
9/7, 14/9	6.092
11/10, 20/11	6.181
16/13, 13/8	6.531
9/8, 16/9	7.855
12/11, 11/6	8.186
15/11, 22/15	10.108
13/11, 22/13	10.790
11/9, 18/11	12.114

Table of 68edo intervals

Step	Five limit	Seven limit	Eleven limit	Thirteen limit
1	?	64/63	55/54	55/54
2	81/80	49/48	49/48	40/39
3	?	28/27	28/27	28/27
4	25/24	25/24	22/21	22/21
5	?	21/20	21/20	21/20
6	16/15	16/15	16/15	16/15
7	?	15/14	15/14	14/13
8	27/25	27/25	27/25	13/12
9	?	35/32	11/10	11/10
10	10/9	10/9	10/9	10/9
11	?	28/25	28/25	28/25
12	9/8	9/8	9/8	9/8
13	?	8/7	8/7	8/7
14	125/108	125/108	63/55	15/13
15	?	7/6	7/6	7/6
16	32/27	32/27	32/27	32/27
17	?	25/21	25/21	13/11
18	6/5	6/5	6/5	6/5
19	?	128/105	11/9	11/9
20	100/81	49/40	49/40	16/13
21	?	56/45	27/22	26/21
22	5/4	5/4	5/4	5/4
23	?	63/50	63/50	33/26
24	32/25	32/25	14/11	14/11
25	?	9/7	9/7	9/7
26	125/96	64/49	55/42	13/10
27	?	21/16	21/16	21/16
28	4/3	4/3	4/3	4/3
29	?	75/56	66/49	35/26
30	27/20	27/20	27/20	27/20
31	?	48/35	11/8	11/8
32	25/18	25/18	25/18	18/13
33	?	7/5	7/5	7/5
34	45/32	45/32	45/32	45/32
35	?	10/7	10/7	10/7
36	36/25	36/25	36/25	13/9
37	?	35/24	16/11	16/11
38	40/27	40/27	40/27	40/27
39	?	112/75	49/33	49/33
40	3/2	3/2	3/2	3/2
41	?	32/21	32/21	32/21
42	125/81	49/32	49/32	20/13
43	?	14/9	14/9	14/9
44	25/16	25/16	11/7	11/7
45	?	63/40	63/40	52/33
46	8/5	8/5	8/5	8/5
47	?	45/28	44/27	21/13
48	81/50	49/30	49/30	13/8
49	?	105/64	18/11	18/11
50	5/3	5/3	5/3	5/3
51	?	42/25	42/25	22/13
52	27/16	27/16	27/16	27/16
53	?	12/7	12/7	12/7
54	125/72	125/72	110/63	26/15
55	?	7/4	7/4	7/4
56	16/9	16/9	16/9	16/9
57	?	25/14	25/14	25/14
58	9/5	9/5	9/5	9/5
59	?	64/35	11/6	11/6
60	50/27	50/27	50/27	24/13
61	?	28/15	28/15	13/7
62	15/8	15/8	15/8	15/8
63	?	40/21	40/21	40/21
64	48/25	48/25	21/11	21/11
65	?	27/14	27/14	27/14
66	125/64	49/25	49/25	39/20
67	?	63/32	63/32	63/32
68	2/1	2/1	2/1	2/1