68edo

← 67edo

68edo

69edo →

Prime factorization	22 × 17
Step size	17.6471 ¢
Fifth	40\68 (705.882 ¢) (→ 10\17)
Semitones (A1:m2)	8:4 (141.2 ¢ : 70.59 ¢)
Consistency limit	9
Distinct consistency limit	9

The 68 equal divisions of the octave (68edo), or the 68(-tone) equal temperament (68tet, 68et) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 68 equally-sized steps. Each step represents a frequency ratio of 17.65 cents.

Theory

68edo's step 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.

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, hemiwürschmidt, 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.

The 3rd degree of 68edo can be used as a generator for stretched 23edo. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65 cents).

Intervals

Degrees	Cents	Approximate Ratios
0	0.00	1/1
1	17.65	64/63, 126/125, 225/224
2	35.29	81/80, 49/48, 50/49
3	52.94	28/27, 36/35, 33/32
4	70.59	25/24, 22/21
5	88.24	21/20, 19/18, 20/19
6	105.88	16/15, 17/16, 18/17
7	123.53	15/14, 14/13
8	141.18	13/12
9	158.82	12/11, 11/10
10	176.47	10/9
11	194.12	28/25, 19/17
12	211.76	9/8
13	229.41	8/7
14	247.06	15/13
15	264.71	7/6
16	282.35	20/17
17	300.00	13/11, 19/16
18	317.65	6/5
19	335.29	11/9, 40/33, 17/14
20	352.94	16/13, 39/32
21	370.59	27/22, 26/21, 21/17
22	388.24	5/4
23	405.88	24/19, 19/15
24	423.53	14/11
25	441.18	9/7
26	458.82	13/10, 17/13
27	476.47	21/16
28	494.12	4/3
29	511.76	75/56
30	529.41	27/20, 19/14
31	547.06	11/8, 15/11
32	564.71	25/18, 18/13, 26/19
33	582.35	7/5
34	600.00	17/12, 24/17
35	617.65	10/7
36	635.29	36/25, 13/9, 19/13
37	652.94	16/11, 22/15
38	670.59	40/27, 28/19
39	688.24	112/75
40	705.88	3/2
41	723.53	32/21
42	741.18	16/13, 26/17
43	758.82	14/9
44	776.47	11/7
45	794.12	19/12, 30/19
46	811.76	8/5
47	829.41	44/27, 21/13, 34/21
48	847.06	13/8, 64/39
49	864.71	18/11, 33/20, 28/17
50	882.35	5/3
51	900.00	22/13, 32/19
52	917.65	17/10
53	935.29	12/7
54	952.94	26/15
55	970.59	7/4
56	988.24	16/9
57	1005.88	25/14, 34/19
58	1023.53	9/5
59	1041.18	11/6, 20/11
60	1058.82	24/13
61	1076.47	28/15, 13/7
62	1094.12	15/8, 32/17, 17/9
63	1111.76	40/21, 36/19, 19/10
64	1129.41	48/25, 21/11
65	1147.06	27/14, 35/18, 64/33
66	1164.71	160/81, 96/49, 49/25
67	1182.35	63/32, 125/64, 448/225
68	1200.00	2/1

Scales

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

Superpyth quarter octave: 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1

Flattone: 10 10 9 10 10 10 9

Inverse: 8 8 14 8 8 8 14

Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7