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

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 in the 11-limit; though it's certainly usable for that purpose, it does not represent the 11-limit diamond consistently. However, 68edo maps many higher primes better than it does 11 (specifically 13 and 23 inherited from 17edo, 17 inherited from 34edo, and 19 and 31 new to 68edo), notably being consistent in the entire no-11s 25-odd limit.

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 3rd, 5th and 7th harmonics all sharp.

The 3rd degree of 68edo can be used as a generator for stretched 23edo, which also acts as the quartkeenlig temperament tempering out the quartisma, 385/384 and 6250/6237. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65 cents). It also works as a 22L 1s MOS of the quartkeenlig temperament.

The 5th degree of 68edo can be used as a generator for 88cET.

Prime harmonics

Approximation of prime harmonics in 68edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+3.93	+1.92	+1.76	-4.26	+6.53	+0.93	+2.49	+7.02	-6.05	+2.02
	Relative (%)	+0.0	+22.3	+10.9	+10.0	-24.1	+37.0	+5.3	+14.1	+39.8	-34.3	+11.5
Steps (reduced)		68 (0)	108 (40)	158 (22)	191 (55)	235 (31)	252 (48)	278 (6)	289 (17)	308 (36)	330 (58)	337 (65)

Subsets and supersets

Since 68 factors into 2² × 17, 68edo has subset edos 2, 4, 17, and 34.

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

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3.5.7	245/243, 2048/2025, 2401/2400	[⟨68 108 158 191]]	-0.983	0.915	5.19
2.3.5.7.11	121/120, 176/175, 245/243, 1375/1372	[⟨68 108 158 191 235]]	-0.541	1.206	6.84
2.3.5.7.11.13	121/120, 176/175, 196/195, 245/243, 275/273	[⟨68 108 158 191 235 252]]	-0.745	1.191	6.75
2.3.5.7.11.13.17	121/120, 136/135, 154/153, 176/175, 196/195, 275/273	[⟨68 108 158 191 235 252 278]]	-0.671	1.118	6.34
2.3.5.7.11.13.17.19	121/120, 136/135, 154/153, 190/189, 176/175, 196/195, 275/273	[⟨68 108 158 191 235 252 278 289]]	-0.661	1.046	5.93

Scales

• Negative semitone: 14 14 -1 14 14 14 -1 (E is sharper than F, and B is sharper than C)
• Deeptone[7]: 10 10 9 10 10 10 9
• Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7
• 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
• Quartkeenlig[23] (Stretched 23edo): 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2

Music

The Mercury Tree

• Grown Apart from Self Similar (2023)