68edo

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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

68 equal divisions of the octave (abbreviated 68edo or 68ed2), also called 68-tone equal temperament (68tet) or 68 equal temperament (68et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 68 equal parts of about 17.6 ¢ each. Each step represents a frequency ratio of 2^1/68, or the 68th root of 2.

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

Notation

Ups and downs notation

68edo can be notated using ups and downs notation using Helmholtz–Ellis accidentals:

Step Offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
Sharp Symbol
Flat Symbol

Sagittal notation

This notation uses the same sagittal sequence as 75-EDO.

Evo flavor

Revo flavor

Evo-SZ flavor

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

Approximation to JI

Zeta peak index

Tuning			Strength			Closest edo		Integer limit
ZPI	Steps per octave	Step size (cents)	Height	Integral	Gap	Edo	Octave (cents)	Consistent	Distinct
354zpi	68.0493056282519	17.6342725163943	7.666604	1.254592	17.034505	68edo	1199.13053111481	10	10

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

See also: 34edo #Scales and 17edo #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)