35ed4

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35ed4

37ed4 →

Prime factorization	5 × 7
Step size	68.5714 ¢
Octave	18\35ed4 (1234.29 ¢)
Twelfth	28\35ed4 (1920 ¢) (→ 4\5ed4)
Consistency limit	1
Distinct consistency limit	1

35 equal divisions of the 4th harmonic (abbreviated 35ed4) is a nonoctave tuning system that divides the interval of 4/1 into 35 equal parts of about 68.6 ¢ each. Each step represents a frequency ratio of 4^1/35, or the 35th root of 4.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	68.6	26/25, 27/26, 28/27
2	137.1	27/25
3	205.7
4	274.3
5	342.9	11/9, 17/14
6	411.4	14/11
7	480
8	548.6
9	617.1
10	685.7
11	754.3	17/11
12	822.9	21/13
13	891.4	5/3
14	960	26/15
15	1028.6
16	1097.1	17/9
17	1165.7
18	1234.3
19	1302.9
20	1371.4	11/5
21	1440
22	1508.6
23	1577.1
24	1645.7
25	1714.3	27/10
26	1782.9	14/5
27	1851.4
28	1920
29	1988.6
30	2057.1	23/7
31	2125.7	17/5
32	2194.3
33	2262.9
34	2331.4
35	2400

Harmonics

35ed4 can be approached as a 4.3.10.7 subgroup tuning. Alternatively it can be approached as a 2.3.5.11.13.17 tuning because the error on all those primes is in the same direction and by similar amounts, so intervals built from them turn out better than one would expect from the raw numbers.

Approximation of harmonics in 35ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+34.3	+18.0	+0.0	+25.1	-16.2	-8.8	+34.3	-32.5	-9.2	+31.5	+18.0
	Relative (%)	+50.0	+26.3	+0.0	+36.6	-23.7	-12.9	+50.0	-47.4	-13.4	+46.0	+26.3
Steps (reduced)		18 (18)	28 (28)	35 (0)	41 (6)	45 (10)	49 (14)	53 (18)	55 (20)	58 (23)	61 (26)	63 (28)

Approximation of harmonics in 35ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+16.6	+25.5	-25.4	+0.0	+32.2	+1.8	-23.2	+25.1	+9.2	-2.7	-11.1
	Relative (%)	+24.2	+37.1	-37.1	+0.0	+46.9	+2.6	-33.9	+36.6	+13.4	-4.0	-16.2
Steps (reduced)		65 (30)	67 (32)	68 (33)	70 (0)	72 (2)	73 (3)	74 (4)	76 (6)	77 (7)	78 (8)	79 (9)