96ed5

← 95ed5

96ed5

97ed5 →

Prime factorization	25 × 3
Step size	29.0241 ¢
Octave	41\96ed5 (1189.99 ¢)
Twelfth	66\96ed5 (1915.59 ¢) (→ 11\16ed5)
Consistency limit	2
Distinct consistency limit	2

96 equal divisions of the 5th harmonic (abbreviated 96ed5) is a nonoctave tuning system that divides the interval of 5/1 into 96 equal parts of about 29 ¢ each. Each step represents a frequency ratio of 5^1/96, or the 96th root of 5.

Theory

This non-octave, non-tritave scale features a well-balanced harmonic series segment from 5 to 9, and performs exceptionally well across all prime harmonics from 5 to 23, with the exception of 19.

This system can be approximated as 41.34495 EDO, meaning each step of 96ed5 corresponds roughly to three steps of 124edo, or 124ed8.

96ed5 sets a height record on the Riemann zeta function with primes 2 and 3 removed, approximating 41.3478 EDO. This record remains unbeaten until approximately 98.62575 EDO (~229ed5).

Additionally, 96ed5 is related to 186zpi.

Harmonic series

Approximation of harmonics in 96ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	Absolute (¢)	-10.0	+13.6	+9.0	+0.0	+3.6	-2.0	-1.0	-1.8	-10.0	-0.9	-6.4	+0.2	-12.0	+13.6	-11.0
	Relative (%)	-34.5	+47.0	+31.0	+0.0	+12.5	-7.0	-3.5	-6.0	-34.5	-3.0	-22.0	+0.6	-41.5	+47.0	-38.0
Steps (reduced)		41 (41)	66 (66)	83 (83)	96 (0)	107 (11)	116 (20)	124 (28)	131 (35)	137 (41)	143 (47)	148 (52)	153 (57)	157 (61)	162 (66)	165 (69)

Approximation of harmonics in 96ed5

Harmonic		17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
Error	Absolute (¢)	+0.1	-11.8	+10.7	+9.0	+11.6	-10.9	-0.8	+12.6	+0.0	-9.9	+11.9	+7.0	+4.3	+3.6	+4.9	+8.0
	Relative (%)	+0.4	-40.5	+37.0	+31.0	+40.0	-37.5	-2.6	+43.5	+0.0	-33.9	+40.9	+24.0	+14.7	+12.5	+16.9	+27.5
Steps (reduced)		169 (73)	172 (76)	176 (80)	179 (83)	182 (86)	184 (88)	187 (91)	190 (94)	192 (0)	194 (2)	197 (5)	199 (7)	201 (9)	203 (11)	205 (13)	207 (15)

Intervals

Steps	Cents	8.9.5.7.11.13.17.23 ratios
0	0	1/1
1	29	56/55, 64/63, 65/64, 119/117, 121/119, 637/625
2	58	65/63, 121/117, 125/121, 175/169
3	87.1	81/77, 104/99, 343/325, 637/605
4	116.1	77/72, 91/85
5	145.1	25/23
6	174.1
7	203.2	9/8, 55/49
8	232.2	8/7, 143/125
9	261.2	99/85
10	290.2	13/11, 77/65
11	319.3
12	348.3	11/9, 104/85, 175/143
13	377.3
14	406.3
15	435.4	9/7
16	464.4	17/13
17	493.4	65/49, 121/91
18	522.4	23/17, 169/125
19	551.5	11/8, 125/91
20	580.5	7/5, 169/121
21	609.5
22	638.5	13/9, 175/121
23	667.6	25/17
24	696.6
25	725.6	35/23
26	754.6	17/11
27	783.7	11/7
28	812.7	8/5
29	841.7	13/8, 125/77
30	870.7	91/55
31	899.7
32	928.8	245/143
33	957.8	40/23
34	986.8	23/13
35	1015.8	9/5
36	1044.9
37	1073.9	13/7, 121/65
38	1102.9	17/9
39	1131.9	25/13
40	1161	45/23, 49/25
41	1190
42	1219	245/121, 343/169
43	1248	35/17
44	1277.1	23/11
45	1306.1	17/8, 49/23
46	1335.1
47	1364.1	11/5, 169/77
48	1393.2
49	1422.2	25/11
50	1451.2
51	1480.2	40/17
52	1509.3	55/23
53	1538.3	17/7, 56/23
54	1567.3	121/49
55	1596.3
56	1625.3	23/9, 125/49
57	1654.4	13/5
58	1683.4	45/17
59	1712.4	35/13
60	1741.4	63/23
61	1770.5	25/9, 64/23
62	1799.5	65/23
63	1828.5	23/8, 49/17
64	1857.5	143/39
65	1886.6
66	1915.6	275/91
67	1944.6	40/13, 77/25, 169/55
68	1973.6	25/8, 72/23
69	2002.7	35/11
70	2031.7	55/17
71	2060.7	23/7, 56/17
72	2089.7	77/23
73	2118.8	17/5
74	2147.8	45/13, 121/35, 169/49
75	2176.8	81/23
76	2205.8	25/7
77	2234.9	40/11, 91/25
78	2263.9	63/17, 85/23
79	2292.9	49/13, 64/17
80	2321.9	65/17
81	2351	35/9
82	2380	91/23
83	2409
84	2438	45/11, 143/35
85	2467
86	2496.1	55/13, 72/17
87	2525.1	56/13, 99/23
88	2554.1	35/8
89	2583.1	40/9, 49/11
90	2612.2	77/17, 104/23
91	2641.2	23/5
92	2670.2
93	2699.2	81/17
94	2728.3	63/13, 121/25, 169/35
95	2757.3	64/13
96	2786.3	5/1

Optimization

The local maxima for the finite Euler product over the primes 5.7.11.13.17.23 is 29.0283 cents.

Approximation of harmonics in optimized 96ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	Absolute (¢)	-9.8	+13.9	+9.3	+0.4	+4.1	-1.5	-0.5	-1.2	-9.4	-0.3	-5.8	+0.8	-11.4	+14.3	-10.3
	Relative (%)	-33.9	+47.9	+32.2	+1.4	+14.0	-5.3	-1.7	-4.1	-32.5	-0.9	-19.9	+2.8	-39.2	+49.3	-35.6
Step		41	66	83	96	107	116	124	131	137	143	148	153	157	162	165

Approximation of harmonics in optimized 96ed5

Harmonic		17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
Error	Absolute (¢)	+0.8	-11.0	+11.5	+9.8	+12.4	-10.1	+0.0	+13.4	+0.8	-9.0	+12.7	+7.8	+5.1	+4.5	+5.8	+8.9
	Relative (%)	+2.8	-38.0	+39.5	+33.6	+42.6	-34.8	+0.1	+46.2	+2.8	-31.1	+43.8	+26.9	+17.6	+15.4	+19.9	+30.5
Step		169	172	176	179	182	184	187	190	192	194	197	199	201	203	205	207

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	29
2	58	30/29, 31/30
3	87.1	41/39
4	116.1	31/29
5	145.1	25/23, 37/34
6	174.1	21/19
7	203.2
8	232.2
9	261.2	36/31, 43/37
10	290.2	13/11
11	319.3
12	348.3
13	377.3	41/33
14	406.3	43/34
15	435.4
16	464.4	17/13
17	493.4
18	522.4	23/17
19	551.5
20	580.5	7/5
21	609.5	37/26
22	638.5
23	667.6	25/17
24	696.6
25	725.6	35/23, 38/25
26	754.6	17/11
27	783.7	11/7
28	812.7
29	841.7
30	870.7	38/23, 43/26
31	899.7	37/22, 42/25
32	928.8
33	957.8	33/19
34	986.8	23/13
35	1015.8
36	1044.9	42/23
37	1073.9	13/7
38	1102.9
39	1131.9	25/13
40	1161	43/22
41	1190
42	1219
43	1248	35/17
44	1277.1	23/11
45	1306.1
46	1335.1
47	1364.1	11/5
48	1393.2	38/17
49	1422.2	25/11
50	1451.2
51	1480.2
52	1509.3
53	1538.3	17/7
54	1567.3	42/17
55	1596.3
56	1625.3
57	1654.4	13/5
58	1683.4	37/14
59	1712.4	35/13
60	1741.4	41/15
61	1770.5
62	1799.5
63	1828.5
64	1857.5	38/13
65	1886.6
66	1915.6
67	1944.6	43/14
68	1973.6
69	2002.7	35/11
70	2031.7	42/13
71	2060.7	23/7
72	2089.7
73	2118.8	17/5
74	2147.8	38/11
75	2176.8
76	2205.8	25/7
77	2234.9
78	2263.9	37/10
79	2292.9
80	2321.9	42/11
81	2351
82	2380
83	2409
84	2438
85	2467
86	2496.1
87	2525.1	43/10
88	2554.1
89	2583.1
90	2612.2
91	2641.2	23/5
92	2670.2
93	2699.2
94	2728.3	29/6
95	2757.3
96	2786.3	5/1