97ed9

97ed9 is an equal-step tuning system created by dividing the interval of 9/1 into 97 equal parts.

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97ed9

99ed9 →

Prime factorization	97 (prime)
Step size	39.2156 ¢
Octave	31\97ed9 (1215.68 ¢)
Twelfth	49\97ed9 (1921.56 ¢)
Consistency limit	2
Distinct consistency limit	2

This system can be approximated as 30.6001 EDO, meaning each step of 97ed9 corresponds closely to five steps of 153edo.

97ed9 is a non-octave, non-tritave scale.

Theory

97ed9 features a well-balanced harmonic series segment from 4 to 9 and another from 39 to 50. It performs well across all prime harmonics from 5 to 19, with the exception of 13, which is slightly flat.

97ed9 sets a height record on the Riemann zeta function with primes 2 and 3 removed, approximating 30.59745 EDO. This record remains unbeaten until approximately 41.3478 EDO.

Additionally, 97ed9 is close to 125zpi (see Zeta peak index).

Harmonic series

2 to 19

Approximation of harmonics in 97ed9

Harmonic		2	3	4	5	6	7	8	9	10
Error	Absolute (¢)	+15.7	+19.6	-7.9	-2.0	-3.9	+3.7	+7.8	+0.0	+13.7
	Relative (%)	+40.0	+50.0	-20.0	-5.1	-10.0	+9.5	+20.0	+0.0	+34.9
Steps (reduced)		31 (31)	49 (49)	61 (61)	71 (71)	79 (79)	86 (86)	92 (92)	97 (0)	102 (5)

(contd.)

Harmonic		11	12	13	14	15	16	17	18	19
Error	Absolute (¢)	+5.5	+11.8	-9.2	+19.4	+17.6	-15.7	-3.0	+15.7	+0.5
	Relative (%)	+14.1	+30.0	-23.4	+49.5	+44.9	-40.0	-7.7	+40.0	+1.3
Steps (reduced)		106 (9)	110 (13)	113 (16)	117 (20)	120 (23)	122 (25)	125 (28)	128 (31)	130 (33)

36 to 53

Approximation of harmonics in 97ed9

Harmonic		36	37	38	39	40	41	42	43	44
Error	Absolute (¢)	-7.9	-16.1	+16.2	+10.4	+5.8	+2.3	-0.2	-1.7	-2.3
	Relative (%)	-20.0	-41.0	+41.3	+26.6	+14.9	+5.8	-0.5	-4.4	-5.9
Steps (reduced)		158 (61)	159 (62)	161 (64)	162 (65)	163 (66)	164 (67)	165 (68)	166 (69)	167 (70)

(contd.)

Harmonic		45	46	47	48	49	50	51	52	53
Error	Absolute (¢)	-2.0	-0.8	+1.1	+3.9	+7.4	+11.7	+16.6	-17.0	-10.8
	Relative (%)	-5.1	-2.2	+2.9	+10.0	+18.9	+29.7	+42.3	-43.4	-27.5
Steps (reduced)		168 (71)	169 (72)	170 (73)	171 (74)	172 (75)	173 (76)	174 (77)	174 (77)	175 (78)

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	39.2
2	78.4	22/21
3	117.6	15/14, 31/29
4	156.9
5	196.1
6	235.3	39/34
7	274.5	34/29, 41/35
8	313.7
9	352.9	38/31
10	392.2
11	431.4
12	470.6
13	509.8
14	549
15	588.2
16	627.4
17	666.7	25/17
18	705.9
19	745.1
20	784.3	11/7
21	823.5	37/23
22	862.7
23	902
24	941.2	43/25
25	980.4
26	1019.6
27	1058.8	35/19
28	1098
29	1137.3
30	1176.5
31	1215.7
32	1254.9	31/15
33	1294.1
34	1333.3	41/19
35	1372.5
36	1411.8	43/19
37	1451
38	1490.2	26/11
39	1529.4
40	1568.6
41	1607.8	38/15, 43/17
42	1647.1
43	1686.3
44	1725.5
45	1764.7
46	1803.9
47	1843.1	29/10
48	1882.3
49	1921.6
50	1960.8	31/10
51	2000
52	2039.2
53	2078.4
54	2117.6	17/5
55	2156.9
56	2196.1
57	2235.3
58	2274.5
59	2313.7	19/5
60	2352.9
61	2392.1
62	2431.4
63	2470.6
64	2509.8
65	2549
66	2588.2
67	2627.4
68	2666.7	14/3
69	2705.9
70	2745.1	44/9
71	2784.3	5/1
72	2823.5
73	2862.7
74	2902
75	2941.2
76	2980.4
77	3019.6
78	3058.8	41/7
79	3098
80	3137.2
81	3176.5
82	3215.7
83	3254.9
84	3294.1
85	3333.3
86	3372.5
87	3411.8
88	3451	22/3
89	3490.2	15/2
90	3529.4
91	3568.6
92	3607.8
93	3647
94	3686.3
95	3725.5	43/5
96	3764.7
97	3803.9