97ed9

Theory

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

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

This non-octave, non-tritave scale 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 15

Approximation of harmonics in 97ed9

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

(contd.)

Harmonic		9	10	11	12	13	14	15
Error	Absolute (¢)	+0.0	+13.7	+5.5	+11.8	-9.2	+19.4	+17.6
	Relative (%)	+0.0	+34.9	+14.1	+30.0	-23.4	+49.5	+44.9
Steps (reduced)		97 (0)	102 (5)	106 (9)	110 (13)	113 (16)	117 (20)	120 (23)

38 to 52

Approximation of harmonics in 97ed9

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

(contd.)

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