9ed7/6

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9ed7/6

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Prime factorization	32
Step size	29.6523 ¢
Octave	40\9ed7/6 (1186.09 ¢) (semiconvergent)
Twelfth	64\9ed7/6 (1897.75 ¢)
Consistency limit	3
Distinct consistency limit	3

9 equal divisions of 7/6 (abbreviated 9ed7/6) is a nonoctave tuning system that divides the interval of 7/6 into 9 equal parts of about 29.7 ¢ each. Each step represents a frequency ratio of (7/6)^1/9, or the 9th root of 7/6.

Harmonics

Approximation of harmonics in 9ed7/6

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-13.9	-4.2	+1.8	+1.0	+11.5	+11.5	-12.1	-8.4	-12.9	+0.0	-2.4
	Relative (%)	-46.9	-14.2	+6.2	+3.4	+38.9	+38.9	-40.7	-28.4	-43.5	+0.0	-8.0
Steps (reduced)		40 (4)	64 (1)	81 (0)	94 (4)	105 (6)	114 (6)	121 (4)	128 (2)	134 (8)	140 (5)	145 (1)

Approximation of harmonics in 9ed7/6 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+7.3	-2.4	-3.2	+3.7	-12.3	+7.3	+2.7	+2.8	+7.3	-13.9	-1.9
	Relative (%)	+24.7	-8.0	-10.8	+12.4	-41.6	+24.7	+9.1	+9.6	+24.7	-46.9	-6.4
Steps (reduced)		150 (6)	154 (1)	158 (5)	162 (0)	165 (3)	169 (7)	172 (1)	175 (4)	178 (7)	180 (0)	183 (3)

Note that this tuning is very close to 81ed4 and thus to every other step of 81edo, although by patent val mapping the double octave (~4/1) is mapped inconsistently to 80\9ed7/6, thus requiring constitution of ~4/1 as 2♭ × 2♯ to yield the direct approximation 81\9ed7/6.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	29.7
2	59.3	20/19, 22/21
3	89	18/17
4	118.6	12/11, 14/13, 15/14, 19/18, 21/20
5	148.3
6	177.9	10/9, 11/10, 21/19
7	207.6	17/15, 19/17
8	237.2	9/8, 15/13, 22/19
9	266.9	20/17

Instruments

A Lumatone mapping for 9ed7/6 is available.

Music

Bryan Deister

• 9ed(7/6) improv (2025)

See also

• 81edo — related (but not relative) edo
• 81ed4 — relative ed4