23ed6

← 22ed6

23ed6

24ed6 →

Prime factorization	23 (prime)
Step size	134.868¢
Octave	9\23ed6 (1213.81¢)
Twelfth	14\23ed6 (1888.15¢)
Consistency limit	8
Distinct consistency limit	5

23 equal divisions of the 6th harmonic (abbreviated 23ed6) is a nonoctave tuning system that divides the interval of 6/1 into 23 equal parts of about 135⁠ ⁠¢ each. Each step represents a frequency ratio of 6^1/23, or the 23rd root of 6.

23ed6 is similar to 9edo, but has the 6/1 tuned just instead of the octave, which stretches the octave by 13.8 ¢. It also approximates Pelog tunings in Indonesian gamelan music very well, since Pelog is well-approximated by 9edo.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	134.9	13/12, 14/13
2	269.7	7/6
3	404.6	14/11, 19/15, 24/19
4	539.5	11/8, 15/11, 19/14
5	674.3	22/15
6	809.2	8/5, 19/12
7	944.1	12/7, 19/11
8	1078.9	13/7, 15/8
9	1213.8
10	1348.7	13/6, 24/11
11	1483.5
12	1618.4	23/9
13	1753.3	11/4
14	1888.1
15	2023	16/5
16	2157.9	7/2
17	2292.7	15/4
18	2427.6
19	2562.5	22/5
20	2697.4	19/4
21	2832.2
22	2967.1
23	3102	6/1

Harmonics

Approximation of harmonics in 23ed6

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+13.8	-13.8	+27.6	+45.9	+0.0	+2.9	+41.4	-27.6	+59.7	+29.6	+13.8
	Relative (%)	+10.2	-10.2	+20.5	+34.0	+0.0	+2.1	+30.7	-20.5	+44.3	+21.9	+10.2
Steps (reduced)		9 (9)	14 (14)	18 (18)	21 (21)	23 (0)	25 (2)	27 (4)	28 (5)	30 (7)	31 (8)	32 (9)

Approximation of harmonics in 23ed6

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+10.1	+16.7	+32.1	+55.2	-49.7	-13.8	+27.5	-61.3	-10.9	+43.4	-33.6
	Relative (%)	+7.5	+12.4	+23.8	+41.0	-36.9	-10.2	+20.4	-45.5	-8.1	+32.2	-24.9
Steps (reduced)		33 (10)	34 (11)	35 (12)	36 (13)	36 (13)	37 (14)	38 (15)	38 (15)	39 (16)	40 (17)	40 (17)