23ed6

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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	6

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.

Theory

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

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 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+10.1	+16.7	+32.1	+55.2	-49.7	-13.8	+27.5	-61.3	-10.9	+43.4	-33.6	+27.6
	Relative (%)	+7.5	+12.4	+23.8	+41.0	-36.9	-10.2	+20.4	-45.5	-8.1	+32.2	-24.9	+20.5
Steps (reduced)		33 (10)	34 (11)	35 (12)	36 (13)	36 (13)	37 (14)	38 (15)	38 (15)	39 (16)	40 (17)	40 (17)	41 (18)

Subsets and supersets

23ed6 is the 9th prime ed6, following 19ed6 and before 29ed6, so it does not contain any nontrivial subset ed6's.

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

See also

• 9edo – relative edo
• 14edt – relative edt
• 32ed12 – relative ed12