26ed5

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26ed5

27ed5 →

Prime factorization	2 × 13
Step size	107.166 ¢
Octave	11\26ed5 (1178.83 ¢)
Twelfth	18\26ed5 (1928.99 ¢) (→ 9\13ed5)
Consistency limit	3
Distinct consistency limit	3

26 equal divisions of the 5th harmonic (abbreviated 26ed5) is a nonoctave tuning system that divides the interval of 5/1 into 26 equal parts of about 107 ¢ each. Each step represents a frequency ratio of 5^1/26, or the 26th root of 5.

26ed5 is a strong tuning for the obscure, complex subgroup 5.6.41.67.97.103.151.181.193.

Alternatively, it can be used as a mediocre but workable tuning for the simpler, less unwieldy subgroup 5.6.11.17.41. Most of these harmonics are tuned sharp, so 26ed5 can be made to work better but compressing 26ed5’s equave, making 5/1 slightly flat but the other harmonics more in-tune. This can elevate 26ed5 from mediocre to pretty decent.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	107.2	18/17
2	214.3	17/15, 25/22
3	321.5	6/5, 23/19
4	428.7	23/18
5	535.8	15/11
6	643
7	750.2	17/11, 20/13, 23/15
8	857.3	18/11
9	964.5	7/4
10	1071.7	13/7, 24/13
11	1178.8
12	1286	19/9, 21/10, 23/11
13	1393.2
14	1500.3
15	1607.5
16	1714.7
17	1821.8	20/7
18	1929
19	2036.2	13/4
20	2143.3	24/7
21	2250.5	11/3
22	2357.7
23	2464.8	25/6
24	2572	22/5
25	2679.1
26	2786.3	5/1

Harmonics

Approximation of harmonics in 26ed5

Harmonic		2	3	4	5	6	7	8	9
Error	Absolute (¢)	-21.2	+27.0	-42.3	+0.0	+5.9	-46.7	+43.6	-53.1
	Relative (%)	-19.8	+25.2	-39.5	+0.0	+5.5	-43.6	+40.7	-49.6
Steps (reduced)		11 (11)	18 (18)	22 (22)	26 (0)	29 (3)	31 (5)	34 (8)	35 (9)

Approximation of prime harmonics in 26ed5

Harmonic		11	13	17	19	23	29	31	37	41	43	47
Error	Absolute (¢)	+28.2	-46.7	+24.7	+46.5	+37.2	-42.6	-50.9	-35.7	+0.9	+25.6	-21.2
	Relative (%)	+26.3	-43.6	+23.0	+43.3	+34.7	-39.8	-47.5	-33.3	+0.8	+23.9	-19.8
Steps (reduced)		39 (13)	41 (15)	46 (20)	48 (22)	51 (25)	54 (2)	55 (3)	58 (6)	60 (8)	61 (9)	62 (10)

Approximation of prime harmonics in 26ed5

Harmonic		59	61	67	71	73	79	83	89	97	101	103
Error	Absolute (¢)	+13.8	-43.9	+8.0	+14.8	-33.3	+44.2	-41.3	+52.2	+10.4	+47.6	+13.6
	Relative (%)	+12.9	-41.0	+7.4	+13.8	-31.1	+41.3	-38.5	+48.7	+9.7	+44.4	+12.7
Steps (reduced)		66 (14)	66 (14)	68 (16)	69 (17)	69 (17)	71 (19)	71 (19)	73 (21)	74 (22)	75 (23)	75 (23)

Approximation of prime harmonics in 26ed5

Harmonic		107	109	113	127	131	137	139	149	151	157	163
Error	Absolute (¢)	-52.3	+22.8	-39.6	-27.5	+26.0	-51.5	+30.5	+17.4	-5.6	+34.1	-30.9
	Relative (%)	-48.8	+21.3	-37.0	-25.6	+24.3	-48.1	+28.5	+16.3	-5.3	+31.8	-28.8
Steps (reduced)		75 (23)	76 (24)	76 (24)	78 (0)	79 (1)	79 (1)	80 (2)	81 (3)	81 (3)	82 (4)	82 (4)

Approximation of prime harmonics in 26ed5

Harmonic		167	173	179	181	191	193	197	199	211	223	227
Error	Absolute (¢)	+34.3	-26.8	+21.4	+2.1	+16.2	-1.8	-37.4	+52.3	-49.1	-37.6	+38.7
	Relative (%)	+32.0	-25.0	+19.9	+2.0	+15.1	-1.7	-34.9	+48.8	-45.8	-35.1	+36.2
Steps (reduced)		83 (5)	83 (5)	84 (6)	84 (6)	85 (7)	85 (7)	85 (7)	86 (8)	86 (8)	87 (9)	88 (10)