41ed4

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41ed4

43ed4 →

Prime factorization	41 (prime)
Step size	58.5366 ¢
Octave	21\41ed4 (1229.27 ¢)
Twelfth	32\41ed4 (1873.17 ¢)
Consistency limit	1
Distinct consistency limit	1

41 equal divisions of the 4th harmonic (abbreviated 41ed4) is a nonoctave tuning system that divides the interval of 4/1 into 41 equal parts of about 58.5⁠ ⁠¢ each. Each step represents a frequency ratio of 4^1/41, or the 41st root of 4. It corresponds to 20.5edo, or every second step of 41edo.

The Kite Guitar (see also KiteGuitar.com and Kite Tuning) is a guitar fretting using every other step of 41edo, i.e. 41ed4 or "20½-edo". However, the interval between two adjacent open strings is always an odd number of 41-edosteps. Thus each string only covers 41ed4, but the full edo can be found on every pair of adjacent strings.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	58.5	30/29
2	117.1
3	175.6	21/19
4	234.1
5	292.7	13/11
6	351.2
7	409.8	19/15
8	468.3	17/13
9	526.8	23/17
10	585.4	7/5
11	643.9
12	702.4
13	761
14	819.5
15	878
16	936.6	12/7
17	995.1
18	1053.7	11/6
19	1112.2
20	1170.7
21	1229.3
22	1287.8
23	1346.3
24	1404.9
25	1463.4
26	1522	12/5, 29/12
27	1580.5	5/2
28	1639
29	1697.6
30	1756.1
31	1814.6
32	1873.2
33	1931.7
34	1990.2	19/6
35	2048.8
36	2107.3
37	2165.9	7/2
38	2224.4
39	2282.9
40	2341.5
41	2400

Harmonics

Approximation of harmonics in 41ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+29.3	-28.8	+0.0	+23.4	+0.5	+26.3	+29.3	+1.0	-5.8	+4.8	-28.8
	Relative (%)	+50.0	-49.2	+0.0	+40.0	+0.8	+44.9	+50.0	+1.7	-10.0	+8.2	-49.2
Steps (reduced)		21 (21)	32 (32)	41 (0)	48 (7)	53 (12)	58 (17)	62 (21)	65 (24)	68 (27)	71 (30)	73 (32)

Approximation of harmonics in 41ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+8.3	-3.0	-5.3	+0.0	+12.1	-28.3	-4.8	+23.4	-2.5	-24.5	+15.6
	Relative (%)	+14.1	-5.1	-9.1	+0.0	+20.7	-48.3	-8.3	+40.0	-4.3	-41.8	+26.7
Steps (reduced)		76 (35)	78 (37)	80 (39)	82 (0)	84 (2)	85 (3)	87 (5)	89 (7)	90 (8)	91 (9)	93 (11)