45ed4

45 equal divisions of the 4th harmonic (abbreviated 45ed4) is a nonoctave tuning system that divides the interval of 4/1 into 45 equal parts of about 53.3 ¢ each. Each step represents a frequency ratio of 4^1/45, or the 45th root of 4. It is a tuning system equvalent to 22.5edo, representing every second step of 45edo.

← 43ed4

45ed4

47ed4 →

Prime factorization	32 × 5
Step size	53.3333 ¢
Octave	23\45ed4 (1226.67 ¢)
Twelfth	36\45ed4 (1920 ¢) (→ 4\5ed4)
Consistency limit	1
Distinct consistency limit	1

Its step size is very close to the Alpharabian quarter tone 33/32. In addition, it tempers out the tritrizo comma in the 4.7/6 subgroup, and along with it the quartisma in the 33/32.7/6 subgroup.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	53.3
2	106.7
3	160	11/10, 23/21
4	213.3	17/15, 26/23
5	266.7
6	320
7	373.3	26/21, 31/25
8	426.7
9	480
10	533.3	15/11, 19/14
11	586.7	7/5
12	640
13	693.3
14	746.7
15	800
16	853.3	23/14
17	906.7
18	960
19	1013.3
20	1066.7	13/7
21	1120	21/11
22	1173.3
23	1226.7
24	1280	21/10, 23/11
25	1333.3
26	1386.7	20/9, 29/13
27	1440	23/10
28	1493.3	26/11
29	1546.7	22/9
30	1600
31	1653.3	13/5
32	1706.7
33	1760
34	1813.3
35	1866.7
36	1920
37	1973.3
38	2026.7
39	2080	10/3
40	2133.3
41	2186.7
42	2240
43	2293.3
44	2346.7
45	2400

Harmonics

Approximation of harmonics in 45ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+26.7	+18.0	+0.0	-13.0	-8.6	-8.8	+26.7	-17.2	+13.7	+8.7	+18.0
	Relative (%)	+50.0	+33.8	+0.0	-24.3	-16.2	-16.5	+50.0	-32.3	+25.7	+16.3	+33.8
Steps (reduced)		23 (23)	36 (36)	45 (0)	52 (7)	58 (13)	63 (18)	68 (23)	71 (26)	75 (30)	78 (33)	81 (36)

Approximation of harmonics in 45ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-13.9	+17.8	+5.1	+0.0	+1.7	+9.4	+22.5	-13.0	+9.2	-18.0	+11.7
	Relative (%)	-26.0	+33.5	+9.5	+0.0	+3.2	+17.7	+42.2	-24.3	+17.3	-33.7	+22.0
Steps (reduced)		83 (38)	86 (41)	88 (43)	90 (0)	92 (2)	94 (4)	96 (6)	97 (7)	99 (9)	100 (10)	102 (12)