45ed4

← 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

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.

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	53.333
2	106.667
3	160	11/10, 23/21
4	213.333	17/15, 26/23
5	266.667
6	320
7	373.333	26/21, 31/25
8	426.667
9	480
10	533.333	15/11, 19/14
11	586.667	7/5
12	640
13	693.333
14	746.667
15	800
16	853.333	23/14
17	906.667
18	960
19	1013.333
20	1066.667	13/7
21	1120	21/11
22	1173.333
23	1226.667
24	1280	21/10, 23/11
25	1333.333
26	1386.667	20/9, 29/13
27	1440	23/10
28	1493.333	26/11
29	1546.667	22/9
30	1600
31	1653.333	13/5
32	1706.667
33	1760
34	1813.333
35	1866.667
36	1920
37	1973.333
38	2026.667
39	2080	10/3
40	2133.333
41	2186.667
42	2240
43	2293.333
44	2346.667
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)