22ed5/2

22 equal divisions of 5/2 (abbreviated 22ed5/2) is a nonoctave tuning system that divides the interval of 5/2 into 22 equal parts of about 72.1 ¢ each. Each step represents a frequency ratio of (5/2)^1/22, or the 22nd root of 5/2.

← 21ed5/2

22ed5/2

23ed5/2 →

Prime factorization	2 × 11
Step size	72.1052 ¢
Octave	17\22ed5/2 (1225.79 ¢)
Twelfth	26\22ed5/2 (1874.73 ¢) (→ 13\11ed5/2)
Consistency limit	2
Distinct consistency limit	2

Theory

22ed5/2 has a vague similarity to 17edo but they cannot be considered truly equivalent due to the severe octave stretch of 25.8 cents. It does not approximate harmonics well, but it approximates several half-integer (or /2 in primodality terms) ratios well, in particular it has decent approximations of 7/2, 11/2 and 13/2, and could be analyzed as a 5/2.7/2.11/2.13/2 subgroup temperament. It supports frosthog temperament, equating 11/7 to the square root of 5/2.

Intervals

Steps	Cents	Approximate ratios*
0	0.000	1/1
1	72.105	26/25
2	144.210	625/572
3	216.316	25/22
4	288.421	13/11
5	360.525	154/125
6	432.631	625/484, 875/676
7	504.736	35/26, 121/91
8	576.842	7/5
9	648.947	182/125
10	721.052	275/182

* Based on treating 22ed5/2 as a 5/2.7/2.11/2.13/2 subgroup temperament.

Harmonics

Approximation of harmonics in 22ed5/2

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+25.8	-27.2	-20.5	+25.8	-1.4	+20.1	+5.3	+17.7	-20.5	+30.8	+24.4
	Relative (%)	+35.8	-37.8	-28.5	+35.8	-2.0	+27.9	+7.3	+24.5	-28.5	+42.7	+33.8
Steps (reduced)		17 (17)	26 (4)	33 (11)	39 (17)	43 (21)	47 (3)	50 (6)	53 (9)	55 (11)	58 (14)	60 (16)

Approximation of harmonics in 22ed5/2

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+30.0	-26.2	-1.4	+31.0	-1.8	-28.7	+22.0	+5.3	-7.1	-15.5	-20.4
	Relative (%)	+41.6	-36.3	-2.0	+43.1	-2.5	-39.7	+30.4	+7.3	-9.9	-21.5	-28.3
Steps (reduced)		62 (18)	63 (19)	65 (21)	67 (1)	68 (2)	69 (3)	71 (5)	72 (6)	73 (7)	74 (8)	75 (9)