15ed5

← 14ed5

15ed5

16ed5 →

Prime factorization	3 × 5
Step size	185.754¢
Octave	6\15ed5 (1114.53¢) (→2\5ed5)
Twelfth	10\15ed5 (1857.54¢) (→2\3ed5)
Consistency limit	3
Distinct consistency limit	3

15 equal divisions of the 5th harmonic (abbreviated 15ed5) is a nonoctave tuning system that divides the interval of 5/1 into 15 equal parts of about 186 ¢ each. Each step represents a frequency ratio of 5^1/15, or the 15th root of 5. A multiple of 5ed5 It approximates the 7th, 12th, 13th, 18th, 31st and 43rd harmonics with some accuracy (but especially the 31st!). Hyperpyth analogues of blackwood of course are warranted.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	185.754	10/9, 11/10, 19/17, 21/19
2	371.508	21/17
3	557.263	15/11
4	743.017	14/9, 17/11, 23/15
5	928.771	17/10, 19/11
6	1114.525	17/9, 19/10, 21/11
7	1300.28	15/7, 19/9, 21/10
8	1486.034	7/3
9	1671.788	13/5
10	1857.542
11	2043.297	23/7
12	2229.051	11/3
13	2414.805
14	2600.559	9/2
15	2786.314	5/1

Harmonics

Approximation of harmonics in 15ed5

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-85.5	-44.4	+14.8	+0.0	+55.9	-25.2	-70.7	-88.8	-85.5	-64.7	-29.6
	Relative (%)	-46.0	-23.9	+8.0	+0.0	+30.1	-13.6	-38.0	-47.8	-46.0	-34.8	-15.9
Steps (reduced)		6 (6)	10 (10)	13 (13)	15 (0)	17 (2)	18 (3)	19 (4)	20 (5)	21 (6)	22 (7)	23 (8)

Approximation of harmonics in 15ed5

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+17.6	+75.0	-44.4	+29.6	-75.3	+11.5	-82.1	+14.8	-69.7	+35.6	-41.4
	Relative (%)	+9.5	+40.4	-23.9	+15.9	-40.6	+6.2	-44.2	+8.0	-37.5	+19.1	-22.3
Steps (reduced)		24 (9)	25 (10)	25 (10)	26 (11)	26 (11)	27 (12)	27 (12)	28 (13)	28 (13)	29 (14)	29 (14)