14ed13

← 13ed13

14ed13

15ed13 →

Prime factorization	2 × 7
Step size	317.181¢
Octave	4\14ed13 (1268.72¢) (→2\7ed13)
Twelfth	6\14ed13 (1903.08¢) (→3\7ed13)
Consistency limit	7
Distinct consistency limit	3

14 equal divisions of the 13th harmonic (abbreviated 14ed13) is a nonoctave tuning system that divides the interval of 13/1 into 14 equal parts of about 317 ¢ each. Each step represents a frequency ratio of 13^1/14, or the 14th root of 13. 14ed13 is related to cata temperament, with one step approximating 6/5 and six approximating 3/1. One step of this tuning is extremely close to 60 steps of 227edo, which supports cata with a nearly justly tuned 13th harmonic.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	317.2	6/5, 23/19
2	634.4	10/7, 13/9, 19/13
3	951.5	7/4, 12/7, 19/11
4	1268.7	21/10, 23/11
5	1585.9	5/2
6	1903.1	3/1
7	2220.3	18/5
8	2537.4	13/3
9	2854.6	21/4
10	3171.8	19/3
11	3489	15/2
12	3806.2	9/1
13	4123.3
14	4440.5

Harmonics

Approximation of harmonics in 14ed13

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+69	+1	+137	+68	+70	+120	-111	+2	+137	-28	+139
	Relative (%)	+21.7	+0.4	+43.3	+21.5	+22.0	+37.9	-35.0	+0.7	+43.2	-8.8	+43.7
Steps (reduced)		4 (4)	6 (6)	8 (8)	9 (9)	10 (10)	11 (11)	11 (11)	12 (12)	13 (13)	13 (13)	14 (0)

Approximation of harmonics in 14ed13

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+0	-128	+69	-42	-147	+71	-23	-111	+121	+41	-36
	Relative (%)	+0.0	-40.4	+21.9	-13.3	-46.4	+22.4	-7.1	-35.1	+38.2	+12.8	-11.4
Steps (reduced)		14 (0)	14 (0)	15 (1)	15 (1)	15 (1)	16 (2)	16 (2)	16 (2)	17 (3)	17 (3)	17 (3)

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