19ed5/3

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19ed5/3

20ed5/3 →

Prime factorization	19 (prime)
Step size	46.5452¢
Octave	26\19ed5/3 (1210.18¢)
Twelfth	41\19ed5/3 (1908.35¢)
Consistency limit	6
Distinct consistency limit	6

19 equal divisions of 5/3 (abbreviated 19ed5/3) is a nonoctave tuning system that divides the interval of 5/3 into 19 equal parts of about 46.5 ¢ each. Each step represents a frequency ratio of (5/3)^1/19, or the 19th root of 5/3. This tuning tempers out the syntonic comma.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	46.5
2	93.1	19/18, 20/19, 22/21, 23/22
3	139.6	14/13
4	186.2	10/9
5	232.7	23/20
6	279.3	13/11
7	325.8	6/5, 17/14, 23/19
8	372.4	5/4, 21/17
9	418.9	14/11, 19/15, 23/18, 24/19
10	465.5	17/13
11	512	4/3
12	558.5
13	605.1
14	651.6	22/15
15	698.2	3/2
16	744.7	17/11, 23/15
17	791.3	11/7, 19/12
18	837.8	21/13
19	884.4	5/3

Harmonics

Approximation of harmonics in 19ed5/3

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+10.2	+6.4	+20.4	+6.4	+16.6	-17.6	-16.0	+12.8	+16.6	-8.8	-19.8
	Relative (%)	+21.9	+13.7	+43.7	+13.7	+35.6	-37.8	-34.4	+27.5	+35.6	-18.9	-42.5
Steps (reduced)		26 (7)	41 (3)	52 (14)	60 (3)	67 (10)	72 (15)	77 (1)	82 (6)	86 (10)	89 (13)	92 (16)

Approximation of harmonics in 19ed5/3

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-18.7	-7.4	+12.8	-5.8	-17.7	+23.0	+22.5	-19.8	-11.2	+1.4	+17.5
	Relative (%)	-40.2	-15.9	+27.5	-12.6	-38.0	+49.4	+48.3	-42.5	-24.0	+3.0	+37.6
Steps (reduced)		95 (0)	98 (3)	101 (6)	103 (8)	105 (10)	108 (13)	110 (15)	111 (16)	113 (18)	115 (1)	117 (3)

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