19ed5/4

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19ed5/4

20ed5/4 →

Prime factorization	19 (prime)
Step size	20.3323¢
Octave	59\19ed5/4 (1199.61¢) (convergent)
Twelfth	94\19ed5/4 (1911.24¢)
Consistency limit	7
Distinct consistency limit	4

19 equal divisions of 5/4 (abbreviated 19ed5/4) is a nonoctave tuning system that divides the interval of 5/4 into 19 equal parts of about 20.3 ¢ each. Each step represents a frequency ratio of (5/4)^1/19, or the 19th root of 5/4.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	20.332
2	40.665
3	60.997
4	81.329	19/18, 20/19, 23/22, 24/23
5	101.662	16/15, 17/16
6	121.994	13/12, 15/14
7	142.326	14/13, 23/21
8	162.658	11/10, 12/11
9	182.991	21/19
10	203.323	17/15, 19/17
11	223.655	8/7
12	243.988	22/19, 23/20
13	264.32	7/6, 15/13
14	284.652	13/11, 20/17
15	304.985	19/16
16	325.317	6/5, 17/14, 23/19
17	345.649
18	365.981	16/13
19	386.314	5/4

Harmonics

Approximation of harmonics in 19ed5/4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-0.39	+9.28	-0.79	-0.79	+8.89	+6.34	-1.18	-1.77	-1.18	-3.53	+8.49
	Relative (%)	-1.9	+45.6	-3.9	-3.9	+43.7	+31.2	-5.8	-8.7	-5.8	-17.4	+41.8
Steps (reduced)		59 (2)	94 (18)	118 (4)	137 (4)	153 (1)	166 (14)	177 (6)	187 (16)	196 (6)	204 (14)	212 (3)

Approximation of harmonics in 19ed5/4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-8.09	+5.94	+8.49	-1.58	-4.87	-2.16	+5.89	-1.58	-4.72	-3.92	+0.45
	Relative (%)	-39.8	+29.2	+41.8	-7.8	-24.0	-10.6	+29.0	-7.8	-23.2	-19.3	+2.2
Steps (reduced)		218 (9)	225 (16)	231 (3)	236 (8)	241 (13)	246 (18)	251 (4)	255 (8)	259 (12)	263 (16)	267 (1)