33ed5/4

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33ed5/4

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Prime factorization	3 × 11
Step size	11.7065¢
Octave	103\33ed5/4 (1205.77¢)
Twelfth	162\33ed5/4 (1896.45¢) (→54\11ed5/4)
Consistency limit	2
Distinct consistency limit	2

33 equal divisions of 5/4 (abbreviated 33ed5/4) is a nonoctave tuning system that divides the interval of 5/4 into 33 equal parts of about 11.7 ¢ each. Each step represents a frequency ratio of (5/4)^1/33, or the 33rd root of 5/4.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	11.706
2	23.413
3	35.119
4	46.826
5	58.532
6	70.239	23/22, 26/25
7	81.945	24/23
8	93.652	18/17, 19/18
9	105.358
10	117.065
11	128.771
12	140.478	25/23
13	152.184	12/11
14	163.891	11/10, 23/21
15	175.597	21/19
16	187.304	19/17
17	199.01
18	210.717	26/23
19	222.423	17/15
20	234.13	23/20
21	245.836	15/13
22	257.542
23	269.249	7/6
24	280.955	13/11
25	292.662
26	304.368	25/21
27	316.075	6/5
28	327.781	17/14
29	339.488	23/19
30	351.194
31	362.901	21/17
32	374.607	26/21
33	386.314

Harmonics

Approximation of harmonics in 33ed5/4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+5.77	-5.51	-0.17	-0.17	+0.26	+2.64	+5.59	+0.69	+5.59	+4.48	-5.68
	Relative (%)	+49.3	-47.0	-1.5	-1.5	+2.2	+22.5	+47.8	+5.9	+47.8	+38.3	-48.5
Steps (reduced)		103 (4)	162 (30)	205 (7)	238 (7)	265 (1)	288 (24)	308 (11)	325 (28)	341 (11)	355 (25)	367 (4)

Approximation of harmonics in 33ed5/4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-3.77	-3.30	-5.68	-0.34	+0.06	-5.24	-5.20	-0.34	-2.87	-1.46	+3.53
	Relative (%)	-32.2	-28.2	-48.5	-2.9	+0.5	-44.8	-44.4	-2.9	-24.5	-12.5	+30.2
Steps (reduced)		379 (16)	390 (27)	400 (4)	410 (14)	419 (23)	427 (31)	435 (6)	443 (14)	450 (21)	457 (28)	464 (2)