57edt

← 56edt

57edt

58edt →

Prime factorization	3 × 19
Step size	33.3676 ¢
Octave	36\57edt (1201.23 ¢) (→ 12\19edt)
Consistency limit	9
Distinct consistency limit	9

57 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 57edt or 57ed3), is a nonoctave tuning system that divides the interval of 3/1 into 57 equal parts of about 33.4 ¢ each. Each step represents a frequency ratio of 3^1/57, or the 57th root of 3.

Theory

57edt is related to 36edo (sixth-tone tuning), but with the 3/1 rather than the 2/1 being just. This stretches the octave by about 1.2347 cents. It is consistent to the 9-integer-limit, whereas 36edo is only consistent up to the 8-integer-limit due to a discrepancy with approximating 9/5, although 57edt barely manages to achieve this since it almost completely misses 5/1.

Harmonics

Approximation of harmonics in 57edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+1.2	+0.0	+2.5	+16.6	+1.2	+1.3	+3.7	+0.0	-15.6	-13.7	+2.5
	Relative (%)	+3.7	+0.0	+7.4	+49.7	+3.7	+3.9	+11.1	+0.0	-46.6	-41.2	+7.4
Steps (reduced)		36 (36)	57 (0)	72 (15)	84 (27)	93 (36)	101 (44)	108 (51)	114 (0)	119 (5)	124 (10)	129 (15)

Approximation of harmonics in 57edt

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-2.6	+2.5	+16.6	+4.9	+0.1	+1.2	+7.7	-14.3	+1.3	-12.5	+10.6
	Relative (%)	-7.9	+7.6	+49.7	+14.8	+0.3	+3.7	+23.2	-42.9	+3.9	-37.5	+31.9
Steps (reduced)		133 (19)	137 (23)	141 (27)	144 (30)	147 (33)	150 (36)	153 (39)	155 (41)	158 (44)	160 (46)	163 (49)

Intervals

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	33.4	22.8
2	66.7	45.6	27/26, 28/27
3	100.1	68.4	18/17
4	133.5	91.2
5	166.8	114	32/29
6	200.2	136.8	9/8
7	233.6	159.6	8/7
8	266.9	182.5	7/6
9	300.3	205.3	19/16, 31/26
10	333.7	228.1	17/14, 23/19
11	367	250.9	21/17, 26/21
12	400.4	273.7	24/19, 29/23, 34/27
13	433.8	296.5	9/7
14	467.1	319.3	17/13, 21/16
15	500.5	342.1	4/3
16	533.9	364.9
17	567.2	387.7	18/13, 32/23
18	600.6	410.5	17/12, 24/17
19	634	433.3	13/9
20	667.4	456.1	28/19
21	700.7	478.9	3/2
22	734.1	501.8	26/17, 29/19
23	767.5	524.6	14/9
24	800.8	547.4	27/17
25	834.2	570.2	21/13, 34/21
26	867.6	593	28/17
27	900.9	615.8	32/19
28	934.3	638.6	12/7
29	967.7	661.4	7/4
30	1001	684.2
31	1034.4	707
32	1067.8	729.8	13/7
33	1101.1	752.6	17/9
34	1134.5	775.4	27/14
35	1167.9	798.2
36	1201.2	821.1	2/1
37	1234.6	843.9
38	1268	866.7	27/13
39	1301.3	889.5	17/8
40	1334.7	912.3	13/6
41	1368.1	935.1
42	1401.4	957.9	9/4
43	1434.8	980.7	16/7
44	1468.2	1003.5	7/3
45	1501.5	1026.3	19/8, 31/13
46	1534.9	1049.1	17/7
47	1568.3	1071.9
48	1601.6	1094.7
49	1635	1117.5	18/7
50	1668.4	1140.4	21/8, 34/13
51	1701.7	1163.2	8/3
52	1735.1	1186
53	1768.5	1208.8
54	1801.9	1231.6	17/6
55	1835.2	1254.4	26/9
56	1868.6	1277.2
57	1902	1300	3/1

See also

• 21edf – relative edf
• 36edo – relative edo
• 93ed6 – relative ed6
• 101ed7 – relative ed7
• 129ed12 – relative ed12, close to the zeta-optimized tuning for 36edo