38edt

← 37edt

38edt

39edt →

Prime factorization	2 × 19
Step size	50.0514 ¢
Octave	24\38edt (1201.23 ¢) (→ 12\19edt)
Consistency limit	6
Distinct consistency limit	6

38 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 38edt or 38ed3), is a nonoctave tuning system that divides the interval of 3/1 into 38 equal parts of about 50.1 ¢ each. Each step represents a frequency ratio of 3^1/38, or the 38th root of 3.

Theory

38edt is closely related to 24edo (quarter-tone tuning), but with the perfect twelfth rather than the octave being just, which stretches the octave by about 1.23 cents. Like 24edo, 38edt is consistent to the 6-integer-limit.

Harmonics

Approximation of harmonics in 38edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+1.2	+0.0	+2.5	+16.6	+1.2	-15.4	+3.7	+0.0	+17.8	+3.0	+2.5
	Relative (%)	+2.5	+0.0	+4.9	+33.1	+2.5	-30.7	+7.4	+0.0	+35.6	+5.9	+4.9
Steps (reduced)		24 (24)	38 (0)	48 (10)	56 (18)	62 (24)	67 (29)	72 (34)	76 (0)	80 (4)	83 (7)	86 (10)

Approximation of harmonics in 38edt (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+14.1	-14.1	+16.6	+4.9	+0.1	+1.2	+7.7	+19.0	-15.4	+4.2	-22.7	+3.7
	Relative (%)	+28.1	-28.3	+33.1	+9.9	+0.2	+2.5	+15.5	+38.0	-30.7	+8.4	-45.4	+7.4
Steps (reduced)		89 (13)	91 (15)	94 (18)	96 (20)	98 (22)	100 (24)	102 (26)	104 (28)	105 (29)	107 (31)	108 (32)	110 (34)

Subsets and supersets

Since 38 factors into primes as 2 × 19, 38edt contains subset edts 2edt and 19edt.

Intervals

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	50.1	34.2
2	100.1	68.4	17/16, 18/17, 19/18
3	150.2	102.6	12/11, 23/21
4	200.2	136.8	9/8, 19/17
5	250.3	171.1	15/13, 22/19
6	300.3	205.3	19/16
7	350.4	239.5	11/9, 27/22
8	400.4	273.7	24/19, 29/23
9	450.5	307.9	13/10, 22/17
10	500.5	342.1	4/3
11	550.6	376.3	11/8, 26/19
12	600.6	410.5	17/12, 24/17
13	650.7	444.7	16/11, 19/13
14	700.7	478.9	3/2
15	750.8	513.2	17/11, 20/13
16	800.8	547.4	19/12, 27/17
17	850.9	581.6	18/11
18	900.9	615.8	27/16
19	951	650	19/11, 26/15
20	1001	684.2	16/9
21	1051.1	718.4	11/6
22	1101.1	752.6	17/9
23	1151.2	786.8
24	1201.2	821.1	2/1
25	1251.3	855.3
26	1301.3	889.5	17/8
27	1351.4	923.7	24/11
28	1401.4	957.9	9/4
29	1451.5	992.1
30	1501.5	1026.3	19/8
31	1551.6	1060.5	22/9, 27/11
32	1601.6	1094.7
33	1651.7	1128.9	13/5
34	1701.7	1163.2	8/3
35	1751.8	1197.4	11/4
36	1801.9	1231.6	17/6
37	1851.9	1265.8
38	1902	1300	3/1

See also

• 14edf – relative edf
• 24edo – relative edo
• 56ed5 – relative ed5
• 62ed6 – relative ed6
• 83ed11 – relative ed11
• 86ed12 – relative ed12
• 198ed304 – close to the zeta-optimized tuning for 24edo