84edt

← 83edt

84edt

85edt →

Prime factorization	22 × 3 × 7
Step size	22.6423 ¢
Octave	53\84edt (1200.04 ¢) (convergent)
Consistency limit	10
Distinct consistency limit	10

84 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 84edt or 84ed3), is a nonoctave tuning system that divides the interval of 3/1 into 84 equal parts of about 22.6 ¢ each. Each step represents a frequency ratio of 3^1/84, or the 84th root of 3.

Theory

84edt is practically identical to 53edo, but with the 3/1 rather than the 2/1 being just. The octave is about 0.0430 cents stretched. Like 53edo, 84edt is consistent to the 10-integer-limit.

Harmonics

Approximation of harmonics in 84edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+0.04	+0.00	+0.09	-1.31	+0.04	+4.88	+0.13	+0.00	-1.27	-7.77	+0.09
	Relative (%)	+0.2	+0.0	+0.4	-5.8	+0.2	+21.6	+0.6	+0.0	-5.6	-34.3	+0.4
Steps (reduced)		53 (53)	84 (0)	106 (22)	123 (39)	137 (53)	149 (65)	159 (75)	168 (0)	176 (8)	183 (15)	190 (22)

Approximation of harmonics in 84edt (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	-2.63	+4.92	-1.31	+0.17	+8.43	+0.04	-2.99	-1.22	+4.88	-7.73	+5.88	+0.13
	Relative (%)	-11.6	+21.7	-5.8	+0.8	+37.2	+0.2	-13.2	-5.4	+21.6	-34.1	+26.0	+0.6
Steps (reduced)		196 (28)	202 (34)	207 (39)	212 (44)	217 (49)	221 (53)	225 (57)	229 (61)	233 (65)	236 (68)	240 (72)	243 (75)

Subsets and supersets

84 is a largely composite number. Since it factors into primes as 2² × 3 × 7, 84edt has subset edts 2, 3, 4, 6, 7, 12, 14, 21, 28, 42.

Intervals

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	22.6	15.5
2	45.3	31	37/36, 38/37, 39/38, 40/39, 41/40
3	67.9	46.4	25/24, 26/25, 27/26
4	90.6	61.9	19/18, 20/19, 39/37
5	113.2	77.4	16/15
6	135.9	92.9	13/12, 27/25, 40/37
7	158.5	108.3	23/21, 34/31
8	181.1	123.8	10/9
9	203.8	139.3	9/8
10	226.4	154.8	33/29, 41/36
11	249.1	170.2	15/13, 37/32
12	271.7	185.7	41/35
13	294.4	201.2	32/27
14	317	216.7	6/5
15	339.6	232.1	28/23, 39/32
16	362.3	247.6	16/13, 37/30
17	384.9	263.1	5/4
18	407.6	278.6	19/15
19	430.2	294	32/25, 41/32
20	452.8	309.5	13/10
21	475.5	325	25/19, 29/22
22	498.1	340.5	4/3
23	520.8	356	23/17, 27/20
24	543.4	371.4	26/19, 37/27, 41/30
25	566.1	386.9	18/13, 25/18
26	588.7	402.4	38/27
27	611.3	417.9	27/19, 37/26
28	634	433.3	13/9, 36/25
29	656.6	448.8	19/13
30	679.3	464.3	34/23, 37/25, 40/27
31	701.9	479.8	3/2
32	724.6	495.2	35/23, 38/25, 41/27
33	747.2	510.7	20/13, 37/24
34	769.8	526.2	25/16, 39/25
35	792.5	541.7	30/19
36	815.1	557.1	8/5
37	837.8	572.6	13/8
38	860.4	588.1	23/14
39	883.1	603.6	5/3
40	905.7	619	27/16
41	928.3	634.5	41/24
42	951	650	26/15
43	973.6	665.5
44	996.3	681	16/9
45	1018.9	696.4	9/5
46	1041.5	711.9	31/17
47	1064.2	727.4	24/13, 37/20
48	1086.8	742.9	15/8
49	1109.5	758.3	19/10
50	1132.1	773.8	25/13
51	1154.8	789.3	37/19, 39/20
52	1177.4	804.8
53	1200	820.2	2/1
54	1222.7	835.7
55	1245.3	851.2	37/18, 39/19, 41/20
56	1268	866.7	25/12, 27/13
57	1290.6	882.1	19/9, 40/19
58	1313.3	897.6	32/15
59	1335.9	913.1	13/6
60	1358.5	928.6
61	1381.2	944	20/9
62	1403.8	959.5	9/4
63	1426.5	975	41/18
64	1449.1	990.5	30/13, 37/16
65	1471.8	1006
66	1494.4	1021.4
67	1517	1036.9	12/5
68	1539.7	1052.4	39/16
69	1562.3	1067.9	32/13, 37/15
70	1585	1083.3	5/2
71	1607.6	1098.8	38/15
72	1630.2	1114.3	41/16
73	1652.9	1129.8	13/5
74	1675.5	1145.2	29/11
75	1698.2	1160.7	8/3
76	1720.8	1176.2	27/10
77	1743.5	1191.7	41/15
78	1766.1	1207.1	25/9, 36/13
79	1788.7	1222.6
80	1811.4	1238.1	37/13
81	1834	1253.6	26/9
82	1856.7	1269	38/13
83	1879.3	1284.5
84	1902	1300	3/1

See also

• 9ed9/8 – relative ed9/8
• 31edf – relative edf
• 53edo – relative edo
• 137ed6 – relative ed6