123edt

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123edt

124edt →

Prime factorization	3 × 41
Step size	15.463¢
Octave	78\123edt (1206.12¢) (→26\41edt)
Consistency limit	2
Distinct consistency limit	2

123 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 123edt or 123ed3), is a nonoctave tuning system that divides the interval of 3/1 into 123 equal parts of about 15.5⁠ ⁠¢ each. Each step represents a frequency ratio of 3^1/123, or the 123rd root of 3.

Intervals

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	15.5	10.6
2	30.9	21.1
3	46.4	31.7
4	61.9	42.3
5	77.3	52.8	23/22, 45/43
6	92.8	63.4	19/18, 39/37
7	108.2	74	33/31, 49/46, 50/47
8	123.7	84.6	29/27, 44/41
9	139.2	95.1
10	154.6	105.7	47/43
11	170.1	116.3	43/39
12	185.6	126.8	39/35, 49/44
13	201	137.4	46/41
14	216.5	148	17/15
15	231.9	158.5
16	247.4	169.1	15/13
17	262.9	179.7	50/43
18	278.3	190.2	27/23
19	293.8	200.8
20	309.3	211.4	49/41
21	324.7	222	35/29, 41/34, 47/39
22	340.2	232.5	45/37
23	355.7	243.1	27/22, 43/35
24	371.1	253.7	26/21, 31/25
25	386.6	264.2
26	402	274.8	29/23
27	417.5	285.4
28	433	295.9
29	448.4	306.5	35/27
30	463.9	317.1	17/13
31	479.4	327.6	29/22, 33/25
32	494.8	338.2
33	510.3	348.8	47/35
34	525.7	359.3
35	541.2	369.9	26/19, 41/30
36	556.7	380.5
37	572.1	391.1
38	587.6	401.6
39	603.1	412.2
40	618.5	422.8	10/7
41	634	433.3	49/34
42	649.4	443.9
43	664.9	454.5	22/15
44	680.4	465	37/25, 43/29
45	695.8	475.6
46	711.3	486.2
47	726.8	496.7	35/23
48	742.2	507.3
49	757.7	517.9
50	773.2	528.5
51	788.6	539	41/26
52	804.1	549.6	35/22, 43/27
53	819.5	560.2
54	835	570.7	34/21, 47/29
55	850.5	581.3	49/30
56	865.9	591.9
57	881.4	602.4
58	896.9	613
59	912.3	623.6	22/13, 39/23
60	927.8	634.1
61	943.2	644.7	50/29
62	958.7	655.3	47/27
63	974.2	665.9
64	989.6	676.4	23/13, 39/22
65	1005.1	687
66	1020.6	697.6
67	1036	708.1
68	1051.5	718.7
69	1067	729.3	50/27
70	1082.4	739.8	43/23
71	1097.9	750.4	49/26
72	1113.3	761
73	1128.8	771.5
74	1144.3	782.1
75	1159.7	792.7	41/21, 43/22
76	1175.2	803.3
77	1190.7	813.8
78	1206.1	824.4
79	1221.6	835
80	1237	845.5	45/22, 47/23
81	1252.5	856.1
82	1268	866.7
83	1283.4	877.2	21/10
84	1298.9	887.8
85	1314.4	898.4	47/22
86	1329.8	908.9	41/19
87	1345.3	919.5	37/17, 50/23
88	1360.7	930.1
89	1376.2	940.7
90	1391.7	951.2
91	1407.1	961.8
92	1422.6	972.4	25/11
93	1438.1	982.9	39/17
94	1453.5	993.5	44/19
95	1469	1004.1
96	1484.5	1014.6
97	1499.9	1025.2	50/21
98	1515.4	1035.8
99	1530.8	1046.3	46/19
100	1546.3	1056.9	22/9
101	1561.8	1067.5	37/15
102	1577.2	1078
103	1592.7	1088.6
104	1608.2	1099.2	43/17
105	1623.6	1109.8	23/9
106	1639.1	1120.3	49/19
107	1654.5	1130.9	13/5
108	1670	1141.5
109	1685.5	1152	45/17
110	1700.9	1162.6
111	1716.4	1173.2	35/13
112	1731.9	1183.7	49/18
113	1747.3	1194.3
114	1762.8	1204.9
115	1778.3	1215.4
116	1793.7	1226	31/11
117	1809.2	1236.6	37/13
118	1824.6	1247.2	43/15
119	1840.1	1257.7
120	1855.6	1268.3
121	1871	1278.9
122	1886.5	1289.4
123	1902	1300	3/1

Harmonics

Approximation of harmonics in 123edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+6.12	+0.00	-3.23	-2.96	+6.12	+2.12	+2.89	+0.00	+3.15	-7.22	-3.23
	Relative (%)	+39.6	+0.0	-20.9	-19.2	+39.6	+13.7	+18.7	+0.0	+20.4	-46.7	-20.9
Steps (reduced)		78 (78)	123 (0)	155 (32)	180 (57)	201 (78)	218 (95)	233 (110)	246 (0)	258 (12)	268 (22)	278 (32)

Approximation of harmonics in 123edt

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-2.63	-7.23	-2.96	-6.45	-3.17	+6.12	+5.29	-6.19	+2.12	-1.10	-0.74
	Relative (%)	-17.0	-46.7	-19.2	-41.7	-20.5	+39.6	+34.2	-40.0	+13.7	-7.1	-4.8
Steps (reduced)		287 (41)	295 (49)	303 (57)	310 (64)	317 (71)	324 (78)	330 (84)	335 (89)	341 (95)	346 (100)	351 (105)