124edo

← 123edo

124edo

125edo →

Prime factorization	22 × 31
Step size	9.67742 ¢
Fifth	73\124 (706.452 ¢)
Semitones (A1:m2)	15:7 (145.2 ¢ : 67.74 ¢)
Dual sharp fifth	73\124 (706.452 ¢)
Dual flat fifth	72\124 (696.774 ¢) (→ 18\31)
Dual major 2nd	21\124 (203.226 ¢)
Consistency limit	5
Distinct consistency limit	5

Theory

124edo is closely related to 31edo, but the patent vals differ on the mapping for 3. The equal temperament tempers out 2048/2025 (diaschisma) and [-6 -24 19⟩ in the 5-limit. Using the patent val, it tempers out 3136/3125, 4000/3969, and 33614/32805 in the 7-limit; 385/384, 1232/1215, 1331/1323, and 3773/3750 in the 11-limit; 196/195, 364/363, 572/567, 625/624, and 1001/1000 in the 13-limit. Note that although its sharp fifth is slightly closer to just, both fifths are about equally off in both directions, and its 9th harmonic is especially accurate as a result, so it can be considered a dual-fifth system, in which it performs very well in the 2.9.5.7.11.13.17.19.23.37 subgroup (the dual-fifth no-31's 37-limit), which is arguably the right way to analyze its approximations of JI. Also interesting is that one may want to double the number of notes to add a fifth closer to just, but this causes the relative errors of other primes to double leading to inconsistencies, so its most reasonable and capable conceptualization seems to be that of a dual-fifth system.

Odd harmonics

Approximation of odd harmonics in 124edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25
Error	Absolute (¢)	+4.50	+0.78	-1.08	-0.68	+0.29	+1.41	-4.40	+1.50	+2.49	+3.41	+0.76	+1.57
	Relative (%)	+46.5	+8.1	-11.2	-7.1	+3.0	+14.5	-45.4	+15.5	+25.7	+35.3	+7.8	+16.2
Steps (reduced)		197 (73)	288 (40)	348 (100)	393 (21)	429 (57)	459 (87)	484 (112)	507 (11)	527 (31)	545 (49)	561 (65)	576 (80)

Approximation of odd harmonics in 124edo (continued)

Harmonic		27	29	31	33	35	37	39	41	43	45	47	49
Error	Absolute (¢)	+3.81	-3.77	-3.10	+4.79	-0.30	+0.27	-3.77	-3.26	+1.39	+0.10	+2.24	-2.17
	Relative (%)	+39.4	-39.0	-32.0	+49.5	-3.1	+2.8	-39.0	-33.6	+14.3	+1.0	+23.1	-22.4
Steps (reduced)		590 (94)	602 (106)	614 (118)	626 (6)	636 (16)	646 (26)	655 (35)	664 (44)	673 (53)	681 (61)	689 (69)	696 (76)

Intervals

124 EDO Table of Intervals

Step	Cents	Ratio	JI Ratio Approximations
0	0.0	1.0	1/1
1	9.6774	1.0056
2	19.3548	1.0112	65/64
3	29.0323	1.0169	65/64
4	38.7097	1.0226	65/64, 33/32
5	48.3871	1.0283	33/32
6	58.0645	1.0341	33/32, 24/23
7	67.7419	1.0399	24/23, 23/22, 67/64, 22/21, 33/32
8	77.4194	1.0457	23/22, 67/64, 22/21, 24/23, 21/20, 20/19
9	87.0968	1.0516	20/19, 21/20, 19/18, 22/21, 67/64, 23/22, 18/17, 24/23
10	96.7742	1.0575	18/17, 19/18, 20/19, 17/16, 21/20, 16/15
11	106.4516	1.0634	17/16, 16/15, 18/17, 19/18, 15/14
12	116.129	1.0694	15/14, 16/15, 17/16, 14/13, 69/64
13	125.8065	1.0754	14/13, 69/64, 15/14, 13/12, 16/15
14	135.4839	1.0814	13/12, 69/64, 14/13, 25/23, 12/11
15	145.1613	1.0875	25/23, 12/11, 13/12, 35/32, 23/21, 69/64
16	154.8387	1.0936	35/32, 23/21, 12/11, 11/10, 25/23
17	164.5161	1.0997	11/10, 23/21, 21/19, 35/32, 12/11, 71/64
18	174.1935	1.1059	21/19, 71/64, 10/9, 11/10
19	183.871	1.1121	10/9, 71/64, 19/17, 21/19
20	193.5484	1.1183	19/17, 9/8, 10/9, 71/64
21	203.2258	1.1246	9/8, 26/23, 19/17, 17/15
22	212.9032	1.1309	26/23, 17/15, 25/22, 9/8, 73/64
23	222.5806	1.1372	25/22, 73/64, 17/15, 8/7, 26/23
24	232.2581	1.1436	8/7, 73/64, 23/20, 25/22, 15/13, 17/15
25	241.9355	1.15	23/20, 15/13, 37/32, 8/7, 22/19, 73/64
26	251.6129	1.1564	37/32, 22/19, 15/13, 23/20, 7/6
27	261.2903	1.1629	7/6, 22/19, 37/32, 75/64, 15/13
28	270.9677	1.1694	75/64, 7/6, 27/23, 20/17
29	280.6452	1.176	20/17, 27/23, 75/64, 13/11, 7/6
30	290.3226	1.1826	13/11, 19/16, 20/17, 25/21, 27/23, 75/64
31	300.0	1.1892	25/21, 19/16, 13/11, 6/5
32	309.6774	1.1959	6/5, 25/21, 77/64, 19/16
33	319.3548	1.2026	77/64, 6/5, 23/19
34	329.0323	1.2093	23/19, 17/14, 77/64, 28/23, 6/5, 39/32
35	338.7097	1.2161	28/23, 17/14, 39/32, 23/19, 11/9, 27/22
36	348.3871	1.2229	11/9, 39/32, 27/22, 28/23, 16/13, 17/14
37	358.0645	1.2298	16/13, 27/22, 79/64, 21/17, 11/9, 26/21, 39/32
38	367.7419	1.2367	21/17, 26/21, 79/64, 16/13, 27/22
39	377.4194	1.2436	26/21, 5/4, 21/17, 79/64
40	387.0968	1.2506	5/4
41	396.7742	1.2576	24/19, 5/4, 81/64, 19/15
42	406.4516	1.2646	81/64, 24/19, 19/15, 14/11
43	416.129	1.2717	14/11, 19/15, 23/18, 81/64, 24/19, 41/32
44	425.8065	1.2788	23/18, 41/32, 14/11, 9/7
45	435.4839	1.286	9/7, 41/32, 22/17, 23/18, 83/64
46	445.1613	1.2932	22/17, 83/64, 13/10, 9/7, 30/23
47	454.8387	1.3005	13/10, 83/64, 30/23, 22/17, 17/13, 21/16
48	464.5161	1.3078	17/13, 30/23, 21/16, 13/10, 25/19, 83/64
49	474.1935	1.3151	25/19, 21/16, 17/13, 30/23
50	483.871	1.3225	85/64, 25/19, 21/16, 4/3
51	493.5484	1.3299	85/64, 4/3
52	503.2258	1.3373	4/3, 43/32, 85/64
53	512.9032	1.3448	43/32, 27/20, 23/17, 4/3, 19/14
54	522.5806	1.3524	23/17, 27/20, 19/14, 87/64, 43/32, 15/11
55	532.2581	1.3599	87/64, 19/14, 15/11, 23/17, 26/19, 27/20
56	541.9355	1.3676	26/19, 15/11, 11/8, 87/64, 19/14
57	551.6129	1.3752	11/8, 26/19, 18/13, 15/11
58	561.2903	1.3829	18/13, 25/18, 89/64, 11/8, 32/23
59	570.9677	1.3907	89/64, 32/23, 25/18, 18/13, 7/5
60	580.6452	1.3985	7/5, 32/23, 45/32, 89/64, 25/18
61	590.3226	1.4063	45/32, 24/17, 7/5, 17/12
62	600.0	1.4142	17/12, 24/17, 27/19, 91/64, 45/32
63	609.6774	1.4221	91/64, 27/19, 17/12, 10/7, 24/17, 33/23
64	619.3548	1.4301	10/7, 33/23, 23/16, 91/64, 27/19
65	629.0323	1.4381	23/16, 33/23, 13/9, 10/7
66	638.7097	1.4462	13/9, 93/64, 16/11, 23/16, 33/23
67	648.3871	1.4543	16/11, 93/64, 19/13, 13/9, 22/15
68	658.0645	1.4624	19/13, 22/15, 47/32, 16/11, 25/17, 93/64, 28/19
69	667.7419	1.4706	25/17, 47/32, 28/19, 22/15, 34/23, 19/13
70	677.4194	1.4789	34/23, 28/19, 95/64, 25/17, 47/32, 22/15
71	687.0968	1.4872	95/64, 34/23, 3/2, 28/19
72	696.7742	1.4955	3/2, 95/64
73	706.4516	1.5039	3/2, 97/64
74	716.129	1.5123	97/64, 35/23, 32/21, 3/2
75	725.8065	1.5208	35/23, 32/21, 97/64, 26/17, 49/32, 23/15
76	735.4839	1.5293	26/17, 49/32, 23/15, 32/21, 35/23, 20/13, 97/64
77	745.1613	1.5379	20/13, 23/15, 49/32, 17/11, 26/17, 99/64, 32/21
78	754.8387	1.5465	99/64, 17/11, 20/13, 14/9, 23/15
79	764.5161	1.5552	14/9, 25/16, 99/64, 17/11, 36/23
80	774.1935	1.5639	36/23, 25/16, 11/7, 14/9, 101/64
81	783.871	1.5727	11/7, 101/64, 30/19, 36/23, 25/16, 19/12
82	793.5484	1.5815	19/12, 30/19, 101/64, 27/17, 35/22, 11/7, 51/32
83	803.2258	1.5904	35/22, 27/17, 51/32, 19/12, 8/5, 30/19, 101/64
84	812.9032	1.5993	8/5, 51/32, 35/22, 103/64, 27/17
85	822.5806	1.6082	103/64, 21/13, 8/5, 34/21, 51/32
86	832.2581	1.6173	34/21, 21/13, 13/8, 103/64
87	841.9355	1.6263	13/8, 34/21, 18/11, 21/13, 105/64
88	851.6129	1.6354	18/11, 105/64, 23/14, 13/8, 28/17, 33/20
89	861.2903	1.6446	23/14, 28/17, 105/64, 33/20, 38/23, 18/11, 53/32
90	870.9677	1.6538	38/23, 53/32, 33/20, 28/17, 23/14, 5/3, 105/64
91	880.6452	1.6631	5/3, 53/32, 107/64, 38/23, 33/20
92	890.3226	1.6724	107/64, 5/3, 32/19, 27/16
93	900.0	1.6818	32/19, 27/16, 107/64, 22/13, 39/23, 5/3
94	909.6774	1.6912	22/13, 27/16, 39/23, 32/19, 17/10, 109/64
95	919.3548	1.7007	17/10, 109/64, 39/23, 22/13, 27/16, 12/7
96	929.0323	1.7102	12/7, 109/64, 55/32, 17/10, 39/23
97	938.7097	1.7198	55/32, 12/7, 19/11, 26/15, 111/64
98	948.3871	1.7295	19/11, 26/15, 111/64, 33/19, 40/23, 55/32, 12/7
99	958.0645	1.7392	40/23, 33/19, 111/64, 26/15, 7/4, 19/11
100	967.7419	1.7489	7/4, 40/23, 33/19, 111/64, 26/15, 30/17
101	977.4194	1.7587	30/17, 113/64, 7/4, 23/13, 39/22
102	987.0968	1.7686	23/13, 113/64, 30/17, 39/22, 16/9, 57/32
103	996.7742	1.7785	16/9, 57/32, 39/22, 25/14, 23/13, 34/19, 113/64, 30/17
104	1006.4516	1.7884	34/19, 25/14, 57/32, 115/64, 16/9, 9/5, 39/22
105	1016.129	1.7985	9/5, 115/64, 34/19, 38/21, 25/14, 29/16
106	1025.8065	1.8086	38/21, 29/16, 9/5, 20/11, 115/64
107	1035.4839	1.8187	20/11, 29/16, 42/23, 38/21, 117/64, 11/6
108	1045.1613	1.8289	117/64, 42/23, 11/6, 20/11, 35/19, 59/32, 29/16
109	1054.8387	1.8391	35/19, 59/32, 11/6, 24/13, 117/64, 42/23
110	1064.5161	1.8495	24/13, 59/32, 35/19, 13/7, 119/64, 11/6
111	1074.1935	1.8598	119/64, 13/7, 28/15, 24/13, 15/8, 59/32
112	1083.871	1.8702	28/15, 15/8, 119/64, 32/17, 13/7
113	1093.5484	1.8807	32/17, 15/8, 17/9, 121/64, 36/19, 28/15
114	1103.2258	1.8913	121/64, 17/9, 36/19, 19/10, 32/17, 40/21, 61/32, 15/8
115	1112.9032	1.9019	19/10, 40/21, 61/32, 36/19, 21/11, 44/23, 121/64, 17/9, 23/12
116	1122.5806	1.9125	44/23, 21/11, 23/12, 61/32, 40/21, 123/64, 25/13, 19/10, 27/14
117	1132.2581	1.9233	25/13, 123/64, 27/14, 23/12, 44/23, 31/16, 21/11, 61/32
118	1141.9355	1.934	31/16, 27/14, 33/17, 35/18, 25/13, 123/64, 39/20, 23/12
119	1151.6129	1.9449	35/18, 33/17, 39/20, 31/16, 125/64, 45/23, 27/14
120	1161.2903	1.9558	45/23, 125/64, 39/20, 35/18, 63/32, 33/17
121	1170.9677	1.9667	63/32, 45/23, 125/64, 39/20, 127/64
122	1180.6452	1.9778	127/64, 63/32
123	1190.3226	1.9889	127/64
124	1200.0	2.0	2/1
JI Ratio Approximations are comprised of 23 limit ratios and the odd harmonics up to 127. The JI Ratio Approximations are stylized as follows to indicate accuracy: Big Bold Underlined: absolute cent error < 1 cent. Big Bold: absolute cent error < 2 cents. Big: absolute cent error < 4 cents. Normal: absolute cent error < 8 cents. Small: absolute cent error < 16 cents.

Intervals

124 EDO Table of Intervals

Step	Cents	Ratio	JI Ratio Approximations	Step
0	0.000		P1	0
1	9.677		^1	1
2	19.355		^^1	2
3	29.032		^^^1	3
4	38.710		^41	4
5	48.387		vvvA1	5
6	58.065	32/31, 31/30, 30/29, 29/28	vvA1, ^^d2	6
7	67.742	28/27, 27/26, 26/25, 25/24	^^^d2	7
8	77.419	24/23, 23/22, 22/21	v4m2	8
9	87.097	21/20, 20/19	vvvm2	9
10	96.774	19/18, 18/17	vvm2	10
11	106.452	17/16	vm2	11
12	116.129	16/15, 31/29, 15/14	m2	12
13	125.806	29/27, 14/13	^m2	13
14	135.484	27/25, 13/12	^^m2	14
15	145.161	25/23	^^^m2	15
16	154.839	12/11, 23/21	~2	16
17	164.516	11/10	vvvM2	17
18	174.194	32/29, 21/19, 31/28	vvM2	18
19	183.871	10/9	vM2	19
20	193.548	29/26, 19/17, 28/25	M2	20
21	203.226	9/8	^M2	21
22	212.903	26/23, 17/15	^^M2	22
23	222.581	25/22	^^^M2	23
24	232.258	8/7	^4M2	24
25	241.935	31/27, 23/20	vvvA2	25
26	251.613	15/13, 22/19	vvA2, ^^d3	26
27	261.290	29/25	^^^d3	27
28	270.968	7/6	v4m3	28
29	280.645	27/23, 20/17	vvvm3	29
30	290.323	13/11, 32/27	vvm3	30
31	300.000	19/16, 25/21, 31/26	vm3	31
32	309.677		m3	32
33	319.355	6/5	^m3	33
34	329.032	29/24, 23/19	^^m3	34
35	338.710	17/14, 28/23	^^^m3	35
36	348.387	11/9	~3	36
37	358.065	27/22, 16/13	vvvM3	37
38	367.742	21/17, 26/21, 31/25	vvM3	38
39	377.419		vM3	39
40	387.097	5/4	M3	40
41	396.774	29/23	^M3	41
42	406.452	24/19, 19/15	^^M3	42
43	416.129	14/11	^^^M3	43
44	425.806	23/18, 32/25	^4M3	44
45	435.484	9/7	vvvA3	45
46	445.161	31/24, 22/17	vvA3, ^^d4	46
47	454.839	13/10	^^^d4	47
48	464.516	30/23, 17/13	v44	48
49	474.194	21/16, 25/19, 29/22	vvv4	49
50	483.871		vv4	50
51	493.548	4/3	v4	51
52	503.226		P4	52
53	512.903	31/23	^4	53
54	522.581	27/20, 23/17	^^4	54
55	532.258	19/14, 15/11	^^^4	55
56	541.935	26/19	~4	56
57	551.613	11/8	vvvA4	57
58	561.290	29/21, 18/13	vvA4	58
59	570.968	25/18, 32/23	vA4	59
60	580.645	7/5	A4	60
61	590.323	31/22	^A4	61
62	600.000	24/17, 17/12	^^A4, vvd5	62
63	609.677	27/19	vd5	63
64	619.355	10/7	d5	64
65	629.032	23/16	^d5	65
66	638.710	13/9, 29/20	^^d5	66
67	648.387	16/11	^^^d5	67
68	658.065	19/13	~5	68
69	667.742	22/15, 25/17, 28/19	vvv5	69
70	677.419	31/21	vv5	70
71	687.097		v5	71
72	696.774		P5	72
73	706.452	3/2	^5	73
74	716.129		^^5	74
75	725.806	32/21	^^^5	75
76	735.484	29/19, 26/17, 23/15	^45	76
77	745.161	20/13	vvvA5	77
78	754.839	17/11, 31/20	vvA5, ^^d6	78
79	764.516	14/9	^^^d6	79
80	774.194	25/16	v4m6	80
81	783.871	11/7	vvvm6	81
82	793.548	30/19, 19/12	vvm6	82
83	803.226	27/17	vm6	83
84	812.903	8/5	m6	84
85	822.581	29/18	^m6	85
86	832.258	21/13	^^m6	86
87	841.935	13/8	^^^m6	87
88	851.613	31/19, 18/11	~6	88
89	861.290	23/14, 28/17	vvvM6	89
90	870.968		vvM6	90
91	880.645	5/3	vM6	91
92	890.323		M6	92
93	900.000	32/19	^M6	93
94	909.677	27/16, 22/13	^^M6	94
95	919.355	17/10	^^^M6	95
96	929.032	29/17, 12/7	^4M6	96
97	938.710	31/18	vvvA6	97
98	948.387	19/11, 26/15	vvA6, ^^d7	98
99	958.065		^^^d7	99
100	967.742	7/4	v4m7	100
101	977.419		vvvm7	101
102	987.097	30/17, 23/13	vvm7	102
103	996.774	16/9	vm7	103
104	1006.452	25/14	m7	104
105	1016.129	9/5	^m7	105
106	1025.806	29/16	^^m7	106
107	1035.484	20/11, 31/17	^^^m7	107
108	1045.161	11/6	~7	108
109	1054.839		vvvM7	109
110	1064.516	24/13	vvM7	110
111	1074.194	13/7	vM7	111
112	1083.871	28/15, 15/8	M7	112
113	1093.548	32/17	^M7	113
114	1103.226	17/9	^^M7	114
115	1112.903	19/10	^^^M7	115
116	1122.581	21/11, 23/12	^4M7	116
117	1132.258	25/13, 27/14	vvvA7	117
118	1141.935	29/15, 31/16	vvA7, ^^d1 +1oct	118
119	1151.613		^^^d1 +1oct	119
120	1161.290		v41 +1oct	120
121	1170.968		vvv1 +1oct	121
122	1180.645		vv1 +1oct	122
123	1190.323		v1 +1oct	123
124	1200.000	2/1	P1 +1oct	124
125	1209.677		^1 +1oct	125
126	1219.355		^^1 +1oct	126
127	1229.032		^^^1 +1oct	127
128	1238.710		^41 +1oct	128
129	1248.387		vvvA1 +1oct	129
130	1258.065	31/15, 29/14	vvA1 +1oct, ^^d2 +1oct	130
131	1267.742	27/13, 25/12	^^^d2 +1oct	131
132	1277.419	23/11	v4m2 +1oct	132
133	1287.097	21/10	vvvm2 +1oct	133
134	1296.774	19/9	vvm2 +1oct	134
135	1306.452	17/8	vm2 +1oct	135
136	1316.129	32/15, 15/7	m2 +1oct	136
137	1325.806	28/13	^m2 +1oct	137
138	1335.484	13/6	^^m2 +1oct	138
139	1345.161		^^^m2 +1oct	139
140	1354.839	24/11	~2 +1oct	140
141	1364.516	11/5	vvvM2 +1oct	141
142	1374.194	31/14	vvM2 +1oct	142
143	1383.871	20/9	vM2 +1oct	143
144	1393.548	29/13	M2 +1oct	144
145	1403.226	9/4	^M2 +1oct	145
146	1412.903		^^M2 +1oct	146
147	1422.581	25/11	^^^M2 +1oct	147
148	1432.258	16/7	^4M2 +1oct	148
149	1441.935	23/10	vvvA2 +1oct	149
150	1451.613	30/13	vvA2 +1oct, ^^d3 +1oct	150
151	1461.290		^^^d3 +1oct	151
152	1470.968	7/3	v4m3 +1oct	152
153	1480.645		vvvm3 +1oct	153
154	1490.323	26/11	vvm3 +1oct	154
155	1500.000	19/8, 31/13	vm3 +1oct	155
156	1509.677		m3 +1oct	156
157	1519.355	12/5	^m3 +1oct	157
158	1529.032	29/12	^^m3 +1oct	158
159	1538.710	17/7	^^^m3 +1oct	159
160	1548.387	22/9	~3 +1oct	160
161	1558.065	27/11, 32/13	vvvM3 +1oct	161
162	1567.742		vvM3 +1oct	162
163	1577.419		vM3 +1oct	163
164	1587.097	5/2	M3 +1oct	164
165	1596.774		^M3 +1oct	165
166	1606.452		^^M3 +1oct	166
167	1616.129	28/11	^^^M3 +1oct	167
168	1625.806	23/9	^4M3 +1oct	168
169	1635.484	18/7	vvvA3 +1oct	169
170	1645.161	31/12	vvA3 +1oct, ^^d4 +1oct	170
171	1654.839	13/5	^^^d4 +1oct	171
172	1664.516		v44 +1oct	172
173	1674.194	21/8, 29/11	vvv4 +1oct	173
174	1683.871		vv4 +1oct	174
175	1693.548	8/3	v4 +1oct	175
176	1703.226		P4 +1oct	176
177	1712.903		^4 +1oct	177
178	1722.581	27/10	^^4 +1oct	178
179	1732.258	19/7, 30/11	^^^4 +1oct	179
180	1741.935		~4 +1oct	180
181	1751.613	11/4	vvvA4 +1oct	181
182	1761.290		vvA4 +1oct	182
183	1770.968	25/9	vA4 +1oct	183
184	1780.645	14/5	A4 +1oct	184
185	1790.323	31/11	^A4 +1oct	185
186	1800.000	17/6	^^A4 +1oct, vvd5 +1oct	186
187	1809.677		vd5 +1oct	187
188	1819.355	20/7	d5 +1oct	188
189	1829.032	23/8	^d5 +1oct	189
190	1838.710	26/9, 29/10	^^d5 +1oct	190
191	1848.387	32/11	^^^d5 +1oct	191
192	1858.065		~5 +1oct	192
193	1867.742		vvv5 +1oct	193
194	1877.419		vv5 +1oct	194
195	1887.097		v5 +1oct	195
196	1896.774		P5 +1oct	196
197	1906.452	3/1	^5 +1oct	197
198	1916.129		^^5 +1oct	198
199	1925.806		^^^5 +1oct	199
200	1935.484		^45 +1oct	200
201	1945.161		vvvA5 +1oct	201
202	1954.839	31/10	vvA5 +1oct, ^^d6 +1oct	202
203	1964.516	28/9	^^^d6 +1oct	203
204	1974.194	25/8	v4m6 +1oct	204
205	1983.871	22/7	vvvm6 +1oct	205
206	1993.548	19/6	vvm6 +1oct	206
207	2003.226		vm6 +1oct	207