124edo

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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

124 equal divisions of the octave (abbreviated 124edo or 124ed2), also called 124-tone equal temperament (124tet) or 124 equal temperament (124et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 124 equal parts of about 9.68 ¢ each. Each step represents a frequency ratio of 2^1/124, or the 124th root of 2.

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	Ups and Downs Notation val <124 196] (124b)
0	0.0	1.0	1/1	P1
1	9.6774	1.0056		^1
2	19.3548	1.0112	65/64	^^1
3	29.0323	1.0169	65/64	^^^1
4	38.7097	1.0226	65/64, 33/32	^41
5	48.3871	1.0283	33/32	vvvA1
6	58.0645	1.0341	33/32, 24/23	vvA1, ^^d2
7	67.7419	1.0399	24/23, 23/22, 67/64, 22/21, 33/32	^^^d2
8	77.4194	1.0457	23/22, 67/64, 22/21, 24/23, 21/20, 20/19	v4m2
9	87.0968	1.0516	20/19, 21/20, 19/18, 22/21, 67/64, 23/22, 18/17, 24/23	vvvm2
10	96.7742	1.0575	18/17, 19/18, 20/19, 17/16, 21/20, 16/15	vvm2
11	106.4516	1.0634	17/16, 16/15, 18/17, 19/18, 15/14	vm2
12	116.129	1.0694	15/14, 16/15, 17/16, 14/13, 69/64	m2
13	125.8065	1.0754	14/13, 69/64, 15/14, 13/12, 16/15	^m2
14	135.4839	1.0814	13/12, 69/64, 14/13, 25/23, 12/11	^^m2
15	145.1613	1.0875	25/23, 12/11, 13/12, 35/32, 23/21, 69/64	^^^m2
16	154.8387	1.0936	35/32, 23/21, 12/11, 11/10, 25/23	~2
17	164.5161	1.0997	11/10, 23/21, 21/19, 35/32, 12/11, 71/64	vvvM2
18	174.1935	1.1059	21/19, 71/64, 10/9, 11/10	vvM2
19	183.871	1.1121	10/9, 71/64, 19/17, 21/19	vM2
20	193.5484	1.1183	19/17, 9/8, 10/9, 71/64	M2
21	203.2258	1.1246	9/8, 26/23, 19/17, 17/15	^M2
22	212.9032	1.1309	26/23, 17/15, 25/22, 9/8, 73/64	^^M2
23	222.5806	1.1372	25/22, 73/64, 17/15, 8/7, 26/23	^^^M2
24	232.2581	1.1436	8/7, 73/64, 23/20, 25/22, 15/13, 17/15	^4M2
25	241.9355	1.15	23/20, 15/13, 37/32, 8/7, 22/19, 73/64	vvvA2
26	251.6129	1.1564	37/32, 22/19, 15/13, 23/20, 7/6	vvA2, ^^d3
27	261.2903	1.1629	7/6, 22/19, 37/32, 75/64, 15/13	^^^d3
28	270.9677	1.1694	75/64, 7/6, 27/23, 20/17	v4m3
29	280.6452	1.176	20/17, 27/23, 75/64, 13/11, 7/6	vvvm3
30	290.3226	1.1826	13/11, 19/16, 20/17, 25/21, 27/23, 75/64	vvm3
31	300.0	1.1892	25/21, 19/16, 13/11, 6/5	vm3
32	309.6774	1.1959	6/5, 25/21, 77/64, 19/16	m3
33	319.3548	1.2026	77/64, 6/5, 23/19	^m3
34	329.0323	1.2093	23/19, 17/14, 77/64, 28/23, 6/5, 39/32	^^m3
35	338.7097	1.2161	28/23, 17/14, 39/32, 23/19, 11/9, 27/22	^^^m3
36	348.3871	1.2229	11/9, 39/32, 27/22, 28/23, 16/13, 17/14	~3
37	358.0645	1.2298	16/13, 27/22, 79/64, 21/17, 11/9, 26/21, 39/32	vvvM3
38	367.7419	1.2367	21/17, 26/21, 79/64, 16/13, 27/22	vvM3
39	377.4194	1.2436	26/21, 5/4, 21/17, 79/64	vM3
40	387.0968	1.2506	5/4	M3
41	396.7742	1.2576	24/19, 5/4, 81/64, 19/15	^M3
42	406.4516	1.2646	81/64, 24/19, 19/15, 14/11	^^M3
43	416.129	1.2717	14/11, 19/15, 23/18, 81/64, 24/19, 41/32	^^^M3
44	425.8065	1.2788	23/18, 41/32, 14/11, 9/7	^4M3
45	435.4839	1.286	9/7, 41/32, 22/17, 23/18, 83/64	vvvA3
46	445.1613	1.2932	22/17, 83/64, 13/10, 9/7, 30/23	vvA3, ^^d4
47	454.8387	1.3005	13/10, 83/64, 30/23, 22/17, 17/13, 21/16	^^^d4
48	464.5161	1.3078	17/13, 30/23, 21/16, 13/10, 25/19, 83/64	v44
49	474.1935	1.3151	25/19, 21/16, 17/13, 30/23	vvv4
50	483.871	1.3225	85/64, 25/19, 21/16, 4/3	vv4
51	493.5484	1.3299	85/64, 4/3	v4
52	503.2258	1.3373	4/3, 43/32, 85/64	P4
53	512.9032	1.3448	43/32, 27/20, 23/17, 4/3, 19/14	^4
54	522.5806	1.3524	23/17, 27/20, 19/14, 87/64, 43/32, 15/11	^^4
55	532.2581	1.3599	87/64, 19/14, 15/11, 23/17, 26/19, 27/20	^^^4
56	541.9355	1.3676	26/19, 15/11, 11/8, 87/64, 19/14	~4
57	551.6129	1.3752	11/8, 26/19, 18/13, 15/11	vvvA4
58	561.2903	1.3829	18/13, 25/18, 89/64, 11/8, 32/23	vvA4
59	570.9677	1.3907	89/64, 32/23, 25/18, 18/13, 7/5	vA4
60	580.6452	1.3985	7/5, 32/23, 45/32, 89/64, 25/18	A4
61	590.3226	1.4063	45/32, 24/17, 7/5, 17/12	^A4
62	600.0	1.4142	17/12, 24/17, 27/19, 91/64, 45/32	^^A4, vvd5
63	609.6774	1.4221	91/64, 27/19, 17/12, 10/7, 24/17, 33/23	vd5
64	619.3548	1.4301	10/7, 33/23, 23/16, 91/64, 27/19	d5
65	629.0323	1.4381	23/16, 33/23, 13/9, 10/7	^d5
66	638.7097	1.4462	13/9, 93/64, 16/11, 23/16, 33/23	^^d5
67	648.3871	1.4543	16/11, 93/64, 19/13, 13/9, 22/15	^^^d5
68	658.0645	1.4624	19/13, 22/15, 47/32, 16/11, 25/17, 93/64, 28/19	~5
69	667.7419	1.4706	25/17, 47/32, 28/19, 22/15, 34/23, 19/13	vvv5
70	677.4194	1.4789	34/23, 28/19, 95/64, 25/17, 47/32, 22/15	vv5
71	687.0968	1.4872	95/64, 34/23, 3/2, 28/19	v5
72	696.7742	1.4955	3/2, 95/64	P5
73	706.4516	1.5039	3/2, 97/64	^5
74	716.129	1.5123	97/64, 35/23, 32/21, 3/2	^^5
75	725.8065	1.5208	35/23, 32/21, 97/64, 26/17, 49/32, 23/15	^^^5
76	735.4839	1.5293	26/17, 49/32, 23/15, 32/21, 35/23, 20/13, 97/64	^45
77	745.1613	1.5379	20/13, 23/15, 49/32, 17/11, 26/17, 99/64, 32/21	vvvA5
78	754.8387	1.5465	99/64, 17/11, 20/13, 14/9, 23/15	vvA5, ^^d6
79	764.5161	1.5552	14/9, 25/16, 99/64, 17/11, 36/23	^^^d6
80	774.1935	1.5639	36/23, 25/16, 11/7, 14/9, 101/64	v4m6
81	783.871	1.5727	11/7, 101/64, 30/19, 36/23, 25/16, 19/12	vvvm6
82	793.5484	1.5815	19/12, 30/19, 101/64, 27/17, 35/22, 11/7, 51/32	vvm6
83	803.2258	1.5904	35/22, 27/17, 51/32, 19/12, 8/5, 30/19, 101/64	vm6
84	812.9032	1.5993	8/5, 51/32, 35/22, 103/64, 27/17	m6
85	822.5806	1.6082	103/64, 21/13, 8/5, 34/21, 51/32	^m6
86	832.2581	1.6173	34/21, 21/13, 13/8, 103/64	^^m6
87	841.9355	1.6263	13/8, 34/21, 18/11, 21/13, 105/64	^^^m6
88	851.6129	1.6354	18/11, 105/64, 23/14, 13/8, 28/17, 33/20	~6
89	861.2903	1.6446	23/14, 28/17, 105/64, 33/20, 38/23, 18/11, 53/32	vvvM6
90	870.9677	1.6538	38/23, 53/32, 33/20, 28/17, 23/14, 5/3, 105/64	vvM6
91	880.6452	1.6631	5/3, 53/32, 107/64, 38/23, 33/20	vM6
92	890.3226	1.6724	107/64, 5/3, 32/19, 27/16	M6
93	900.0	1.6818	32/19, 27/16, 107/64, 22/13, 39/23, 5/3	^M6
94	909.6774	1.6912	22/13, 27/16, 39/23, 32/19, 17/10, 109/64	^^M6
95	919.3548	1.7007	17/10, 109/64, 39/23, 22/13, 27/16, 12/7	^^^M6
96	929.0323	1.7102	12/7, 109/64, 55/32, 17/10, 39/23	^4M6
97	938.7097	1.7198	55/32, 12/7, 19/11, 26/15, 111/64	vvvA6
98	948.3871	1.7295	19/11, 26/15, 111/64, 33/19, 40/23, 55/32, 12/7	vvA6, ^^d7
99	958.0645	1.7392	40/23, 33/19, 111/64, 26/15, 7/4, 19/11	^^^d7
100	967.7419	1.7489	7/4, 40/23, 33/19, 111/64, 26/15, 30/17	v4m7
101	977.4194	1.7587	30/17, 113/64, 7/4, 23/13, 39/22	vvvm7
102	987.0968	1.7686	23/13, 113/64, 30/17, 39/22, 16/9, 57/32	vvm7
103	996.7742	1.7785	16/9, 57/32, 39/22, 25/14, 23/13, 34/19, 113/64, 30/17	vm7
104	1006.4516	1.7884	34/19, 25/14, 57/32, 115/64, 16/9, 9/5, 39/22	m7
105	1016.129	1.7985	9/5, 115/64, 34/19, 38/21, 25/14, 29/16	^m7
106	1025.8065	1.8086	38/21, 29/16, 9/5, 20/11, 115/64	^^m7
107	1035.4839	1.8187	20/11, 29/16, 42/23, 38/21, 117/64, 11/6	^^^m7
108	1045.1613	1.8289	117/64, 42/23, 11/6, 20/11, 35/19, 59/32, 29/16	~7
109	1054.8387	1.8391	35/19, 59/32, 11/6, 24/13, 117/64, 42/23	vvvM7
110	1064.5161	1.8495	24/13, 59/32, 35/19, 13/7, 119/64, 11/6	vvM7
111	1074.1935	1.8598	119/64, 13/7, 28/15, 24/13, 15/8, 59/32	vM7
112	1083.871	1.8702	28/15, 15/8, 119/64, 32/17, 13/7	M7
113	1093.5484	1.8807	32/17, 15/8, 17/9, 121/64, 36/19, 28/15	^M7
114	1103.2258	1.8913	121/64, 17/9, 36/19, 19/10, 32/17, 40/21, 61/32, 15/8	^^M7
115	1112.9032	1.9019	19/10, 40/21, 61/32, 36/19, 21/11, 44/23, 121/64, 17/9, 23/12	^^^M7
116	1122.5806	1.9125	44/23, 21/11, 23/12, 61/32, 40/21, 123/64, 25/13, 19/10, 27/14	^4M7
117	1132.2581	1.9233	25/13, 123/64, 27/14, 23/12, 44/23, 31/16, 21/11, 61/32	vvvA7
118	1141.9355	1.934	31/16, 27/14, 33/17, 35/18, 25/13, 123/64, 39/20, 23/12	vvA7, ^^d1 +1 oct
119	1151.6129	1.9449	35/18, 33/17, 39/20, 31/16, 125/64, 45/23, 27/14	^^^d1 +1 oct
120	1161.2903	1.9558	45/23, 125/64, 39/20, 35/18, 63/32, 33/17	v41 +1 oct
121	1170.9677	1.9667	63/32, 45/23, 125/64, 39/20, 127/64	vvv1 +1 oct
122	1180.6452	1.9778	127/64, 63/32	vv1 +1 oct
123	1190.3226	1.9889	127/64	v1 +1 oct
124	1200.0	2.0	2/1	P1 +1 oct
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.