124edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 123edo124edo125edo →
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 21/124, 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.