124edo
| ← 123edo | 124edo | 125edo → |
124edo is the equal division of the octave into 124 parts of 9.6774 cents each. It is closely related to 31edo, but the patent vals differ on the mapping for 3. It tempers out 2048/2025 (diaschisma) and 19073486328125/18075490334784 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-fifths system, in which it performs very well in the 2.9.5.7.11.13.17.19.23.37 subgroup (AKA 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.
Harmonics
| 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) | |
| 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
| 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 |
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JI Ratio Approximations are comprised of 23 limit ratios and the odd harmonics up to 127.
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