89edf
89edf is the equal division of the just perfect fifth into eighty nine parts of 7.887 cents each, corresponding to 152.1465 edo.
Theory
89edf is similar to Carlos Gamma in that it includes a just perfect fifth and an incredibly precise approximation of the major and minor thirds, and is next in line in the sequence {1, 2, 5, 7, 9, 11, 20, 89, 109, 129, 149} of divisions of the perfect fifth to approximate 5/4 with increasing accuracy.
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.16 | -1.16 | -2.16 | -1.02 | -2.69 | -0.07 | +0.84 | -2.42 | -1.93 | -0.98 | +1.86 |
| Relative (%) | -14.7 | -14.7 | -27.3 | -12.9 | -34.0 | -0.9 | +10.7 | -30.7 | -24.4 | -12.5 | +23.6 | |
| Steps (reduced) |
152 (63) |
241 (63) |
353 (86) |
427 (71) |
526 (81) |
563 (29) |
622 (88) |
646 (23) |
688 (65) |
739 (27) |
754 (42) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.15 | -1.05 | +3.26 | -0.88 | -3.81 | -0.19 | -2.69 | +0.52 | +2.66 | +1.89 | -0.77 |
| Relative (%) | +40.0 | -13.3 | +41.3 | -11.1 | -48.3 | -2.4 | -34.1 | +6.6 | +33.7 | +24.0 | -9.8 | |
| Steps (reduced) |
793 (81) |
815 (14) |
826 (25) |
845 (44) |
871 (70) |
895 (5) |
902 (12) |
923 (33) |
936 (46) |
942 (52) |
959 (69) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 7.9 | |
| 2 | 15.8 | |
| 3 | 23.7 | |
| 4 | 31.5 | |
| 5 | 39.4 | 42/41 |
| 6 | 47.3 | 36/35, 39/38 |
| 7 | 55.2 | 33/32 |
| 8 | 63.1 | 27/26, 28/27, 29/28 |
| 9 | 71 | 24/23, 25/24 |
| 10 | 78.9 | 22/21, 23/22 |
| 11 | 86.8 | 20/19, 21/20, 41/39 |
| 12 | 94.6 | 19/18 |
| 13 | 102.5 | 35/33 |
| 14 | 110.4 | 16/15 |
| 15 | 118.3 | 15/14 |
| 16 | 126.2 | 14/13, 29/27 |
| 17 | 134.1 | 27/25, 41/38 |
| 18 | 142 | 25/23, 38/35 |
| 19 | 149.9 | 12/11 |
| 20 | 157.7 | 23/21, 34/31, 35/32 |
| 21 | 165.6 | 11/10 |
| 22 | 173.5 | 21/19 |
| 23 | 181.4 | 10/9 |
| 24 | 189.3 | 29/26, 39/35 |
| 25 | 197.2 | 28/25 |
| 26 | 205.1 | 9/8 |
| 27 | 213 | 26/23 |
| 28 | 220.8 | 25/22 |
| 29 | 228.7 | 8/7 |
| 30 | 236.6 | 39/34 |
| 31 | 244.5 | 23/20, 38/33 |
| 32 | 252.4 | 22/19 |
| 33 | 260.3 | |
| 34 | 268.2 | 7/6 |
| 35 | 276 | 27/23, 34/29, 41/35 |
| 36 | 283.9 | 33/28 |
| 37 | 291.8 | 13/11, 32/27 |
| 38 | 299.7 | 19/16, 25/21 |
| 39 | 307.6 | 37/31 |
| 40 | 315.5 | 6/5 |
| 41 | 323.4 | 35/29, 41/34 |
| 42 | 331.3 | 23/19, 40/33 |
| 43 | 339.1 | 28/23 |
| 44 | 347 | 11/9 |
| 45 | 354.9 | 27/22 |
| 46 | 362.8 | |
| 47 | 370.7 | 26/21 |
| 48 | 378.6 | 41/33 |
| 49 | 386.5 | 5/4 |
| 50 | 394.4 | |
| 51 | 402.2 | 24/19, 29/23 |
| 52 | 410.1 | 19/15, 33/26 |
| 53 | 418 | 14/11 |
| 54 | 425.9 | 23/18, 32/25 |
| 55 | 433.8 | 9/7 |
| 56 | 441.7 | |
| 57 | 449.6 | 35/27 |
| 58 | 457.5 | 30/23 |
| 59 | 465.3 | 17/13, 38/29 |
| 60 | 473.2 | 21/16, 25/19 |
| 61 | 481.1 | 33/25 |
| 62 | 489 | |
| 63 | 496.9 | 4/3 |
| 64 | 504.8 | |
| 65 | 512.7 | 35/26, 39/29 |
| 66 | 520.6 | 23/17, 27/20 |
| 67 | 528.4 | 19/14 |
| 68 | 536.3 | 15/11 |
| 69 | 544.2 | 26/19 |
| 70 | 552.1 | 11/8 |
| 71 | 560 | 29/21 |
| 72 | 567.9 | 25/18 |
| 73 | 575.8 | 39/28 |
| 74 | 583.6 | 7/5 |
| 75 | 591.5 | 38/27 |
| 76 | 599.4 | 41/29 |
| 77 | 607.3 | 27/19 |
| 78 | 615.2 | 10/7 |
| 79 | 623.1 | 33/23 |
| 80 | 631 | 23/16, 36/25 |
| 81 | 638.9 | 13/9, 42/29 |
| 82 | 646.7 | 16/11 |
| 83 | 654.6 | 19/13, 35/24 |
| 84 | 662.5 | 22/15, 41/28 |
| 85 | 670.4 | 28/19 |
| 86 | 678.3 | 34/23, 40/27 |
| 87 | 686.2 | |
| 88 | 694.1 | |
| 89 | 702 | 3/2 |
Music
- 89ed(3/2) groove (2026)