61edo
| ← 60edo | 61edo | 62edo → |
61 equal divisions of the octave (abbreviated 61edo or 61ed2), also called 61-tone equal temperament (61tet) or 61 equal temperament (61et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 61 equal parts of about 19.672 ¢ each. Each step represents a frequency ratio of 21/61, or the 61st root of 2.
Theory
61edo provides the optimal patent val for the freivald (24 & 37) temperament in the 7-, 11- and 13-limit.
61edo is the 18th prime edo, after of 59edo and before of 67edo.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +6.24 | +7.13 | -4.89 | -7.19 | -0.50 | +5.37 | -6.30 | -6.59 | -2.43 | +1.35 | +1.23 |
| relative (%) | +32 | +36 | -25 | -37 | -3 | +27 | -32 | -34 | -12 | +7 | +6 | |
| Steps (reduced) |
97 (36) |
142 (20) |
171 (49) |
193 (10) |
211 (28) |
226 (43) |
238 (55) |
249 (5) |
259 (15) |
268 (24) |
276 (32) | |
Table of intervals
| Steps | Cents | Ups and downs notation | Approximate ratios |
|---|---|---|---|
| 0 | 0 | D | 1/1 |
| 1 | 19.6721 | ^D, vvEb | 64/63, 66/65 |
| 2 | 39.3443 | ^^D, vEb | 40/39 |
| 3 | 59.0164 | ^3D, Eb | 33/32, 65/63 |
| 4 | 78.6885 | ^4D, v7E | 21/20, 22/21 |
| 5 | 98.3607 | ^5D, v6E | 35/33, 55/52 |
| 6 | 118.033 | ^6D, v5E | |
| 7 | 137.705 | ^7D, v4E | 13/12, 27/25 |
| 8 | 157.377 | D#, v3E | 11/10, 12/11, 35/32 |
| 9 | 177.049 | ^D#, vvE | 10/9, 72/65 |
| 10 | 196.721 | ^^D#, vE | |
| 11 | 216.393 | E | |
| 12 | 236.066 | ^E, vvF | 8/7, 55/48, 63/55 |
| 13 | 255.738 | ^^E, vF | 64/55 |
| 14 | 275.41 | F | |
| 15 | 295.082 | ^F, vvGb | 13/11 |
| 16 | 314.754 | ^^F, vGb | 6/5, 65/54, 77/64 |
| 17 | 334.426 | ^3F, Gb | 40/33, 63/52 |
| 18 | 354.098 | ^4F, v7G | 16/13 |
| 19 | 373.77 | ^5F, v6G | 26/21 |
| 20 | 393.443 | ^6F, v5G | 5/4, 44/35, 63/50 |
| 21 | 413.115 | ^7F, v4G | 14/11, 33/26, 80/63 |
| 22 | 432.787 | F#, v3G | 50/39 |
| 23 | 452.459 | ^F#, vvG | 13/10 |
| 24 | 472.131 | ^^F#, vG | 21/16, 55/42, 72/55 |
| 25 | 491.803 | G | 4/3 |
| 26 | 511.475 | ^G, vvAb | 35/26 |
| 27 | 531.148 | ^^G, vAb | 65/48 |
| 28 | 550.82 | ^3G, Ab | 11/8, 48/35 |
| 29 | 570.492 | ^4G, v7A | 18/13, 25/18 |
| 30 | 590.164 | ^5G, v6A | 55/39 |
| 31 | 609.836 | ^6G, v5A | 78/55 |
| 32 | 629.508 | ^7G, v4A | 13/9, 36/25 |
| 33 | 649.18 | G#, v3A | 16/11, 35/24 |
| 34 | 668.852 | ^G#, vvA | |
| 35 | 688.525 | ^^G#, vA | 52/35 |
| 36 | 708.197 | A | 3/2 |
| 37 | 727.869 | ^A, vvBb | 32/21, 55/36 |
| 38 | 747.541 | ^^A, vBb | 20/13 |
| 39 | 767.213 | ^3A, Bb | 39/25 |
| 40 | 786.885 | ^4A, v7B | 11/7, 52/33, 63/40 |
| 41 | 806.557 | ^5A, v6B | 8/5, 35/22 |
| 42 | 826.23 | ^6A, v5B | 21/13 |
| 43 | 845.902 | ^7A, v4B | 13/8 |
| 44 | 865.574 | A#, v3B | 33/20 |
| 45 | 885.246 | ^A#, vvB | 5/3 |
| 46 | 904.918 | ^^A#, vB | 22/13 |
| 47 | 924.59 | B | |
| 48 | 944.262 | ^B, vvC | 55/32 |
| 49 | 963.934 | ^^B, vC | 7/4 |
| 50 | 983.607 | C | |
| 51 | 1003.28 | ^C, vvDb | |
| 52 | 1022.95 | ^^C, vDb | 9/5, 65/36 |
| 53 | 1042.62 | ^3C, Db | 11/6, 20/11, 64/35 |
| 54 | 1062.3 | ^4C, v7D | 24/13, 50/27 |
| 55 | 1081.97 | ^5C, v6D | |
| 56 | 1101.64 | ^6C, v5D | 66/35 |
| 57 | 1121.31 | ^7C, v4D | 21/11, 40/21 |
| 58 | 1140.98 | C#, v3D | 64/33 |
| 59 | 1160.66 | ^C#, vvD | 39/20 |
| 60 | 1180.33 | ^^C#, vD | 63/32, 65/33 |
| 61 | 1200 | D | 2/1 |
Miscellany
Mnemonic descriptive poem
These 61 equal divisions of the octave,
though rare are assuredly a ROCK-tave (har har),
while the 3rd and 5th harmonics are about six cents sharp,
(and the flattish 15th poised differently on the harp),
the 7th and 11th err by less, around three,
and thus mayhap, a good orgone tuning found to be;
slightly sharp as well, is the 13th harmonic's place,
but the 9th and 17th lack near so much grace,
interestingly the 19th is good but a couple cents flat,
and the 21st and 23rd are but a cent or two sharp!
—by Peter Kosmorsky