61edo
| ← 60edo | 61edo | 62edo → |
Theory
61edo provides the optimal patent val for the freivald (24 & 37) temperament in the 7-, 11- and 13-limit.
Introductory 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!
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 (%) | +31.7 | +36.2 | -24.9 | -36.5 | -2.5 | +27.3 | -32.0 | -33.5 | -12.4 | +6.9 | +6.3 | |
| Steps (reduced) |
97 (36) |
142 (20) |
171 (49) |
193 (10) |
211 (28) |
226 (43) |
238 (55) |
249 (5) |
259 (15) |
268 (24) |
276 (32) | |
Subsets and supersets
61edo is the 18th prime edo, after 59edo and before 67edo.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 19.7 | ^D, vvE♭ | |
| 2 | 39.3 | ^^D, vE♭ | |
| 3 | 59 | 29/28, 32/31 | ^3D, E♭ |
| 4 | 78.7 | 22/21, 23/22 | ^4D, ^E♭ |
| 5 | 98.4 | 35/33 | v3D♯, ^^E♭ |
| 6 | 118 | 31/29 | vvD♯, ^3E♭ |
| 7 | 137.7 | 13/12 | vD♯, v4E |
| 8 | 157.4 | 23/21, 34/31, 35/32 | D♯, v3E |
| 9 | 177 | 21/19, 31/28 | ^D♯, vvE |
| 10 | 196.7 | 19/17 | ^^D♯, vE |
| 11 | 216.4 | 26/23 | E |
| 12 | 236.1 | ^E, vvF | |
| 13 | 255.7 | 22/19 | ^^E, vF |
| 14 | 275.4 | 34/29 | F |
| 15 | 295.1 | 19/16 | ^F, vvG♭ |
| 16 | 314.8 | 6/5 | ^^F, vG♭ |
| 17 | 334.4 | 17/14, 23/19 | ^3F, G♭ |
| 18 | 354.1 | ^4F, ^G♭ | |
| 19 | 373.8 | 26/21 | v3F♯, ^^G♭ |
| 20 | 393.4 | vvF♯, ^3G♭ | |
| 21 | 413.1 | 14/11, 33/26 | vF♯, v4G |
| 22 | 432.8 | F♯, v3G | |
| 23 | 452.5 | 13/10 | ^F♯, vvG |
| 24 | 472.1 | 21/16 | ^^F♯, vG |
| 25 | 491.8 | G | |
| 26 | 511.5 | 35/26 | ^G, vvA♭ |
| 27 | 531.1 | 19/14 | ^^G, vA♭ |
| 28 | 550.8 | 11/8 | ^3G, A♭ |
| 29 | 570.5 | 25/18, 32/23 | ^4G, ^A♭ |
| 30 | 590.2 | 31/22 | v3G♯, ^^A♭ |
| 31 | 609.8 | vvG♯, ^3A♭ | |
| 32 | 629.5 | 23/16 | vG♯, v4A |
| 33 | 649.2 | 16/11, 35/24 | G♯, v3A |
| 34 | 668.9 | 28/19 | ^G♯, vvA |
| 35 | 688.5 | ^^G♯, vA | |
| 36 | 708.2 | A | |
| 37 | 727.9 | 29/19, 32/21, 35/23 | ^A, vvB♭ |
| 38 | 747.5 | 20/13 | ^^A, vB♭ |
| 39 | 767.2 | ^3A, B♭ | |
| 40 | 786.9 | 11/7 | ^4A, ^B♭ |
| 41 | 806.6 | 35/22 | v3A♯, ^^B♭ |
| 42 | 826.2 | 21/13 | vvA♯, ^3B♭ |
| 43 | 845.9 | 31/19 | vA♯, v4B |
| 44 | 865.6 | 28/17, 33/20 | A♯, v3B |
| 45 | 885.2 | 5/3 | ^A♯, vvB |
| 46 | 904.9 | 32/19 | ^^A♯, vB |
| 47 | 924.6 | 29/17 | B |
| 48 | 944.3 | 19/11 | ^B, vvC |
| 49 | 963.9 | ^^B, vC | |
| 50 | 983.6 | 23/13 | C |
| 51 | 1003.3 | 34/19 | ^C, vvD♭ |
| 52 | 1023 | ^^C, vD♭ | |
| 53 | 1042.6 | 31/17 | ^3C, D♭ |
| 54 | 1062.3 | 24/13 | ^4C, ^D♭ |
| 55 | 1082 | v3C♯, ^^D♭ | |
| 56 | 1101.6 | vvC♯, ^3D♭ | |
| 57 | 1121.3 | 21/11 | vC♯, v4D |
| 58 | 1141 | 31/16 | C♯, v3D |
| 59 | 1160.7 | ^C♯, vvD | |
| 60 | 1180.3 | ^^C♯, vD | |
| 61 | 1200 | 2/1 | D |