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 19.7 ¢ each. Each step represents a frequency ratio of 21/61, or the 61st root of 2.
Theory
61edo is only consistent to the 5-odd-limit. Its 3rd and 5th harmonics are sharp of just by more than 6 cents, and the 7th and 11th, though they err by less, are on the flat side. This limits its harmonic inventory. However, it does possess reasonably good approximations of 21/16 and 23/16, only a bit more than one cent off in each case.
As an equal temperament, 61et is characterized by tempering out 20000/19683 (tetracot comma) and 262144/253125 (passion comma) in the 5-limit. In the 7-limit, the patent val ⟨61 97 142 171] supports valentine (15 & 46), and is the optimal patent val for freivald (24 & 37) in the 7-, 11- and 13-limit. The 61d val ⟨61 97 142 172] is a great tuning for modus and quasisuper, and is a simple but out-of-tune edo tuning for parakleismic.
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. 183edo, which triples it, corrects its approximation to many of the lower harmonics.
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 |
Notation
Ups and downs notation
61edo can be notated using ups and downs notation using Helmholtz–Ellis accidentals:
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol | 
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| Flat symbol |  |
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Sagittal notation
This notation uses the same sagittal sequence as 54edo.
Evo flavor
Revo flavor
Evo-SZ flavor
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [97 -61⟩ | [⟨61 97]] | −1.97 | 1.97 | 10.0 |
| 2.3.5 | 20000/19683, 262144/253125 | [⟨61 97 142]] | −2.33 | 1.69 | 8.59 |
| 2.3.5.7 | 64/63, 2430/2401, 3125/3087 | [⟨61 97 142 172]] (61d) | −3.06 | 1.93 | 9.84 |
| 2.3.5.7 | 126/125, 1029/1024, 2240/2187 | [⟨61 97 142 171]] (61) | −1.32 | 2.29 | 11.7 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 1 | 2\61 | 39.3 | 40/39 | Hemivalentine (61) |
| 1 | 3\61 | 59.0 | 28/27 | Dodecacot (61de…) |
| 1 | 4\61 | 78.7 | 22/21 | Valentine (61) |
| 1 | 5\61 | 98.4 | 16/15 | Passion (61de…) / passionate (61) |
| 1 | 7\61 | 137.7 | 13/12 | Quartemka (61) |
| 1 | 9\61 | 177.0 | 10/9 | Modus (61de) / wollemia (61e) |
| 1 | 11\61 | 236.1 | 8/7 | Slendric (61) |
| 1 | 16\61 | 314.8 | 6/5 | Parakleismic (61d) |
| 1 | 23\61 | 452.5 | 13/10 | Maja (61d) |
| 1 | 25\61 | 491.8 | 4/3 | Quasisuper (61d) |
| 1 | 28\61 | 550.8 | 11/8 | Freivald (61) |
* Octave-reduced form, reduced to the first half-octave
Instruments
A Lumatone mapping for 61edo has now been demonstrated (see the Valentine mapping for full gamut coverage).
See also
Introductory poem
Peter Kosmorsky wrote a poem on 61edo; see the 61edo poem.