59edo
| ← 58edo | 59edo | 60edo → |
59 equal divisions of the octave (abbreviated 59edo or 59ed2), also called 59-tone equal temperament (59tet) or 59 equal temperament (59et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 59 equal parts of about 20.3 ¢ each. Each step represents a frequency ratio of 21/59, or the 59th root of 2.
Theory
59edo's best fifth is stretched about 9.91 cents from the just interval, and yet its 5/4 is nearly pure (stretched only 0.127 ¢), as the denominator of a convergent to log25. It is a good porcupine tuning, giving in fact the optimal patent val for 11-limit porcupine. This patent val tempers out 250/243 in the 5-limit, 64/63 and 16875/16807 in the 7-limit, and 55/54, 100/99 and 176/175 in the 11-limit.
Using the flat fifth instead of the sharp one allows for the 12 & 35 temperament, which is a kind of bizarre cousin to garibaldi with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for flattertone temperament in the 59bcd val, a variant of meantone with very flat fifths.
As every other step of 118edo, 59edo is an excellent tuning for the 2.9.5.21.11 11-limit 2*59 subgroup, on which it takes the same tuning and tempers out the same commas. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the 50 & 59 temperament with a subminor third generator provides an interesting temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +9.91 | +0.13 | +7.45 | -0.52 | -2.17 | -6.63 | +10.04 | -3.26 | +7.57 | -2.98 | +2.23 | +0.25 | +9.39 |
| Relative (%) | +48.7 | +0.6 | +36.6 | -2.6 | -10.6 | -32.6 | +49.3 | -16.0 | +37.2 | -14.7 | +11.0 | +1.2 | +46.2 | |
| Steps (reduced) |
94 (35) |
137 (19) |
166 (48) |
187 (10) |
204 (27) |
218 (41) |
231 (54) |
241 (5) |
251 (15) |
259 (23) |
267 (31) |
274 (38) |
281 (45) | |
Octave stretch
59edo’s approximations of 3/1, 7/1 and 11/1 are improved by 93edt, a stretched-octave version of 59edo. The trade-off is a slightly worse 2/1 and 5/1. The tunings 203ed11 or 59ed257/128 could also be used as they are near identical to 93edt.
If one prefers compressed octaves, then 166ed7 is a viable option. It improves upon 59edo’s 3/1, 7/1 and 13/1 at the cost of a slightly worse 2/1 and 5/1, but substantially worse 11/1. It is a more accurate choice than 93edt in the 7-limit or no-11s 13-limit due to its better 2/1, but less accurate than 93edt in the 11-limit due to its heavily damaged 11/1.
There are also some nearby Zeta peak index (ZPI) tunings which can be used to improve 59edo’s JI approximations: 293zpi, 294zpi, 295zpi, 296zpi and 297zpi. The main Zeta peak index page details all five tunings.
Subsets and supersets
59edo is the 17th prime edo, following 53edo and before 61edo. As noted above, 118edo is a superset that yields most of the same tuning properties, but it also adds a near-just third harmonic to enable strong full 11-limit tuning.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 34\59) |
Ups and downs notation (Dual sharp fifth 35\59) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 20.3 | ^D, E♭♭♭♭ | ^D, vE♭ | |
| 2 | 40.7 | D♯, vE♭♭♭ | ^^D, E♭ | |
| 3 | 61 | 29/28, 30/29 | ^D♯, E♭♭♭ | ^3D, ^E♭ |
| 4 | 81.4 | 23/22 | D𝄪, vE♭♭ | ^4D, ^^E♭ |
| 5 | 101.7 | 17/16 | ^D𝄪, E♭♭ | v4D♯, ^3E♭ |
| 6 | 122 | 15/14 | D♯𝄪, vE♭ | v3D♯, ^4E♭ |
| 7 | 142.4 | 25/23 | ^D♯𝄪, E♭ | vvD♯, v4E |
| 8 | 162.7 | 11/10, 34/31 | D𝄪𝄪, vE | vD♯, v3E |
| 9 | 183.1 | E | D♯, vvE | |
| 10 | 203.4 | ^E, F♭♭♭ | ^D♯, vE | |
| 11 | 223.7 | 25/22, 33/29 | E♯, vF♭♭ | E |
| 12 | 244.1 | 23/20 | ^E♯, F♭♭ | ^E, vF |
| 13 | 264.4 | 7/6 | E𝄪, vF♭ | F |
| 14 | 284.7 | 13/11, 20/17, 33/28 | ^E𝄪, F♭ | ^F, vG♭ |
| 15 | 305.1 | 31/26 | E♯𝄪, vF | ^^F, G♭ |
| 16 | 325.4 | 29/24 | F | ^3F, ^G♭ |
| 17 | 345.8 | ^F, G♭♭♭♭ | ^4F, ^^G♭ | |
| 18 | 366.1 | F♯, vG♭♭♭ | v4F♯, ^3G♭ | |
| 19 | 386.4 | 5/4 | ^F♯, G♭♭♭ | v3F♯, ^4G♭ |
| 20 | 406.8 | 19/15, 24/19 | F𝄪, vG♭♭ | vvF♯, v4G |
| 21 | 427.1 | 32/25 | ^F𝄪, G♭♭ | vF♯, v3G |
| 22 | 447.5 | 22/17 | F♯𝄪, vG♭ | F♯, vvG |
| 23 | 467.8 | 17/13 | ^F♯𝄪, G♭ | ^F♯, vG |
| 24 | 488.1 | F𝄪𝄪, vG | G | |
| 25 | 508.5 | G | ^G, vA♭ | |
| 26 | 528.8 | 19/14, 34/25 | ^G, A♭♭♭♭ | ^^G, A♭ |
| 27 | 549.2 | 11/8 | G♯, vA♭♭♭ | ^3G, ^A♭ |
| 28 | 569.5 | 32/23 | ^G♯, A♭♭♭ | ^4G, ^^A♭ |
| 29 | 589.8 | 31/22 | G𝄪, vA♭♭ | v4G♯, ^3A♭ |
| 30 | 610.2 | ^G𝄪, A♭♭ | v3G♯, ^4A♭ | |
| 31 | 630.5 | 23/16 | G♯𝄪, vA♭ | vvG♯, v4A |
| 32 | 650.8 | 16/11 | ^G♯𝄪, A♭ | vG♯, v3A |
| 33 | 671.2 | 25/17, 28/19 | G𝄪𝄪, vA | G♯, vvA |
| 34 | 691.5 | A | ^G♯, vA | |
| 35 | 711.9 | ^A, B♭♭♭♭ | A | |
| 36 | 732.2 | 26/17, 29/19 | A♯, vB♭♭♭ | ^A, vB♭ |
| 37 | 752.5 | 17/11 | ^A♯, B♭♭♭ | ^^A, B♭ |
| 38 | 772.9 | 25/16 | A𝄪, vB♭♭ | ^3A, ^B♭ |
| 39 | 793.2 | 19/12, 30/19 | ^A𝄪, B♭♭ | ^4A, ^^B♭ |
| 40 | 813.6 | 8/5 | A♯𝄪, vB♭ | v4A♯, ^3B♭ |
| 41 | 833.9 | ^A♯𝄪, B♭ | v3A♯, ^4B♭ | |
| 42 | 854.2 | A𝄪𝄪, vB | vvA♯, v4B | |
| 43 | 874.6 | B | vA♯, v3B | |
| 44 | 894.9 | ^B, C♭♭♭ | A♯, vvB | |
| 45 | 915.3 | 17/10, 22/13 | B♯, vC♭♭ | ^A♯, vB |
| 46 | 935.6 | 12/7 | ^B♯, C♭♭ | B |
| 47 | 955.9 | 33/19 | B𝄪, vC♭ | ^B, vC |
| 48 | 976.3 | ^B𝄪, C♭ | C | |
| 49 | 996.6 | B♯𝄪, vC | ^C, vD♭ | |
| 50 | 1016.9 | C | ^^C, D♭ | |
| 51 | 1037.3 | 20/11, 31/17 | ^C, D♭♭♭♭ | ^3C, ^D♭ |
| 52 | 1057.6 | C♯, vD♭♭♭ | ^4C, ^^D♭ | |
| 53 | 1078 | 28/15 | ^C♯, D♭♭♭ | v4C♯, ^3D♭ |
| 54 | 1098.3 | 32/17 | C𝄪, vD♭♭ | v3C♯, ^4D♭ |
| 55 | 1118.6 | ^C𝄪, D♭♭ | vvC♯, v4D | |
| 56 | 1139 | 29/15 | C♯𝄪, vD♭ | vC♯, v3D |
| 57 | 1159.3 | ^C♯𝄪, D♭ | C♯, vvD | |
| 58 | 1179.7 | C𝄪𝄪, vD | ^C♯, vD | |
| 59 | 1200 | 2/1 | D | D |
Notation
Sagittal notation
Best fifth notation
This notation uses the same sagittal sequence as 66-EDO.
Evo flavor
Revo 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.
Second-best fifth notation
This notation uses the same sagittal sequence as EDOs 45 and 52.
Evo flavor
Revo flavor
Evo-SZ flavor
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.
Instruments
- Lumatone
See Lumatone mapping for 59edo.
Music
- "too powerful if i had social skills" from Melancholie (2023) – Spotify | Bandcamp | YouTube
- "Stay Away From The Fog" from Void (2025) – Spotify | Bandcamp | YouTube
- Chinchillian Fugue – first mode of the Porcupine[7] scale in 59edo