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.
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