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 major third is nearly pure (stretched only 0.127 cents), 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. 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.
Using the flat fifth instead of the sharp one allows for the 12&35 temperament, which is a kind of bizarre cousin to garibaldi temperament 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 flattone temperament in the 59bc val, a variant of meantone with flat fifths.
59edo is the 17th prime edo.
| 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 (%) | +49 | +1 | +37 | -3 | -11 | -33 | +49 | -16 | +37 | -15 | +11 | +1 | +46 | |
| 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) | |
Intervals
| Steps | Cents | Ups and downs notation (dual flat fifth 34\59) |
Ups and downs notation (dual sharp fifth 35\59) |
Approximate ratios |
|---|---|---|---|---|
| 0 | 0 | D | D | 1/1 |
| 1 | 20.339 | ↑D, E♭♭♭♭ | ↑D, ↓E♭ | 65/64, 78/77 |
| 2 | 40.678 | D♯, ↓E♭♭♭ | ↑↑D, E♭ | 40/39, 49/48, 77/75 |
| 3 | 61.0169 | ↑D♯, E♭♭♭ | ↑3D, ↓8E | 26/25, 33/32, 80/77 |
| 4 | 81.3559 | D𝄪, ↓E♭♭ | ↑4D, ↓7E | |
| 5 | 101.695 | ↑D𝄪, E♭♭ | ↑5D, ↓6E | 35/33, 55/52 |
| 6 | 122.034 | D♯𝄪, ↓E♭ | ↑6D, ↓5E | 15/14 |
| 7 | 142.373 | ↑D♯𝄪, E♭ | ↑7D, ↓4E | 49/45 |
| 8 | 162.712 | D𝄪𝄪, ↓E | ↑8D, ↓3E | 11/10, 35/32 |
| 9 | 183.051 | E | D♯, ↓↓E | 39/35 |
| 10 | 203.39 | ↑E, F♭♭♭ | ↑D♯, ↓E | 28/25, 44/39 |
| 11 | 223.729 | E♯, ↓F♭♭ | E | 8/7, 25/22 |
| 12 | 244.068 | ↑E♯, F♭♭ | ↑E, ↓F | |
| 13 | 264.407 | E𝄪, ↓F♭ | F | 7/6, 64/55 |
| 14 | 284.746 | ↑E𝄪, F♭ | ↑F, ↓G♭ | 13/11, 33/28 |
| 15 | 305.085 | E♯𝄪, ↓F | ↑↑F, G♭ | |
| 16 | 325.424 | F | ↑3F, ↓8G | 40/33, 77/64 |
| 17 | 345.763 | ↑F, G♭♭♭♭ | ↑4F, ↓7G | 39/32, 60/49 |
| 18 | 366.102 | F♯, ↓G♭♭♭ | ↑5F, ↓6G | 16/13 |
| 19 | 386.441 | ↑F♯, G♭♭♭ | ↑6F, ↓5G | 5/4 |
| 20 | 406.78 | F𝄪, ↓G♭♭ | ↑7F, ↓4G | |
| 21 | 427.119 | ↑F𝄪, G♭♭ | ↑8F, ↓3G | 32/25, 50/39, 77/60 |
| 22 | 447.458 | F♯𝄪, ↓G♭ | F♯, ↓↓G | 13/10 |
| 23 | 467.797 | ↑F♯𝄪, G♭ | ↑F♯, ↓G | |
| 24 | 488.136 | F𝄪𝄪, ↓G | G | 33/25 |
| 25 | 508.475 | G | ↑G, ↓A♭ | 66/49, 75/56 |
| 26 | 528.814 | ↑G, A♭♭♭♭ | ↑↑G, A♭ | 49/36 |
| 27 | 549.153 | G♯, ↓A♭♭♭ | ↑3G, ↓8A | 11/8, 48/35 |
| 28 | 569.492 | ↑G♯, A♭♭♭ | ↑4G, ↓7A | 39/28 |
| 29 | 589.831 | G𝄪, ↓A♭♭ | ↑5G, ↓6A | 7/5, 55/39 |
| 30 | 610.169 | ↑G𝄪, A♭♭ | ↑6G, ↓5A | 10/7, 78/55 |
| 31 | 630.508 | G♯𝄪, ↓A♭ | ↑7G, ↓4A | 56/39 |
| 32 | 650.847 | ↑G♯𝄪, A♭ | ↑8G, ↓3A | 16/11, 35/24 |
| 33 | 671.186 | G𝄪𝄪, ↓A | G♯, ↓↓A | 65/44, 72/49 |
| 34 | 691.525 | A | ↑G♯, ↓A | 49/33 |
| 35 | 711.864 | ↑A, B♭♭♭♭ | A | 50/33 |
| 36 | 732.203 | A♯, ↓B♭♭♭ | ↑A, ↓B♭ | 75/49 |
| 37 | 752.542 | ↑A♯, B♭♭♭ | ↑↑A, B♭ | 20/13, 77/50 |
| 38 | 772.881 | A𝄪, ↓B♭♭ | ↑3A, ↓8B | 25/16, 39/25 |
| 39 | 793.22 | ↑A𝄪, B♭♭ | ↑4A, ↓7B | |
| 40 | 813.559 | A♯𝄪, ↓B♭ | ↑5A, ↓6B | 8/5, 77/48 |
| 41 | 833.898 | ↑A♯𝄪, B♭ | ↑6A, ↓5B | 13/8 |
| 42 | 854.237 | A𝄪𝄪, ↓B | ↑7A, ↓4B | 49/30, 64/39 |
| 43 | 874.576 | B | ↑8A, ↓3B | 33/20 |
| 44 | 894.915 | ↑B, C♭♭♭ | A♯, ↓↓B | |
| 45 | 915.254 | B♯, ↓C♭♭ | ↑A♯, ↓B | 22/13, 56/33 |
| 46 | 935.593 | ↑B♯, C♭♭ | B | 12/7, 55/32 |
| 47 | 955.932 | B𝄪, ↓C♭ | ↑B, ↓C | |
| 48 | 976.271 | ↑B𝄪, C♭ | C | 7/4, 44/25 |
| 49 | 996.61 | B♯𝄪, ↓C | ↑C, ↓D♭ | 25/14, 39/22 |
| 50 | 1016.95 | C | ↑↑C, D♭ | 70/39 |
| 51 | 1037.29 | ↑C, D♭♭♭♭ | ↑3C, ↓8D | 20/11, 64/35 |
| 52 | 1057.63 | C♯, ↓D♭♭♭ | ↑4C, ↓7D | |
| 53 | 1077.97 | ↑C♯, D♭♭♭ | ↑5C, ↓6D | 28/15 |
| 54 | 1098.31 | C𝄪, ↓D♭♭ | ↑6C, ↓5D | 66/35 |
| 55 | 1118.64 | ↑C𝄪, D♭♭ | ↑7C, ↓4D | |
| 56 | 1138.98 | C♯𝄪, ↓D♭ | ↑8C, ↓3D | 25/13, 64/33, 77/40 |
| 57 | 1159.32 | ↑C♯𝄪, D♭ | C♯, ↓↓D | 39/20 |
| 58 | 1179.66 | C𝄪𝄪, ↓D | ↑C♯, ↓D | 77/39 |
| 59 | 1200 | D | D | 2/1 |
Instruments
Lumatone
See Lumatone mapping for 59edo