59edo
| ← 58edo | 59edo | 60edo → |
59 equal divisions of the octave (59edo), or 59-tone equal temperament (59tet), 59 equal temperament (59et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 59 equal parts of about 20.3 ¢ each.
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, 55/54, 64/63 |
| 1 | 20.339 | ^D, Ebbbb | ^D, vEb | 50/49, 65/63, 65/64, 78/77 |
| 2 | 40.678 | D#, vEbbb | ^^D, Eb | 28/27, 40/39, 49/48, 56/55, 66/65, 77/75 |
| 3 | 61.0169 | ^D#, Ebbb | ^3D, v8E | 22/21, 25/24, 26/25, 33/32, 36/35, 45/44, 80/77, 81/80 |
| 4 | 81.3559 | Dx, vEbb | ^4D, v7E | 52/49 |
| 5 | 101.695 | ^Dx, Ebb | ^5D, v6E | 16/15, 21/20, 27/26, 35/33, 55/52, 77/72 |
| 6 | 122.034 | D#x, vEb | ^6D, v5E | 13/12, 15/14, 81/77 |
| 7 | 142.373 | ^D#x, Eb | ^7D, v4E | 14/13, 49/45 |
| 8 | 162.712 | Dxx, vE | ^8D, v3E | 10/9, 11/10, 12/11, 27/25, 35/32 |
| 9 | 183.051 | E | D#, vvE | 39/35, 54/49, 55/49 |
| 10 | 203.39 | ^E, Fbbb | ^D#, vE | 28/25, 44/39, 49/44, 72/65 |
| 11 | 223.729 | E#, vFbb | E | 8/7, 9/8, 25/22, 52/45, 55/48 |
| 12 | 244.068 | ^E#, Fbb | ^E, vF | 65/56 |
| 13 | 264.407 | Ex, vFb | F | 7/6, 15/13, 32/27, 63/55, 64/55 |
| 14 | 284.746 | ^Ex, Fb | ^F, vGb | 13/11, 25/21, 33/28, 65/54, 75/64, 81/70 |
| 15 | 305.085 | E#x, vF | ^^F, Gb | 77/65 |
| 16 | 325.424 | F | ^3F, v8G | 6/5, 11/9, 40/33, 77/64 |
| 17 | 345.763 | ^F, Gbbbb | ^4F, v7G | 26/21, 39/32, 60/49 |
| 18 | 366.102 | F#, vGbbb | ^5F, v6G | 16/13, 49/40, 56/45, 63/52 |
| 19 | 386.441 | ^F#, Gbbb | ^6F, v5G | 5/4, 27/22, 44/35, 80/63 |
| 20 | 406.78 | Fx, vGbb | ^7F, v4G | 49/39 |
| 21 | 427.119 | ^Fx, Gbb | ^8F, v3G | 14/11, 32/25, 33/26, 35/27, 50/39, 63/50, 77/60 |
| 22 | 447.458 | F#x, vGb | F#, vvG | 9/7, 13/10, 55/42, 64/49, 81/64 |
| 23 | 467.797 | ^F#x, Gb | ^F#, vG | 65/49 |
| 24 | 488.136 | Fxx, vG | G | 4/3, 21/16, 33/25, 72/55 |
| 25 | 508.475 | G | ^G, vAb | 65/48, 66/49, 75/56 |
| 26 | 528.814 | ^G, Abbbb | ^^G, Ab | 35/26, 49/36 |
| 27 | 549.153 | G#, vAbbb | ^3G, v8A | 11/8, 15/11, 25/18, 27/20, 48/35 |
| 28 | 569.492 | ^G#, Abbb | ^4G, v7A | 39/28 |
| 29 | 589.831 | Gx, vAbb | ^5G, v6A | 7/5, 18/13, 55/39, 64/45, 77/54 |
| 30 | 610.169 | ^Gx, Abb | ^6G, v5A | 10/7, 13/9, 45/32, 78/55 |
| 31 | 630.508 | G#x, vAb | ^7G, v4A | 56/39 |
| 32 | 650.847 | ^G#x, Ab | ^8G, v3A | 16/11, 22/15, 35/24, 36/25, 40/27, 63/44, 75/52 |
| 33 | 671.186 | Gxx, vA | G#, vvA | 52/35, 65/44, 72/49, 81/56 |
| 34 | 691.525 | A | ^G#, vA | 49/33, 77/52 |
| 35 | 711.864 | ^A, Bbbbb | A | 3/2, 32/21, 50/33, 55/36 |
| 36 | 732.203 | A#, vBbbb | ^A, vBb | 65/42, 75/49 |
| 37 | 752.542 | ^A#, Bbbb | ^^A, Bb | 14/9, 20/13, 49/32, 77/50 |
| 38 | 772.881 | Ax, vBbb | ^3A, v8B | 11/7, 25/16, 39/25, 52/33, 54/35 |
| 39 | 793.22 | ^Ax, Bbb | ^4A, v7B | 78/49 |
| 40 | 813.559 | A#x, vBb | ^5A, v6B | 8/5, 35/22, 44/27, 63/40, 77/48 |
| 41 | 833.898 | ^A#x, Bb | ^6A, v5B | 13/8, 45/28, 80/49 |
| 42 | 854.237 | Axx, vB | ^7A, v4B | 21/13, 49/30, 64/39 |
| 43 | 874.576 | B | ^8A, v3B | 5/3, 18/11, 33/20, 81/50 |
| 44 | 894.915 | ^B, Cbbb | A#, vvB | 81/49 |
| 45 | 915.254 | B#, vCbb | ^A#, vB | 22/13, 42/25, 56/33, 77/45 |
| 46 | 935.593 | ^B#, Cbb | B | 12/7, 26/15, 27/16, 55/32, 75/44 |
| 47 | 955.932 | Bx, vCb | ^B, vC | |
| 48 | 976.271 | ^Bx, Cb | C | 7/4, 16/9, 44/25, 45/26 |
| 49 | 996.61 | B#x, vC | ^C, vDb | 25/14, 39/22, 65/36 |
| 50 | 1016.95 | C | ^^C, Db | 49/27, 70/39 |
| 51 | 1037.29 | ^C, Dbbbb | ^3C, v8D | 9/5, 11/6, 20/11, 50/27, 64/35 |
| 52 | 1057.63 | C#, vDbbb | ^4C, v7D | 13/7 |
| 53 | 1077.97 | ^C#, Dbbb | ^5C, v6D | 24/13, 28/15 |
| 54 | 1098.31 | Cx, vDbb | ^6C, v5D | 15/8, 40/21, 52/27, 66/35 |
| 55 | 1118.64 | ^Cx, Dbb | ^7C, v4D | 49/26 |
| 56 | 1138.98 | C#x, vDb | ^8C, v3D | 21/11, 25/13, 35/18, 48/25, 64/33, 77/40 |
| 57 | 1159.32 | ^C#x, Db | C#, vvD | 27/14, 39/20, 55/28, 65/33 |
| 58 | 1179.66 | Cxx, vD | ^C#, vD | 49/25, 77/39 |
| 59 | 1200 | D | D | 2/1, 55/27, 63/32 |