217edo
| ← 216edo | 217edo | 218edo → |
217 equal divisions of the octave (abbreviated 217edo or 217ed2), also called 217-tone equal temperament (217tet) or 217 equal temperament (217et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 217 equal parts of about 5.53 ¢ each. Each step represents a frequency ratio of 21/217, or the 217th root of 2.
Theory
217edo is a strong 19-limit system, the smallest distinctly consistent in the 19-odd-limit and consistent to the 21-odd-limit as well as the no-23 31-odd-limit. It shares the same 5th and 7th harmonics with 31edo (217 = 7 × 31), as well as the 11/9 interval (supporting the birds temperament). However, compared to 31edo, its patent val differ on the mappings for 3, 11, 13, 17 and 19 – in fact, this edo has a very accurate 13th harmonic, as well as the 19/15 interval. It can also be used in the 23-limit. The only inconsistently mapped intervals in the 23-odd-limit are 23/14, 23/21, and their octave complements.
The equal temperament tempers out the parakleisma, [8 14 -13⟩, and the escapade comma, [32 -7 -9⟩ in the 5-limit; 3136/3125, 4375/4374, 10976/10935 and 823543/819200 in the 7-limit; 441/440, 4000/3993, 5632/5625, and 16384/16335 in the 11-limit; 364/363, 676/675, 1001/1000, 1575/1573, 2080/2079 and 4096/4095 in the 13-limit; 595/594, 833/832, 936/935, 1156/1155, 1225/1224, 1701/1700 in the 17-limit; 343/342, 476/475, 969/968, 1216/1215, 1445/1444, 1521/1520 and 1540/1539 in the 19-limit. It allows minor minthmic chords, werckismic chords, and sinbadmic chords in the 13-odd-limit, in addition to island chords and nicolic chords in the 15-odd-limit. It provides the optimal patent val for the 11- and 13-limit arch and the 11- and 13-limit cotoneum.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | +0.35 | +0.78 | -1.08 | +1.68 | +0.03 | +0.11 | +1.10 | +2.14 | -1.01 | -0.34 |
| relative (%) | +0 | +6 | +14 | -20 | +30 | +0 | +2 | +20 | +39 | -18 | -6 | |
| Steps (reduced) |
217 (0) |
344 (127) |
504 (70) |
609 (175) |
751 (100) |
803 (152) |
887 (19) |
922 (54) |
982 (114) |
1054 (186) |
1075 (207) | |
Approximation to JI
Selected just intervals
The following table shows how 23-odd-limit intervals are represented in 217edo. Prime harmonics are in bold; inconsistent intervals are in italic.
| Interval, complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 13/8, 16/13 | 0.025 | 0.5 |
| 19/15, 30/19 | 0.028 | 0.5 |
| 9/5, 10/9 | 0.085 | 1.5 |
| 17/13, 26/17 | 0.088 | 1.6 |
| 17/16, 32/17 | 0.114 | 2.1 |
| 17/12, 24/17 | 0.235 | 4.3 |
| 19/10, 20/19 | 0.321 | 5.8 |
| 13/12, 24/13 | 0.324 | 5.9 |
| 3/2, 4/3 | 0.349 | 6.3 |
| 19/18, 36/19 | 0.406 | 7.3 |
| 5/3, 6/5 | 0.434 | 7.8 |
| 23/22, 44/23 | 0.463 | 8.4 |
| 15/11, 22/15 | 0.545 | 9.9 |
| 19/11, 22/19 | 0.573 | 10.4 |
| 17/9, 18/17 | 0.585 | 10.6 |
| 17/10, 20/17 | 0.669 | 12.1 |
| 13/9, 18/13 | 0.673 | 12.2 |
| 9/8, 16/9 | 0.698 | 12.6 |
| 21/16, 32/21 | 0.735 | 13.3 |
| 19/12, 24/19 | 0.755 | 13.7 |
| 13/10, 20/13 | 0.758 | 13.7 |
| 21/13, 26/21 | 0.760 | 13.7 |
| 5/4, 8/5 | 0.783 | 14.2 |
| 21/17, 34/21 | 0.849 | 15.3 |
| 11/10, 20/11 | 0.894 | 16.2 |
| 11/9, 18/11 | 0.979 | 17.7 |
| 19/17, 34/19 | 0.991 | 17.9 |
| 23/15, 30/23 | 1.008 | 18.2 |
| 17/15, 30/17 | 1.018 | 18.4 |
| 23/19, 38/23 | 1.036 | 18.7 |
| 19/13, 26/19 | 1.079 | 19.5 |
| 7/4, 8/7 | 1.084 | 19.6 |
| 19/16, 32/19 | 1.104 | 20.0 |
| 15/13, 26/15 | 1.107 | 20.0 |
| 13/7, 14/13 | 1.109 | 20.1 |
| 15/8, 16/15 | 1.132 | 20.5 |
| 17/14, 28/17 | 1.198 | 21.7 |
| 11/6, 12/11 | 1.328 | 24.0 |
| 23/20, 40/23 | 1.357 | 24.5 |
| 7/6, 12/7 | 1.433 | 25.9 |
| 23/18, 36/23 | 1.442 | 26.1 |
| 21/20, 40/21 | 1.518 | 27.4 |
| 17/11, 22/17 | 1.564 | 28.3 |
| 13/11, 22/13 | 1.652 | 29.9 |
| 11/8, 16/11 | 1.677 | 30.3 |
| 9/7, 14/9 | 1.782 | 32.2 |
| 23/12, 24/23 | 1.791 | 32.4 |
| 21/19, 38/21 | 1.839 | 33.3 |
| 7/5, 10/7 | 1.867 | 33.8 |
| 23/17, 34/23 | 2.027 | 36.6 |
| 23/13, 26/23 | 2.115 | 38.2 |
| 23/16, 32/23 | 2.140 | 38.7 |
| 19/14, 28/19 | 2.188 | 39.6 |
| 15/14, 28/15 | 2.216 | 40.1 |
| 21/11, 22/21 | 2.412 | 43.6 |
| 11/7, 14/11 | 2.761 | 49.9 |
| 23/21, 42/23 | 2.875 | 52.0 |
| 23/14, 28/23 | 3.224 | 58.3 |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [344 -217⟩ | [⟨217 344]] | -0.110 | 0.1101 | 1.99 |
| 2.3.5 | [8 14 -13⟩, [32 -7 -9⟩ | [⟨217 344 504]] | -0.186 | 0.1398 | 2.53 |
| 2.3.5.7 | 3136/3125, 4375/4374, 823543/819200 | [⟨217 344 504 609]] | -0.043 | 0.2757 | 4.99 |
| 2.3.5.7.11 | 441/440, 3136/3125, 4000/3993, 4375/4374 | [⟨217 344 504 609 751]] | -0.131 | 0.3034 | 5.49 |
| 2.3.5.7.11.13 | 364/363, 441/440, 676/675, 3136/3125, 4375/4374 | [⟨217 344 504 609 751 803]] | -0.111 | 0.2808 | 5.08 |
| 2.3.5.7.11.13.17 | 364/363, 441/440, 595/594, 676/675, 1156/1155, 3136/3125 | [⟨217 344 504 609 751 803 887]] | -0.099 | 0.2616 | 4.73 |
| 2.3.5.7.11.13.17.19 | 343/342, 364/363, 441/440, 476/475, 595/594, 676/675, 1216/1215 | [⟨217 344 504 609 751 803 887 922]] | -0.119 | 0.2504 | 4.53 |
| 2.3.5.7.11.13.17.19.23 | 343/342, 364/363, 392/391, 441/440, 476/475, 507/506, 595/594, 676/675 | [⟨217 344 504 609 751 803 887 922 982]] | -0.158 | 0.2610 | 4.72 |
- 217et has lower relative errors than any previous equal temperaments in the 19- and 23-limit. It is the first to beat 72 in the 19-limit and 193 in the 23-limit. The next equal temperament that does better in either subgroup is 243e for absolute error and 270 for relative error.
- 23-limit is not the subgroup it does the best, with the no-23 29- and 31-limit approximated even better.
- It is also prominent in the 17-limit, with a lower absolute error than any previous equal temperaments, beating 183 and superseded by 224.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 3\217 | 16.59 | 100/99 | Quincy |
| 1 | 5\217 | 27.65 | 64/63 | Arch |
| 1 | 9\217 | 49.77 | 36/35 | Hemiquindromeda |
| 1 | 10\217 | 55.30 | 16875/16384 | Escapade |
| 1 | 18\217 | 99.54 | 18/17 | Quintagar / quintoneum / quinsandra |
| 1 | 30\217 | 165.90 | 11/10 | Satin |
| 1 | 33\217 | 182.49 | 10/9 | Mitonic / mineral |
| 1 | 57\217 | 315.21 | 6/5 | Parakleismic / paralytic |
| 1 | 86\217 | 475.58 | 320/243 | Vulture |
| 1 | 90\217 | 497.70 | 4/3 | Cotoneum |
| 1 | 101\217 | 558.53 | 112/81 | Condor |
| 7 | 94\217 (1\217) |
519.82 (5.53) |
27/20 (325/324) |
Brahmagupta |
| 31 | 90\217 (1\217) |
497.70 (5.53) |
4/3 (243/242) |
Birds |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct