103edo
| ← 102edo | 103edo | 104edo → |
103 equal divisions of the octave (abbreviated 103edo or 103ed2), also called 103-tone equal temperament (103tet) or 103 equal temperament (103et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 103 equal parts of about 11.7 ¢ each. Each step represents a frequency ratio of 21/103, or the 103rd root of 2.
Theory
In 103edo, all intervals within the 17-odd-limit are consistent, with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%). Its closest zeta peak index, 596zpi, stretches the octave by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.
103edo is a good miracle tuning, especially for the 7-limit, and for benediction and hemisecordite, two of the 13-limit extensions of miracle. It tempers out 78732/78125 in the 5-limit; 225/224, 1029/1024, and 2401/2400 in the 7-limit; 243/242, 441/440, and 540/539 in the 11-limit; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for marvel temperament as well as benediction and hemisecordite.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -2.93 | -1.85 | -1.84 | -3.75 | -1.69 | -0.10 | +5.40 | +0.85 | -4.33 | -3.29 |
| Relative (%) | +0.0 | -25.1 | -15.9 | -15.8 | -32.1 | -14.5 | -0.9 | +46.3 | +7.3 | -37.2 | -28.2 | |
| Steps (reduced) |
103 (0) |
163 (60) |
239 (33) |
289 (83) |
356 (47) |
381 (72) |
421 (9) |
438 (26) |
466 (54) |
500 (88) |
510 (98) | |
Subsets and supersets
103edo is the 27th prime edo, following 101edo and before 107edo.
Intervals
Approximation to JI
Interval mappings
The following tables show how 15-odd-limit intervals are represented in 103edo. Prime harmonics are in bold; inconsistent intervals are in italics.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 7/5, 10/7 | 0.012 | 0.1 |
| 13/7, 14/13 | 0.143 | 1.2 |
| 13/10, 20/13 | 0.155 | 1.3 |
| 11/6, 12/11 | 0.819 | 7.0 |
| 15/11, 22/15 | 1.028 | 8.8 |
| 5/3, 6/5 | 1.078 | 9.3 |
| 7/6, 12/7 | 1.090 | 9.4 |
| 13/12, 24/13 | 1.233 | 10.6 |
| 13/8, 16/13 | 1.693 | 14.5 |
| 7/4, 8/7 | 1.836 | 15.8 |
| 5/4, 8/5 | 1.848 | 15.9 |
| 11/10, 20/11 | 1.897 | 16.3 |
| 11/7, 14/11 | 1.910 | 16.4 |
| 13/11, 22/13 | 2.052 | 17.6 |
| 11/9, 18/11 | 2.107 | 18.1 |
| 3/2, 4/3 | 2.926 | 25.1 |
| 15/14, 28/15 | 2.938 | 25.2 |
| 15/13, 26/15 | 3.081 | 26.4 |
| 11/8, 16/11 | 3.745 | 32.1 |
| 9/5, 10/9 | 4.004 | 34.4 |
| 9/7, 14/9 | 4.016 | 34.5 |
| 13/9, 18/13 | 4.159 | 35.7 |
| 15/8, 16/15 | 4.774 | 41.0 |
| 9/8, 16/9 | 5.799 | 49.8 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 7/5, 10/7 | 0.012 | 0.1 |
| 13/7, 14/13 | 0.143 | 1.2 |
| 13/10, 20/13 | 0.155 | 1.3 |
| 11/6, 12/11 | 0.819 | 7.0 |
| 15/11, 22/15 | 1.028 | 8.8 |
| 5/3, 6/5 | 1.078 | 9.3 |
| 7/6, 12/7 | 1.090 | 9.4 |
| 13/12, 24/13 | 1.233 | 10.6 |
| 13/8, 16/13 | 1.693 | 14.5 |
| 7/4, 8/7 | 1.836 | 15.8 |
| 5/4, 8/5 | 1.848 | 15.9 |
| 11/10, 20/11 | 1.897 | 16.3 |
| 11/7, 14/11 | 1.910 | 16.4 |
| 13/11, 22/13 | 2.052 | 17.6 |
| 11/9, 18/11 | 2.107 | 18.1 |
| 3/2, 4/3 | 2.926 | 25.1 |
| 15/14, 28/15 | 2.938 | 25.2 |
| 15/13, 26/15 | 3.081 | 26.4 |
| 11/8, 16/11 | 3.745 | 32.1 |
| 9/5, 10/9 | 4.004 | 34.4 |
| 9/7, 14/9 | 4.016 | 34.5 |
| 13/9, 18/13 | 4.159 | 35.7 |
| 15/8, 16/15 | 4.774 | 41.0 |
| 9/8, 16/9 | 5.852 | 50.2 |
Zeta peak index
| Tuning | Strength | Octave (cents) | Integer limit | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per 8ve |
Step size (cents) |
Height | Integral | Gap | Size | Stretch | Consistent | Distinct | |
| Tempered | Pure | |||||||||
| 596zpi | 102.93663 | 11.657658 | 8.54351 | 5.620365 | 1.340775 | 18.270998 | 1200.738751 | 0.738751 | 15 | 15 |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-163 103⟩ | [⟨103 166]] | +0.923 | 0.924 | 7.92 |
| 2.3.5 | 78732/78125, 34171875/33554432 | [⟨103 166 239]] | +0.881 | 0.757 | 6.49 |
| 2.3.5.7 | 225/224, 1029/1024, 78732/78125 | [⟨103 166 239 289]] | +0.824 | 0.663 | 5.68 |
| 2.3.5.7.11 | 225/224, 243/242, 385/384, 43923/43750 | [⟨103 166 239 289 356]] | +0.876 | 0.602 | 5.16 |
| 2.3.5.7.11.13 | 225/224, 243/242, 351/350, 385/384, 847/845 | [⟨103 166 239 289 356 381]] | +0.806 | 0.571 | 4.90 |
| 2.3.5.7.11.13.17 | 225/224, 243/242, 273/272, 351/350, 375/374, 847/845 | [⟨103 166 239 289 356 381 421]] | +0.694 | 0.595 | 5.10 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 3\103 | 34.951 | 1990656/1953125 | Gammy |
| 1 | 5\103 | 58.252 | 27/26 | Hemisecordite |
| 1 | 9\103 | 104.854 | 17/16 | Septendesemi |
| 1 | 10\103 | 116.505 | 15/14~16/15 | Miracle / benediction |
| 1 | 16\103 | 186.408 | 10/9 | Mintone |
| 1 | 20\103 | 233.010 | 8/7 | Slendric |
| 1 | 21\103 | 244.660 | 15/13 | Subsemifourth |
| 1 | 26\103 | 303.013 | 25/21 | Quinmite |
| 1 | 31\103 | 361.165 | 16/13 | Phicordial |
| 1 | 37\103 | 431.06 | 77/60 | Lockerbie |
| 1 | 38\103 | 442.708 | 162/125 | Sensei |
| 1 | 39\103 | 454.369 | 13/10 | Fibo |
| 1 | 40\103 | 466.019 | 55/42 | Hemiseptisix |
| 1 | 42\103 | 489.320 | 65/49 | Catafourth |
| 1 | 45\103 | 524.272 | 65/48 | Widefourth |
| 1 | 47\103 | 547.573 | 11/8 | Heinz |
| 1 | 48\103 | 559.223 | 242/175 | Tritriple |
| 1 | 50\103 | 582.524 | 7/5 | Neptune |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct