241edo
| ← 240edo | 241edo | 242edo → |
241 equal divisions of the octave (abbreviated 241edo or 241ed2), also called 241-tone equal temperament (241tet) or 241 equal temperament (241et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 241 equal parts of about 4.98 ¢ each. Each step represents a frequency ratio of 21/241, or the 241st root of 2.
Theory
241edo is distinctly consistent in the 15-odd-limit. It has a sharp tendency, with prime harmonics 3 through 13 all tuned sharp. The equal temperament tempers out 78732/78125 in the 5-limit, 19683/19600 and 3136/3125 in the 7-limit, 540/539, 43923/43904, 65536/65219, and 151263/151250 in the 11-limit, and 351/350, 676/675, 729/728, 1001/1000 and 2080/2079 in the 13-limit. It provides the optimal patent val for subpental.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | +0.12 | +2.07 | +2.13 | +1.38 | +0.97 | -0.39 | +1.24 | -0.89 | +1.13 | +0.19 |
| relative (%) | +0 | +2 | +42 | +43 | +28 | +19 | -8 | +25 | -18 | +23 | +4 | |
| Steps (reduced) |
241 (0) |
382 (141) |
560 (78) |
677 (195) |
834 (111) |
892 (169) |
985 (21) |
1024 (60) |
1090 (126) |
1171 (207) |
1194 (230) | |
Subsets and supersets
241edo is the 53rd prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [382 -241⟩ | [⟨241 382]] | -0.038 | 0.038 | 0.76 |
| 2.3.5 | 78732/78125, [56 -28 -5⟩ | [⟨241 382 560]] | -0.322 | 0.403 | 8.10 |
| 2.3.5.7 | 3136/3125, 19683/19600, 829940/823543 | [⟨241 382 560 677]] | -0.431 | 0.397 | 7.97 |
| 2.3.5.7.11 | 540/539, 3136/3125, 8019/8000, 15488/15435 | [⟨241 382 560 677 834]] | -0.425 | 0.355 | 7.14 |
| 2.3.5.7.11.13 | 351/350, 540/539, 676/675, 3136/3125, 10648/10647 | [⟨241 382 560 677 834 892]] | -0.397 | 0.330 | 6.63 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 20\241 | 99.59 | 200/189 | Quintagar / quinsandric |
| 1 | 50\241 | 248.96 | [-26 18 -1⟩ | Monzismic |
| 1 | 76\241 | 378.42 | 56/45 | Subpental |
| 1 | 89\241 | 443.15 | 162/125 | Sensipent |
| 1 | 100\241 | 497.93 | 4/3 | Gary |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct