742edo
| ← 741edo | 742edo | 743edo → |
742 equal divisions of the octave (abbreviated 742edo or 742ed2), also called 742-tone equal temperament (742tet) or 742 equal temperament (742et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 742 equal parts of about 1.617 ¢ each. Each step represents a frequency ratio of 21/742, or the 742nd root of 2.
Theory
742edo is a very strong 19-limit system and a zeta peak edo, and is distinctly consistent in the 21-odd-limit. The equal temperament tempers out the vishnuzma and the fortune comma in the 5-limit, supporting vishnu and fortune; 2401/2400 in the 7-limit, 9801/9800 in the 11-limit, 4096/4095, 6656/6655, 10648/10647 in the 13-limit, 1701/1700, 2058/2057, 2601/2600, 4914/4913, 5832/5831 in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.000 | -0.068 | +0.209 | -0.093 | +0.165 | +0.443 | +0.166 | +0.061 | -0.781 | +0.611 | -0.022 |
| relative (%) | +0 | -4 | +13 | -6 | +10 | +27 | +10 | +4 | -48 | +38 | -1 | |
| Steps (reduced) |
742 (0) |
1176 (434) |
1723 (239) |
2083 (599) |
2567 (341) |
2746 (520) |
3033 (65) |
3152 (184) |
3356 (388) |
3605 (637) |
3676 (708) | |
Subsets and supersets
Since 742 factors into 2 × 7 × 53, 742edo has subset edos 2, 7, 14, 53, 106, and 371, of which 7edo, 14edo and 53edo are very notable. It supports silicon (224 & 518) with period 14 in the 13-limit, and iodine (159 & 583f) with period 53 in the 17-limit.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [23 6 -14⟩, [-84 53⟩ | [⟨742 1176 1723]] | -0.0157 | 0.0555 | 3.43 |
| 2.3.5.7 | 2401/2400, 14348907/14336000, [23 6 -14⟩ | [⟨742 1176 1723 2083]] | -0.0035 | 0.0525 | 3.24 |
| 2.3.5.7.11 | 2401/2400, 9801/9800, 172032/171875, 1240029/1239040 | [⟨742 1176 1723 2083 2567]] | -0.0123 | 0.0501 | 3.10 |
| 2.3.5.7.11.13 | 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325 | [⟨742 1176 1723 2083 2567 2746]] | -0.0302 | 0.0608 | 3.76 |
| 2.3.5.7.11.13.17 | 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655 | [⟨742 1176 1723 2083 2567 2746 3033]] | -0.0317 | 0.0564 | 3.49 |
| 2.3.5.7.11.13.17.19 | 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211 | [⟨742 1176 1723 2083 2567 2746 3033 3152]] | -0.0295 | 0.0531 | 3.28 |
| 2.3.5.7.11.13.17.19.23 | 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211 | [⟨742 1176 1723 2083 2567 2746 3033 3152 3357]] (742i) | -0.0468 | 0.0699 | 4.32 |
- 742et has a lower 19-limit relative error than any previous equal temperaments. It is only bettered by 935 in terms of absolute error, and by 1178 in terms of relative error.
- 742et (742i val) is also notable in the 17- and 23-limit, where it has lower absolute errors than any previous equal temperaments. In the 17-limit it beats 581 and is bettered by 764; in the 23-limit it beats 718 and is bettered by 814.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 137\742 | 221.563 | 8388608/7381125 | Fortune |
| 1 | 303\742 | 490.026 | 896/675 | Surmarvelpyth |
| 2 | 44\742 | 71.159 | 25/24 | Vishnu |
| 14 | 434\742 (10\742) |
701.886 (16.173) |
3/2 (105/104) |
Silicon |
| 53 | 239\742 (1\742) |
386.523 (1.617) |
5/4 (32805/32768) |
Mercator |
| 53 | 565\742 (5\742) |
913.746 (8.086) |
441/260 (196/195) |
Iodine |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct