742edo: Difference between revisions
| Line 45: | Line 45: | ||
|- | |- | ||
|14 | |14 | ||
|434\ | |434\74<br>(10\742) | ||
|701.886 (16.173) | |701.886<br>(16.173) | ||
|3/2 (105/104) | |3/2<br>(105/104) | ||
|Silicon | |Silicon | ||
|- | |- | ||
|53 | |53 | ||
|565\742 (5\742) | |565\742<br>(5\742) | ||
|913.746 (8.086) | |913.746<br>(8.086) | ||
|441/260 | |441/260 | ||
|Iodine | |Iodine | ||
|}<!-- 3-digit number --> | |}<!-- 3-digit number --> | ||
[[Category:Zeta]] | [[Category:Zeta]] | ||
Revision as of 13:03, 6 September 2022
| ← 741edo | 742edo | 743edo → |
Theory
742edo is a very strong 19-limit system and a zeta peak tuning, and is uniquely consistent in the 21-odd-limit. It has a lower 19-limit relative error than any edo until 1178. It tempers out 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.
742 = 2 × 7 × 53, so it notably contains 53edo and 14edo. It supports silicon temperament (224 & 742) with period 14 in the 13-limit, and iodine temperament (159 & 742) with period 53 in the 17-limit.
In 5-limit, 742edo supports vishnu and fortune.
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 (%) | +0.0 | -4.2 | +12.9 | -5.7 | +10.2 | +27.4 | +10.3 | +3.8 | -48.3 | +37.8 | -1.4 | |
| Steps (reduced) |
742 (0) |
1176 (434) |
1723 (239) |
2083 (599) |
2567 (341) |
2746 (520) |
3033 (65) |
3152 (184) |
3356 (388) |
3605 (637) |
3676 (708) | |
Rank-two temperaments by generator
| Periods
per octave |
Generator
(reduced) |
Cents
(reduced) |
Associated
ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 137\742 | 221.563 | 8388608/7381125 | Fortune |
| 2 | 44\742 | 71.159 | 25/24 | Vishnu |
| 14 | 434\74 (10\742) |
701.886 (16.173) |
3/2 (105/104) |
Silicon |
| 53 | 565\742 (5\742) |
913.746 (8.086) |
441/260 | Iodine |