742edo
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 (%) | +0.0 | -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) | |
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
Template:Comma basis begin |- | 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 Template:Comma basis end
- 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
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf