742edo: Difference between revisions
i'll fill the optimal stretch and stuff later once i have time |
|||
| Line 32: | Line 32: | ||
|6115295232/6103515625, {{monzo|-61 59 -14}} | |6115295232/6103515625, {{monzo|-61 59 -14}} | ||
|[{{val|742 1176 1723}}] | |[{{val|742 1176 1723}}] | ||
| | |<nowiki>-0.015690</nowiki> | ||
| | |0.055470 | ||
| | |3.43 | ||
|- | |- | ||
|2.3.5.7 | |2.3.5.7 | ||
|2401/2400, 14348907/14336000, 6115295232/6103515625 | |2401/2400, 14348907/14336000, 6115295232/6103515625 | ||
|[{{val|742 1176 1723 2083}}] | |[{{val|742 1176 1723 2083}}] | ||
| | | -0.003508 | ||
| | |0.052468 | ||
| | |3.24 | ||
|- | |- | ||
|2.3.5.7.11 | |2.3.5.7.11 | ||
|9801/9800, 2401/2400, 172032/171875, 1240029/1239040 | |9801/9800, 2401/2400, 172032/171875, 1240029/1239040 | ||
|[{{val|742 1176 1723 2083 2567}}] | |[{{val|742 1176 1723 2083 2567}}] | ||
| | | -0.012319 | ||
| | |0.050128 | ||
| | |3.10 | ||
|- | |- | ||
|2.3.5.7.11.13 | |2.3.5.7.11.13 | ||
|4096/4095, 9801/9800, 2401/2400, 67392/67375, 59535/59488 | |4096/4095, 9801/9800, 2401/2400, 67392/67375, 59535/59488 | ||
|[{{val|742 1176 1723 2083 2567 2746}}] | |[{{val|742 1176 1723 2083 2567 2746}}] | ||
| | | -0.030205 | ||
| | |0.060773 | ||
| | |3.76 | ||
|- | |||
|2.3.5.7.11.13.17 | |||
|1701/1700, 2058/2057, 2601/2600, 8624/8619, 12376/12375, 14400/14399 | |||
|[{{val|742 1176 1723 2083 2567 2746 3033}}] | |||
|<nowiki>-0.031687</nowiki> | |||
|0.056382 | |||
|3.49 | |||
|} | |} | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
Revision as of 20:20, 12 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 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.
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.
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) | |
Regular temperament properties
742et has a lower 19-limit relative error than any previous equal temperaments. It is only bettered by 1178et.
| Subgroup | Comma list | Mapping | Optimal
8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 6115295232/6103515625, [-61 59 -14⟩ | [⟨742 1176 1723]] | -0.015690 | 0.055470 | 3.43 |
| 2.3.5.7 | 2401/2400, 14348907/14336000, 6115295232/6103515625 | [⟨742 1176 1723 2083]] | -0.003508 | 0.052468 | 3.24 |
| 2.3.5.7.11 | 9801/9800, 2401/2400, 172032/171875, 1240029/1239040 | [⟨742 1176 1723 2083 2567]] | -0.012319 | 0.050128 | 3.10 |
| 2.3.5.7.11.13 | 4096/4095, 9801/9800, 2401/2400, 67392/67375, 59535/59488 | [⟨742 1176 1723 2083 2567 2746]] | -0.030205 | 0.060773 | 3.76 |
| 2.3.5.7.11.13.17 | 1701/1700, 2058/2057, 2601/2600, 8624/8619, 12376/12375, 14400/14399 | [⟨742 1176 1723 2083 2567 2746 3033]] | -0.031687 | 0.056382 | 3.49 |
Rank-2 temperaments
| 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\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 |