User:Overthink/41edo well temperament
A highly notable tuning system is 41edo, which is distinctly consistent and consistent to distance 2 in the 9-odd-limit, consistent to the 15-odd-limit, and is overall great for its size in all prime limits up to 13. However, the accuracy of some intervals is debateable, so we will construct a well temperament in order to fix this issue.
The chain of fifths
Our well temperament will be based on a chain of fifths, and we will target harmonics 5, 7, 11, and 13. In 41edo, these prime harmonics are mapped as follows (corresponding to andromeda temperament if the fifths are all the same):
| Prime | Fifths down |
|---|---|
| 5 | -8 |
| 7 | -14 |
| 11 | -18 |
| 13 | -21 |
Prime 5 is closest to the root on the circle of fifths; being only 8 fifths down due to 41edo tempering out the schisma. We build our well temperament down the chain of fifths to reach a more accurate prime 5, using slightly flat fifths of 701.792 ¢, or 1/12 schisma flat of just. We continue this chain of schismic fifths down until we reach -12 fifths, or an exact 160/81. We then stack down 12 parapyth fifths of 704.002 ¢ to obtain accurate approximations of harmonics 7, 11, and 13. Finally, the remaining fifths are all 1/12-schisma flat, giving a total of 29 schismic fifths and 12 parapyth fifths.
Intervals
| Steps | Cents (ET) | Cents (WT) | Nearest Ratio |
|---|---|---|---|
| 0 | 0.000 | 0.000 | 1/1 |
| 1 | 29.268 | 21.506 | 81/80 |
| 2 | 58.537 | 58.483 | 33/32 |
| 3 | 87.805 | 91.039 | 135/128 |
| 4 | 117.073 | 112.545 | 16/15 |
| 5 | 146.341 | 138.472 | 13/12 |
| 6 | 175.610 | 182.078 | 10/9 |
| 7 | 204.878 | 203.584 | 9/8 |
| 8 | 234.146 | 225.091 | 256/225 |
| 9 | 263.415 | 266.487 | 7/6 |
| 10 | 292.683 | 294.623 | 32/27 |
| 11 | 321.951 | 316.130 | 6/5 |
| 12 | 351.220 | 346.476 | 11/9 |
| 13 | 380.488 | 385.662 | 5/4 |
| 14 | 409.756 | 407.169 | 81/64 |
| 15 | 439.024 | 428.675 | 32/25 |
| 16 | 468.293 | 474.492 | 21/16 |
| 17 | 497.561 | 498.208 | 4/3 |
| 18 | 526.829 | 519.714 | 27/20 |
| 19 | 556.098 | 554.480 | 11/8 |
| 20 | 585.366 | 589.247 | 45/32 |
| 21 | 614.634 | 610.753 | 64/45 |
| 22 | 643.902 | 634.469 | 13/9 |
| 23 | 673.171 | 680.286 | 40/27 |
| 24 | 702.439 | 701.792 | 3/2 |
| 25 | 731.707 | 723.298 | 243/160 |
| 26 | 760.976 | 762.485 | 14/9 |
| 27 | 790.244 | 792.831 | 128/81 |
| 28 | 819.512 | 814.338 | 8/5 |
| 29 | 848.780 | 842.474 | 13/8 |
| 30 | 878.049 | 883.870 | 5/3 |
| 31 | 907.317 | 905.377 | 27/16 |
| 32 | 936.585 | 926.883 | 128/75 |
| 33 | 965.854 | 970.489 | 7/4 |
| 34 | 995.122 | 996.416 | 16/9 |
| 35 | 1024.390 | 1017.922 | 9/5 |
| 36 | 1053.659 | 1050.478 | 11/6 |
| 37 | 1082.927 | 1087.455 | 15/8 |
| 38 | 1112.195 | 1108.961 | 243/128 |
| 39 | 1141.463 | 1130.467 | 48/25 |
| 40 | 1170.732 | 1178.494 | 160/81 |
| 41 | 1200.000 | 1200.000 | 2/1 |
As you can see, this well temperament approximates many 5-limit ratios of low-to-medium complexity with accuracy, while also having a good approximation of the higher limits.