17edt
Properties[edit]
17edt divides 3, the tritave, into 17 equal parts of 111.880 cents each, corresponding to 10.726 edo. It tempers out 245/243 and 16807/15625 in the 7-limit, 77/75 and 1331/1323 in the 11-limit, and 175/169 and 121/117 in the 13-limit. It supports the no-twos temperament tempering out 245/243 and 77/75, which in terms of tritave patent vals could be written 17&21.
17edt is the sixth zeta peak tritave division.
Discussion[edit]
17edt is closely related to 13edt, the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly in return for gaining a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step. However, by the coincidence of the 11-limit commas 17edt tempers out, 5/3 and 11/9 are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of 55/27 (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents),
leading to an excellent approximation of 11/7 (only .6 cents flat) and these sum to 605/189-1.7 cents, which is also a 16/5 which is only .3 cents flat (in addition to equaling 256).
Intervals[edit]
| degree of 17edt | note name | cents value | cents value octave reduced |
| 0 | C | 0 | |
| 1 | Db = B# | 111.9 | |
| 2 | Eb = C# | 223.8 | |
| 3 | D | 335.6 | |
| 4 | E | 447.5 | |
| 5 | F = D# | 559.4 | |
| 6 | Gb = E# | 671.3 | |
| 7 | Hb = F# | 783.2 | |
| 8 | G | 895.1 | |
| 9 | H | 1006.9 | |
| 10 | Jb = G# | 1118.8 | |
| 11 | Ab = H# | 1230.7 | 30.7 |
| 12 | J | 1342.6 | 142.6 |
| 13 | A | 1454.5 | 254.5 |
| 14 | Bb = J# | 1566.3 | 366.3 |
| 15 | Cb = A# | 1678.2 | 478.2 |
| 16 | B | 1790.1 | 590.1 |
| 17 | C | 1902.0 | 702.0 |
| 18 | 2013.9 | 813.9 | |
| 19 | 2125.8 | 925.8 | |
| 20 | 2237.6 | 1037.6 | |
| 21 | 2349.5 | 1149.5 | |
| 22 | 2461.4 | 61.4 | |
| 23 | 2573.2 | 173.2 | |
| 24 | 2685.2 | 285.2 | |
| 25 | 2797.1 | 397.1 | |
| 26 | 2908.9 | 508.9 | |
| 27 | 3020.8 | 620.8 | |
| 28 | 3132.7 | 732.7 | |
| 29 | 3244.6 | 844.6 | |
| 30 | 3356.5 | 956.5 | |
| 31 | 3468.3 | 1068.3 | |
| 32 | 3580.2 | 1180.2 | |
| 33 | 3692.1 | 92.1 | |
| 34 | 3804.0 | 204.0 |
- Notes named so that C D E F G H J A B C = Lambda mode
It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar and how their approximations of 9/7 are off by such similar amounts in opposite directions (17edo -11.6 cents and 17edt +12.4 cents).
Z function[edit]
Below is a plot of the no-twos Z function in the vicinity of 17edt.
