17edt
| ← 16edt | 17edt | 18edt → |
17EDT is equal division of the third harmonic into 17 parts of 111.880 cents each (corresponding to 10.726 EDO).
Properties
In the no-twos subgroup, 17EDT 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 b17&b21.
17EDT is the sixth zeta peak tritave division.
Discussion
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 .15 cents sharp (in addition to equaling 256).
Intervals
| degree | note name | cents value | hekts | cents value octave reduced |
|---|---|---|---|---|
| 0 | C | 0 | ||
| 1 | Db = B# | 111.9 | 76.5 | |
| 2 | Eb = C# | 223.8 | 152.9 | |
| 3 | D | 335.6 | 229.4 | |
| 4 | E | 447.5 | 305.9 | |
| 5 | F = D# | 559.4 | 382.35 | |
| 6 | Gb = E# | 671.3 | 458.8 | |
| 7 | Hb = F# | 783.2 | 535.3 | |
| 8 | G | 895.1 | 611.8 | |
| 9 | H | 1006.9 | 688.2 | |
| 10 | Jb = G# | 1118.8 | 764.7 | |
| 11 | Ab = H# | 1230.7 | 841.2 | 30.7 |
| 12 | J | 1342.6 | 917.65 | 142.6 |
| 13 | A | 1454.4 | 994.1 | 254.4 |
| 14 | Bb = J# | 1566.3 | 1070.6 | 366.3 |
| 15 | Cb = A# | 1678.2 | 1147.1 | 478.2 |
| 16 | B | 1790.1 | 1223.5 | 590.1 |
| 17 | C | 1902 | 1300 | 702 |
| 18 | 2013.8 | 1376.5 | 813.8 | |
| 19 | 2125.7 | 1452.9 | 925.7 | |
| 20 | 2237.6 | 1529.4 | 1037.6 | |
| 21 | 2349.5 | 1605.9 | 1149.5 | |
| 22 | 2461.35 | 1682.35 | 61.35 | |
| 23 | 2573.2 | 1758.8 | 173.2 | |
| 24 | 2685.1 | 1835.3 | 285.1 | |
| 25 | 2797 | 1911.8 | 397 | |
| 26 | 2908.9 | 1988.2 | 508.9 | |
| 27 | 3020.75 | 2064.7 | 620.75 | |
| 28 | 3132.6 | 2141.2 | 732.6 | |
| 29 | 3244.5 | 2217.65 | 844.5 | |
| 30 | 3356.4 | 2294.1 | 956.4 | |
| 31 | 3468.3 | 2370.6 | 1068.3 | |
| 32 | 3580.15 | 2447.1 | 1180.15 | |
| 33 | 3692 | 2523.5 | 92 | |
| 34 | 3803.9 | 2600 | 203.9 | |
- 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).
Regular temperament
17EDT is also be thought of as a generator of the vavoom temperament. As one degree of 17EDT is very close to 16/15, an unnoticeable comma [-68 18 17⟩ is tempered out in the vavoom temperament.
- Vavoom (118&783)
5-limit
Comma: [-68 18 17⟩
Mapping: [⟨1 0 4], ⟨0 17 -18]]
POTE generator: ~16/15 = 111.876
Optimal ET sequence: 11, 32, 43, 75, 118, 429, 547, 665, 783, 901, 1684
Badness: 0.098376
Z function
Below is a plot of the no-twos Z function in the vicinity of 17EDT.
