17edt: Difference between revisions
No edit summary |
No edit summary |
||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
'''17EDT''' is [[EDT|equal division of the third harmonic]] into 17 parts of 111.880 cents each (corresponding to 10.726 [[EDO]]). | '''17EDT''' is the [[EDT|equal division of the third harmonic]] into 17 parts of 111.880 cents each (corresponding to 10.726 [[EDO]]). | ||
== Properties == | == Properties == | ||
In the no-twos subgroup, | Following [[13edt]], 17edt is the first EDT that can reasonably be described as a [[3.5.7 subgroup]] temperament, though one that sacrifices much accuracy compared to 13edt, compensating for that by representing primes 11 and 17. 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 [[1/7]] (only .6 cents flat). 17edt's step is also, notably, only 0.15 cents sharp of [[16/15]]. | ||
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 [[support]]s 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 [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|zeta peak tritave division]]. | 17EDT is the sixth [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|zeta peak tritave division]]. | ||
| Line 10: | Line 12: | ||
== Discussion == | == Discussion == | ||
17edt is closely related to [[13edt]], the Bohlen-Pierce division, because they share the feature of tempering out 245/243, so that the interval of [[9/7]] stacked twice results in [[5/3]]. Therefore, like 13edt, 17edt's 9/7 generates an enneatonic Lambda ([[4L 5s (3/1-equivalent)|4L 5s]]) scale. If 13edt can be considered an analogue of [[12edo]] as the basic tuning of this scale, 17edt is an analogue of [[17edo]] as the hard 3:1 tuning. While the approximation of 5/3 and 7/3 is less good than that of 13edt, this scale has 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, as it is equated to [[25/21]] by virtue of tempering out 77/75. | |||
17edt is also very notable in that its generator spans 4 steps, meaning that it is divisible into 2; and the interval [[27/7]] spans 21 steps, meaning it is divisible into 3. These lead to weak extensions of [[BPS]] (the temperament defined in 3.5.7 by tempering out 245/243) known as [[Dubhe]] (which splits the 9/7 generator into two intervals of [[17/15]], tempering out [[2025/2023]] as the additional comma in 3.5.7.17), and [[Mintra]] (which splits 27/7 into three intervals of 11/7, tempering out [[1331/1323]] as the additional comma in 3.5.7.11), respectively. 17edt in fact supports basic tunings of Dubhe[9] (which is [[8L 1s (3/1-equivalent)|8L 1s]]) and Mintaka[12] (which is [[5L 7s (3/1-equivalent)|5L 7s]], i.e. a macro-chromatic scale). Therefore 17edt is important as the smallest nontrivial tuning to support each, and it is remarkable for providing such an efficient intersection of temperaments in the 3.5.7.11.17 subgroup, despite being an extremal tuning of most of these and losing much accuracy compared to more optimal tunings. | |||
== Intervals == | == Intervals == | ||