17edt: Difference between revisions

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== Properties ==

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]].

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 [[11/7]] (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.

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17EDT is the sixth [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|zeta peak tritave division]].

== Discussion ==

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 == Properties ==
-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]].
+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 [[11/7]] (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&amp;b21.
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 EDT is the sixth [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|zeta peak tritave division]].
 {{Harmonics in equal|17|3|1|intervals=prime|columns=15}}
 == Discussion ==