17edt: Difference between revisions

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== Discussion ==

17EDT is closely related to [[13edt|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).

17EDT is closely related to [[13edt|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 ==

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== 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 {{monzo|-68 18 17}} is tempered out in the vavoom 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 {{monzo|-68 18 17}} is tempered out in the vavoom temperament.

~~'''~~Vavoom (118&783)</font~~>'''<br~~>

'''~~~~5-limit~~~~'''

; Vavoom (118&783)

'''5-limit'''

Comma: {{monzo|-68 18 17}}

Mapping: [{{val|1 0 4}}, {{val|0 17 -18}}]

@@ Line 7: / Line 7: @@
 == Discussion ==
-EDT is closely related to [[13edt|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).
+EDT is closely related to [[13edt|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 ==
@@ Line 231: / Line 231: @@
 == Regular temperament ==
-EDT 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 {{monzo|-68 18 17}} is tempered out in the vavoom temperament.
+EDT 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 {{monzo|-68 18 17}} is tempered out in the vavoom temperament.
-'''<font style="font-size: 1.35em">Vavoom (118&amp;783)</font>'''<br>
-'''<font style="font-size: 1.2em">5-limit</font>'''<br>
+; <font style="font-size:1.15em">Vavoom (118&amp;783)</font>
+'''5-limit'''<br>
 Comma: {{monzo|-68 18 17}}<br>
 Mapping: [{{val|1 0 4}}, {{val|0 17 -18}}]<br>