17edo: Difference between revisions

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== Theory ==

17edo can plausibly be treated as a 2.3.25.7.11.13.23 subgroup temperament, for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because ~~the 3, 7, 11, and 13~~ are all sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]].

17edo's perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to an expressive [[diatonic scale]]. Meanwhile, it approximates [[harmonic]]s [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] to reasonable degrees, despite completely missing harmonic [[5/1|5]]. Thus it can plausibly be treated as a 2.3.25.7.11.13.23 [[subgroup temperament]], for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because these harmonics are all tempered sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]].

As a no-fives system, it is best used with timbres in which harmonic multiples of 5 are attenuated or absent. Also, the standard major triad is quite dissonant as the major third is closer to 9/7 than the traditional 5/4.

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=== Subsets and supersets ===

17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. [[34edo]], which doubles it, provides a good correction to the harmonic 5.

== Intervals ==

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 == Theory ==
-edo can plausibly be treated as a 2.3.25.7.11.13.23 subgroup temperament, for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because the 3, 7, 11, and 13 are all sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]].
+edo's perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to an expressive [[diatonic scale]]. Meanwhile, it approximates [[harmonic]]s [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] to reasonable degrees, despite completely missing harmonic [[5/1|5]]. Thus it can plausibly be treated as a 2.3.25.7.11.13.23 [[subgroup temperament]], for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because these harmonics are all tempered sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]].
 As a no-fives system, it is best used with timbres in which harmonic multiples of 5 are attenuated or absent. Also, the standard major triad is quite dissonant as the major third is closer to 9/7 than the traditional 5/4.
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 === Subsets and supersets ===
 edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. [[34edo]], which doubles it, provides a good correction to the harmonic 5.
 == Intervals ==