17edo: Difference between revisions

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== Theory ==

17edo's [[3/2|perfect fifth]] is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to a [[5L 2s|diatonic]] scale with more constrasting large and small steps, so it can be seen as a tuning that emphasizes the hardness of [[Pythagorean tuning]] rather than mellowing it out as in [[meantone]]. 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 temperament of the 2.3.25.7.11.13.23 [[subgroup]], for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers).

17edo is the next smallest edo to have a [[5L 2s|diatonic]] [[3/2|fifth]] after [[12edo]], and is quite popular for that reason. 17edo's [[3/2|perfect fifth]] is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to a [[5L 2s|diatonic]] scale with more constrasting large and small steps, so it can be seen as a tuning that emphasizes the hardness of [[Pythagorean tuning]] rather than mellowing it out as in [[meantone]]. 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 temperament of the 2.3.25.7.11.13.23 [[subgroup]], 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 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.

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 == Theory ==
-edo's [[3/2|perfect fifth]] is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to a [[5L 2s|diatonic]] scale with more constrasting large and small steps, so it can be seen as a tuning that emphasizes the hardness of [[Pythagorean tuning]] rather than mellowing it out as in [[meantone]]. 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 temperament of the 2.3.25.7.11.13.23 [[subgroup]], for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers).
+edo is the next smallest edo to have a [[5L 2s|diatonic]] [[3/2|fifth]] after [[12edo]], and is quite popular for that reason. 17edo's [[3/2|perfect fifth]] is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to a [[5L 2s|diatonic]] scale with more constrasting large and small steps, so it can be seen as a tuning that emphasizes the hardness of [[Pythagorean tuning]] rather than mellowing it out as in [[meantone]]. 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 temperament of the 2.3.25.7.11.13.23 [[subgroup]], 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 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.