17edo: Difference between revisions

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== Theory ==

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]]. It completely misses [[harmonic]] [[5/1|5]], with it being about halfway between its steps, but it approximates harmonics [[7/1|7]], [[11/1|11]], ~~and~~ [[13/1|13]] ~~decently~~, ~~as well as harmonic~~ [[23/1|23]], with a sharp tuning for all of them. Thus it can plausibly be treated as a temperament of the 2.3.7.11.13(.23.25) [[subgroup]], for which it is quite accurate for its size.

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]]. It completely misses [[harmonic]] [[5/1|5]], with it being about halfway between its steps, but it approximates harmonics [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] decently, with a sharp tuning for all of them. Thus it can plausibly be treated as a temperament of the 2.3.7.11.13.23.25 [[subgroup]], for which it is quite accurate for its size.

It notably tempers out [[64/63]], which equates the harmonic seventh [[7/4]] with the pythagorean minor seventh [[16/9]], while its patent val does not temper out [[81/80]]. This makes 17edo by default a [[superpythagorean]] system rather than a [[meantone]] one. Other commas it tempers out include [[78/77]], [[99/98]], [[144/143]], [[169/168]], [[243/242]], and many more, each of which has its own effect on the structure of 17edo. If one wants to approximate JI with prime 5, then 17edo would not be the best option, and it would be better to use other systems like [[19edo]], [[22edo]], [[27edo]], or [[31edo]] instead. That said, the 17c val does temper out 81/80 (while improving upon 15-odd-limit consistency as shown below in [[#Approximation to JI]]), while still tempering out 64/63, thus putting it on the meantone spectrum with the [[dominant]] extension.

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 == Theory ==
-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]]. It completely misses [[harmonic]] [[5/1|5]], with it being about halfway between its steps,  but it approximates harmonics [[7/1|7]], [[11/1|11]], and [[13/1|13]] decently, as well as harmonic [[23/1|23]], with a sharp tuning for all of them. Thus it can plausibly be treated as a temperament of the 2.3.7.11.13(.23.25) [[subgroup]], for which it is quite accurate for its size.
+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]]. It completely misses [[harmonic]] [[5/1|5]], with it being about halfway between its steps,  but it approximates harmonics [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] decently, with a sharp tuning for all of them. Thus it can plausibly be treated as a temperament of the 2.3.7.11.13.23.25 [[subgroup]], for which it is quite accurate for its size.
 It notably tempers out [[64/63]], which equates the harmonic seventh [[7/4]] with the pythagorean minor seventh [[16/9]], while its patent val does not temper out [[81/80]]. This makes 17edo by default a [[superpythagorean]] system rather than a [[meantone]] one. Other commas it tempers out include [[78/77]], [[99/98]], [[144/143]], [[169/168]], [[243/242]], and many more, each of which has its own effect on the structure of 17edo. If one wants to approximate JI with prime 5, then 17edo would not be the best option, and it would be better to use other systems like [[19edo]], [[22edo]], [[27edo]], or [[31edo]] instead. That said, the 17c val does temper out 81/80 (while improving upon 15-odd-limit consistency as shown below in [[#Approximation to JI]]), while still tempering out 64/63, thus putting it on the meantone spectrum with the [[dominant]] extension.