17edo: Difference between revisions

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== Theory ==

17edo is the next smallest edo to have a [[5L 2s|diatonic]] [[3/2|perfect fifth]] after [[12edo]], and is quite popular for that reason. The perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of 12edo's, lending itself to a 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 [[5/4]] and [[6/5]] both being ~~about halfway between its steps,~~ but it approximates harmonics [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] acceptably, with a sharp tuning for all of them. It can thus be treated as a temperament of the 2.3.7.11.13.23.25-[[subgroup]] or any of its subsets, where it is quite accurate for its size.

17edo is the next smallest edo to have a [[5L 2s|diatonic]] [[3/2|perfect fifth]] after [[12edo]], and is quite popular for that reason. The perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of 12edo's, lending itself to a 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]]. Because it tempers [[25/24]], it completely misses [[harmonic]] [[5/1|5]], with [[5/4]] and [[6/5]] both being tempered together into a neutral third; but it approximates harmonics [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] acceptably, with a sharp tuning for all of them. It can thus be treated as a temperament of the 2.3.7.11.13.23.25-[[subgroup]] or any of its subsets, where it is quite accurate for its size.

A notable [[comma]] it [[tempering out|tempers out]] is [[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 [[wart|17c]] [[val]] does temper out 81/80 (while improving consistency as shown below in [[#Approximation to JI]]), while still tempering out 64/63, thus placing it on the meantone spectrum with the [[dominant (temperament)|dominant]] [[extension]].

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17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. It does not contain any nontrivial subset edos, though it contains [[17ed4]] and [[17ed8]]. 17ed8, built by taking every third step of 17edo, is a system where all odd harmonics up to the 21st are mapped exactly as in 17edo, except for the 11th. Beyond that, the 27th, 31st, 35th, and 39th harmonics are likewise mapped identically.

[[34edo]], which doubles 17edo, provides a ~~good~~ correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.

[[34edo]], which doubles 17edo, provides a great correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.

== Intervals ==

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== Tuning by ear ==

17edo is very close to a circle of seventeen [[25/24]] chromatic semitones: (25/24)<sup>17</sup> is only 1.43131 cents sharp of an octave. This means that if you can tune seventeen 25/24's accurately (by say, tuning 5/4 up, 3/2 down and 5/4 up, taking care to minimize the error at each step), you have a shot at approximating 17edo within melodic just noticeable difference.

Despite tempering [[25/24]] by patent val, 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones: (25/24)<sup>17</sup> is only [[Septendecima|1.43131 cents sharp of an octave]]. This means that if you can tune seventeen 25/24's accurately (by say, tuning 5/4 up, 3/2 down and 5/4 up, taking care to minimize the error at each step), you have a shot at approximating 17edo within melodic just noticeable difference.

== Regular temperament properties ==

@@ Line 10: / Line 10: @@
 == Theory ==
-edo is the next smallest edo to have a [[5L 2s|diatonic]] [[3/2|perfect fifth]] after [[12edo]], and is quite popular for that reason. The perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of 12edo's, lending itself to a 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 [[5/4]] and [[6/5]] both being about halfway between its steps,  but it approximates harmonics [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] acceptably, with a sharp tuning for all of them. It can thus be treated as a temperament of the 2.3.7.11.13.23.25-[[subgroup]] or any of its subsets, where it is quite accurate for its size.
+edo is the next smallest edo to have a [[5L 2s|diatonic]] [[3/2|perfect fifth]] after [[12edo]], and is quite popular for that reason. The perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of 12edo's, lending itself to a 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]]. Because it tempers [[25/24]], it completely misses [[harmonic]] [[5/1|5]], with [[5/4]] and [[6/5]] both being tempered together into a neutral third; but it approximates harmonics [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] acceptably, with a sharp tuning for all of them. It can thus be treated as a temperament of the 2.3.7.11.13.23.25-[[subgroup]] or any of its subsets, where it is quite accurate for its size.
 A notable [[comma]] it [[tempering out|tempers out]] is [[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 [[wart|17c]] [[val]] does temper out 81/80 (while improving consistency as shown below in [[#Approximation to JI]]), while still tempering out 64/63, thus placing it on the meantone spectrum with the [[dominant (temperament)|dominant]] [[extension]].
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 edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. It does not contain any nontrivial subset edos, though it contains [[17ed4]] and [[17ed8]]. 17ed8, built by taking every third step of 17edo, is a system where all odd harmonics up to the 21st are mapped exactly as in 17edo, except for the 11th. Beyond that, the 27th, 31st, 35th, and 39th harmonics are likewise mapped identically.
-[[34edo]], which doubles 17edo, provides a good correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.
+[[34edo]], which doubles 17edo, provides a great correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.
 == Intervals ==
@@ Line 508: / Line 508: @@
 == Tuning by ear ==
-edo is very close to a circle of seventeen [[25/24]] chromatic semitones: (25/24)<sup>17</sup> is only 1.43131 cents sharp of an octave. This means that if you can tune seventeen 25/24's accurately (by say, tuning 5/4 up, 3/2 down and 5/4 up, taking care to minimize the error at each step), you have a shot at approximating 17edo within melodic just noticeable difference.
+Despite tempering [[25/24]] by patent val, 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones: (25/24)<sup>17</sup> is only [[Septendecima|1.43131 cents sharp of an octave]]. This means that if you can tune seventeen 25/24's accurately (by say, tuning 5/4 up, 3/2 down and 5/4 up, taking care to minimize the error at each step), you have a shot at approximating 17edo within melodic just noticeable difference.
 == Regular temperament properties ==