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]], [[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.

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 [[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.

~~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.

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]] [[extension]].

=== As a means of extending harmony ===

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Another approach is also possible. In the 5-limit, the major triad can be constructed by octave-reducing odd harmonics 1, 3, and 5, giving us 4:5:6, with the minor triad being its utonal inversion. A similar construction of septimal chords gives us [[6:7:8]] and its inversion [[21:24:28]], which are built with the intervals [[7/6]] and [[8/7]]. These intervals contrast by [[49/48]], similarly to how 5-limit thirds contrast by [[25/24]]. There are some issues, however. For example, the 6:7:8 chord has the root on the top rather than the bottom, and the notes may clash from being too close to each other. However, the wide voicing of these chords, those being 4:7:12 and 7:12:21, solve both of these issues. These triads span a twelfth, realized in 17edo as 0–14–27 and 0–13–27, respectively. In terms of the [[chain of fifths]], these chords are simpler in [[archy]] than the 5-limit triads in meantone, with [[64/63]] being tempered out rather than [[81/80]]. Extensions of these chords include 0–12–14–27, representing 8:13:14:24, and 0–13–15–27, representing 7:12:13:21.

Since the intervals of the 2.3.7-subgroup cluster around [[5edo]], a pentatonic system of interval classification may be preferred over the heptatonic one, with 7/6 ~~being~~ a major interval and 8/7 ~~being~~ a minor one.

Since the intervals of the 2.3.7-subgroup cluster around [[5edo]], a pentatonic system of interval classification may be preferred over the heptatonic one, with 7/6 becoming a major interval and 8/7 becoming a minor one.

Of course, scales generated by the perfect fifth aren't the only scales 17edo contains. Another type of scale is [[neutral third scales]], which are generated by half a fifth (5\17), and take the MOS patterns [[4L 3s]] (mosh) and [[7L 3s]] (dicoid). Other notable scales include that of [[bleu]] (generated by 2\17), and [[skwares]] (generated by 6\17). Non-MOS scales also exist; a more complete list can be found in the [[#Scales]] section.

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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]], [[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.
+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 [[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.
-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.
+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]] [[extension]].
 === As a means of extending harmony ===
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 Another approach is also possible. In the 5-limit, the major triad can be constructed by octave-reducing odd harmonics 1, 3, and 5, giving us 4:5:6, with the minor triad being its utonal inversion. A similar construction of septimal chords gives us [[6:7:8]] and its inversion [[21:24:28]], which are built with the intervals [[7/6]] and [[8/7]]. These intervals contrast by [[49/48]], similarly to how 5-limit thirds contrast by [[25/24]]. There are some issues, however. For example, the 6:7:8 chord has the root on the top rather than the bottom, and the notes may clash from being too close to each other. However, the wide voicing of these chords, those being 4:7:12 and 7:12:21, solve both of these issues. These triads span a twelfth, realized in 17edo as 0–14–27 and 0–13–27, respectively. In terms of the [[chain of fifths]], these chords are simpler in [[archy]] than the 5-limit triads in meantone, with [[64/63]] being tempered out rather than [[81/80]]. Extensions of these chords include 0–12–14–27, representing 8:13:14:24, and 0–13–15–27, representing 7:12:13:21.
-Since the intervals of the 2.3.7-subgroup cluster around [[5edo]], a pentatonic system of interval classification may be preferred over the heptatonic one, with 7/6 being a major interval and 8/7 being a minor one.
+Since the intervals of the 2.3.7-subgroup cluster around [[5edo]], a pentatonic system of interval classification may be preferred over the heptatonic one, with 7/6 becoming a major interval and 8/7 becoming a minor one.
 Of course, scales generated by the perfect fifth aren't the only scales 17edo contains. Another type of scale is [[neutral third scales]], which are generated by half a fifth (5\17), and take the MOS patterns [[4L 3s]] (mosh) and [[7L 3s]] (dicoid). Other notable scales include that of [[bleu]] (generated by 2\17), and [[skwares]] (generated by 6\17). Non-MOS scales also exist; a more complete list can be found in the [[#Scales]] section.