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. It notably tempers out [[64/63]], which equates the harmonic seventh [[7/4]] with the pythagorean minor seventh [[16/9]], ~~meaning~~ 17edo ~~is 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.

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. 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 defaulta [[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 [[17edo#Approximation_to_JI|Approximation to JI]]), while still tempering out 64/63, thus putting it on the meantone spectrum with the [[Meantone_family#Dominant|dominant]] extension.

The diatonic [[major triad]], which is 0–6–10 steps, is quite [[dissonant]] compared to [[4:5:6]], as the major third is over 37 cents sharp from the traditional [[5/4]], and is instead closer to [[9/7]] or [[14/11]]. Instead, a different construction based on the [[2.3.7 subgroup|2.3.7-subgroup]] may be preferred. One approach is for the tonic chords of 17edo to be considered the tetrad [[6:7:8:9]] and its utonal inversion (representing [[14:16:18:21]] as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w|mu chord}} of {{w|Steely Dan}} fame). These are realized in 17edo as 0–4–7–10 and 0–3–6–10, respectively. Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add the 0–3–7–10 (a sus2-4 chord). These three chords comprise the three ways to divide the 17edo perfect fifth into two whole tones and one subminor third. In the diatonic major scale, the 6:7:8:9 chord occurs on II, III, and VI, while its inversion occurs on I, IV, and V. Compared to meantone, major and minor swap places in a sense (though in a different way from in [[mavila]]). Chromatic alterations of them also exist, for example, the 0–3–7–10 chord may be altered to 0–2–7–10 (which approximates 12:13:16:18) or 0–3–8–10 (which approximates 8:9:11:12). The 0–3–8–10 chord is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 0–3–6–10.

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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. It notably tempers out [[64/63]], which equates the harmonic seventh [[7/4]] with the pythagorean minor seventh [[16/9]], meaning 17edo is 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.
+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. 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 defaulta [[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 [[17edo#Approximation_to_JI|Approximation to JI]]), while still tempering out 64/63, thus putting it on the meantone spectrum with the [[Meantone_family#Dominant|dominant]] extension.
 The diatonic [[major triad]], which is 0–6–10 steps, is quite [[dissonant]] compared to [[4:5:6]], as the major third is over 37 cents sharp from the traditional [[5/4]], and is instead closer to [[9/7]] or [[14/11]]. Instead, a different construction based on the [[2.3.7 subgroup|2.3.7-subgroup]] may be preferred. One approach is for the tonic chords of 17edo to be considered the tetrad [[6:7:8:9]] and its utonal inversion (representing [[14:16:18:21]] as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the {{w|mu chord}} of {{w|Steely Dan}} fame). These are realized in 17edo as 0–4–7–10 and 0–3–6–10, respectively. Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add the 0–3–7–10 (a sus2-4 chord). These three chords comprise the three ways to divide the 17edo perfect fifth into two whole tones and one subminor third. In the diatonic major scale, the 6:7:8:9 chord occurs on II, III, and VI, while its inversion occurs on I, IV, and V. Compared to meantone, major and minor swap places in a sense (though in a different way from in [[mavila]]). Chromatic alterations of them also exist, for example, the 0–3–7–10 chord may be altered to 0–2–7–10 (which approximates 12:13:16:18) or 0–3–8–10 (which approximates 8:9:11:12). The 0–3–8–10 chord is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 0–3–6–10.