17edo: Difference between revisions

Line 15:

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

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~~.

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]] follows naturally from its [[support]] of [[superpyth]], and may be preferred. Such chords include the tetrads [[6:7:8:9]] and its utonal inverse, realized in 17edo as 0–4–7–10 and 0–3–6–10, respectively, in addition to the sus2-4 chord, realized as 0–3–7–10. Possible chromatic alterations include but are not limited to an approximation of 12:13:16:18, 0–2–7–10, and an approximation of 8:9:11:12, 0–3–7–10. Similarly, the fourth-spanning triad [[6:7:8]] and its inverse can be used, with their wide voicing realized in 17edo as 0–14–27 and 0–13–27, respectively. Extensions of these chords include 0–12–14–27, representing 8:13:14:24, and 0–13–15–27, representing 7:12:13:21.

~~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 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.

Of course, scales generated by the perfect fifth are not 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.

Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.

@@ Line 15: / Line 15: @@
 === As a means of extending harmony ===
-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.
+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]] follows naturally from its [[support]] of [[superpyth]], and may be preferred. Such chords include the tetrads [[6:7:8:9]] and its utonal inverse, realized in 17edo as 0–4–7–10 and 0–3–6–10, respectively, in addition to the sus2-4 chord, realized as 0–3–7–10. Possible chromatic alterations include but are not limited to an approximation of 12:13:16:18, 0–2–7–10, and an approximation of 8:9:11:12, 0–3–7–10. Similarly, the fourth-spanning triad [[6:7:8]] and its inverse can be used, with their wide voicing realized in 17edo as 0–14–27 and 0–13–27, respectively. Extensions of these chords include 0–12–14–27, representing 8:13:14:24, and 0–13–15–27, representing 7:12:13:21.
-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 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.
+Of course, scales generated by the perfect fifth are not 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.
 Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.