17edo: Difference between revisions

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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. Notable commas it tempers out include [[64/63]], which equates the harmonic seventh [[7/4]] with the pythagorean minor seventh [[16/9]], as well as [[78/77]], [[99/98]], [[144/143]], [[169/168]], [[243/242]], and many others, each of which has its own effect on the structure of 17edo.

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 involving septimal intervals 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. 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 involving septimal intervals may be preferred.

The 5-limit ~~triads~~ can be constructed by octave-reducing odd harmonics 1, 3, and 5, giving us 4:5:6, ~~and taking~~ 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]]. ~~The~~ wide voicing of these chords ~~are~~ 4:7:12 and 7:12:21. These triads span a twelfth, 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.

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. 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.

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. Extensions of these chords include 0–12–14–27, representing 8:13:14:24, and 0–13–15–27, representing 7:12:13:21.

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

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 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. Notable commas it tempers out include [[64/63]], which equates the harmonic seventh [[7/4]] with the pythagorean minor seventh [[16/9]], as well as [[78/77]], [[99/98]], [[144/143]], [[169/168]], [[243/242]], and many others, each of which has its own effect on the structure of 17edo.
-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 involving septimal intervals 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. 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 involving septimal intervals may be preferred.
-The 5-limit triads can be constructed by octave-reducing odd harmonics 1, 3, and 5, giving us 4:5:6, and taking 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]]. The wide voicing of these chords are 4:7:12 and 7:12:21. These triads span a twelfth, 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.
+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. 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.
+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. Extensions of these chords include 0–12–14–27, representing 8:13:14:24, and 0–13–15–27, representing 7:12:13:21.
 Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.