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Undo revision 198797 by VectorGraphics (talk) "neogothic" is not notable enough for it to be the first thing mentioned and arguably doesnt even apply to 17edo
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== Theory ==
== Theory ==
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. Meanwhile, it approximates [[harmonic]]s [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] to reasonable degrees, despite completely missing harmonic [[5/1|5]]. Thus it can plausibly be treated as a 2.3.25.7.11.13.23 [[subgroup temperament]], for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because these harmonics are all tempered sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]].


=== Neogothic harmony ===
Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.  
The most common appeal of 17edo is that it contains a [[5L 2s|diatonic]] scale with more contrasting large and small steps. 17edo is the simplest tuning for [[Neogothic major and minor|neogothic]] harmony (though its fifth is about 4 cents sharp of just, in the "shrub region" rather than the more "proper" [[gentle region]]), and it can be seen as a tuning that emphasizes the hardness of Pythagorean tuning rather than mellowing it out as in meantone, and at the same time approaching concordant intervals such as 14/11 and 13/11 with the thirds.  


=== Prime harmonics ===
The standard major triad is quite dissonant as the major third is closer to [[9/7]] than the traditional [[5/4]]. Instead, the tonic chords of 17edo could be considered to be 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 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 (which is a sus4 with added second, or sus2 with added fourth). 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.  
As for harmonics, 7, 11, 13, and 23 are approximated to reasonable degrees, while the 5th harmonic is missing. As such, it can be treated as a 2.3.11.13 system, or more expansively a 2.3.25.7.11.13.23 temperament (allowing the extremely accurate mapping of 25/24 to be taken advantage of without direct approximation). Although the ratios of 7 are not as well represented, they are closer than in 12edo, and in fact 17edo supports [[superpyth]] temperament.
 
Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.  


=== Superpyth temperament and fifth-spanning tetrads ===
Another construction of septimal chords involves 4:7:12 and its inversion 7:12:21. These triads span a twelfth, realized in 17edo as 0–14–27 and 0–13–27, respectively. To this we may add 0–12–14–27, representing 8:13:14:24, or 0–13–15–27, representing 7:12:13:21.  
The standard major triad can be seen as supermajor, as the major third is closer to [[9/7]] than the traditional [[5/4]]. As such, a perfect fourth can be introduced to the minor chord 0-4-10 to form the tetrad 0-4-7-10, which represents the harmonic series segment 6:7:8:9 as a stable chord. This also implies the tetrad's reflection 0-3-6-10 (a major chord with an added major second), representing 14:16:18:21 as [[64/63]] is tempered out. The latter resembles the {{w|mu chord}} of Steely Dan fame. 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 (the 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.  


Another construction of septimal chords involves 4:7:12 and its inversion 7:12:21. These triads span a twelfth, realized in 17edo as 0–14–27 and 0–13–27, respectively. To this we may add 0–12–14–27, representing 8:13:14:24, or 0–13–15–27, representing 7:12:13:21. <!-- explain relative error in an odd limit -->
In the no-5 [[13-odd-limit]], 17edo maintains the smallest relative error{{clarify}} of any edo until [[166edo]]. <!-- explain relative error in an odd limit -->


=== Odd harmonics ===
=== Odd harmonics ===
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