3.5.11 subgroup: Difference between revisions

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== Chords and harmony ==

Simple [[isoharmonic]] chords are largely absent from this subgroup, but non-isoharmonic chords can be used instead. An obvious place to start would be with the chord 3:5:11, which tritave-reduces to 9:11:15, analogous to how 2:3:5 octave-reduces to [[4:5:6]]. This chord is actually surprisingly vaguely familiar, at least compared to the Bohlen-Pierce [[3:5:7]] triad, because 9:11:15 (1-[[11/9]]-[[5/3]]) somewhat resembles 12:15:20 (1-[[5/4]]-[[5/3]]), which is an octave-equivalent inversion of the familiar [[10:12:15]] minor triad, although there is still a very large difference between [[11/9]] and [[5/4]] such that 9:11:15 still clearly sounds xenharmonic. 9:11:15 also has a utonal counterpart, 33:45:55 (which resembles 3:4:5), and a simpler, but very wide, tritave-equivalent inversion, 5:9:11. The tetrad 9:11:15:25 adds [[25/9]] or [[5/3]]<sup>2</sup> above the root. If you have another tritave above the root then this will become 9:11:15:25:27 with a dissonant [[27/25]] semitone between the 25/9 and 27/9 (3/1), which is a similar situation to the [[16/15]] at the top of the 5-limit major seventh chord [[8:10:12:15]].

Simple [[isoharmonic]] chords are largely absent from this subgroup, but non-isoharmonic chords can be used instead. An obvious place to start would be with the chord 3:5:11, which tritave-reduces to 9:11:15, analogous to how 2:3:5 octave-reduces to [[4:5:6]]. This chord is actually surprisingly vaguely familiar, at least compared to the Bohlen-Pierce [[3:5:7]] triad, because 9:11:15 (1-[[11/9]]-[[5/3]]) somewhat resembles 12:15:20 (1-[[5/4]]-[[5/3]]), which is an octave-equivalent inversion of the familiar [[10:12:15]] minor triad, although there is still a very large difference between [[11/9]] and [[5/4]] such that 9:11:15 still clearly sounds xenharmonic. 9:11:15 also has a utonal counterpart, 33:45:55 (which resembles 3:4:5), and a simpler, but very wide, tritave-equivalent inversion, 5:9:11. The tetrad 9:11:15:25 adds [[25/9]] or [[5/3]]<sup>2</sup> above the root. If you have another tritave above the root then this will become 9:11:15:25:27 with a dissonant [[27/25]] semitone between the 25/9 and 27/9 (3/1), which is a similar situation to the [[16/15]] at the top of the 5-limit major seventh chord [[8:10:12:15]]. In the 3.5.11.13 subgroup, 9:11:15 can be extended to the isoharmonic chord 9:11:13:15, which is sort of a [[diminished seventh chord]].

~~In the 3.5.11.13 subgroup, 9:11:15 can be extended to the isoharmonic chord 9:11:13:15, which is sort of a [[diminished seventh chord]].~~

''One should bear in mind that, in the world of tritave equivalence, ratios of 3 in the denominator are fungible instead of those of 2.'' For example, while the octave-reduced fifth harmonic is 5:4, the tritave-reduced fifth harmonic would be 5:3 instead, which would be a "major sixth" by conventional [[pitch class|pitch-class]] terminology. The tritave-reduced 11th harmonic is 11:9.

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 == Chords and harmony ==
-Simple [[isoharmonic]] chords are largely absent from this subgroup, but non-isoharmonic chords can be used instead. An obvious place to start would be with the chord 3:5:11, which tritave-reduces to 9:11:15, analogous to how 2:3:5 octave-reduces to [[4:5:6]]. This chord is actually surprisingly vaguely familiar, at least compared to the Bohlen-Pierce [[3:5:7]] triad,  because 9:11:15 (1-[[11/9]]-[[5/3]]) somewhat resembles 12:15:20 (1-[[5/4]]-[[5/3]]), which is an octave-equivalent inversion of the familiar [[10:12:15]] minor triad, although there is still a very large difference between [[11/9]] and [[5/4]] such that 9:11:15 still clearly sounds xenharmonic. 9:11:15 also has a utonal counterpart, 33:45:55 (which resembles 3:4:5), and a simpler, but very wide, tritave-equivalent inversion, 5:9:11. The tetrad 9:11:15:25 adds [[25/9]] or [[5/3]]<sup>2</sup> above the root. If you have another tritave above the root then this will become 9:11:15:25:27 with a dissonant [[27/25]] semitone between the 25/9 and 27/9 (3/1), which is a similar situation to the [[16/15]] at the top of the 5-limit major seventh chord [[8:10:12:15]].
+Simple [[isoharmonic]] chords are largely absent from this subgroup, but non-isoharmonic chords can be used instead. An obvious place to start would be with the chord 3:5:11, which tritave-reduces to 9:11:15, analogous to how 2:3:5 octave-reduces to [[4:5:6]]. This chord is actually surprisingly vaguely familiar, at least compared to the Bohlen-Pierce [[3:5:7]] triad,  because 9:11:15 (1-[[11/9]]-[[5/3]]) somewhat resembles 12:15:20 (1-[[5/4]]-[[5/3]]), which is an octave-equivalent inversion of the familiar [[10:12:15]] minor triad, although there is still a very large difference between [[11/9]] and [[5/4]] such that 9:11:15 still clearly sounds xenharmonic. 9:11:15 also has a utonal counterpart, 33:45:55 (which resembles 3:4:5), and a simpler, but very wide, tritave-equivalent inversion, 5:9:11. The tetrad 9:11:15:25 adds [[25/9]] or [[5/3]]<sup>2</sup> above the root. If you have another tritave above the root then this will become 9:11:15:25:27 with a dissonant [[27/25]] semitone between the 25/9 and 27/9 (3/1), which is a similar situation to the [[16/15]] at the top of the 5-limit major seventh chord [[8:10:12:15]]. In the 3.5.11.13 subgroup, 9:11:15 can be extended to the isoharmonic chord 9:11:13:15, which is sort of a [[diminished seventh chord]].
- In the 3.5.11.13 subgroup, 9:11:15 can be extended to the isoharmonic chord 9:11:13:15, which is sort of a [[diminished seventh chord]].
 ''One should bear in mind that, in the world of tritave equivalence, ratios of 3 in the denominator are fungible instead of those of 2.'' For example, while the octave-reduced fifth harmonic is 5:4, the tritave-reduced fifth harmonic would be 5:3 instead, which would be a "major sixth" by conventional [[pitch class|pitch-class]] terminology. The tritave-reduced 11th harmonic is 11:9.

3.5.11 subgroup: Difference between revisions

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