3.5.11 subgroup: Difference between revisions
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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]](:16). 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]]. | 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]](:16). 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]]. | ||
It is interesting to note that if you run with the analogy of 9:11:15 as a "minor triad" equivalent and 33:44:55 as a "major triad" equivalent, [[9/7]] acts as a sort of "neutral third" equivalent, being near √[[5/3]] ([[BPS]] temperament realizes this). | It is interesting to note that if you run with the analogy of 9:11:15 as a "minor triad" equivalent and 33:44:55 as a "major triad" equivalent, [[9/7]] acts as a sort of "neutral third" equivalent, being near √([[5/3]]) ([[BPS]] temperament realizes this). | ||
''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. | ''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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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{{Todo|inline=1|text=add more temperaments maybe?}} | |||
==== Delta centauri ==== | ==== Delta centauri ==== | ||
{{Main|Delta centauri}} | {{Main|Delta centauri}} | ||