3.5.11 subgroup: Difference between revisions
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== Chords and harmony == | == 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]](: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). | |||
''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 ==== | |||
{{Main|Delta centauri}} | |||
[[Delta centauri]] (b28 & b71) is a [[microtemperament]] that is generated by [[~]][[5/3]], and 9 generators stacked represent [[11/9]]. It tempers out the [[4-cent comma]] (1953125/1948617). The best tuning is to flatten 5/3 by around half a cent, though just 5/3 or slightly sharp 5/3 will work fine too. For a microtemperament, it is quite low in [[complexity]], and can be used with a 13-note {{MOS scalesig|2L 11s<3/1>|link=1}} or 15-note {{MOS scalesig|13L 2s<3/1>|link=1}} (these are melodically comparable to 8-10 note octave-equivalent scales), making it a very [[efficiency|efficient]] temperament. Due to the ~3/1 period and ~5/3 generator, there is one-to-one correspondence between the [[3.5 subgroup]] and mapped intervals. It also has the advantage (or disadvantage, depending on your point of view) that 11/9 is simple enough to appear in the 13-note and 15-note MOS scales but the quasi-octave 243/121 is too complex, being at -18 generators. | |||
==== Deneb ==== | |||
{{Main|Deneb}} | |||
[[Deneb]] (b11 & b28) tempers out the comma [[6655/6561]] and equates three [[11/9]]'s with [[9/5]]. It is supported by [[11edt]] and the 11-note {{mos scalesig|6L 5s<3/1>|link=1}} scale can be used with diatonic-like melodic properties. In some tunings, 5 generators up is an accurate [[8/3]]–if this is included, rendering it a 3.5.8.11 or 8.3.5.11 temperament, then it is actually related to [[hitchcock]]. [[28edt]] is notable for being the unique intersection of deneb and delta centauri. The 3.5.7.11 temperament [[Mintra]], although it's mainly known as a strong extension of [[Mintaka]], is additionally a weak extension of Deneb and a weak extension of [[BPS]], combining all three temperaments into an efficient no-2s 11-limit temperament. | |||
==== Polaris ==== | |||
[[Polaris]] (b17 & b32) constructs an ~11/9 from stacking ~5/3 generators, similar to delta centauri, but here ~5/3 is tuned quite sharp so that -6 5/3s stacked (or 6 [[9/5]]s stacked) tritave-reduced is ~11/9. | |||
[[Category:Just intonation subgroups|#]] | [[Category:Just intonation subgroups|#]] | ||
[[Category:Rank-3 temperaments|#]] | [[Category:Rank-3 temperaments|#]] | ||
[[Category:11-limit|#]] | [[Category:11-limit|#]] | ||