3.5.11 subgroup: Difference between revisions

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The '''3.5.11 subgroup''' is a [[non-octave]] [[just intonation subgroup]] consisting of [[rational interval]]s where 3, 5, and 11 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 3, 5, and 11. This is an infinite set. Some examples of intervals in this subgroup are [[5/3]], [[11/9]], [[15/11]], [[33/25]], and so on. Due to the lack of octaves, it would most obviously be approached by using [[3/1|tritave]]-equivalence. It could perhaps be considered the simplest place to look in the no-2s world after the [[3.5.7 subgroup]]~~–it~~ is to the [[3.5.7 subgroup]] as the [[2.3.7 subgroup]] is to the [[5-limit]].

The '''3.5.11 subgroup''' is a [[non-octave]] [[just intonation subgroup]] consisting of [[rational interval]]s where 3, 5, and 11 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 3, 5, and 11. This is an infinite set. Some examples of intervals in this subgroup are [[5/3]], [[11/9]], [[15/11]], [[33/25]], and so on. Due to the lack of octaves, it would most obviously be approached by using [[3/1|tritave]]-equivalence. It could perhaps be considered the simplest place to look in the no-2's world after the [[3.5.7 subgroup]] – it is to the [[3.5.7 subgroup]] as the [[2.3.7 subgroup]] is to the [[5-limit]].

The 3.5.11 subgroup is a [[retraction]] of the [[11-limit]], obtained by removing primes 2 and 7. Its simplest [[expansion]] is the [[3.5.11.13 subgroup]], which adds prime [[13/1|13]]. It can also be retracted to the [[3.5 subgroup]] by removing prime 11.

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One important feature of subgroups involving 3 and 11 is the quasi-octave at the interval designated 243/121, which is probably meriting of special treatment in terms of consonance and dissonance. Although, this will not be as much of a concern in 3.5.11 [[subgroup temperament]]s where 11 is complex, or temperaments where 11 isn't very accurate (which will distort the 243/121 interval into being a much more out-of-tune quasi-octave).

When [[tritave]] equivalence is assumed, an interval can be taken as representing that interval in every possible tritave-equivalent voicing. This leaves primes 5 and 11, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]].

When tritave equivalence is assumed, an interval can be taken as representing that interval in every possible tritave-equivalent voicing. This leaves primes 5 and 11, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]].

~~== Introduction~~ to tritave equivalence ==

For an introduction to tritave equivalence, see [[3/1 #As an interval of equivalence]].

~~See~~ [[~~EDT~~ #~~Introduction to tritave~~ equivalence]].

== 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]].

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 33:45:55 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.

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-The '''3.5.11 subgroup''' is a [[non-octave]] [[just intonation subgroup]] consisting of [[rational interval]]s where 3, 5, and 11 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 3, 5, and 11. This is an infinite set. Some examples of intervals in this subgroup are [[5/3]], [[11/9]], [[15/11]], [[33/25]], and so on. Due to the lack of octaves, it would most obviously be approached by using  [[3/1|tritave]]-equivalence. It could perhaps be considered the simplest place to look in the no-2s world after the [[3.5.7 subgroup]]–it is to the [[3.5.7 subgroup]] as the [[2.3.7 subgroup]] is to the [[5-limit]].
+The '''3.5.11 subgroup''' is a [[non-octave]] [[just intonation subgroup]] consisting of [[rational interval]]s where 3, 5, and 11 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 3, 5, and 11. This is an infinite set. Some examples of intervals in this subgroup are [[5/3]], [[11/9]], [[15/11]], [[33/25]], and so on. Due to the lack of octaves, it would most obviously be approached by using [[3/1|tritave]]-equivalence. It could perhaps be considered the simplest place to look in the no-2's world after the [[3.5.7 subgroup]] – it is to the [[3.5.7 subgroup]] as the [[2.3.7 subgroup]] is to the [[5-limit]].
 The 3.5.11 subgroup is a [[retraction]] of the [[11-limit]], obtained by removing primes 2 and 7. Its simplest [[expansion]] is the [[3.5.11.13 subgroup]], which adds prime [[13/1|13]]. It can also be retracted to the [[3.5 subgroup]] by removing prime 11.
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 One important feature of subgroups involving 3 and 11 is the quasi-octave at the interval designated 243/121, which is probably meriting of special treatment in terms of consonance and dissonance. Although, this will not be as much of a concern in 3.5.11 [[subgroup temperament]]s where 11 is complex, or temperaments where 11 isn't very accurate (which will distort the 243/121 interval into being a much more out-of-tune quasi-octave).
-When [[tritave]] equivalence is assumed, an interval can be taken as representing that interval in every possible tritave-equivalent voicing. This leaves primes 5 and 11, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]].
+When tritave equivalence is assumed, an interval can be taken as representing that interval in every possible tritave-equivalent voicing. This leaves primes 5 and 11, which can be represented in a 2-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]].
-== Introduction to tritave equivalence ==
+For an introduction to tritave equivalence, see [[3/1 #As an interval of equivalence]].
-See [[EDT #Introduction to tritave equivalence]].
 == 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]].
-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 33:45:55 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.

3.5.11 subgroup: Difference between revisions

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