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.

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 [[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.

~~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]].~~

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).

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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.
+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 [[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.
-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]].
 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).

3.5.11 subgroup: Difference between revisions

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