2.5.7 subgroup: Difference between revisions

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The '''2.5.7 subgroup''', or the '''no-threes 7-limit''' ~~('''yaza nowa''' in [[color notation]])~~ is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 5, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 5, and 7. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[7/4]], [[8/7]], [[7/5]], [[28/25]], [[35/32]], and so on.

The '''2.5.7 subgroup''', or the '''no-threes 7-limit''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 5, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 5, and 7. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[7/4]], [[8/7]], [[7/5]], [[28/25]], [[35/32]], and so on.

The 2.5.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 3. Its simplest expansion is the [[2.5.7.11 subgroup]], which adds prime 11.

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When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 5 and 7, 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]].

In [[color notation]], this subgroup may be called '''yaza nowa''', which means that it is the intersection of 2.3.5 and 2.3.7 ("yaza"), but without 3 ("nowa").

== Properties ==

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-The '''2.5.7 subgroup''', or the '''no-threes 7-limit''' ('''yaza nowa''' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 5, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 5, and 7. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[7/4]], [[8/7]], [[7/5]], [[28/25]], [[35/32]], and so on.
+The '''2.5.7 subgroup''', or the '''no-threes 7-limit'''  is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 5, and 7 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 5, and 7. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[5/4]], [[7/4]], [[8/7]], [[7/5]], [[28/25]], [[35/32]], and so on.
 The 2.5.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 3. Its simplest expansion is the [[2.5.7.11 subgroup]], which adds prime 11.
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 When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 5 and 7, 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]].
+In [[color notation]], this subgroup may be called '''yaza nowa''', which means that it is the intersection of 2.3.5 and 2.3.7 ("yaza"), but without 3 ("nowa").
 == Properties ==