2.3.7 subgroup: Difference between revisions

Line 1:

The '''2.3.7 subgroup'''<ref>Sometimes incorrectly named '''2.3.7-limit''' or '''2.3.7-prime limit'''; a [[prime limit]] is a subgroup spanned by all primes up to a given prime, ~~which defines the~~ limit.</ref> is a [[just intonation subgroup]] consisting of rational ~~intervals~~ where 2, 3, and 7 are the only allowable prime ~~factors~~, so that every such interval may be written as a ratio of integers which are products of 2, 3, 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 [[3/2]], [[7/4]], [[7/6]], [[9/7]], [[9/8]], [[21/16]], and so on.

The '''2.3.7 subgroup'''<ref>Sometimes incorrectly named '''2.3.7-limit''' or '''2.3.7-prime limit'''; a [[prime limit]] is a subgroup spanned by all primes up to a given prime, and "limit" alone is ambiguous.</ref> is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 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, 3, 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 [[3/2]], [[7/4]], [[7/6]], [[9/7]], [[9/8]], [[21/16]], and so on.

~~The~~ 1.3.7~~-odd-limit refers to a constraint on the selection of~~ [[~~Just intonation|just~~]] ~~[[Interval class|~~intervals~~]] for a scale or composition such that~~ 3 and 7 are the only allowable odd numbers, either for the intervals of the scale, or the ratios between successive or simultaneously sounding notes of the composition. The complete list of 1.3.7~~-odd-limit intervals~~ within the octave is [[1/1]], [[8/7]], [[7/6]], [[4/3]], [[3/2]], [[12/7]], [[7/4]], and [[2/1]]~~, which is known as the 1.3.7-limit tonality diamond~~.

A notable subset of the 2.3.7 subgroup is the 1.3.7 [[tonality diamond]], comprised of all intervals in which 1, 3 and 7 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. It is an infinite set, but finite within any given boundaries. The complete list of intervals in the 1.3.7 tonality diamond within the octave is [[1/1]], [[8/7]], [[7/6]], [[4/3]], [[3/2]], [[12/7]], [[7/4]], and [[2/1]].

~~The phrase "2~~.3.7~~-limit just intonation" usually refers~~ to the ~~2.3.~~7 ~~subgroup and includes primes 2~~, 3, and 7. When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 and 7, which can be represented in 2-dimensional [[~~Harmonic~~ lattice diagram~~|lattice diagrams~~]], each prime represented by a different dimension.

Another such subset is the 1.3.7.9 tonality diamond, which adds the following intervals to the previous list: [[9/8]], [[9/7]], [[14/9]], and [[16/9]].

When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 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]].

== Scales ==

Line 12:

Line 14:

== Rank-2 temperaments ==

{{Main|Tour of regular temperaments#Clans defined by a 2.3.7 (za) comma}}

== Notes ==

Line 17:

Line 21:

[[Category:7-limit]]

[[Category:Rank 3]]

~~[[Category:Stub]]~~

@@ Line 1: / Line 1: @@
-The '''2.3.7 subgroup'''<ref>Sometimes incorrectly named '''2.3.7-limit''' or '''2.3.7-prime limit'''; a [[prime limit]] is a subgroup spanned by all primes up to a given prime, which defines the limit.</ref> is a [[just intonation subgroup]] consisting of rational intervals where 2, 3, and 7 are the only allowable prime factors, so that every such interval may be written as a ratio of integers which are products of 2, 3, 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 [[3/2]], [[7/4]], [[7/6]], [[9/7]], [[9/8]], [[21/16]], and so on.
+The '''2.3.7 subgroup'''<ref>Sometimes incorrectly named '''2.3.7-limit''' or '''2.3.7-prime limit'''; a [[prime limit]] is a subgroup spanned by all primes up to a given prime, and "limit" alone is ambiguous.</ref> is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 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, 3, 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 [[3/2]], [[7/4]], [[7/6]], [[9/7]], [[9/8]], [[21/16]], and so on.
-The 1.3.7-odd-limit refers to a constraint on the selection of [[Just intonation|just]] [[Interval class|intervals]] for a scale or composition such that 3 and 7 are the only allowable odd numbers, either for the intervals of the scale, or the ratios between successive or simultaneously sounding notes of the composition. The complete list of 1.3.7-odd-limit intervals within the octave is [[1/1]], [[8/7]], [[7/6]], [[4/3]], [[3/2]], [[12/7]], [[7/4]], and [[2/1]], which is known as the 1.3.7-limit tonality diamond.
+A notable subset of the 2.3.7 subgroup is the 1.3.7 [[tonality diamond]], comprised of all intervals in which 1, 3 and 7 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. It is an infinite set, but finite within any given boundaries. The complete list of intervals in the 1.3.7 tonality diamond within the octave is [[1/1]], [[8/7]], [[7/6]], [[4/3]], [[3/2]], [[12/7]], [[7/4]], and [[2/1]].
-The phrase "2.3.7-limit just intonation" usually refers to the 2.3.7 subgroup and includes primes 2, 3, and 7. When octave equivalence is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 and 7, which can be represented in 2-dimensional [[Harmonic lattice diagram|lattice diagrams]], each prime represented by a different dimension.
+Another such subset is the 1.3.7.9 tonality diamond, which adds the following intervals to the previous list: [[9/8]], [[9/7]], [[14/9]], and [[16/9]].
+When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3 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]].
 == Scales ==
@@ Line 12: / Line 14: @@
 == Rank-2 temperaments ==
+{{Main|Tour of regular temperaments#Clans defined by a 2.3.7 (za) comma}}
 == Notes ==
 <references />
@@ Line 17: / Line 21: @@
 [[Category:7-limit]]
 [[Category:Rank 3]]
-[[Category:Stub]]