2.3.7 subgroup: Difference between revisions

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The 2.3.7-limit or 2.3.7-prime-limit 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, 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 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.

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== Edos ==

== Rank-2 temperaments ==

== Notes ==

[[Category:7-limit]]

[[Category:Rank 3]]

[[Category:Stub]]

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-The 2.3.7-limit or 2.3.7-prime-limit 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, 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 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.
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 == Edos ==
 == Rank-2 temperaments ==
+== Notes ==
+<references />
 [[Category:7-limit]]
 [[Category:Rank 3]]
 [[Category:Stub]]