2.3.7 subgroup: Difference between revisions

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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" used alone usually implies prime limit.</ref>~~, called~~ '''za''' in [[color notation]], 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 '''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" used alone usually implies prime limit.</ref> ('''za''' in [[color notation]]) 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 2.3.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 5. Its simplest expansion is the [[2.3.7.11 subgroup]], which adds prime 11.

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. 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]].

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[[Category:7-limit]]

[[Category:Rank 3]]

[[Category:Lists of scales]]

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-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" used alone usually implies prime limit.</ref>, called '''za''' in [[color notation]], 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 '''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" used alone usually implies prime limit.</ref> ('''za''' in [[color notation]]) 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 2.3.7 subgroup is a retraction of the [[7-limit]], obtained by removing prime 5. Its simplest expansion is the [[2.3.7.11 subgroup]], which adds prime 11.
 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. 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]].
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 [[Category:7-limit]]
 [[Category:Rank 3]]
+[[Category:Lists of scales]]