2.3.7 subgroup: Difference between revisions
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The 2.3.7-limit or 2.3.7-prime | 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. | 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 == | == Edos == | ||
== Rank-2 temperaments == | == Rank-2 temperaments == | ||
== Notes == | |||
<references /> | |||
[[Category:7-limit]] | [[Category:7-limit]] | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||
[[Category:Stub]] | [[Category:Stub]] | ||
Revision as of 21:44, 26 May 2023
The 2.3.7 subgroup[1] 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 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.
The phrase "2.3.7-limit just intonation" usually refers to the 2.3.7-prime-limit 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 lattice diagrams, each prime represented by a different dimension.
Scales
Edos
Rank-2 temperaments
Notes
- ↑ 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.