2.3.7.11 subgroup: Difference between revisions
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The '''2.3.7.11 subgroup''' (''' | The '''2.3.7.11 subgroup''' ('''zala''' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 7, and 11 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, 7, and 11. 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]], [[9/7]], [[21/16]], [[11/9]], [[22/21]], and so on. | ||
The 2.3.7.11 subgroup is a retraction of the [[11-limit]], obtained by removing prime 5. Its simplest expansion is the 2.3.7.11.13 subgroup, which adds prime 13. It can also be retracted to the [[2.3.7 subgroup]] by removing prime 11. | The 2.3.7.11 subgroup is a retraction of the [[11-limit]], obtained by removing prime 5. Its simplest expansion is the 2.3.7.11.13 subgroup, which adds prime 13. It can also be retracted to the [[2.3.7 subgroup]] by removing prime 11. | ||
A notable subset of the 2.3.7.11 subgroup is the 1 | A notable subset of the 2.3.7.11 subgroup is the {1, 3, 7, 9, 11} [[tonality diamond]], comprising all intervals in which 1, 3, 7, 9, and 11 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 this tonality diamond within the octave is [[1/1]], [[12/11]], [[9/8]], [[8/7]], [[7/6]], [[11/9]], [[14/11]], [[9/7]], [[4/3]], [[11/8]], [[16/11]], [[3/2]], [[14/9]], [[11/7]], [[18/11]], [[12/7]], [[7/4]], [[16/9]], [[11/6]], and [[2/1]]. | ||
When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3, 7, and 11, which can be represented in a 3-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]]. | When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3, 7, and 11, which can be represented in a 3-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]]. | ||
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== Regular temperaments == | == Regular temperaments == | ||
=== Rank-1 temperaments (edos) === | === Rank-1 temperaments (edos) === | ||
The 2.3.7.11 subgroup is relatively well approximated by the following edos (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs| '''5''', 9, 10, 12, 14, '''17''', 31 | The 2.3.7.11 subgroup is relatively well approximated by the following edos (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs| '''5''', 9, 10, 12, 14, '''17''', 31, '''41''', 58, 63, '''72''', 94, 118, 130, '''135''', 342, … }} | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||