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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* A septimal flavor of Sazkar, which has a minor third above the 1/1 ascending but a tone descending: thus ascending 1/1-7/6-11/9-4/3-3/2-27/16-11/6-2/1, and descending 1/1-9/8-11/9-4/3-3/2-27/16-11/6-2/1 | * A septimal flavor of Sazkar, which has a minor third above the 1/1 ascending but a tone descending: thus ascending 1/1-7/6-11/9-4/3-3/2-27/16-11/6-2/1, and descending 1/1-9/8-11/9-4/3-3/2-27/16-11/6-2/1 | ||
** Margo Schulter adds: "I should caution that an Arab Rast, of which Sazkar is an offshoot, might usually have a Zalzalian third more like 27/22 or 16/13 rather than 11/9, so this may be more of a new tuning than a traditional Arab Sazkar." | ** Margo Schulter adds: "I should caution that an Arab Rast, of which Sazkar is an offshoot, might usually have a Zalzalian third more like 27/22 or 16/13 rather than 11/9, so this may be more of a new tuning than a traditional Arab Sazkar." | ||
* The 1-3-7-9-11 [[ | * The 1-3-7-9-11 [[Combination product set|2(5 Dekany]] within a 12-tone [[Periodic_scale#Constant Structure|constant structure]] (see [http://anaphoria.com/dekanyconstantstructures.pdf link]): 1*3, 9*11, 3*9, 1*7, 3*7*11, 7*9, 3*11, 1*9, 3*9*11, 7*11, 3*7, 1*11. | ||
* (with prime 13 added) A 12f, {352/351, 364/363} 2.3.7.11.13 elf transversal: 28/27-9/8-13/11-9/7-4/3-11/8-3/2-14/9-22/13-16/9-27/14-2 | * (with prime 13 added) A 12f, {352/351, 364/363} 2.3.7.11.13 elf transversal: 28/27-9/8-13/11-9/7-4/3-11/8-3/2-14/9-22/13-16/9-27/14-2 | ||
== 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, '''41''', 58, 63, '''72''', 94, 118, 130, '''135''', 342, … }} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
[[Supra]], which extends [[archy]], provides a simple yet high-damage approximation to the subgroup. It is generated by a perfect fifth, tuned a little sharp so that two make [[8/7]][[~]][[9/8]] and six make [[16/11]], tempering out [[64/63]] and [[99/98]]. Alternatively, [[suhajira]] can be considered an extension of archy that adds neutral intervals, with the perfect fifth split into two neutral third generators, each representing [[11/9]]~[[27/22]], thus tempering out 64/63 and [[243/242]]. | |||
[[Skwares]] takes the 11/9~27/22 neutral third, adds an octave to it and splits it in halves for ~[[11/7]], tempering out 99/98 and 243/242. | |||
[[Radon]], which adds prime 11 to [[slendric]] by tempering out [[896/891]], provides a more complex entry, well represented by 41edo and 46edo. A similar temperament at this level is [[hemif]], which tempers out 243/242 and 896/891 and can be tuned to 58edo. In both cases, the diatonic major third represents [[14/11]]. | |||
On the high-accuracy side, [[gary]] is an important temperament that finds 7 and 11 far into the [[chain of fifths]]. | |||
=== Rank-3 temperaments === | |||
[[Parapyth]] equates [[28/27]] with [[33/32]] and uses this interval as a spacer added to the chain of fifths to finds intervals of 7 and 11. | |||
[[Symbiotian]], which makes 33/32 and 28/27 sum to the [[Pythagorean apotome]], is an efficient high-accuracy counterpart of parapyth. | |||
[[Olympian]] equates 33/32 with a stack of two 64/63's. It is very accurate yet still easy to notate on the staff, using the septimal comma as spacer added to the chain of fifths to finds intervals of 7 and 11. | |||
== Music == | == Music == | ||
; [[Domin]] | |||
* [https://www.youtube.com/watch?v=Hr4WtKdvs_4 ''Zutsz (2.3.7.11 JI)''] (2024) | |||
* [https://www.youtube.com/watch?v=jjD9IGhCVFQ ''Tzakh (2.3.7.11 JI)''] (2024) | |||
[[Category: | [[Category:Just intonation subgroups|#]] | ||
[[Category: | [[Category:Rank-4 temperaments|#]] | ||
[[Category: | [[Category:11-limit|#]] | ||
[[Category:Lists of scales]] | [[Category:Lists of scales|#]] | ||