2.3.7.11 subgroup: Difference between revisions

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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

** 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 [[~~Combination_product_sets~~|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.

* 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

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=== Rank-2 temperaments ===

~~{{Main|Tour~~ of ~~regular temperaments#Temperaments defined~~ by an 11-~~limit~~ comma}}

[[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]].

~~{{Todo|inline~~=~~1|expand}}~~

[[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 ==

; [[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:Just intonation subgroups|#]]

@@ Line 16: / Line 16: @@
 * 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."
-* The 1-3-7-9-11 [[Combination_product_sets|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.
+* 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
@@ Line 24: / Line 24: @@
 === Rank-2 temperaments ===
-{{Main|Tour of regular temperaments#Temperaments defined by an 11-limit comma}}
+[[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]].
-{{Todo|inline=1|expand}}
+[[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 ==
+; [[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:Just intonation subgroups|#]]