2.3.5.11 subgroup: Difference between revisions

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The '''2.3.5.11 subgroup''' (~~AKA~~ ''yala'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 5, 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, 5 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 [[5/4]], [[3/2]], [[11/8]], [[11/9]], [[27/22]], and so on.

The '''2.3.5.11 subgroup''' (a.k.a. ''yala'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 5, 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, 5 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 [[5/4]], [[3/2]], [[11/8]], [[11/9]], [[27/22]], and so on.

In can be thought as either an extension of [[Alpharabian tuning]] with the familiar 5-limit chords and stuctures, or a retraction of the 11-limit by removing prime 7. It can be similar to the [[2.3.5.13 subgroup]], specially considering neutral interval pairs such as 39/32 ~ 11/9 and 16/13 ~ 27/22, which are connected by the small comma of [[352/351]].

In can be thought as either an extension of [[Alpharabian tuning]] with the familiar 5-limit chords and stuctures, or a retraction of the 11-limit by removing prime 7. It can be similar to the [[2.3.5.13 subgroup]], specially considering neutral interval pairs such as 39/32~11/9 and 16/13~27/22, which are connected by the small comma of [[352/351]].

== Regular temperaments ==

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[[Schismic]] provides two reasonable aproximations to the 2.3.5.11 subgroup, one by finding 11/8 at the triple-augmented second (+23 fifths) through the [[cassandra]] mapping, and another by finding 11/8 at the quadruple-diminished seventh (-30 fifths) through the [[helenus]] mapping. Helenus, {{nowrap| 53 & 65 }}, provides a much better approximation to the subgroup, as both 5 and 11 are generated in the same direction.

[[Gravity]] also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at -6 gens, 5/4 at -17 gens and 11/8 at -15 gens. It is the unique temperament in the 2.3.5.11 subgroup equating [[~~81/80|S9 =~~ 81/80]], [[~~100/99|S10 =~~ 100/99]], and [[~~121/120|S11 =~~ 121/120]], thus tempering out [[243/242]], [[4000/3993]], and [[8019/8000]]. [[65edo]] is the unique intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup.

[[Gravity]] also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at -6 gens, 5/4 at -17 gens and 11/8 at -15 gens. It is the unique temperament in the 2.3.5.11 subgroup equating [[81/80]] ({{S|9}}), [[100/99]] ({{S|10}}), and [[121/120]] ({{S|11}}), thus [[tempering out]] [[243/242]], [[4000/3993]], and [[8019/8000]]. [[65edo]] is the unique intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup.

=== Rank-3 temperaments ===

[[Vishdel]] provides a low-complexity, accurate temperament, but for those searching a much higher accuracy system, [[tritomere]] is among the best rank-3 temperaments for this case, having tremendous accuracy with manageable complexity, tempering out the difference between three [[~~243/242|rastmas~~]] and one [[~~81/80|~~syntonic comma]] (0.08 cents). Its boundary of usability begins at [[152edo|152]] and [[159edo]], the latter inheriting the marvelous fifths from 53edo, one that [[Aura]] has shown great interest in. Bigger edos that support this excellent temperament include [[342edo]], [[494edo]], [[677edo]], [[1171edo]], among others.

[[Vishdel]] provides a low-complexity, accurate temperament, but for those searching a much higher accuracy system, [[tritomere]] is among the best rank-3 temperaments for this case, having tremendous accuracy with manageable complexity, tempering out the difference between three [[rastma]]s and one [[syntonic comma]] (0.08 cents). Its boundary of usability begins at [[152edo|152]] and [[159edo]], the latter inheriting the marvelous fifths from 53edo, one that [[Aura]] has shown great interest in. Bigger edos that support this excellent temperament include [[342edo]], [[494edo]], [[677edo]], [[1171edo]], among others.

[[Category:Just intonation subgroups|#]]

[[Category:Rank-4 temperaments|#]]

[[Category:11-limit|#]]

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-The '''2.3.5.11 subgroup''' (AKA ''yala'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 5, 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, 5 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 [[5/4]], [[3/2]], [[11/8]], [[11/9]], [[27/22]], and so on.
+The '''2.3.5.11 subgroup''' (a.k.a. ''yala'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 5, 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, 5 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 [[5/4]], [[3/2]], [[11/8]], [[11/9]], [[27/22]], and so on.
-In can be thought as either an extension of [[Alpharabian tuning]] with the familiar 5-limit chords and stuctures, or a retraction of the 11-limit by removing prime 7. It can be similar to the [[2.3.5.13 subgroup]], specially considering neutral interval pairs such as 39/32 ~ 11/9 and 16/13 ~ 27/22, which are connected by the small comma of [[352/351]].
+In can be thought as either an extension of [[Alpharabian tuning]] with the familiar 5-limit chords and stuctures, or a retraction of the 11-limit by removing prime 7. It can be similar to the [[2.3.5.13 subgroup]], specially considering neutral interval pairs such as 39/32~11/9 and 16/13~27/22, which are connected by the small comma of [[352/351]].
 == Regular temperaments ==
@@ Line 10: / Line 10: @@
 [[Schismic]] provides two reasonable aproximations to the 2.3.5.11 subgroup, one by finding 11/8 at the triple-augmented second (+23 fifths) through the [[cassandra]] mapping, and another by finding 11/8 at the quadruple-diminished seventh (-30 fifths) through the [[helenus]] mapping. Helenus, {{nowrap| 53 & 65 }}, provides a much better approximation to the subgroup, as both 5 and 11 are generated in the same direction.
-[[Gravity]] also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at -6 gens, 5/4 at -17 gens and 11/8 at -15 gens. It is the unique temperament in the 2.3.5.11 subgroup equating [[81/80|S9 = 81/80]], [[100/99|S10 = 100/99]], and [[121/120|S11 = 121/120]], thus tempering out [[243/242]], [[4000/3993]], and [[8019/8000]]. [[65edo]] is the unique intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup.
+[[Gravity]] also provides a very natural approximation to the 2.3.5.11 subgroup, having ~40/27 as the generator, and finding 3/2 at -6 gens, 5/4 at -17 gens and 11/8 at -15 gens. It is the unique temperament in the 2.3.5.11 subgroup equating [[81/80]] ({{S|9}}), [[100/99]] ({{S|10}}), and [[121/120]] ({{S|11}}), thus [[tempering out]] [[243/242]], [[4000/3993]], and [[8019/8000]]. [[65edo]] is the unique intersection of schismic (helenus) and gravity, and thus has, for its size, great approximations to the subgroup.
 === Rank-3 temperaments ===
-[[Vishdel]] provides a low-complexity, accurate temperament, but for those searching a much higher accuracy system, [[tritomere]] is among the best rank-3 temperaments for this case, having tremendous accuracy with manageable complexity, tempering out the difference between three [[243/242|rastmas]] and one [[81/80|syntonic comma]] (0.08 cents). Its boundary of usability begins at [[152edo|152]] and [[159edo]], the latter inheriting the marvelous fifths from 53edo, one that [[Aura]] has shown great interest in. Bigger edos that support this excellent temperament include [[342edo]], [[494edo]], [[677edo]], [[1171edo]], among others.
+[[Vishdel]] provides a low-complexity, accurate temperament, but for those searching a much higher accuracy system, [[tritomere]] is among the best rank-3 temperaments for this case, having tremendous accuracy with manageable complexity, tempering out the difference between three [[rastma]]s and one [[syntonic comma]] (0.08 cents). Its boundary of usability begins at [[152edo|152]] and [[159edo]], the latter inheriting the marvelous fifths from 53edo, one that [[Aura]] has shown great interest in. Bigger edos that support this excellent temperament include [[342edo]], [[494edo]], [[677edo]], [[1171edo]], among others.
 [[Category:Just intonation subgroups|#]]
 [[Category:Rank-4 temperaments|#]]
 [[Category:11-limit|#]]