3.5 subgroup: Difference between revisions

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The 3.5 subgroup is a [[retraction]] of the [[5-limit]], obtained by removing prime 2. Its simplest [[expansion]] is the [[3.5.7 subgroup]], which adds prime [[7/1|7]].

If used with [[3/1|tritave]]-equivalence, it is an infinite chain of stacking the classical major sixth [[5/3]] with tritave-reduction, analogous to how [[Pythagorean tuning]] (also known as the 2.3 subgroup or 3-limit) involves stacking [[2/1]] with octave-reduction. It can be regarded as an application of the [[Pythagorean ~~tuning]]~~ principle (stacking the smallest prime larger than the equave) to tritave-equivalence. The 3.5 subgroup is related to tritave-equivalent [[rank-2 temperament]]s generated by ~[[5/3]], such as 3.5.7 [[Arcturus]] and 3.5.11 [[Delta Centauri]]. The famous rank 2 Bohlen-Pierce/sensamagic temperament, however, is generated by [[7/3]].

If used with [[3/1|tritave]]-equivalence, it is an infinite chain of stacking the classical major sixth [[5/3]] with tritave-reduction, analogous to how [[Pythagorean tuning]] (also known as the 2.3 subgroup or 3-limit) involves stacking [[2/1]] with octave-reduction. It can be regarded as an application of the Pythagorean principle (stacking the smallest prime harmonic larger than the [[equave]]) to tritave-equivalence. The 3.5 subgroup is related to tritave-equivalent [[rank-2 temperament]]s generated by ~[[5/3]], such as 3.5.7 [[Arcturus]] and 3.5.11 [[Delta Centauri]]. The famous rank 2 Bohlen-Pierce/sensamagic temperament, however, is generated by [[7/3]].

== MOS scales ==

As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.

The non-trivial MOS scales that are produced are: 2L 1s, 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s,

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 The 3.5 subgroup is a [[retraction]] of the [[5-limit]], obtained by removing prime 2. Its simplest [[expansion]] is the [[3.5.7 subgroup]], which adds prime [[7/1|7]].
-If used with [[3/1|tritave]]-equivalence, it is an infinite chain of stacking the classical major sixth [[5/3]] with tritave-reduction, analogous to how [[Pythagorean tuning]] (also known as the 2.3 subgroup or 3-limit) involves stacking [[2/1]] with octave-reduction. It can be regarded as an application of the [[Pythagorean tuning]] principle (stacking the smallest prime larger than the equave) to tritave-equivalence. The 3.5 subgroup is related to tritave-equivalent [[rank-2 temperament]]s generated by ~[[5/3]], such as 3.5.7 [[Arcturus]] and 3.5.11 [[Delta Centauri]]. The famous rank 2 Bohlen-Pierce/sensamagic temperament, however, is generated by [[7/3]].
+If used with [[3/1|tritave]]-equivalence, it is an infinite chain of stacking the classical major sixth [[5/3]] with tritave-reduction, analogous to how [[Pythagorean tuning]] (also known as the 2.3 subgroup or 3-limit) involves stacking [[2/1]] with octave-reduction. It can be regarded as an application of the Pythagorean principle (stacking the smallest prime harmonic larger than the [[equave]]) to tritave-equivalence. The 3.5 subgroup is related to tritave-equivalent [[rank-2 temperament]]s generated by ~[[5/3]], such as 3.5.7 [[Arcturus]] and 3.5.11 [[Delta Centauri]]. The famous rank 2 Bohlen-Pierce/sensamagic temperament, however, is generated by [[7/3]].
 == MOS scales ==
 As it is a [[rank-2]] system, stacking 5/3 with tritave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.
-The non-trivial MOS scales that are produced are: 2L 1s, 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s,
+The non-trivial MOS scales that are produced are: [[2L 1s (3/1-equivalent)|2L 1s<3/1>]], [[2L 3s (3/1-equivalent)|2L 3s<3/1>]], [[2L 5s (3/1-equivalent))|2L 5s<3/1>]], [[2L 7s (3/1-equivalent)|2L 7s<3/1>]], [[2L 9s (3/1-equivalent)|2L 9s<3/1>]], [[2L 11s (3/1-equivalent)|2L 11s<3/1>]], [[13L 2s (3/1-equivalent)|13L 2s<3/1>]], [[15L 13s (3/1-equivalent)|15L 13s<3/1>]], [[28L 15s (3/1-equivalent)|28L 15s<3/1>]]