3.5 subgroup: Difference between revisions
CompactStar (talk | contribs) Created page with "The '''3.5 subgroup''' is a non-octave just intonation subgroup consisting of rational intervals where 3 and 5 are the only allowable prime factors, so that every such interval may be written as a ratio of integers which are products of 3 and 5. This is an infinite set. Some examples of intervals in this subgroup are 5/3, 9/5, 27/25, and so on. The 3.5 subgroup is a retraction of the 5-limit, obtained by removing prime 2. Its simples..." Tags: Mobile edit Mobile web edit |
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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]]. | 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 tritave-equivalence, it is an infinite chain of stacking 5/3 with tritave-reduction, analogous to how [[Pythagorean tuning]] involves stacking [[2/1]] with octave-reduction. It can be regarded as an application of the [[Pythagorean tuning]] principle (the stacking the smallest prime larger than the equave) to tritave-equivalence. The 3.5 subgroup is related to tritave-equivalent]] [[rank-2 temperaments]] generated by ~[[5/3]], such as [[Arcturus]] and [[Delta Centauri]]. The famous rank 2 Bohlen-Pierce/sensamagic temperament, however, is generated by [[7/3]]. | If used with tritave-equivalence, it is an infinite chain of stacking the classical major sixth [[5/3]] with tritave-reduction, analogous to how [[Pythagorean tuning]] involves stacking [[2/1]] with octave-reduction. It can be regarded as an application of the [[Pythagorean tuning]] principle (the stacking the smallest prime larger than the equave) to tritave-equivalence. The 3.5 subgroup is related to tritave-equivalent]] [[rank-2 temperaments]] generated by ~[[5/3]], such as [[Arcturus]] and [[Delta Centauri]]. The famous rank 2 Bohlen-Pierce/sensamagic temperament, however, is generated by [[7/3]]. | ||
Revision as of 04:21, 19 September 2026
The 3.5 subgroup is a non-octave just intonation subgroup consisting of rational intervals where 3 and 5 are the only allowable prime factors, so that every such interval may be written as a ratio of integers which are products of 3 and 5. This is an infinite set. Some examples of intervals in this subgroup are 5/3, 9/5, 27/25, and so on.
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.
If used with tritave-equivalence, it is an infinite chain of stacking the classical major sixth 5/3 with tritave-reduction, analogous to how Pythagorean tuning involves stacking 2/1 with octave-reduction. It can be regarded as an application of the Pythagorean tuning principle (the stacking the smallest prime larger than the equave) to tritave-equivalence. The 3.5 subgroup is related to tritave-equivalent]] rank-2 temperaments generated by ~5/3, such as Arcturus and Delta Centauri. The famous rank 2 Bohlen-Pierce/sensamagic temperament, however, is generated by 7/3.