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]]. | 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 | 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 [[3/2]] 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 == | == MOS scales == | ||
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[[Category:Just intonation subgroups|#]] | |||
[[Category:5-limit|#]] | |||
Latest revision as of 06:58, 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 (also known as the 2.3 subgroup or 3-limit) involves stacking 3/2 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 temperaments 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 scales, 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<3/1>, 2L 3s<3/1>, 2L 5s<3/1>, 2L 7s<3/1>, 2L 9s<3/1>, 2L 11s<3/1>, 13L 2s<3/1>, 15L 13s<3/1>, 28L 15s<3/1>. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents) (incidentally, this fact is realized by the bug temperament, and removing octaves from bug results in 2edt as a 3.5 equal temperament).
Generator chain
| Number of generators | Interval | Cents |
|---|---|---|
| 0 | 1/1 | 0.000 |
| 1 | 5/3 | 884.359 |
| 2 | 25/9 | 1768.717 |
| 3 | 125/81 | 751.121 |
| 4 | 625/343 | 1635.480 |