3.5 subgroup: Difference between revisions
CompactStar (talk | contribs) No edit summary |
CompactStar (talk | contribs) No edit summary |
||
| Line 8: | Line 8: | ||
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 a version of the pentatonic, diatonic and chromatic 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 a version of the pentatonic, diatonic and chromatic scales. | ||
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>]]. 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 | 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>]]. 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 a 3.5 [[equal temperament]] interpretation of [[2edt]]). | ||
== Generator chain == | == Generator chain == | ||
| Line 36: | Line 36: | ||
|[[625/343]] | |[[625/343]] | ||
|1635.480 | |1635.480 | ||
|- | |||
|5 | |||
|[[3125/2187]] | |||
|617.884 | |||
|- | |||
|6 | |||
|[[15625/6561]] | |||
|1502.242 | |||
|- | |||
|7 | |||
|[[78125/59049]] | |||
|484.646 | |||
|- | |||
|8 | |||
|[[390625/177147]] | |||
|1369.005 | |||
|- | |||
|9 | |||
|[[390625/177147]] | |||
|351.408 | |||
|- | |||
|10 | |||
|[[9765625/4782969]] | |||
|1235.767 | |||
|- | |||
|11 | |||
|[[48828125/43046721]]] | |||
|218.171 | |||
|- | |||
|12 | |||
|[[244140625/129140163]] | |||
|1102.530 | |||
|- | |||
|13 | |||
|[[1220703125/1162261467]] | |||
|84.933 | |||
|- | |||
|14 | |||
|[[6103515625/3486784401]] | |||
|969.219 | |||
|- | |||
|15 | |||
|[[30517578125/10460353203]] | |||
|1853.651 | |||
|} | |} | ||
[[Category:Just intonation subgroups|#]] | [[Category:Just intonation subgroups|#]] | ||
[[Category:5-limit|#]] | [[Category:5-limit|#]] | ||
Revision as of 03:52, 2 October 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 a version of the 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 a 3.5 equal temperament interpretation of 2edt).
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 |
| 5 | 3125/2187 | 617.884 |
| 6 | 15625/6561 | 1502.242 |
| 7 | 78125/59049 | 484.646 |
| 8 | 390625/177147 | 1369.005 |
| 9 | 390625/177147 | 351.408 |
| 10 | 9765625/4782969 | 1235.767 |
| 11 | 48828125/43046721] | 218.171 |
| 12 | 244140625/129140163 | 1102.530 |
| 13 | 1220703125/1162261467 | 84.933 |
| 14 | 6103515625/3486784401 | 969.219 |
| 15 | 30517578125/10460353203 | 1853.651 |