3.5 subgroup
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>, etc. 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). All of the scales with less than 11 notes are very hard (lopsided L/s ratio) due to the proximity to 2edt, making them difficult to use melodically, so it's more practical to use an 11-note or higher scale.
The 13-note MOS scale is also called Trithagorean by Jake Freivald (perhaps could be called Trithagorean[13] to specify, like saying Pythagorean[7] or Pythagorean[12]).
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 | 625/243 | 1635.480 |
| 5 | 3125/2187 | 617.884 |
| 6 | 15625/6561 | 1502.242 |
| 7 | 78125/59049 | 484.646 |
| 8 | 390625/177147 | 1369.005 |
| 9 | 1953125/1594323 | 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 |
| Number of generators | Interval | Cents |
|---|---|---|
| 0 | 1/1 | 0.000 |
| -1 | 9/5 | 1017.596 |
| -2 | 27/25 | 133.238 |
| -3 | 243/125 | 1150.834 |
| -4 | 729/625 | 266.475 |
| -5 | 6561/3125 | 1284.071 |
| -6 | 19683/15625 | 399.713 |
| -7 | 177147/78125 | 1417.309 |
| -8 | 531441/390625 | 532.950 |
| -9 | 4782969/1953125 | 1550.547 |
| -10 | 14348907/9765625 | 666.188 |
| -11 | 129140163/48828125 | 1683.784 |
| -12 | 387420489/244140625 | 799.425 |
| -13 | 3486784401/1220703125 | 1817.021 |
| -14 | 10460353203/6103515625 | 932.663 |
| -15 | 31381059609/30517578125 | 48.304 |