3.5 subgroup: Difference between revisions
CompactStar (talk | contribs) |
CompactStar (talk | contribs) |
||
| (12 intermediate revisions by the same user not shown) | |||
| Line 4: | Line 4: | ||
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]]. | 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 == | ||
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 a 3.5 [[equal temperament]] interpretation of [[2edt]]). | 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>]], 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 == | == Generator chain == | ||
| Line 35: | Line 38: | ||
|- | |- | ||
|4 | |4 | ||
|[[625/ | |[[625/243]] | ||
|1635.480 | |1635.480 | ||
|- | |- | ||
| Line 55: | Line 58: | ||
|- | |- | ||
|9 | |9 | ||
|[[ | |[[1953125/1594323]] | ||
|351.408 | |351.408 | ||
|- | |- | ||
| Line 63: | Line 66: | ||
|- | |- | ||
|11 | |11 | ||
|[[48828125/43046721 | |[[48828125/43046721]] | ||
|218.171 | |218.171 | ||
|- | |- | ||
| Line 113: | Line 116: | ||
|[[6561/3125]] | |[[6561/3125]] | ||
|1284.071 | |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 | |||
|} | |} | ||
</div> | </div> | ||