3.5 subgroup: Difference between revisions
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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 [[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 == | ||
<div style="display: inline-grid; margin-right: 25px;"> | |||
{|class="wikitable" | {|class="wikitable" | ||
|- | |- | ||
| Line 34: | Line 38: | ||
|- | |- | ||
|4 | |4 | ||
|[[625/ | |[[625/243]] | ||
|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 | |||
|[[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 | |||
|} | |} | ||
</div> | |||
[[ | <div style="display: inline-grid; margin-right: 25px;"> | ||
[[ | {|class="wikitable" | ||
|- | |||
!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 | |||
|} | |||
</div> | |||
Latest revision as of 23:20, 8 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>, 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 |