16ed5/3
16ed5/3 (or less accurately 16edVI) is the equal division of the just major sixth into sixteen parts of 55.2724 cents each, corresponding to 21.7106 edo. It is very closely related to the escapade temperament.
It very accurately approximates a number of low complexity just intervals, such as: 4/3 (<1¢), 5/4 (<1¢), 11/8 (<2¢), 11/10 (<1¢), 16/15 (<2¢), and 25/16 (<2¢). It also approximates the just fifth and octave to within 20¢, making it a flexible non-octave scale. Notably, having a period of 5/3, the diatonic minor third (6/5) is the period-reduced diatonic octave. This means both are approximated identically (16¢ sharp).
Intervals
16ed5/3 can be notated using steps 7 (~5/4) and 9 (~4/3) as generators, as these are accurate to within 0.6¢. The resulting scale is a heptatonic 2L 5s (similar to the octave repeating antidiatonic).
| Degree | Cents | Approximate intervals | Mos-interval | Diatonic interval | Notation |
|---|---|---|---|---|---|
| 0 | 0.0000 | 1 | unison | unison | A |
| 1 | 55.2724 | 36/35, 33/32, 31/30 | aug unison | quatertone | A# |
| 2 | 110.5448 | 16/15 | min mos2nd | minor second | Bb |
| 3 | 165.8173 | 11/10 | maj mos2nd | neutral second | B |
| 4 | 221.0897 | 8/7, 17/15 | min mos3rd | major second | Cb |
| 5 | 276.3621 | 75/64, 7/6, 20/17 | maj mos3rd | subminor third | C |
| 6 | 331.6345 | 6/5, 40/33, 17/14 | dim mos4th | minor third | Db |
| 7 | 386.9069 | 5/4 | perf mos4th | major third | D |
| 8 | 442.1794 | 9/7, 22/17 | aug mos4th | supermajor third | D# |
| 9 | 497.4517 | 4/3 | perf mos5th | just fourth | E |
| 10 | 552.7242 | 25/18, 11/8, 18/13 | aug mos5th | wide fourth | E# |
| 11 | 607.9966 | 64/45, 10/7, 17/12 | min mos6th | large tritone | Fb |
| 12 | 663.2690 | 72/49, 22/15 | maj mos6th | narrow fifth | F |
| 13 | 718.5415 | 3/2, 50/33 | min mos7th | acute fifth | F# |
| 14 | 773.8129 | 25/16 | maj mos7th | subminor sixth | G |
| 15 | 829.0863 | 8/5, 13/8 | dim mos8ave | minor sixth | G# |
| 16 | 884.3587 | 5/3 | mosoctave | major sixth | A |
| 17 | 939.6311 | 12/7, 19/11 | aug mos8ave | supermajor sixth | A# |
| 18 | 994.9035 | 16/9 | min mos9th | minor seventh | Bb |
| 19 | 1050.1760 | 11/6 | maj mos9th | neutral seventh | B |
| 20 | 1105.4484 | 40/21, 17/9 | min mos10th | major seventh | Cb |
| 21 | 1160.7208 | 35/18, 43/22 | maj mos10th | narrow octave | C |
| 22 | 1215.9932 | 2/1 | dim mos11th | octave | C# |
These intervals are close to a few other related non-octave scales:
| 16ed16\22 | 7ed5/4 | 16ed5/3 | 9ed4/3 | 43ed4 | 16ed16\21 | |
|---|---|---|---|---|---|---|
| 1 | 54.54545 | 55.188 | 55.2724 | 55.338 | 55.81395 | 57.1429 |
| 2 | 109.0909 | 110.375 | 110.5448 | 110.677 | 111.6729 | 114.2857 |
| 3 | 163.6364 | 165.563 | 165.8173 | 166.015 | 167.4419 | 171.4286 |
| 4 | 218.1818 | 220.751 | 221.0897 | 221.353 | 223.2558 | 228.5714 |
| 5 | 272.7273 | 275.938 | 276.3621 | 276.692 | 279.0698 | 285.7143 |
| 6 | 327.2727 | 331.126 | 331.6345 | 332.030 | 334.8837 | 342.8571 |
| 7 | 381.8182 | 386.314 | 386.9069 | 387.368 | 390.6977 | 400 |
| 8 | 436.3636 | 441.501 | 442.1794 | 442.707 | 446.5116 | 457.1429 |
| 9 | 490.9091 | 496.689 | 497.4517 | 498.045 | 502.3256 | 514.2857 |
| 10 | 545.54545 | 551.877 | 552.7242 | 553.383 | 558.1395 | 571.4286 |
| 11 | 600 | 607.064 | 607.9966 | 608.722 | 613.9535 | 628.5714 |
| 12 | 654.54545 | 662.252 | 663.269 | 664.060 | 669.7674 | 685.7143 |
| 13 | 709.0909 | 717.440 | 718.54145 | 719.398 | 725.5814 | 742.8571 |
| 14 | 763.6364 | 772.627 | 773.8129 | 774.737 | 781.39535 | 800 |
| 15 | 818.1818 | 827.815 | 829.0863 | 830.075 | 837.7209 | 857.1429 |
| 16 | 872.7273 | 883.003 | 884.3587 | 885.413 | 893.0233 | 914.2857 |
MOS Scales
16edVI supports the same MOS scales as 16edo, as such it contains the following scales:
| Periods
per octave |
Generator | Pattern |
|---|---|---|
| 1 | 1\16 | 1L ns (pathological) |
| 1 | 3\16 | 1L 4s, 5L 1s |
| 1 | 5\16 | 3L 4s, 3L 7s |
| 1 | 7\16 | 2L 5s, 7L 2s |
| 2 | 1\16 | 2L 8s, 2L 10s, 2L 12s |
| 2 | 3\16 | 4L 2s, 6L 4s |
| 4 | 1\16 | 4L 4s, 4L 8s |
For the 2L 5s scale, the genchain is this:
| B# | F# | C# | G# | D# | A# | E# | B | F | C | G | D | A | E | Bb | Fb | Cb | Gb | Db | Ab | Eb | Bbb | Fbb | Cbb | Gbb |
| A2 | A6 | A3 | A7 | A4 | A1 | A5 | M2 | M6 | M3 | M7 | P4 | P1 | P5 | m2 | m6 | m3 | m7 | d4 | d1 | d5 | d2 | d6 | d3 | d7 |
Temperaments
The 2L 5s scale is generated by a very accurate 4/3, such that two of them wind up on a near exact 16/9, which period-reduces to 16/15 (the minor mossecond). This interval taken 2 times is approximated by an 8/7, and taken 4 times is approximated by a 6/5 (or 2/1 in the next mosoctave). These 2 equivalencies result in two tempered commas: the marvel comma - 225/224 ((16/15)2=(8/7)), and the diaschisma - 2048/2025 ((16/15)3=(6/5)). The diaschisma can also be tempered by taking 5 generators to mean a 3/2 ((4/3)5=(3/2)·(5/3)2), while the marvel comma can also be tempered with a stack of 3 generators, making a 10/7 ((4/3)3=(10/7)·(5/3)). The tempered marvel comma also means that the two large tritones (pental and septimal) are addressed by the same scale step. The tempered diaschisma, on the other hand, means that both pental tritones are also addressed by the same scale step. I propose the name tristone for the basic temperament, as 3 semitones make a period-reduced octave, and it alludes to the tritone tempering.