16ed5/3
| ← 15ed5/3 | 16ed5/3 | 17ed5/3 → |
16ed5/3 is the equal division of the just major sixth into sixteen parts of 55.2724 cents each, corresponding to 21.7106edo. It is very closely related to the escapade temperament. It is vaguely equivalent to 22edo.
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 17¢, 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).
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +16.0 | -22.7 | -23.3 | -22.7 | -6.7 | +2.8 | -7.3 | +9.9 | -6.7 | -5.9 | +9.3 |
| relative (%) | +29 | -41 | -42 | -41 | -12 | +5 | -13 | +18 | -12 | -11 | +17 | |
| Steps (reduced) |
22 (6) |
34 (2) |
43 (11) |
50 (2) |
56 (8) |
61 (13) |
65 (1) |
69 (5) |
72 (8) |
75 (11) |
78 (14) | |
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). It can also be notated using the fifth-generated Blackcomb temperament as discussed in #Temperaments, which lines up quite nicely with diatonic notation, aside from the "minor second" being in neutral second range and "perfect fourth" being in superfourth range.
| Degree | Cents | Approximate intervals | 2L 5s<5/3> mos-interval | Diatonic interval | 2L 5s<5/3> notation | 1L 4s<5/3> (Blackcomb[5]) interval | 1L 4s<5/3> (Blackcomb[5]) notation |
|---|---|---|---|---|---|---|---|
| 0 | 0.0000 | 1 | unison | unison | E | unison | C |
| 1 | 55.2724 | 36/35, 33/32, 31/30 | aug unison | quartertone | E# | aug unison | C# |
| 2 | 110.5448 | 16/15, (21/20) | min mos2nd | minor second | Fb | double-aug unison, dim second | Cx, Dbb |
| 3 | 165.8173 | 11/10 | maj mos2nd | neutral second | F | minor second | Db |
| 4 | 221.0897 | 8/7, 17/15 | min mos3rd | major second | F#/Gb | major second | D |
| 5 | 276.3621 | 75/64, 7/6, 20/17 | maj mos3rd | subminor third | G | aug second | D# |
| 6 | 331.6345 | 6/5, 40/33, 17/14 | dim mos4th | minor third | G#/Ab | minor third | Eb |
| 7 | 386.9069 | 5/4 | perf mos4th | major third | A | major third | E |
| 8 | 442.1794 | 9/7, 35/27, 22/17 | aug mos4th | supermajor third | A#/Bb | aug third | E# |
| 9 | 497.4517 | 4/3 | perf mos5th | just fourth | B | dim fourth | Fb |
| 10 | 552.7242 | 25/18, 11/8, 18/13 | aug mos5th | wide fourth | B# | perfect fourth | F |
| 11 | 607.9966 | 64/45, 10/7, 17/12 | min mos6th | large tritone | Cb | aug fourth | F# |
| 12 | 663.2690 | 72/49, 22/15 | maj mos6th | narrow fifth | C | dim fifth | Gb |
| 13 | 718.5415 | 3/2, 50/33 | min mos7th | acute fifth | C#/Db | perfect fifth | G |
| 14 | 773.8129 | 25/16 | maj mos7th | subminor sixth | D | aug fifth | G# |
| 15 | 829.0863 | 8/5, 13/8 | dim mos8ave | minor sixth | D#/Eb | dim sixth | Cb |
| 16 | 884.3587 | 5/3 | mosoctave | major sixth | E | perfect sixth | C |
| 17 | 939.6311 | 12/7, 19/11 | aug mos8ave | supermajor sixth | E# | aug sixth | C# |
| 18 | 994.9035 | 16/9, (7/4) | min mos9th | minor seventh | Fb | double-aug sixth, dim seventh | Cx, Dbb |
| 19 | 1050.1760 | 11/6 | maj mos9th | neutral seventh | F | minor seventh | Db |
| 20 | 1105.4484 | 40/21, (27/14), 17/9 | min mos10th | major seventh | F#/Gb | major seventh | D |
| 21 | 1160.7208 | 35/18, 43/22 | maj mos10th | narrow octave | G | aug seventh | D# |
| 22 | 1215.9932 | 2/1 | dim mos11th | octave | G#/Ab | minor octave | Eb |
These intervals are close to a few other related scales:
| 22edo | 7ed5/4 | 23ed18\17 | 16ed5/3 | 9ed4/3 | 43ed4 | 34edt | 21edo | |
|---|---|---|---|---|---|---|---|---|
| 1 | 54.54545 | 55.188 | 55.2429 | 55.2724 | 55.338 | 55.8140 | 55.9399 | 57.1429 |
| 2 | 109.0909 | 110.375 | 110.4859 | 110.5448 | 110.677 | 111.6729 | 111.8797 | 114.2857 |
| 3 | 163.6364 | 165.563 | 165.7288 | 165.8173 | 166.015 | 167.4419 | 167.8196 | 171.4286 |
| 4 | 218.1818 | 220.751 | 220.9718 | 221.0897 | 221.353 | 223.2558 | 223.7594 | 228.5714 |
| 5 | 272.7273 | 275.938 | 276.2147 | 276.3621 | 276.692 | 279.0698 | 279.6993 | 285.7143 |
| 6 | 327.2727 | 331.126 | 331.4576 | 331.6345 | 332.030 | 334.8837 | 335.6391 | 342.8571 |
| 7 | 381.8182 | 386.314 | 386.7006 | 386.9069 | 387.368 | 390.6977 | 391.5790 | 400 |
| 8 | 436.3636 | 441.501 | 441.9435 | 442.1794 | 442.707 | 446.5116 | 447.5188 | 457.1429 |
| 9 | 490.9091 | 496.689 | 497.1865 | 497.4517 | 498.045 | 502.3256 | 503.4587 | 514.2857 |
| 10 | 545.5455 | 551.877 | 552.4294 | 552.7242 | 553.383 | 558.1395 | 559.3985 | 571.4286 |
| 11 | 600 | 607.064 | 607.6723 | 607.9966 | 608.722 | 613.9535 | 615.3384 | 628.5714 |
| 12 | 654.5455 | 662.252 | 662.9153 | 663.269 | 664.060 | 669.7674 | 671.2782 | 685.7143 |
| 13 | 709.0909 | 717.440 | 718.1582 | 718.5415 | 719.398 | 725.5814 | 727.2181 | 742.8571 |
| 14 | 763.6364 | 772.627 | 773.4011 | 773.8129 | 774.737 | 781.3954 | 783.1579 | 800 |
| 15 | 818.1818 | 827.815 | 828.6441 | 829.0863 | 830.075 | 837.7209 | 839.0978 | 857.1429 |
| 16 | 872.7273 | 883.003 | 883.8870 | 884.3587 | 885.413 | 893.0233 | 895.0376 | 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:
| F# | C# | G# | D# | A# | E# | B# | F | C | G | D | A | E | B | Fb | Cb | Gb | Db | Ab | Eb | Db | Fbb | Cbb | Gbb | Dbb |
| 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 |
Commas
Depending on your mapping, 16ed5/3 can be said to temper a number of commas, including the diaschisma, the marvel comma, Archytas' comma, and the jubilisma, all discussed in the temperaments section. In addition, being an even division of the 5/3, it tempers the sensamagic comma, as the half mosoctave is midway between 9/7 and 35/27. This is analogous to the tritone in 2n edo systems. The keema is tempered due to the septimal interpretation of the diatonic sevenths, and the mothwellsma is tempered by two major mos3rds (7/6) resulting in an augmented mos5th (11/8).
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 3 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.
Both of the 7-limit approaches also temper Archytas' comma as a result of equating the 16/9 with 7/4, and the jubilisma (50/49) due to tritone equivalence. These are relatively large commas, given the step size (about half, and 7/11ths respectively).
This shows the close relationships with srutal and pajara octave temperaments. In 16ed5/3's case, there is a close equivalence to 22edo's pajara tuning.
As 3 semitones make a period-reduced octave, and it alludes to tritone tempering, I propose the name tristone for the basic diaschismic temperament, based on the 16/15 to 6/5 relationship, as well as the following variants and extensions:
Tristone
Subgroup: 5/3.20/9.10/3
Comma list: 2048/2025
POL2 generator: ~5/4 = 389.8224
Mapping: [⟨1 2 5], ⟨0 -1 -6]]
TE complexity: 1.988720
RMS error: 2.228679 cents
Optimal ET sequence: 9ed5/3, 16ed5/3, 25ed5/3
Tridistone
Subgroup: 5/3.20/9.10/3.1000/189
Comma list: 2048/2025, 225/224, 64/63, 50/49
POL2 generator: ~5/4 = 389.6140
Mapping: [⟨1 2 5 5], ⟨0 -1 -6 -4]]
TE complexity: 1.724923
RMS error: 8.489179 cents
Optimal ET sequence: 9ed5/3, 16ed5/3
Metatristone
Subgroup: 5/3.20/9.5/2
Comma list: 2048/2025
POL2 generator: ~5/4 = 390.5180
Mapping: [⟨1 2 4], ⟨0 -1 -5]]
TE complexity: 2.192193
RMS error: 2.021819 cents
Optimal ET sequence: 9ed5/3, 16ed5/3, 25ed5/3
Metatridistone
Subgroup: 5/3.20/9.5/2.250/63
Comma list: 2048/2025, 225/224, 64/63, 50/49
POL2 generator: ~5/4 = 390.5430
Mapping: [⟨1 2 4 4], ⟨0 -1 -5 -3]]
TE complexity: 1.895168
RMS error: 7.910273 cents
Optimal ET sequence: 9ed5/3, 16ed5/3
16ed5/3 also supports Blackcomb temperament which is built on 5/4 and 3/2 in a very similar way to octave-repeating meantone but is less accurate. Blackcomb tempers out the comma 250/243, the amount by which 3 3/2's exceed 5/4 sixth-reduced, in the 5/3.2.3 subgroup (equal to the 5-limit).