16ed5/3

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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 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).

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 E
1 55.2724 36/35, 33/32, 31/30 aug unison quatertone E#
2 110.5448 16/15, (21/20) min mos2nd minor second Fb
3 165.8173 11/10 maj mos2nd neutral second F
4 221.0897 8/7, 17/15 min mos3rd major second F#/Gb
5 276.3621 75/64, 7/6, 20/17 maj mos3rd subminor third G
6 331.6345 6/5, 40/33, 17/14 dim mos4th minor third G#/Ab
7 386.9069 5/4 perf mos4th major third A
8 442.1794 9/7, 35/27, 22/17 aug mos4th supermajor third A#/Bb
9 497.4517 4/3 perf mos5th just fourth B
10 552.7242 25/18, 11/8, 18/13 aug mos5th wide fourth B#
11 607.9966 64/45, 10/7, 17/12 min mos6th large tritone Cb
12 663.2690 72/49, 22/15 maj mos6th narrow fifth C
13 718.5415 3/2, 50/33 min mos7th acute fifth C#/Db
14 773.8129 25/16 maj mos7th subminor sixth D
15 829.0863 8/5, 13/8 dim mos8ave minor sixth D#/Eb
16 884.3587 5/3 mosoctave major sixth E
17 939.6311 12/7, 19/11 aug mos8ave supermajor sixth E#
18 994.9035 16/9, (7/4) min mos9th minor seventh Fb
19 1050.1760 11/6 maj mos9th neutral seventh F
20 1105.4484 40/21, (27/14), 17/9 min mos10th major seventh F#/Gb
21 1160.7208 35/18, 43/22 maj mos10th narrow octave G
22 1215.9932 2/1 dim mos11th octave G#/Ab

These intervals are close to a few other related non-octave 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 (245/243), as the half mosoctave is midway between 9/7 and 35/27. This is analogous to the tritone in 2n edo systems. The keema (875/864) is tempered due to the septimal interpretation of the diatonic sevenths, and the Motwellsma (99/98) 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 (64/63) 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

Vals: 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

Vals: 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

Vals: 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

Vals: 9ed5/3, 16ed5/3