16ed5/3: Difference between revisions
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! Cents | ! Cents | ||
! Approximate intervals | ! Approximate intervals | ||
! | ! 2L 5s<5/3> mos-interval | ||
! Diatonic interval | ! Diatonic interval | ||
! 2L 5s<5/3> notation | ! 2L 5s<5/3> notation | ||
! 1L 4s<5/3> ([[Blackcomb]]) notation | |||
|- | |- | ||
| '''0''' | | '''0''' | ||
| Line 20: | Line 21: | ||
| '''unison''' | | '''unison''' | ||
| '''E''' | | '''E''' | ||
| '''C''' | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 25: | Line 27: | ||
| 36/35, 33/32, 31/30 | | 36/35, 33/32, 31/30 | ||
| aug unison | | aug unison | ||
| | | quartertone | ||
| E# | | E# | ||
| C# | |||
|- | |- | ||
| 2 | | 2 | ||
| Line 34: | Line 37: | ||
| minor second | | minor second | ||
| Fb | | Fb | ||
| Cx, Dbb | |||
|- | |- | ||
| 3 | | 3 | ||
| Line 41: | Line 45: | ||
| neutral second | | neutral second | ||
| F | | F | ||
| Db | |||
|- | |- | ||
| 4 | | 4 | ||
| Line 48: | Line 53: | ||
| major second | | major second | ||
| F#/Gb | | F#/Gb | ||
| D | |||
|- | |- | ||
| 5 | | 5 | ||
| Line 55: | Line 61: | ||
| subminor third | | subminor third | ||
| G | | G | ||
| D# | |||
|- | |- | ||
| 6 | | 6 | ||
| Line 62: | Line 69: | ||
| minor third | | minor third | ||
| G#/Ab | | G#/Ab | ||
|- | | Eb | ||
|- | |||
| 7 | | 7 | ||
| ''386.9069'' | | ''386.9069'' | ||
| Line 69: | Line 77: | ||
| major third | | major third | ||
| A | | A | ||
| E | |||
|- | |- | ||
| 8 | | 8 | ||
| Line 76: | Line 85: | ||
| supermajor third | | supermajor third | ||
| A#/Bb | | A#/Bb | ||
|- | |- | ||
| 9 | | 9 | ||
| ''497.4517'' | | ''497.4517'' | ||
| Line 83: | Line 92: | ||
| just fourth | | just fourth | ||
| B | | B | ||
| E#, Fb | |||
|- | |- | ||
| 10 | | 10 | ||
| Line 90: | Line 100: | ||
| wide fourth | | wide fourth | ||
| B# | | B# | ||
| F | |||
|- | |- | ||
| 11 | | 11 | ||
| Line 97: | Line 108: | ||
| large tritone | | large tritone | ||
| Cb | | Cb | ||
| F# | |||
|- | |- | ||
| 12 | | 12 | ||
| Line 104: | Line 116: | ||
| narrow fifth | | narrow fifth | ||
| C | | C | ||
| Gb | |||
|- | |- | ||
| 13 | | 13 | ||
| Line 111: | Line 124: | ||
| acute fifth | | acute fifth | ||
| C#/Db | | C#/Db | ||
| G | |||
|- | |- | ||
| 14 | | 14 | ||
| Line 118: | Line 132: | ||
| subminor sixth | | subminor sixth | ||
| D | | D | ||
| G# | |||
|- | |- | ||
| 15 | | 15 | ||
| Line 125: | Line 140: | ||
| minor sixth | | minor sixth | ||
| D#/Eb | | D#/Eb | ||
|- | | Cb | ||
|- | |||
| '''16''' | | '''16''' | ||
| '''884.3587''' | | '''884.3587''' | ||
| Line 132: | Line 148: | ||
| '''major sixth''' | | '''major sixth''' | ||
| '''E''' | | '''E''' | ||
| '''C''' | |||
|- | |- | ||
| 17 | | 17 | ||
| Line 139: | Line 156: | ||
| supermajor sixth | | supermajor sixth | ||
| E# | | E# | ||
| | |||
|- | |- | ||
| 18 | | 18 | ||
| Line 146: | Line 164: | ||
| minor seventh | | minor seventh | ||
| Fb | | Fb | ||
| | |||
|- | |- | ||
| 19 | | 19 | ||
| Line 153: | Line 172: | ||
| neutral seventh | | neutral seventh | ||
| F | | F | ||
| | |||
|- | |- | ||
| 20 | | 20 | ||
| Line 160: | Line 180: | ||
| major seventh | | major seventh | ||
| F#/Gb | | F#/Gb | ||
| | |||
|- | |- | ||
| 21 | | 21 | ||
| Line 167: | Line 188: | ||
| narrow octave | | narrow octave | ||
| G | | G | ||
| | |||
|- | |- | ||
| 22 | | 22 | ||
| Line 174: | Line 196: | ||
| octave | | octave | ||
| G#/Ab | | G#/Ab | ||
| | |||
|} | |} | ||
These intervals are close to a few other related | These intervals are close to a few other related scales: | ||
{| class="wikitable left-all" | {| class="wikitable left-all" | ||
! | ! | ||
Revision as of 00:23, 10 March 2023
| ← 15ed5/3 | 16ed5/3 | 17ed5/3 → |
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 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).
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.
| Degree | Cents | Approximate intervals | 2L 5s<5/3> mos-interval | Diatonic interval | 2L 5s<5/3> notation | 1L 4s<5/3> (Blackcomb) notation |
|---|---|---|---|---|---|---|
| 0 | 0.0000 | 1 | unison | unison | E | C |
| 1 | 55.2724 | 36/35, 33/32, 31/30 | aug unison | quartertone | E# | C# |
| 2 | 110.5448 | 16/15, (21/20) | min mos2nd | minor second | Fb | Cx, Dbb |
| 3 | 165.8173 | 11/10 | maj mos2nd | neutral second | F | Db |
| 4 | 221.0897 | 8/7, 17/15 | min mos3rd | major second | F#/Gb | D |
| 5 | 276.3621 | 75/64, 7/6, 20/17 | maj mos3rd | subminor third | G | D# |
| 6 | 331.6345 | 6/5, 40/33, 17/14 | dim mos4th | minor third | G#/Ab | Eb |
| 7 | 386.9069 | 5/4 | perf mos4th | major third | A | E |
| 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 | E#, Fb |
| 10 | 552.7242 | 25/18, 11/8, 18/13 | aug mos5th | wide fourth | B# | F |
| 11 | 607.9966 | 64/45, 10/7, 17/12 | min mos6th | large tritone | Cb | F# |
| 12 | 663.2690 | 72/49, 22/15 | maj mos6th | narrow fifth | C | Gb |
| 13 | 718.5415 | 3/2, 50/33 | min mos7th | acute fifth | C#/Db | G |
| 14 | 773.8129 | 25/16 | maj mos7th | subminor sixth | D | G# |
| 15 | 829.0863 | 8/5, 13/8 | dim mos8ave | minor sixth | D#/Eb | Cb |
| 16 | 884.3587 | 5/3 | mosoctave | major sixth | E | C |
| 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 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:
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).
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