38edo
| ← 37edo | 38edo | 39edo → |
38 equal divisions of the octave (abbreviated 38edo or 38ed2), also called 38-tone equal temperament (38tet) or 38 equal temperament (38et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 38 equal parts of about 31.6 ¢ each. Each step represents a frequency ratio of 21/38, or the 38th root of 2.
Theory
Since 38 = 2*19, it can be thought of as two parallel 19edos. While the halving of the step size lowers consistency and leaves it only mediocre in terms of overall relative error, the fact that the 3rd & 5th harmonics are flat by almost exactly the same amount, while the 11th is double that means there are quite a few near perfect composite ratios, such as the the 6/5 it shares with 19edo, plus 11/9, 15/11 & 25/22, (and their inversions) while a single step nears 55/54. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30. It tempers out the same 5-limit commas as 19edo, namely 81/80, 3125/3072 and 15625/15552. In the 7-limit, we can add 50/49, and tempering out 81/80 and 50/49 gives injera temperament, for which 38 is the optimal patent val. In the 11-limit, we can add 121/120 and 176/175.
In 38df, every prime interval from 3 to 19 is characterized by a flat intonation. Furthermore, the mapping of all 19-odd-limit intervals in 38df aligns with their closest approximations in 38edo, excepting for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their secondary optimal steps within 38df. In other words, all 19-odd-limit intervals are consistent within the 38df val ⟨38 60 88 106 131 140 155 161].
The harmonic series from 1 to 20 is approximated within 38df by the sequence: 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3
[Harmonic series 2-20 in 38df]
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.0 | -7.2 | -7.4 | +10.1 | -14.5 | +12.1 | -10.2 | -13.3 | +3.3 | +12.5 | -8.2 |
| relative (%) | +0 | -23 | -23 | +32 | -46 | +38 | -32 | -42 | +10 | +40 | -26 | |
| Steps (reduced) |
38 (0) |
60 (22) |
88 (12) |
107 (31) |
131 (17) |
141 (27) |
155 (3) |
161 (9) |
172 (20) |
185 (33) |
188 (36) | |
Intervals
| Step | Cents | 19-odd-limit ratios,
treated as 38df |
Ups and Downs Notation* | ||
|---|---|---|---|---|---|
| 0 | 0.0000 | Perfect 1sn | P1 | D | |
| 1 | 31.5789 | Up 1sn | ^1 | ^D | |
| 2 | 63.1579 | Aug 1sn, dim 2nd | A1, d2 | D# | |
| 3 | 94.7368 | 20/19, 19/18, 18/17, 17/16 | Upaug 1sn, downminor 2nd | ^A1, vm2 | ^D#, vEb |
| 4 | 126.3157 | 16/15, 15/14, 14/13, 13/12 | Minor 2nd | m2 | Eb |
| 5 | 157.8947 | 12/11, 11/10 | Mid 2nd | ~2 | vE |
| 6 | 189.4737 | 10/9, 19/17, 9/8 | Major 2nd | M2 | E |
| 7 | 221.0526 | 17/15 | Upmajor 2nd | ^M2 | ^E |
| 8 | 252.6316 | 8/7, 15/13, 22/19, 7/6 | Aug 2nd, Dim 3rd | A2, d3 | E#, Fb |
| 9 | 284.2105 | 20/17, 13/11, 19/16 | Downminor 3rd | vm3 | vF |
| 10 | 315.7895 | 6/5 | Minor 3rd | m3 | F |
| 11 | 347.3684 | 17/14, 11/9 | Mid 3rd | ~3 | ^F |
| 12 | 378.9474 | 16/13, 5/4 | Major 3rd | M3 | F# |
| 13 | 410.5263 | 24/19, 19/15, 14/11 | Upmajor 3rd, Downdim 4th | ^M3, vd4 | ^F#, vGb |
| 14 | 442.1053 | 9/7, 22/17, 13/10 | Aug 3rd, dim 4th | A3, d4 | Gb |
| 15 | 473.6843 | 17/13 | Down 4th | v4 | vG |
| 16 | 505.2632 | 4/3 | Perfect 4th | P4 | G |
| 17 | 536.8421 | 19/14, 15/11, 26/19, 11/8 | Up 4th | ^4 | ^G |
| 18 | 568.4211 | 18/13, 7/5 | Aug 4th | A4 | G# |
| 19 | 600.0000 | 24/17, 17/12 | Upaug 4th, downdim 5th | ^A4, vd5 | ^G#, vAb |
| 20 | 631.5789 | 10/7, 13/9 | Dim 5th | d5 | Ab |
| 21 | 663.1579 | 16/11, 19/13, 22/15, 28/19 | Down 5th | v5 | vA |
| 22 | 694.7368 | 3/2 | Perfect 5th | P5 | A |
| 23 | 726.3157 | 26/17 | Up 5th | ^5 | ^A |
| 24 | 757.8947 | 20/13, 17/11, 14/9 | Aug 5th, dim 6th | A5, d6 | A# |
| 25 | 789.4737 | 11/7, 30/19, 19/12 | Upaug 5th, downminor 6th | ^A5, vm6 | ^A#, vBb |
| 26 | 821.0526 | 8/5, 13/8 | Minor 6th | m6 | Bb |
| 27 | 852.6316 | 18/11, 28/17 | Mid 6th | ~6 | vB |
| 28 | 884.2105 | 5/3 | Major 6th | M6 | B |
| 29 | 915.7895 | 32/19, 22/13, 17/10 | Upmajor 6th | ^M6 | ^B |
| 30 | 947.3684 | 12/7, 19/11, 26/15, 7/4 | Aug 6th, dim 7th | A6, d7 | B#, Cb |
| 31 | 978.9474 | 30/17 | Downminor 7th | vm7 | vC |
| 32 | 1010.5263 | 16/9, 34/19, 9/5 | Minor 7th | m7 | C |
| 33 | 1042.1053 | 20/11, 11/6 | Mid 7th | ~7 | ^C |
| 34 | 1073.6843 | 24/13, 13/7, 28/15, 15/8 | Major 7th | M7 | C# |
| 35 | 1105.2632 | 32/17, 17/9, 36/19, 19/10 | Upmajor 7th, Downdim 8ve | ^M7, vd8 | ^C#, vDb |
| 36 | 1136.8421 | Aug 7th, dim 8ve | A7, d8 | Db | |
| 37 | 1168.4211 | Down 8ve | v8 | vD | |
| 38 | 1200.0000 | Perfect 8ve | P8 | D | |
* Ups and downs may be substituted with semi-sharps and semi-flats, respectively