33ed8/3
| ← 32ed8/3 | 33ed8/3 | 34ed8/3 → |
(semiconvergent)
33 equal divisions of 8/3 (abbreviated 33ed8/3) is a nonoctave tuning system that divides the interval of 8/3 into 33 equal parts of about 51.5 ¢ each. Each step represents a frequency ratio of (8/3)1/33, or the 33rd root of 8/3.
33ed8/3 was first explored by Budjarn Lambeth in his 2026 piece Improvisation in 33ed8/3. Below his music video, he gave the following description of the tuning and why he used it:
"33ed8/3 is a very squished version of 23edo such that the octave equals 1183.5 cents instead of the usual 1200.0.
I came across 33ed8/3 because I tried out the 10 10 3 MOS scale from 23edo using the "platinum" timbre in Scale Workshop, and I realized that the interval near 8/3 was sounding more like an equivalence to my ear than the octave was!
The 8/3 interval was 33 steps of 23edo, so I decided to change tuning to 33ed8/3, and it worked perfectly, the equivalence became even stronger and everything overall sounded "more in-tune".
Before today, I would never have thought that 8/3 could sound like an equivalence. If I had to guess, the platinum timbre is the reason. It has a stretched overtone spectrum, so perhaps it is stretched just the right amount such that the overtone that occurs at 2/1 for a harmonic instrument, occurs at 8/3 for struck platinum?
That's just a guess, I have no idea whether it's true."
Besides 23edo, a closer lookalike to 33ed8/3 is actually 36edt, although it still has significant enough differences to be an audibly different scale - its mappings of every harmonic after the first are different (a different number of steps).
Interval table
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 51.5 | |
| 2 | 102.9 | 18/17 |
| 3 | 154.4 | 23/21 |
| 4 | 205.8 | 26/23 |
| 5 | 257.3 | 22/19 |
| 6 | 308.7 | 6/5, 25/21 |
| 7 | 360.2 | 21/17, 27/22 |
| 8 | 411.6 | 19/15 |
| 9 | 463.1 | 13/10, 17/13 |
| 10 | 514.6 | 23/17 |
| 11 | 566 | 18/13, 25/18 |
| 12 | 617.5 | 10/7 |
| 13 | 668.9 | 22/15, 25/17 |
| 14 | 720.4 | |
| 15 | 771.8 | |
| 16 | 823.3 | 21/13 |
| 17 | 874.8 | |
| 18 | 926.2 | 12/7, 17/10 |
| 19 | 977.7 | 7/4 |
| 20 | 1029.1 | |
| 21 | 1080.6 | 13/7 |
| 22 | 1132 | 23/12, 25/13 |
| 23 | 1183.5 | |
| 24 | 1234.9 | |
| 25 | 1286.4 | 19/9, 21/10 |
| 26 | 1337.9 | 13/6 |
| 27 | 1389.3 | |
| 28 | 1440.8 | 23/10 |
| 29 | 1492.2 | |
| 30 | 1543.7 | 17/7, 22/9 |
| 31 | 1595.1 | 5/2 |
| 32 | 1646.6 | 13/5 |
| 33 | 1698 |
Harmonics
In the 11-odd-limit, 33ed8/3 improves 23edo's tuning of most odd harmonics at the expense of most even harmonics. This makes it potentially a good no-twos subgroup tuning (eg for the 3.5.7.11 subgroup).
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -16.5 | +1.9 | +18.4 | -7.7 | -14.6 | -24.2 | +1.9 | +3.8 | -24.2 | +16.6 | +20.3 |
| Relative (%) | -32.1 | +3.7 | +35.8 | -15.0 | -28.4 | -47.0 | +3.7 | +7.4 | -47.0 | +32.3 | +39.5 | |
| Steps (reduced) |
23 (23) |
37 (4) |
47 (14) |
54 (21) |
60 (27) |
65 (32) |
70 (4) |
74 (8) |
77 (11) |
81 (15) |
84 (18) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -15.3 | +10.8 | -5.8 | -14.6 | -16.6 | -12.7 | -3.4 | +10.7 | -22.3 | +0.1 | -25.4 |
| Relative (%) | -29.8 | +20.9 | -11.2 | -28.4 | -32.3 | -24.7 | -6.6 | +20.9 | -43.3 | +0.2 | -49.4 | |
| Steps (reduced) |
86 (20) |
89 (23) |
91 (25) |
93 (27) |
95 (29) |
97 (31) |
99 (0) |
101 (2) |
102 (3) |
104 (5) |
105 (6) | |
23edo for comparison
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | -23.7 | +0.0 | -21.1 | -23.7 | +22.5 | +0.0 | +4.8 | -21.1 | +22.6 | -23.7 |
| Relative (%) | +0.0 | -45.4 | +0.0 | -40.4 | -45.4 | +43.1 | +0.0 | +9.2 | -40.4 | +43.3 | -45.4 | |
| Steps (reduced) |
23 (0) |
36 (13) |
46 (0) |
53 (7) |
59 (13) |
65 (19) |
69 (0) |
73 (4) |
76 (7) |
80 (11) |
82 (13) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.7 | +22.5 | +7.4 | +0.0 | -0.6 | +4.8 | +15.5 | -21.1 | -1.2 | +22.6 | -2.2 |
| Relative (%) | -11.0 | +43.1 | +14.2 | +0.0 | -1.2 | +9.2 | +29.8 | -40.4 | -2.3 | +43.3 | -4.2 | |
| Steps (reduced) |
85 (16) |
88 (19) |
90 (21) |
92 (0) |
94 (2) |
96 (4) |
98 (6) |
99 (7) |
101 (9) |
103 (11) |
104 (12) | |
36edt for comparison
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +15.1 | +0.0 | -22.6 | +13.8 | +15.1 | +12.4 | -7.4 | +0.0 | -23.9 | +22.4 | -22.6 |
| Relative (%) | +28.7 | +0.0 | -42.7 | +26.1 | +28.7 | +23.5 | -14.0 | +0.0 | -45.3 | +42.4 | -42.7 | |
| Steps (reduced) |
23 (23) |
36 (0) |
45 (9) |
53 (17) |
59 (23) |
64 (28) |
68 (32) |
72 (0) |
75 (3) |
79 (7) |
81 (9) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.6 | -25.3 | +13.8 | +7.7 | +8.4 | +15.1 | -25.6 | -8.8 | +12.4 | -15.3 | +13.4 |
| Relative (%) | -5.0 | -47.8 | +26.1 | +14.6 | +16.0 | +28.7 | -48.5 | -16.6 | +23.5 | -28.9 | +25.4 | |
| Steps (reduced) |
84 (12) |
86 (14) |
89 (17) |
91 (19) |
93 (21) |
95 (23) |
96 (24) |
98 (26) |
100 (28) |
101 (29) |
103 (31) | |
Music
- Improvisation in 33ed8/3 - Budjarn Lambeth (2026)