6ed7/3
6 equal divisions of 7/3 (abbreviated 6ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 6 equal parts of about 244 ¢ each. Each step represents a frequency ratio of (7/3)1/6, or the 6th root of 7/3.
Theory
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +22 | +54 | +45 | -97 | +76 | +54 | +67 | +108 | -75 | +5 | +99 |
| Relative (%) | +9.2 | +22.0 | +18.3 | -39.7 | +31.2 | +22.0 | +27.5 | +44.1 | -30.5 | +2.0 | +40.4 | |
| Steps (reduced) |
5 (5) |
8 (2) |
10 (4) |
11 (5) |
13 (1) |
14 (2) |
15 (3) |
16 (4) |
16 (4) |
17 (5) |
18 (0) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -40 | +76 | -43 | +90 | -15 | -114 | +37 | -52 | +108 | +27 | -50 | +121 |
| Relative (%) | -16.3 | +31.2 | -17.7 | +36.6 | -6.3 | -46.8 | +14.9 | -21.4 | +44.1 | +11.1 | -20.3 | +49.5 | |
| Steps (reduced) |
18 (0) |
19 (1) |
19 (1) |
20 (2) |
20 (2) |
20 (2) |
21 (3) |
21 (3) |
22 (4) |
22 (4) |
22 (4) |
23 (5) | |
6ed7/3+7edo scale
On the Xenharmonic Alliance Discord in September 2025, Maeve Gutierrez noted that the notes of 3ed7/3 make for a nice chord when played simultaneously, and that 6ed7/3 is a good tuning for using said chord.
Gutierrez also noted that playing 6ed7/3 on one instrument/track simultaneously with 7edo on another (a polymicrotonal approach) makes for some useful effects: "6ed7/3+7edo together gives lots of shimmer to play with+2 different flavours of detuned perfect fifth and fourth".
Lériendil then noted that this 6ed7/3+7edo scale is very closely approximated by 49edo. Budjarn Lambeth expanded on this idea, mentioning that after going 3 octaves up or 3 octaves down from the root note, the discrepancy between the two tunings (6ed7/3 and a stack of 7/3 from 49edo) will be no more than 6 cents.
If one wished to use this 6ed7/3+7edo scale tempered to 49edo, then it would look as follows:
Within 49edo: • 6ed7/3 is the step pattern 10 10 10... • 7edo is the step pattern 7 7 7... Which means that both scales sync up every 70 steps of 49edo, at the interval 1714.286c.
So (tempered to 49edo), the combined 6ed7/3 & 7edo scale is:
- 7\49
- 10\49
- 14\49
- 20\49
- 21\49
- 28\49
- 30\49
- 35\49
- 40\49
- 42\49
- 49\49
- 50\49
- 56\49
- 60\49
- 63\49
- 70\49 (period)
= Lumatone mappings
Mapping the 6ed7/3+7edo scale onto a 2D isomporhic keyboard like the Lumatone, one can use 7\49 for the x-steps and 10\49 for the y-steps or vice versa.
- 6ed7/3 on the x-steps
- 6ed7/3 on the y-steps