31ed6
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Prime factorization
31 (prime)
Step size
100.063¢
Octave
12\31ed6 (1200.76¢)
(convergent)
Twelfth
19\31ed6 (1901.2¢)
(convergent)
Consistency limit
10
Distinct consistency limit
5
| ← 30ed6 | 31ed6 | 32ed6 → |
(convergent)
(convergent)
31 equal divisions of the 6th harmonic (abbreviated 31ed6) is a nonoctave tuning system that divides the interval of 6/1 into 31 equal parts of about 100 ¢ each. Each step represents a frequency ratio of 61/31, or the 31st root of 6.
Theory
31ed6 is not a truly xenharmonic tuning; it is a slightly stretched version (with an octave of 1200.8 cents) of the normal 12edo, similar to 19ed3. It is very nearly identical to 12edo, but with the 6/1 rather than the 2/1 being just.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.8 | -0.8 | +1.5 | +15.5 | +0.0 | +33.3 | +2.3 | -1.5 | +16.2 | -48.7 | +0.8 | -37.8 |
| Relative (%) | +0.8 | -0.8 | +1.5 | +15.4 | +0.0 | +33.3 | +2.3 | -1.5 | +16.2 | -48.7 | +0.8 | -37.7 | |
| Steps (reduced) |
12 (12) |
19 (19) |
24 (24) |
28 (28) |
31 (0) |
34 (3) |
36 (5) |
38 (7) |
40 (9) |
41 (10) |
43 (12) |
44 (13) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -37.8 | +34.1 | +14.7 | +3.0 | -1.9 | -0.8 | +5.7 | +17.0 | +32.6 | -48.0 | -24.9 | +1.5 |
| Relative (%) | -37.7 | +34.1 | +14.7 | +3.0 | -1.9 | -0.8 | +5.7 | +17.0 | +32.5 | -47.9 | -24.9 | +1.5 | |
| Steps (reduced) |
44 (13) |
46 (15) |
47 (16) |
48 (17) |
49 (18) |
50 (19) |
51 (20) |
52 (21) |
53 (22) |
53 (22) |
54 (23) |
55 (24) | |
Subsets and supersets
31ed6 is the 11th prime ed6, following 29ed6 and before 37ed6.