6ed6
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
| ← 5ed6 | 6ed6 | 7ed6 → |
6 equal divisions of the 6th harmonic (abbreviated 6ed6) is a nonoctave tuning system that divides the interval of 6/1 into 6 equal parts of about 517 ¢ each. Each step represents a frequency ratio of 61/6, or the 6th root of 6.
Theory
6ed6 corresponds to 2.3211…edo. It is the generator chain for the harry temperament.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -166 | +166 | +185 | -201 | +0 | +250 | +19 | -185 | +150 | -15 | -166 |
| Relative (%) | -32.1 | +32.1 | +35.8 | -38.9 | +0.0 | +48.4 | +3.7 | -35.8 | +28.9 | -3.0 | -32.1 | |
| Steps (reduced) |
2 (2) |
4 (4) |
5 (5) |
5 (5) |
6 (0) |
7 (1) |
7 (1) |
7 (1) |
8 (2) |
8 (2) |
8 (2) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +212 | +84 | -35 | -147 | -252 | +166 | +72 | -16 | -101 | -181 | -258 | +185 |
| Relative (%) | +41.1 | +16.3 | -6.8 | -28.4 | -48.7 | +32.1 | +14.0 | -3.2 | -19.5 | -35.1 | -50.0 | +35.8 | |
| Steps (reduced) |
9 (3) |
9 (3) |
9 (3) |
9 (3) |
9 (3) |
10 (4) |
10 (4) |
10 (4) |
10 (4) |
10 (4) |
10 (4) |
11 (5) | |
Subsets and supersets
Since 6 factors into primes as 2 × 3, 6ed6 contains 2ed6 and 3ed6 as subset ed6's.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 517 | 15/11, 18/13, 19/14 |
| 2 | 1034 | 11/6, 13/7 |
| 3 | 1551 | 5/2, 12/5 |
| 4 | 2068 | 17/5 |
| 5 | 2585 | 13/3 |
| 6 | 3102 | 6/1 |