4ed5/2
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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. |
4 equal divisions of 5/2 (abbreviated 4ed5/2) is a nonoctave tuning system that divides the interval of 5/2 into 4 equal parts of about 397 ¢ each. Each step represents a frequency ratio of (5/2)1/4, or the 4th root of 5/2.
Theory
4ed5/2 is 3edo with the octave compressed by about 10 cents, or 7ed5 with the 5th harmonic compressed by the same amount. It therefore can be considered as an overcompressed 2.5-subgroup temperament tempering out 128/125.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -10 | +81 | -21 | -10 | +71 | -196 | -31 | +162 | -21 | -186 | +60 |
| Relative (%) | -2.6 | +20.4 | -5.2 | -2.6 | +17.8 | -49.5 | -7.8 | +40.8 | -5.2 | -46.8 | +15.2 | |
| Steps (reduced) |
3 (3) |
5 (1) |
6 (2) |
7 (3) |
8 (0) |
8 (0) |
9 (1) |
10 (2) |
10 (2) |
10 (2) |
11 (3) | |
Subsets and supersets
4ed5/2 is the first composite ed5/2, containing 2ed5/2 as the only nontrivial subset ed5/2.
Intervals
| # | Cents | Approximated ratio |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 397 | 5/4 |
| 2 | 793 | 8/5 |
| 3 | 1190 | 2/1 |
| 4 | 1586 | 5/2 |