Direct approximation
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A direct approximation of an interval in a given edo is the number of edosteps that most closely approximates it, found by rounding to the nearest integer the edo number times the binary logarithm of the interval:
[math]\operatorname {round} (n\log_2(i))[/math]
for ratio i in n-edo.
Examples of direct approximations
Interval, ratio | 12edo | 17edo | 19edo | 26edo |
---|---|---|---|---|
Perfect fifth, 3/2 | 7 | 10 | 11 | 15 |
Just major third, 5/4 | 4 | 5 | 6 | 8 |
Just minor third, 6/5 | 3 | 4 | 5 | 7 |
Harmonic seventh, 7/4 | 10 | 14 | 15 | 21 |
Of these intervals, the fifth plays an important role for characterizing edo systems (as it defines the size of M2, m2, A1). Also, a simple test can show if circle-of-fifths notation can be applied to a given edo system, because for this the sizes of fifth and octave must be relatively prime.