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.