Direct approximation

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

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.

Problems

Unless one sticks to one or two notes at a time, direct approximation is not always practical in harmony. For example, it is impossible to construct a just major triad using the direct approximations of 3/2, 5/4, and 6/5 in 17edo since the step numbers do not add up (5 steps + 4 steps ≠ 10 steps). The closest 3/2 and 5/4 imply the second closest 6/5; the closest 3/2 and 6/5 imply the second closest 5/4; and the closest 5/4 and 6/5 imply the second closest 3/2. We see one of the direct approximations must be given up. This is called inconsistency, and chords like this exists in every edo.

In regular temperament theory, intervals are mapped through vals. Although more complex, it recognizes the fact that intervals like 3/2, 5/4, and 6/5 are related, as the number of steps of one interval is determined once the other two have been determined. The three situations in the above example correspond to using vals ⟨17 27 39], ⟨17 27 40], and ⟨17 26 39], respectively.