# 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

Although direct approximation is perhaps easier to understand than mapping through vals, it is not always practical in harmony. For example, it is impossible to construct a 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).