Thank-you Julian from Wilson's School for
this clear explanation :
Consider the chromatic scale with12 equal intervals, starting with one note (for example C) and ending with the same note, but an octave above (so C again). We are told that the ratio between each of the notes is the same, and we were told in the previous problems that any note an octave above will be 1/2 the note one octave below. Let's use the value 2 for the bottom note of our scale, so the octave above will be 1. We know that there are 12 equal ratios in between these two in the scale, so to evaluate the note n steps lower on the chromatic scale from any position we use the expression 2[(n)/12] . This means take the twelfth root of 2, which gives the multiplier for one step, and raise it to the power of n to find the multiplier for n steps. For example, with the bottom note: 2[12/12] = 2 And with the top note: 2[0/12] = 1 Therefore, to find out the interval of a fifth, which misleadingly has 7 equal ratios (or semitones), we work out: 2[7/12] = 1.498307... A perfect ratio of 3:2 would give the note 1.5 Therefore, the interval of a fifth is less than 3:2 by 0.0016929231 (10dp)