Equal Temperament

\(\frac 1 1\), \(\frac 2 1\) and \(\frac 3 2\) are the most important musical intervals in western music because, assuming the overtones are harmonics, they have greatest consonance. in the ideal scale, any note one octave or one fifth above or below a note of the scale would also be in the scale. we could start from one note and keep adding fifths and octaves until no more notes are missing, but the problem is there'll always be missing notes because there is no solution to \(\frac 2 1 ^m = \frac 3 2 ^n \land m, n \in \mathbb Z\) but \(m = n = 0\)

fortunately, \(\frac 2 1 ^{7} \approx \frac 3 2 ^{12}\); by splitting the octave into \(12\) equal musical intervals, we get perfect octaves and very good approximations to perfect fifths, with \(\frac 3 2 \approx \sqrt[12]2^7\). the idea behind equal temperament is that all simple musical intervals (apart from octaves) are imperfect but close enough to sound good to the human ear

properties

equal temperament allows for perfect transposition

while fifths in \(12\)-tone equal temperament are close to perfect fifths, thirds in \(12\)-tone equal temperament are not so close to perfect thirds

proof \(\frac 3 2 / \sqrt[12]2^7 = 1.0011 \dots\) while \(\frac 5 4 / \sqrt[12]2^4 = \overline{1.0079 \dots}\)