It is a remarkable coincidence that

2

^{7/12}is very close to 3/2.*Harmony*occurs in music when two pitches vibrate at frequencies in small integer ratios.

For instance, the notes of middle C and high C sound good together (

*concordant*) because the latter has TWICE the frequency of the former. Middle C and the G above it sound good together because the frequencies of G and C are in a 3:2 ratio.Well, almost!

In the 16th century the popular method for tuning a piano was to a

*just-toned*scale. What this means is that harmonies with the fundamental note (*tonic*) of the scale were pure; i.e., the frequency ratios were pure integer ratios. But because of this, shifting the melody to other keys would make the music sound different (and bad) because the harmonies in other keys were impure!So, the

*equal-tempered*scale (in common use today), popularized by Bach, sets out to "even out" the badness by making the frequency ratios the same between all 12 notes of the chromatic scale (the white and the black keys on a piano). Thus, harmonies shifted to other keys would sound exactly the same, although a really good ear might be able to tell that the harmonies in the equal-tempered scale are not quite pure.So to divide the ratio 2:1 from high C to middle C into 12 equal parts, we need to make the ratios between successive note frequencies 2

^{1/12}:1. The startling fact that 2^{7/12}is very close to 3/2 ensures that the interval between C and G, which are 7 notes apart in the chromatic scale, sounds "almost" pure! Most people cannot tell the difference!What a harmonious coincidence!!!!

**THANKS FOR READING THIS :)**

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