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The belief by the early church in the perfectibility of the universe led to rules that mostly focused on small integers and simple geometric shapes.
The most absurd instance was the insistence that the earth was the center of the universe and the the orbits of the planets were circles, but something similar happened in music.
By using simple ratios one can get close to the common western scale. The easiest way to demonstrate this oneself is to pluck a stretched string and then stop it length at various points. Stopping at the middle produces an octave (1;2), 3:2 produces a fifth, etc.
The problem is that when one divides the scale into 12 steps the ratios are slightly off. Music theorists struggled with the discrepancy for several hundred years and rather than give into the actual mathematics of the situation they invented all sorts of fudges. These are called "temperaments", and the most famous is the one used by JS Bach in his "Well Tempered Clavier". The correct mathematical solution to dividing an octave into 12 equal parts is that each step's frequency has a ratio to the next of the 12th root of 2 - an irrational number (1.059...).
Since this is an irrational number the intervals that should sound harmonious such as the fifth and third are actually discordant and beat. Western ears no longer notice this, but many other cultures use different musical step sizes and/or have other than 12 notes to the scale.
There are many sites where one can hear examples of this, sometimes the differences are subtle and require retraining one's ears, sometimes (as in Indian and other Asian music) the differences are quite noticeable. Even in western culture jazz and blues musicians don't stick to the standard pitches, but "bend" the notes.
Over the years I've tried different tunings on my harpsichord, some can really throw pieces into a different light. The key that a piece was written in used to make a big difference when instruments didn't used equal note steps, now it is mostly just nostalgia when someone choses a key for a piece.
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