The tempered scale is a construct of our European forefathers of an equal division of the octave into 12 parts. They did this to solve the problems of harmony, yes, but many say also to make their cosmology seem synchronous: 12 signs of the zodiac, 12 months of the year, 12 apostles etc. But in creating this construct (which I believe was brilliant in many respects) they deviated from nature in a very consequential way.
The pure overtone series of the vibrating string (which is the model of nature they started with in the first place) has dramatic tuning discrepancies with the tempered scale. This can be illustrated with a guitar and an electronic tuner by comparing the harmonics of a major 3rd and major 6th with fretted notes of the same intervals. The third is 14 cents sharp of the true 3rd in the overtone series, and the 6th is 16 cents sharp when compared to its pure counterpart (very painful), and those are only two of the intervals with significant problems.
The most common alternative to tempered tuning is called Just Intonation. However, chord harmony that is ubiquitous in tempered tuning is very limited in this system. If one wants to compose something in Just Intonation, one needs to screw on a very different cap (and maybe use some different instruments or gadgets as well).
Entire musical forms that we love and take for granted would never have existed without the equal-tempered scale. I use it myself on a regular basis knowing full well it is a deviation from the natural model it was derived from.
But my ear will always long for the beauty of a true (or just) interval of a third, and lament the compromise of equal-temperament.
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