As the above posts indicate, the issue is a lot more complicated than it first appears. If you dig deeper, the history behind
tuning is a mess, the maths is a mess, and the theory of
consonance - again a mess - and not fully understood.
I'll have to disagree with the view that
equal temperament (
12-ET from now on) is inherently
out of tune. Yes, 1.5 might
look better than the irrational 1.4983..., and 1.25 'looks' better than 1.25992 for the
Major third, but many important numbers in maths are
irrational - like
pi and
e.
To my ear, 1.25992... sounds 'sweeter' than the
Just Intonation equivalent - 1.25.
Don't get me wrong, pure ratios are ideal for creating unique timbres (the basis behind the
harmonic series), but I am
suspicious whether this
ideal should be carried over to
melody and chords.
So if 12-ET isn't based on Just Intonation, then where does the number 12 come from? Put simply;
Why are there 12 notes in the chromatic scale? My
pet theory is that you can perfectly surround a
sphere with 12 equally sized spheres.
Coincidence maybe? You decide...
By the way, the reason why it's
impossible to tune a piano properly isn't because there's anything
fundamentally wrong with 12-ET. It's because the lower notes on the piano form slightly
inharmonic overtones. These overtones will then clash with the higher notes - making the piano itself
imperfect. Because of this, it's necessary to
compromise the tuning by 'stretching' the octaves slightly. However, it's important to note that this type of compromise is
not to be confused with the type of 'compromise' that is usually associated when comparing Just Intonation and Equal Temperament. Yes, confusing I know.
Incidentally, it goes without saying that synthesized sound doesn't suffer from this 'piano problem', as all harmonic partials can be tailored to exact specified frequencies.
Finally, there's a third type of 'compromise'. It's well known that two tones played together will produce extra 'unwanted' frequencies such as
summation tones,
difference tones,
interference beats, not to mention the
phenomenon known as
virtual pitch (which is based on the harmonic series). Anyway, upon closer study, it turns out that both Just Intonation and Equal Temperament (and every other tuning) 'suffer' from these added 'flavourings'. Thankfully, these extra tones are very quiet and are barely (if at all) noticable in music.