What is all the fuss about temperament?
Litres of ink have been spilt on this topic, and I expect that there are many web sites. But it is by no means a purely academic problem, so let me give a quick introduction. For example, suppose a violist tunes up nice pure fifths C3, G3, D4, A4, removing interference beats as he does so. These will give intervals with frequency ratios of 3:2. The violinist tunes in unison to the last three of these notes, ie G3, D4, A4, but then adds E5. So the ratio of the frequency of the violinist’s E string to the violist’s C is (3/2)*(3/2)*(3/2)*(3/2) = 5.063. If they played these open strings together it would sound (to many people, anyway) uncomfortably out of tune. The fifth harmonic of C3 at 5.000 times the fundamental clashes and produces beats with the open E5 at 5.063 times that fundamentall. Of course in practice, while the violist must play the C3 as an open string, the violinist will rarely play E5 on open string, so in musical context the violinist solves it by adjusting the position
