Kristen Nivling
 
    I play the piano, so I take tuning for granted.  I press a C and I hear a C.  I don't have to worry about intonation, like you would on a violin or a cello.  I don't even have to tune every time I start to play.  That is to say, I don't have the option of tuning every time I play.   Growing up I always assumed that an A was an A and an E was an E because that's how it was.  That's not the case.  So I'll try to explain what little I do understand.
 

First of all, what is pitch?
    Sound travels in waves.  As we know, waves all have frequency, velocity, and wavelength.  Pitch is our brains' perceptions of the frequencies of sound waves.

What is an interval?
    In music, an interval is the relationship between two notes.  The distance from one A to the next A is an octave, or an eighth.  Certain intervals are pleasing to the ear.  The best-sounding intervals are called perfect intervals: the 8th, the 1st, the 5th, and the 4th.  Why do these notes sound so good together?  Scientifically, an interval is the ratio of the frequencies of the two notes.  An octave's frequencies have a ratio of 2:1.  The perfect fifth has a ratio of 4:3.
    Part of the reason certain notes sound good together is a concept called overtones.  When one note is played on, for instance, the piano, you are hearing more than just the frequency of the note being played.  You hear the frequencies of other notes at the same time.  Their frequencies are mostly whole-number multiples of that of the note being played.  Two of the loundest (but still imperceptible) overtones are the octave and the fifth of the given note.

How do we make a scale?
    A scale is an attempt to divide the octave into equal and good-sounding pitches.  Over the years this has been done in several ways.  An ideal way to make a scale would be to ascend in perfect fifths or major thirds, since their ratios are so nice.  Unfortunately, if you ascend in either of those manners, you will never arrive at a whole-number ratio.  So there's no perfect way to make a scale.  In centuries past, scales existed where some intervals were nearly perfect, but others were terribly off.

So what do they tune keyboard instruments to now?
    A system called the tempered scale has been devised so that keyboard instruments can be tuned in one way and the relationships betweent the notes, while not ideal, are acceptable to the ear.  The system compromises some intervals, but the octave remains in tune.  What follows is a chart of some intervals, their theoretical and actual ratios with equal temperment.  The chart is courtesy http://www.anu.edu.au/ITA/ACAT/drw/PPofM/scales/scales7.html

So when I play an F, for instance, it's not really an F?
    Nope, it's not an F that would be in ideal relation with the notes around it.  Good string players and singers will play/sing notes differently when they're not being accompanied by a piano.