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I read that the formula relating frequency to the note played was:

F = 440 + 2^(n/12)

Where F is the frequency in hertz of the note played and n is the number of notes from middle A.

It seems strange to me that this relationship is exponential, doesn't it make more sense for the relationship to be linear, so it is easier for the musician to quickly subconsciously predict what each increasing note will sound like?

Is there even any point in this formula at all, or is it just a strange convention? If it is just convention, then where does it originate?

It's because the way the ear actually hears pitch differences (for most people) is based on frequency ratios, rather than absolute frequency differences.

If I played you "twinkle twinkle little star" starting at on a note of 400 Hz, and then played it again with another 300Hz added the frequency of each note, it wouldn't sound like the same tune. However, if we multiplied the frequency of each note in the original by a ratio (say 1.75), it would sound like "the same tune, but higher".

I'm not an expert on the physiology, but I believe there are even physical characteristics of the ear that relate to notes an octave apart (which corresponds to a doubling in frequency) being heard as somewhat equivalent.

By the way, the formula you quote describes how to find note frequencies in an equal temperament system; this system came into common usage over time as a clever compromise that allows many different combinations of notes with 'almost consonant' harmonic relationships to be sounded. There are other systems of temperament possible, and therefore there are also different equations possible, though they will all be broadly logarithmic.

Essentially, it's because we humans perceive pitch on a logarithmic/exponential scale. We hear an octave when the frequency is doubled or halved, not when it has a certain amount added or subtracted to it. Since musicians (well, the western ones, anyway) divide the octave into 12 equal parts, we had to take the 12th root of two as our factor to represent a semitone.

As for the origins of this system, we have to go back all the way to at least the time of Pythagoras, who was one of the first to discover the ratio-based nature of music, and also the harmonic series (which itself is wholly integer multiples of the fundamental). Now, back in Pythagoras' day, there was no such thing as 12-TET - the system we use now and the system described by that handy equation you posted here -, but Pythagoras knew that integer ratios are what drives most harmony, and our modern system of frequencys and their relationships is in large part an approximation of Pythagoras' harmony with some things fixed up. My source for all this is Tom Jackson's Mathematics: An Illustrated History of Numbers. It's a great book, written mostly about math, but math and music are inextricably bound together, and there's a page or two on the origins of music itself.

You can experiment for yourself, too.

1. Go find any tone generator application (this website works).


2. Try picking a note (say, 440hz). Play it, then play another sound simultaneously with a frequency 1.5x the original (660hz). Observe.


3. Clear both tones. Play another tone with a different frequency (say, 500hz). Play a second tone at the same time with 1.5x the frequency of the original (750hz). You should hear a very similar sound, starting on a higher note. This is the result of multiplying the frequency by the same amount.


4. Start again with a 440hz tone. This time, add 220hz to it to produce the second note (which should still end up as 660hz). Play that; obvoiusly, it's the same sound as before.


5. Now, starting on 500hz, add 220hz to your 500hz frequency. Play the 500hz frequency at the same time as your new 720hz frequency.


6. Notice the difference?

Conclusion: Our ears perceive pitch in a logarithmic manner. Therefore, to change a frequency by any amount, one must multiply the frequency by certain factors instead of adding or subtracting. All musical intervals can be represented as a ratio, and multiplying the two frequencies by the same factor produces that same ratio. Adding the same amount to both frequencies does not preserve the ratio.

1. 
   440hz, 660hz
   


2. 
   500hz, 750hz
   


3. 
   440hz, 660hz
   


4. 
   500hz, 720hz
   

And if you haven't already, check out this question.