First used on the clock at Great St. Mary's, Cambridge, these were originally called the Cambridge Chimes and are derived from Handel's 'Messiah'. Now famous because of their adoption at Westminster, they are known as the Westminster Chimes. The bells are tuned to the key of F major with the hour bell (Big Ben) sounding note E.
There are two things to notice here. The first is that there are only 5 unique musical phrases, labelled here 1 - 5. It can be seen from the scheme that the entire sequence is actually played twice within the hour. This means (in some clocks) that the barrel has cams for phrases 1 to 5 and rotates twice every hour. In the case of Big Ben however, the sequence of cams on the barrel repeats three times, so the barrel rotates once every 1.5 hours. This reduces the fall of the weights (although a larger barrel is then required).
The second point of interest is how phrase number 3 calls for the F bell twice in quick succession. This is why the F bell has two hammers since one could not operate twice in such quick succession. (But we have a doubt here, notice how the C bell is also required to sound twice in quick succession on the transition from phrases 4 to 5, arguably more quickly than the F bell is required to repeat within phrase 3. It may be that the C bell is the one with two hammers, can anybody confirm which?)