Where does the musical motif come from that is played by so many bell towers 
around the world, and why is it so widespread? E-c-d-g...g-d-e-c (where g is 
the lowest note and c, d, e represent the fourth, fifth, and sixth above.)

This tune is called Westminster Chimes. Here's a little bit about its history.

http://www.bbc.co.uk/education/beyond/factsheets/makhist/makhist3_prog12b.shtml

"Westminster chimes
Paddy Dowding from Hornchurch asked about the origin of clock chimes, 
especially the Westminster chime. Making History consulted Viscount Alan 
Middleton of the British Horological Institute, and Ranald Clouston, Bells and 
Clocks Consultant to the Council for the Care of Churches. 
The Westminster chime was originally the Cambridge chime, the chime at Great St 
Mary's, Cambridge, where a new clock was installed in 1793. The Revd Dr Joseph 
Jowett, a law professor, was asked to compose a chime but it is usually 
supposed that the composition was by his pupil, William Crotch (1775-1847). 
Crotch was a child prodigy and at the age of 11 was assistant organist at 
King's College, Cambridge. The tune of the chime is said to be based on a 
phrase from Handel's aria 'I know that my Redeemer Liveth'. In 1859 Lord 
Grimthorpe chose Crotch's tune for the new clock and bells in the Palace of 
Westminster. These words have also become attached to the chime: 

Lord through this hour, 
Be Thou our guide 
So, by Thy power 
No foot shall slide 

There are many more chimes. The Whittington chime, for instance, comes from the 
Church of St Mary Le Bow, in Cheapside, London. Dick Whittington, running away, 
heard the Bow bells and turned back - he eventually served three terms as 
London's Lord Mayor. Many churches have their own individual chimes played on 
three, four, six or even ten bells. Making History played the distinctive 
chimes of Merton College, Oxford, and Magdalen College, Oxford. In the late 
nineteenth century town and city halls had chiming clocks installed as a symbol 
of civic dignity 


Further reading 
Eric Bruton, The Illustrated History of Clocks and Watches (Little, Brown & 
Company, 2000) "

http://www.billib.co.uk/content/historyofchimes.asp

"The world's most famous chimes are still a popular favourite. Originally 
derived from Handel's Messiah, they were first fitted to the University Church 
clock of St. Mary's the Great in Cambridge, England. Via their association with 
Big Ben in the Victoria Clock Tower of London's Houses of Parliament, their 
fame has spread worldwide."
[This site also has a recording of the tune in MP3 and WAV formats.]

Very thorough answer.