The `resurrection¿ of medieval music in the last century, after several hundred years of lying dormant on library shelves, has not only profoundly enhanced the range of musical experiences for performers and public, but, perhaps more significantly, it has provided a wealth of new ideas and possibilities for 20th and 21st-century composers seeking new languages, structures and aesthetics. A particular inspiration has been medieval ideas and practices on the question of number and proportion in music. This course explores the importance of number and proportion, tracing the theoretical tradition from ancient and medieval writings on music (Pythagoras, Aristotle, Boethius, Vitruvius etc) and examining how these concepts are reflected in medieval compositional practices from the early 3 and 4-part polyphonic works (composed for the proportionally splendid architecture of Notre Dame in Paris) through to the elaborate proportional rhythmic systems which reached their zenith in the intensely complex rhythmic and metrical music of the medieval avant garde at the end of the 14th century, the so-called ars subtilior. The role of number in 20th-century music will be explored through a series of case studies (including music by Messiaen, Xenakis, Hoyland, Ona, Walther, Nancarrow and others). Issues of number symbolism in music of different periods will also be addressed.