Symmetries overt and hidden in musical form

Some large-scale symmetries in musical form or dimensions seem designed to
draw the listener’s attention: there can be no doubt they were a deliberate element of the musico-expressive concept. Others are more hidden yet minutely exact, leaving questions of whether they happened by chance, intuition or the composer’s unspoken design. Sometimes the numbers involved run so counter to the musical processes inside them as to suggest a sort of structural stubbornness, or compositional irony. A few of them suggest revealing extra-musical links. The examples here, ranging from Schubert to Debussy and Ravel, also show some surprising analogies of process or numbers across ostensibly very different musical works or forms. Some of this material has been explored in my earlier book Debussy in proportion, whose main focus however was on golden section proportions; the present study takes a closer look at the role of symmetries either within or independent of these, as well as relationships between local hypermeter and larger proportions.