Hide and seek

In the same way that one gradually discovers hidden qualities in friends that give them an extra layer of character to appreciate, so with some pieces of music. We’re not talking here of analysing a Mozart aria to establish the DNA of beauty – that would be like cutting the throat of the songbird to find out what makes it sing. Many composers, however, have revelled in building complexities into pieces that remain inaudible to the passive listener, but fascinating for the analytical musicologist.

palindromeTake the third movement of Haydn‘s Symphony No. 47, for example (Naxos 8.554767). It’s cast in three sections, each in simple binary form (AB), with each section repeated (AABB). The expediency of repetition to fill the time available was commonly used in this third movement of  classical period symphonies, and the ear must have sometimes been challenged by the seemingly never-ending repeats. Yet Haydn adds an extra appearance of tunes A and B in this work that is undetectable to the ear: if you play A + B backwards, it is exactly the same as when played forwards; a palindrome.

Such eccentricities weren’t unknown centuries before Haydn’s little tease. Guillaume Dufay was a Franco-Flemish composer crabwho lived from around 1400 to 1474. Among his pieces written for the church is the famous mass titled Missa L’homme armé (Naxos 8.553087), so called because it’s built on the popular tune of the same name. In the Agnus Dei, Dufay applied a riddle in Latin to one of the lines that had to be solved before it could be performed. Translated into English, it reads: “Let the crab proceed in full and return by half.” Given that some crabs walk backwards, can you deduce how the notes and rhythms of the given tenor part should be performed?

stalag-viii-aOlivier Messiaen injected a number of mathematical and structural planks into the fabric of his works that go undetected on the surface but act as a unifying bond for the music. His Quartet for the End of Time (Naxos 8.554824) was written in 1941 when he was a prisoner of war in Stalag VIII A. It was scored for an ensemble of fellow prisoners who had managed to keep hold of their instruments:  violin, cello, clarinet and a decrepit piano shipped in for Messiaen himself to play. Examples of his non-retrogradable rhythms can be found in the work. These rhythms cannot be played backwards because they would then sound the same; the first movement, Liturgie de cristal, even contains two such rhythms that are interlocked.

The 8-movement work is based on a biblical quotation from the Book of Revelations, reflecting the devoutness of Messiaen’s Catholicism. The Holy and Indivisible Trinity of his faith might be encoded in the lengths of some of the movements: I (43 bars), IV (73 bars), VI (109 bars) and VII (97 bars). All these are prime numbers, indivisible except by themselves or one. The first movement’s rhythmic ostinato contains 17 notes, also a prime number. Although these encoded significances would have been undetected at the work’s first performance, given before an audience of 8,000 prisoners from all levels of society, Messiaen later recalled that “Never have I been heard with as much attention and understanding.”

Bar numbers and ratios crop up again in the application of the Golden Section principle to compositions: take a line and divide it into two sections so golden-ratiothat the ratio between the shorter and the longer section is the same as the longer section to the entire length. When the innately satisfying ratio of 0.618 results in both calculations, it is known as the Golden Section, which is found in the dimensions of living things in nature. The question arises: did Mozart use it consciously in some of his works? Take his Piano Sonata No. 1 in C major, K. 279 (Naxos 8.550447). The ‘entire length’ of the first movement is 100 bars. The ‘shorter section’ of the Exposition is 38 bars. The ‘longer section’ of the Development/Recapitulation is 62 bars. The ratio of Exposition to Development/Recapitulation, and the ratio of Development/Recapitulation to the entire movement, comes within a smidgeon of 0.618. Similar results also emerge from the ratios of the second movement.

Completed in 1899, Edward Elgar drew the characteristics of a number of his friends into his Variations on an Original Theme (Naxos 8.553564), but it’s a second, more familiar melody that Elgar said could be played in counterpoint to that original theme that has given the work its more common title of the Enigma Variations: the identity of that second melody has never been discovered so, if you fancy your chances of cracking the riddle, happy sleuthing!