DE MACHAUT – Ma fin est mon commencement (arr. trio)
Ma fin est mon commencement [My end is my beginning] by the late-Medieval French composer and poet Guillaume de Machaut is a cunningly constructed piece. Originally composed as a song for three voices, the music is written in the form of several palindromes. The lowest voice – in this arrangement, the bassoon – first plays its part forward. Then, at the middle of the piece, it reads it backwards, playing an exact mirror of the melodic line. The upper two voices – violin and viola, here – play their parts forward, swapping in the middle to then play each other’s melodies backwards. The result is music of perfect symmetry.
PICFORTH – In Nomine (arr. chamber ensemble)
This ‘In Nomine’ by the English composer Picforth (about which little is known) was written around 1580. Originally composed for a consort of five viols, each part plays notes of only one duration. The cello plays notes lasting eight beats, the bassoon: six, viola: four, and so on. The inspiration here was planetary movement, with each part representing the orbit of a celestial body. Although each instrument is on a different temporal plane – most clearly, the second violin’s triple time against the first violin’s duple – they interlock perfectly. For artists, astronomers, and philosophers in Renaissance Europe, the idea of the ‘harmony of the spheres’ was central. Nature, they felt, existed in perfect balance and harmony; man’s music was our nearest approximation to this heavenly structure.
MOZART – Der Spiegel (2′)
Attributed to Mozart, this playful piece for two violins is a musical curio. A violinist stands on one side of a table, reading the music right way up, while a second stands on the other, reading that same page upside down. Miraculously, the music is a perfectly formed duet, with the music working no matter which way up the page is read.
CAGE – ‘Book One’ from Music of Changes (8′)
The twentieth century composer John Cage – perhaps best known for his work 4’33’’ which consists of no music, only the sounds of the concert hall – was deeply influenced by Zen Buddhism. Starting in the 1950s, Cage began experimenting with ways of removing his ego from the act of composition. In particular, he made use of the ancient Chinese divination text, the I Ching, to create a system of ‘chance music’. In Music of Changes – a groundbreaking piece of indeterminate music from 1951 – Cage uses coin tosses and charts derived from the I Ching to remove his own intentions. Instead, chance procedures are used to decide all aspects of the musical composition from pitch and duration to dynamics and rhythms.
PÄRT – Spiegel im Spiegel (10′)
The Estonian composer Arvo Pärt is best known for his minimal, meditative compositions. Influenced by his own mystical experiences with chant music, he coined a term for this compositional style – tintinnabuli – which describes the simple, bell like textures. In Spiegel im Spiegel [mirror within mirror], you hear this clearly in the piano’s arpeggios. The structure of the piece follows a strict formula, with the title directly reflecting what is happening in the music: each ascending melodic line is followed by a descending mirror phrase. Initially, the melody consists of only two notes, with another note being added with each of the following phrases, creating a seemingly endless continuum. After each distancing, the melody returns to the central pitch of A, which, according to the composer, is like “returning home after being away”. The piano part accompanies the melody part at each step like a “guardian angel” (as Pärt says).
DEBUSSY – Reflets dans l’eau (5′)
Debussy’s Reflets dans l’eau [Reflections in the water] is the first piece from his collection of solo piano works, Images: Volume 1. In it, Debussy conjures a feeling of water – musical gestures emanating outward from the centre of the keyboard, the harmony free-floating, textures reflecting water as both glistening and murky depths. There is another mathematical wonder at work, here. The piece is organised using the golden ratio. The sequence of keys is marked out by the intervals 34, 21, 13 and 8, while the piece’s main climax sits at the ‘phi position’.
MOZART – Sonata in Bb for Bassoon and Cello K.292 (15’)
The origins of Mozart’s Sonata for Bassoon and Cello are shrouded in mystery: no autograph manuscript exists and the work was not published until 1805, fourteen years after Mozart’s death. Along with the Bassoon Concerto it is one of two surviving works that Mozart composed for the instrument. They were possibly composed in 1774 when the 18-year-old Mozart was staying in Munich and had made friends with an amateur bassoonist called Baron Thaddäus von Dürniz. While the bassoon line in the Sonata is the principal part, the cello line is essentially an accompanying bass line rather than an equal partner.
The piece shows the young Mozart’s mastery of musical proportion and balance. The first two movements are in sonata form; contrasting ideas are introduced, developed, and then returned to. The ratios here are those of the golden ratio. Indeed, Mozart’s sister Nannerl noted he was always playing with numbers and even scribbled mathematical equations for probabilities in the margins of some compositions (for example, the Fantasia and Fugue in C Major, K394), some of which mathematicians suggest were Fibonaccci number calculations.
BARTOK – String Quartet No. 4 (25’)
The Hungarian composer Béla Bartók held a long fascination with mathematics and how it related to music. His String Quartet No.4, like the fifth string quartet and several other pieces by the composer, is composed in an arch (or mirror) form. The first movement is thematically related to the last, and the second to the fourth, while the third movement stands alone as a central pivot point. What is more, the outer four movements feature rhythmic sforzandos that cyclically tie them together in terms of climactic areas. The symmetry of the movements isn’t limited only to the themes; the lengths of the movements show symmetry as well. The first, third and fifth movements are approximately six minutes long, whereas the second and fourth are shorter, at about three minutes each. Bartók’s harmony is also mathematically derived. The quartet focusses on the chromatic scale, with the twelve notes divided into symmetrical units, with tonal centres being based on ‘axes of symmetry’. He also incorporates whole-tone, pentatonic, and octatonic scales as subsets of the chromatic scale, exploring their asymmetries.
© Dr Benjamin Tassie 2026
