What the Maths Mean

In​ the 1938 edition of the Oxford Companion to Music, Percy Scholes defined composition as ‘the culmination of the mental and psychological process of a remarkable and inspired personality’. Even today, the notion of the Great Composer – the individual genius, whose inimitable music is an expression of a singular mind – still holds sway: the composers guaranteed to draw a crowd are the select few, a Debussy, a Wagner, a Beethoven. Concert promoters call them ‘bankers’. By contrast, the music discussed in Composers in the Middle Ages must rank as the least successful fare that has ever been put before the public. The men and women who composed it are usually anonymous, and their compositions are likely to be based on mathematical conceits no one could possibly follow in performance. You will hardly ever hear them in a concert hall or church service. Art-house concerts may feature the secular music of the medieval period, attractively informal in style and presentation, but this volume of essays is largely about sacred music and there is nothing informal about that, or about the academics who have devoted their careers to trying to make sense of it.

Composers such as Debussy wrote their music alone. The creative milieu of the Middle Ages was collaborative, with poets also working as musicians and singers. As Margaret Bent writes, ‘the extent to which “poetry” includes music is open, as is the question of how far “singing” includes or implies creation or composition. Poetry, accessible to a wider readership than the specialised notation of music, often had a longer shelf life.’ Mark Everist lists the various competencies that might contribute to the creation of a musical work in this period as ‘writing poetry, creating monophonic music, creating polyphonic music, writing down music of all sorts, singing, writing down words, writing new words to old music and old words to new music, and assimilating pre-existing music and poetry to new contexts’.

The editors claim that this is the first book on the subject of the ‘medieval composer’. They could have helped readers by giving what they understand as the Middle Ages some shape and outlining the major stages in musical style in a period that lasted perhaps a thousand years (if one is generous with the first signs of liturgical chant). The reader is left to work out that the anonymous chant tradition ran roughly from the sixth century until the first named composers – Hildegard of Bingen, Peter Abelard and Héloïse d’Argenteuil – of the early 1100s. Then, during the later 12th century, a group of composers working in Paris, including Léonin and Pérotin (the only two whose names survive), wrote the first polyphony, a transcending moment in the history of Western music. They were followed by Philippe de Vitry and Guillaume de Machaut, whose music, in all its astonishing complexity, eventually fed into the early Renaissance (again no date is given, possibly because the other arts with which music was so involved tended to reach their Renaissance phase before music did).

Legend has it that the founding corpus of liturgical chant was dictated by a dove into the ear of Pope Gregory I (r.590-604), hence the term ‘Gregorian’. It is examined here by Henry Parkes, who argues that the tradition of humble anonymity was derived from a self-deprecating desire not to upstage Gregory. It was only after many centuries that a personality as argumentative as Abelard’s finally made no attempt to conceal his name. Both he and Hildegard were pursued by exceptional fame in their lifetimes, which ensured that much they did was documented. Abelard was the most sought-after theologian and public debater of the 12th century. Hildegard was content to be known as an abbess and a polymath, concerned not only with music but also with medicine, both as a writer and as a practitioner. She was unusual in openly acknowledging the existence of inspiration in her music, so bringing her part way towards the world of the 19th-century composer and qualifying medieval composers for entry into Scholes’s Companion, though with the inconvenient proviso that she thought composition attributable only to God. In the opening of Scivias (1151), she writes: ‘I spoke and wrote these things not by the invention of my heart or that of any other person, but as by the secret mysteries of God I heard and received them in the heavenly places.’ Hildegard died at the age of 81, having written more music than any other identifiable writer of the time.

Both Abelard and Hildegard composed exceptional melodies, usually writing the texts to them too, though whether these monodies can be admitted to the corpus of chant continues to be debated. They are certainly not Gregorian, though the sound and style of 12th-century monody make it difficult for us to be dogmatic about what is and what is not chant. Abelard’s most lasting composition was the hymn ‘O quanta qualia’, which has survived, harmonised, in many hymn books today and is often said to be based on chant. The difference in Abelard’s case is the nature of his texts. This monophonic phase of ‘composition’ included a literary tradition known as the planctus, or lament. In time it came to cover anything that needed moaning about and could take the form of poetry or music, separately or together. The most famous was the ‘Planctus ante nescia’, which has come down to us in nine musical sources and sixteen text manuscripts – an exceptional number and the best proof of the collaborative nature of composition in the period. It was expected that the text of a planctus would be imbued with intense personal emotion, which would make this corpus, if it were better known, easily the rival of the mannerist texts of the Renaissance madrigalists. Abelard, possibly with Héloïse, is credited with writing six of these laments, which were not suited to liturgical worship and took the monophonic tradition in a new, far more personal direction.

It wasn’t until 1400 that composers were regularly named in the sources. But this doesn’t mean there aren’t clues, not least in the music itself, as the tradition of introducing acrostics into texts to identify people – composers, lovers and wives, sponsoring monarchs and aristocrats – bears witness. A motet from 1373, ‘Ferre solet’, not only gives the name of the composer, encoded in the first letter of various lines in the poetry (‘JOHANNES VAVASSORIS’), but by the same process, elaborately concealed, we find the words ‘ANNO DOMINI MILLESIMO TRECENTESIMO SEPTUAGESIMO TERCIO FECIT ISTUM MOTETUM’ (‘he made this motet in the year of our Lord 1373’). The most elaborate acrostic of all, said to be the longest in Western literature, is contained in Boccaccio’s Amorosa visione (c.1342), which in its completed form consists of fifty canti of poetry in terza rima. The initial letters of each terza rima create three complete sonnets, the first of which ends: ‘GIOVANNI E DI BOCCACCIO da CERTALDO’. It also refers to Boccaccio’s presumed lover, Maria d’Aquino, whose name is ciphered through an additional acrostic (the initial letters of the odd-numbered lines), creating an acrostic within an acrostic. There are plenty of names in this repertoire if you know where to look.

This was a culture that delighted in codes, hidden patterns and mathematical complexity. The music which sets these texts could be just as complex as the hidden acrostics, and even more difficult to unravel. In Christian thinking, to arrange or create mathematically was to draw closer to God. St Augustine made this point throughout his career, starting in The Confessions (397) and running to The City of God (c.413-26). He argued that mathematical patterns reveal something of the structure of creation: ‘Wherever you turn, Wisdom speaks to you through the imprint it has stamped upon its works. Look at the sky, the earth and the sea, and at whatever in them shines from above or crawls, flies or swims below. These have form because they have number.’

The first experiments with voice part-writing came in the middle of the 12th century. From the start this involved writing a tenor part, based in chant that followed rigid mathematical formulae, with two parts above, often with very lively rhythms. Until the end of the 14th century, the mathematics tended to be applied only to the tenor; by the 15th century, it had spread to include the upper voices as well. But the tenor part remained the generating force for every sacred composition, often providing the theological ideas that the texts in the upper parts would debate. This technique is known today as isorhythm and it is in contemporary descriptions of isorhythm that the word ‘tenor’ is first found (from the Latin tenere, to hold). The tenor voice held the structure together, and was usually the lowest in the ensemble. In time, contrary to modern usage, the countertenor voice came to sing lower as well as higher than the tenor. Proper bass parts, and even the use of the word ‘bass’, are vanishingly rare at this time.

A useful example of isorhythmic composition can be found in the Kyrie of the Messe de Nostre Dame by Machaut (c.1360), which belongs to the first polyphonic mass cycle. The tenor part is a pre-existing chant melody organised into seven units, all with the same rhythm and the same number of notes (four), though of different pitches, as the chant dictates. The ‘triplum’ and ‘motetus’ parts then weave their counterpoints above the tenor and around each other. It is a simple example of the type. Within a few decades, however, this way of composing had produced some of the most rhythmically difficult music that has ever been written. The tenor parts could easily employ gradual diminution of the note lengths, making the chant quotations harder and harder to follow, while in the upper parts any of the standard notes could be subdivided according to context and whim: a minim could be divided into any number of crotchets, not simply two or four as we would do today, but sometimes an inconvenient number such as seven or nine. The result was a school of composition known as the ars subtilior, which not only drove these possibilities to extraordinary lengths, but also turned their notation into works of art. Baude Cordier’s heart-shaped chanson about love, ‘Belle, Bonne, Sage’, is a classic example (the red notes imply rhythmic alterations).

‘Belle, Bonne, Sage’ by Baude Cordier

The texts of the standard three voices of an isorhythmic motet were also capable of engaging in intimate and involved dialogue with one another, despite the fact that their parts were not connected by any musical motifs (as would soon become commonplace in Renaissance polyphony). Vavassoris’s ‘Ferre solet’ should correctly be listed as ‘Ferre solet/Ana theos de Gracia/Ave Maria’, which tells the listener that the chant and inspiration for the composition is the ‘Ave Maria’. The newly written poem of the triplum praises the redemptive powers of the Virgin in high-register Latin, while the meaning of the motetus’s poem, also devotional in character, is written in a strange dialect.

Composition would take a new direction, one of greater simplicity, with the Renaissance. But there was one last hurrah. Charles W. Warren claims that when Guillaume Dufay was asked to write the ceremonial motet for the dedication of Florence Cathedral, including Brunelleschi’s new dome, he wrote it with the numbers that went into the architecture of the cathedral and its dome as his starting point. If true (and there are dissenters), Dufay was attempting to produce a sounding model of the cathedral, based in precise mathematics – a harmonic organisation of space.

Pope Eugene IV dedicated the cathedral on 25 March 1436 and the papal choir sang Dufay’s ‘Nuper rosarum flores’. As Warren writes, ‘both dome and motet occupy special places in the history of their respective disciplines, the former as the most important achievement of “the father of Renaissance architecture”, and the latter as one of the most impressive occasional pieces ever written.’ Work on the cathedral had begun in 1296 and everything apart from the dome was finished by 1380. After winning a competition, Brunelleschi started work in 1420 on the most daring design anyone had ever seen, featuring a dome within a dome, a double cupola, an unheard-of tour de force of engineering. To write his motet, Dufay took the measurements of the cathedral as it stood, which can be codified as a repeating modular unit of 6:4:2:3, and disposed his tenors, quoting the chant ‘Terribilis est locus iste’, with those proportions in the four sections of his composition. This was not in itself a major undertaking – things like this had been done before. The real subtlety was in writing, against any precedent, two tenor parts which were in canon with each other at the fifth, carrying into them proportions that reflect the thickness of the domes. Brunelleschi built the outer of his two domes because he wanted an elaborate and beautiful shell (‘più magnifica e gonfiata’ – more magnificent and swelling). There is every reason to suppose that Dufay wrote his two tenor parts similarly to make his composition more magnificent and resonant.

It is commonly assumed that audiences are put off by numbers. Nicholas Cook wrote in Music, Imagination and Culture (1990) that

people who go to concerts must sometimes be upset by the lack of correspondence between the manner in which they experience a piece of music and the manner in which it is described in the programme note; for programme notes often dwell on the aesthetic importance of large-scale tonal structures or motivic relationships that are in practice inaudible to most listeners. To be told that the beauty or significance of a piece of music lies in relationships that one cannot hear is to have the aesthetic validity of one’s experience of the music thrown into doubt; and the manner in which music is described by professionals can only create in the untrained listener a sense of inadequacy, a feeling that though he may enjoy the music he cannot claim really to understand it.

Yet these off-putting properties in medieval music aren’t so different from those that have been put before the public in recent times, including serialism. If you wouldn’t choose to go out to hear Stockhausen’s Mikrophonie II, you might for the same reasons avoid Machaut’s ‘De souspirant/Tous corps qui de bien amer/Suspiro’. After all, it’s no easier to ‘hear’ what the Fibonacci sequence is doing in Mikrophonie II than it is to hear what the maths mean in Machaut’s motet. You would be missing something, though. Behind every mathematical puzzle in music there is a solution, and one doesn’t need to work it out mentally to hear it, or at any rate enjoy it. The ‘cancrizans’ or crab canons by Josquin or Bach, which may use retrograde inversion with augmentation, can’t be fully understood just in the hearing – you need, at the very least, a score and some time – but they have an appeal that defies analysis.