I’ve been reading about Renaissance music. People sometimes say that it began in the early 1400s when musicians rebelled against the dry, complicated mathematical structures of late medieval music and switched to a more emotionally expressive style. For example, the *New Oxford History of Music* writes:

The isorhythmic motet, the highest achievement of medieval rationalism, reached its climax during Dufay’s prentice years (c. 1410-20), with works in which the quasi-mathematical construction arouses more admiration than pleasure.

But since I’m a mathematician, this actually got me interested in isorhythmic motets!

I found them hard to understand from written descriptions. Isorhythm involves a rhythmic pattern called a *talea* which is applied to different melodies. Often the *talea* lasts for a different amount of time than the melody, which leads to some interesting effects. For example, here is a melody that lasts for 28 measures divided into 7 *talea* that each last for 4 measures:

This is from a composition written sometime around 1360. Isorhythm gets a lot more complicated than this! The music typically has several parts, in which *talea* get sped up or slowed down independently. But reading about these things didn’t give me much of a feel for what isorhythmic motets actually sound like.

When I listened to some by Guillaume Dufay, they didn’t sound dry at all! For example:

It’s quite thrilling and romantic, actually! Listen to how he uses the leading-tone, the note one half a step below the tonic, to build tension. This is a big medieval thing. If you don’t know what the heck I’m talking about, wait to the very end of the piece! Here Dufay clobbers us with a *loooong* leading-tone, the medieval equivalent of the wail of electric guitar at the end of a classic rock song. He’s really hamming it up.

Now, Dufay is famous for being the first really major Renaissance composer and *breaking* from the medieval traditions, so maybe his isorhythmic motets are more exciting than average. But still, I hear they are intensely mathematical.

In fact there’s an album of Dufay’s isorhythmic motets with the great title *Quadrivium* which features a booklet by a mathematical physicist named Guido Magnano, an expert on general relativity at the University of Turin. On a Russian website I read:

This album by the Italian vocal-instrumental group Cantica Symphonia takes off from the proportional aspect of large Dufay works like the motet

Nuper rosarum flores, long thought to have been based on the proportions of the great cathedral of Florence but recently discovered to have probably been modeled on another building. The album was actually sponsored by the mathematics department of the University of Turin, and an essay by professor Guido Magnano explores the mathematical bases of the musical system Dufay knew. For the average listener the musical manifestations of these principles are going to be hard to hear sitting in front of your stereo; the chief interpretive decision made by Cantica Symphonia is to strive for a transparent texture, judiciously using a small instrumental ensemble to bring out structural details. Save for the fact that the voice parts are sung solo, it’s a Renaissance performance in the classic “pure” mold. For the numerologically inclined or for the serious student of the Renaissance era, the disc is an interdisciplinary goldmine. Recorded in an Italian church, the disc matches its engineering to its aims, and the packaging by the Spanish label Glossa is uncommonly attractive.

I decided I needed to get my hands on that booklet. The album is on YouTube:

Unfortunately the actual CD costs $47.53 on Amazon. Luckily I was able to get it for much less online from Barnes and Noble… thereby procuring the booklet, which is what I really want. This should arrive in a week or so, so with luck I’ll tell you more later. I’m also quite fascinated by Dufay and the whole Franco-Flemish school of Renaissance music that he helped start, and—as usual when I’m just starting to learn about something—I have dreams of blogging about it.

In the meantime, I found out a bit from an interview with Guido Magnano, where he says this:

Your first disc for Glossa,Quadrivium, placed special emphasis on the question of mathematical proportions in Dufay’s motets. If these considerations apply also to the works recorded here, can you provide some examples of how this came through in practice?

Guido Magnano:Mathematical proportions do not occur in medieval and Renaissance music as occasional, accessory stylistic elements: the Pythagorean-Platonic paradigm states that music itself is nothing but “auditory perception of numbers”. The hypothetical relationship between the mensural proportions of the motetNuper rosarum floresand the proportions of Brunelleschi’s Duomo, although fascinating and quite plausible, should not obscure that in other motets, particularly in the later isorythmic motets (Fulgens iubarandMoribus et genere), Dufay attains an even higher degree of formal complexity.The motet

Magnam me gentes, (12:4:2:3) also included in this CD, has a mensural structure very close toNuper(6:4:2:3). Worth noting is that the 15th century humanist Marsilio Ficino introduced a “Platonic-Hermetic” movement, attributing occult significance to numerical relations. Did Dufay himself share these ideas? Do the numerical ratios in his motets hide a symbolic content? Some modern scholars have claimed so, even though the pieces for which numerological interpretations have been proposed were written some thirty years before Ficino’s works, and it is impossible to obtain a conclusive proof that such interpretations reflect Dufay’s intentions.More concretely, one could ask to what extent mathematical proportions can be perceived by the listener. The “mensural proportions” (which are but one example of numerical ratios in this music) are merely changes of meter: in an isorhythmic motet, for instance, the basic sequence of note values (

talea) is repeated with all durations multiplied by a fixed ratio (e.g. 2:1, 1:2 or 2:3). Whenever the change occurs simultaneously in all voices, it can be clearly heard; in other cases, it remains hidden in the polyphonic texture. The mensural proportions also determine the ratio of the lengths of the various sections of the piece, and the choice of appropriate proportions was considered to be essential to the overall structure of the piece, much as in the Pythagorean scale where such ratios (1:2, 2:3, …) determine the consonance of a chord. As Leibniz states three centuries later: “Music is the pleasure the human mind experiences from counting without being aware that it is counting.”

Here is *Nuper rosarum flores* as played by Cantica Symphonia:

This is the isorhythmic motet that Dufay wrote for the consecration of the cathedral in Florence in 1436, the one with Brunelleschi’s famous dome. He was 35 then, living in Florence and working for the Pope. Later he would return to Cambrai, in what is now Northern France.

I’ve visited this cathedral and taken the terrifying tour that lets you climb up to the dome, go *above* it into the rafters between the dome and roof, and then out onto the roof. It’s an amazing structure:

For sheer joy, so far my favorite performance of Dufay’s isorhythmic motets is the album *O gemma lux* by the Huelgas Ensemble:

Here’s the man himself—Guillaume Dufay:

To be honest, nobody is completely sure whether this is Guillaume Dufay or another famous composer of the early 1400s, Gilles Binchois.

It’s more about architectural systems of mathematical proportion than musical, but you might want to check out Rudolf Wittkower’s “Architectural Principles in the Age of Humanism”. I think over the years acceptance of his thesis has waxed and waned, not sure how it’s taken these days.

Thanks! I’ll start by reading that booklet by Guido Magnano when I get my copy of

Quadriviumin the mail, and see how much my interest is piqued.By the way, it turns out Magnano is the one who uploaded this example of isorhythmic composition to the Italian Wikipedia:

Hi John,

Thank you so much for this wonderful discovery. If you’re interested in Renaissance music as well, I would definitely recommend Richafort’s “Requiem in memoriam Josquin Desprez” (also performed by the Huelgas ensemble):

if you haven’t heard of it already. It is a marvellous thing — and one that I would say also has deep mathematical resonances (although that’s just my uninformed impression, I haven’t read on the subject).

Thanks! I’m definitely interested in Renaissance music; you could say I’m trying to understand (and of course enjoy) the Franco-Flemish school of polyphony, that starts roughly with Du Fay and then goes through Ockeghem, Obrecht, Josquin, Manchicourt and Lassus—and others. This school started around 1420 and ended around 1620, so there were a lot of changes. I haven’t listened to anything by Richafort yet!

I should probably get ahold of everything by the Huelgas Ensemble, but I’m not quite serious enough to buy it all at once!

Yes, this requiem by Richafort is great! Thanks!