Bas Cornelissen looks at music as a mathematical puzzle

Bas Cornelissen looks at music as a mathematical puzzle

“I would like to understand how music works, and this sheds light on that,” says Bas Cornelissen. The music researcher received his PhD for his dissertation on February 24 Measuring Musics, a scientifically somewhat unorthodox explosion of ideas about analyzing music. Everything is included, from Gregorian chants to Malian djembe rhythms, from rhythm triangles and melody squares to the call of the zebra finch and the mathematical music of Arvo Pärt. Not everything is equally detailed.

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“I opted for an unconventional form,” Cornelissen admits in his office in the Amsterdam University Theater, where colorful plots hang on the wall and the sound of construction work occasionally drowns out the interview. There is no central thesis or a clear line, and as in a piece of music there are short parts between the longer parts interludes: looser written chapters about smaller projects. It led to discussion among the PhD supervisors, but ultimately approval was given.

The road to get there was just as colorful, says Cornelissen, who, in addition to being a researcher, is also a teacher and is following a master’s degree in early music (main subject in singing) at the Conservatory of Amsterdam. He also performs as a soloist in choral music performances.

“I actually wanted to investigate the cultural evolution of language. In linguistics they do lab experiments and computer simulations of simple model languages, with just a few ‘words’, to understand how they subsequently evolve.”

Science is monomaniac: you can always dig further, you never fully understand anythingBas Cornelissen music researcher

As a child, Cornelissen was quite science-oriented: he was interested in mathematics and physics, as well as music. Language came later. “My only plan has always been to become a scientist. But I found getting a PhD difficult. I actually got completely stuck.”

On a whim, he auditioned at the Utrecht Conservatory at the last minute. “When I was unexpectedly hired, my life was turned upside down. But when I told my supervisor ‘I’m quitting, I’m going to sing’, he said ‘why don’t you do both?’ Which is of course a crazy idea. But it worked for me.”

“Science is monomaniac: you can always dig further, you never fully understand anything, and nothing puts that into perspective. Making music offered balance. You also want to do everything as well as possible, but there is a limit to that. During a performance you do what you can in the time available, but there are always practical limitations. And sometimes you rise above yourself. I like that about it.”

Photos: Dieuwertje Bravenboer

Music collections

After the change, Cornelissen changed his research into the evolution of music. He delved into music collections, including those of church chants that we best know as Gregorian chant.

“Hymns in that tradition have been sung in churches and monasteries since the ninth century. To my surprise, there were huge data sets: tens of thousands of chants. They don’t have a key, like modern Western music, but something called ‘mode’. Someone said: if you really want to understand that, you should spend a few years singing Gregorian chant. I didn’t have that time, so I thought: let’s see if we can characterize it with computers.”

That worked, and indirectly helped to shed light on an academic question: what are the smallest units of evolution in music? Is there some kind of basic unit for the evolution of music, just like the base pairs in DNA or sounds in languages?

Characterizing the mode for the chants was best achieved if you started from word units instead of groups of notes with a fixed length. “This ties in with the discussion among experts about whether the chants are constructed from recycled blocks of notes, often based on words in the text, or whether they from scratch have been composed.”

A interlude the thesis is about rhythm. The promoter Henkjan Honing had previously developed ‘rhythm triangles’, a graphic tool for analyzing rhythms. “You can immediately see whether there is structure in your rhythmic data,” says Cornelissen, who used the triangles to analyze Western piano music and Cuban salsa, as well as the calls of animals, from songbirds to monkeys and whales. “In addition, they are also very beautiful, rich images.”

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Melody squares

To make melodies visible in a similar way, he invented ‘melody squares’. This is where collections of folk music came in handy, coming from Irish, Scottish, German, Luxembourgish and various Chinese and Native American music cultures.

A total of about 15,000 melodies went through Cornelissen’s melody square generator, which produces density plots that clearly differ per tradition. On this basis, Cornelissen constructed a family tree of music genres, based on the differences between the squares. That family tree seems to fit quite well: Irish folk music is on the same branch as the Scottish, far from the Chinese and Native American branches. But Cornelissen is ‘hesitant’ to draw too many conclusions from this. Whether you can trace the real evolution of music traditions in this way remains to be determined by future research.

The last chapter is an odd one out. “In all other chapters I make music measurable with a lot of data. Here I analyze one piece of music by Arvo Pärt.” The popular Estonian composer Pärt (1935) developed a new composition technique, which he tintinnabuli mentioned, which he applied most strictly in his piece of music Summa. Cornelissen: “There is a very clear mathematical structure behind it. Each melody is accompanied by a tintinnabuli voice, an accompaniment voice, for which the choice is very limited.”

Although Pärt is still alive, he refuses to comment on his music. Cornelissen still managed to deduce the tintinnabuli rules by doing a lot of puzzling. “The idea was to see whether you can generate the entire piece based on the rules. Then you get very far, up to 3 percent of all nuts. But at the point where refinements are made, there are sometimes inconsistent notes. What does that mean? Are they individual variations, choices by Pärt? Or, and I find this a funny possibility, was it just a mistake on Pärt’s part? But it is also possible that there is another rule that I have not yet come up with.”

Rhythm triangles

A rhythm triangle shows a rhythm of four beats. There are three time intervals between those four strokes. Those intervals (divided by the total time interval) are plotted in a triangle, with the fast rhythms given a lighter color than the slow ones.

In this way, a monotonous rhythm ends up in the middle of the triangle, a short-short-long rhythm (1-1-2) somewhere between the center and a corner point. Rhythms in many musical genres tend to occur in small integer ratios. Such points are shown with crosses in the plots. The plots can be used to analyze music, or to see what changes when musicians play back rhythms. For example, the triangles of Malian drummers look very different from those of Bulgarians, from a music tradition with many odd rhythms, which is different from Cuban salsa.

Cornelissen: “It is striking that the rhythms of professional drummers are often very precisely just next to the points of small number ratios.” He suspects that the rhythms remain interesting and don’t sound too mechanical.

Melody squares

The melodies are broken up into groups of three notes, and the two intervals in between are plotted on the x and y axes: a rising fourth, five semitones, is +5. Then you move up one note and take the adjacent group of three.

A melody thus becomes a sequence of points in the square. Some intervals occur more often, some rarely or never, depending on the music tradition. Cornelissen: “It shows which routes melodies like to take in different traditions.”

For example, the top right quarter in American music cultures is quite empty, for melodies that indeed often descend. Rising and falling note triplets are common in Scottish and Irish music.

Arvo Parts tintinnabuli

The melody line is the black line in the graph. The possible tintinnabuli accompaniment voices are colored. In the piece Summa the accompaniment notes may only come from one triad. It may only move in adjacent steps, but it must always move, and it must remain under the melody. These strict rules determine almost the entire guidance voice.