Mathematics and music are to some extent related. So, when Aditya Ganesh, amateur musician and Master’s student of Computing Science at the University of Groningen, sat in a lecture on complex behaviour created by a simple formula, he saw the potential for a new music piece. It took him six months to convert a logistic map into a nearly eight minutes’ long composition called ‘Feigenbaum’s Orbit’.
‘Sometimes teaching has an effect,’ tweeted Michael Biehl, associate professor of Computing Science, who taught the lecture. ‘Apparently, my lectures on the #logisticmap and #perioddoubling inspired one of the students to compose and create a nice piece of music that is based on the first #Feigenbaum constant.’
A week later, Ganesh explains this tweet—cryptic for non-mathematicians—in the Bernoulliborg. ‘I am taking the Data Science and Systems Complexity track in this Master’s programme. The lecture was on how a simple non-linear equation, the logistic map, can produce complex chaotic behaviour.’ He shows the accompanying Wikipedia page on his laptop. In an animated graph, data points assemble into a line that bifurcates repeatedly.
‘The line splits into two, then four, then eight, and so on. This is called period doubling,’ says Ganesh. In 1978, physicist Mitchell J. Feigenbaum published a paper in which he described how the interval between bifurcations approaches a single value, now known as the Feigenbaum constant. ‘The starting points for my composition were the period doubling and the first 32 digits of the constant.’ (For those who are interested, this is: 4.6692016091029906718532038204662.)
Ganesh has been composing music for a while now, as a hobby. He uses Waveform, a digital audio workstation that allows him to create a composition on his laptop. The results are posted on his SoundCloud account ‘Wolfy the Terrible’—a nod to Mozart’s nickname ‘Wolfie’. ‘I try to make a new composition every month. But this took me longer, over six months.’ He used blocks of four digits from Feigenbaum’s constant to create a percussion line. ‘The first four digits are 4669—this is translated into four percussion lines in the first block, with four, six, six, and nine beats,’ Ganesh explains.
The period doubling is used by doubling the number of digits in each subsequent block. Therefore, the second block uses a percussion line of 8 digits, the third 16, and the fourth block has 32 percussion digits. Ganesh: ‘Also, each block is repeated before moving on to the next and then reduced again.’ If we assign letters to the four blocks, the entire composition looks like this:
A AAB ABABCD ABCDCD CDD D
Ganesh: ‘The percussion lines will vary a lot, in some sections you have a lot of percussion, while other sections are quieter. And by chance, the first and last group of digits are very similar: 4669 and 4662. This created a truly natural ending to the composition.’
Having percussion is obviously not enough. The beats and the sequence of blocks were determined by the mathematical formula but the composer needed to add melody lines. Ganesh used a chord scheme that he borrowed from Pink Floyd. ‘And I added some ghost tones for richness.’ This was challenging, certainly further on in the composition where he had to add eight or sixteen new blocks.
He is happy with the end result. ‘My wife encouraged me to show the result to Michael Biehl, who then wrote that tweet with a link to the composition in SoundCloud.’ It brought his music to the attention of a wider audience. He is not the first person to turn maths into music: ‘Many composers use mathematical concepts, for example the number pi. You see this a lot in progressive rock. I felt that the logistic map was a very interesting concept and I had to try it. I’m very happy with how it turned out.’
But his main reason for taking an MSc programme in Groningen was not music: ‘I completed an MBA in my own country, India, but after four years in industry, I decided that it was not for me.’ The Data Science and Systems Complexity track of the Master’s programme in Computing Science should shift his professional focus to the boundary between artificial intelligence and security. He has one more year to go—time for a few more compositions.
Aditya Ganesh on SoundCloud
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