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Weird Math – At The Edge Of Infinity And Beyond,
David Darling and Agnito Banerjee, One World Books, 2018.


Other related sources of information:

Fluctuations of Hi-Hat Timing and Dynamics in a Virtuoso Drum Track of a Popular Music Recording, by Esa Räsänen, Otto Pulkkinen, Tuomas Virtanen, Manfred Zollner, and Holger Hennig, PLOS Cognitive Neuroscience Channel, June 3, 2015.


/////// Quoting (Darling & Banerjee, 2018, p. 79):

Fractals crop up not just in spacial patterns but also in temporal ones. Drumming is a case in point. It’s easy to program a computer to generate a rhythmic drum pattern or have a robot musician play one. But there’s something about the sounds produced by professional drummers that distinguishes them from the perfectly steady, impeccably accurate beats of their synthetic counterparts. That ‘something’ is the slight variation in timing and loudness – the little deviations from perfection – which, research has shown, are fractal in nature.

An international team of scientists analysed the drumming of Jeff Porcaro, who played with the band Toto and was famed for his rapid and intricate one-handed playing of the hi-hat cymbals. In both the rhythm and loudness of Porcaro’s hits on the hi-hat, the researchers found self-similar patterns with structures in longer periods of time that echoed structures present in shorter time intervals.

Porcaro’s hits are the sonic equivalent of a fractal coastline, revealing self-similarity at different scale lengths. What’s more, the researchers found that listeners prefer exactly this type of variation, as opposed to precise percussion or that produced more randomly.

The fractal patterns differ from one drummer to another, forming part of what makes their playing distinctive. Similar patterns occur when musicians perform on other instruments and, although subtle, are the minute imperfections that separate human from machine.

/////// End of Quote from (Darling & Banerjee, 2018)


Atonal Music

The Voice of Fibonacci: Maths Meets Music (Jülide Dursun on TEDxYouth):

Fibonacci Melody (Greg Sheehan at TEDxSydney):


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