Kenneth and Andrew Hsu attempted to find fractal patterns in music by establishing scaling relations
f = c*i-D
between the frequency f of the interval
i between successive notes of a musical composition. For each interval i they
plotted Log(f) vs Log(i) and sought a linear trend in the graph. For
most compositions, the range of intervals is small, certainly
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In 1991 they took a different approach. Assuming a fractal structure for music, supported by their earlier work, the Hsus note an implication of self-similarity: "a musical composition could be represented by a music score of a different scale, using half, a quarter, or twice as many notes as were written by the composer." Noting their fractal character implies no definite length for coastlines, they ask "If a coastline has no definite length, could we state that Mozart's music has no definite number of notes or note intervals?" (Recall Emperor Joseph's comment to Mozart, that Abduction from the Seraglio was heavenly, but had too many notes. Perhaps Emperor Joseph anticipated the fractal nature of music. Probably not.)
They tested this hypothesis on Bach's Invention no. 1 in C Major. To understand the motivation, first convert the notes to a graph. Taking f0 = 60 Hz, the note j intervals above f0 has frequency fj given by fj/f0 = (15.9/15)j. In this way, an integer j is assignd to each note, and the graph of a composition is the plot of j (vertically) versus i (horizontally), where f0 is the frequency of the ith note. Such a plot gives a very jagged graph, and the Hsus observed that similar graphs made with every second note, or with every fourth note, and so on, of the original still sound remarkably Bach-like. This suggested a method for making new Bach-like compositions: "start with a 1/32 eduction of Bach and ... build the composition up into an alternative score, according to the theory of music harmony."
Here are the graph of the original, the 1/2, 1/4 and 1/8 reduction of BWV 772 of Bach.
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These ideas attracted sufficient popular attention to be described in the New York Times. Here it is reported the Hsus have patented a fractal music box, a device that filters music from a CD, producing a reduction, or abstract, of the music.
Thinking over this thinning process, Peak and Frame imagined a similar approach they called "Cantoring." Remove the middle third of a composition, then remove the middle thirds of the two remaining thirds, and so on for several levels. Do some adjustment to aleviate dissonances where notes have been removed. Does what remains still sound like Bach? Preliminary student experiments have suggested "yes" for Bach, Beethoven, and Brahms, but "no" for less complex composers, Stephen Foster, for example. Might robustness under Cantoring be a measure of musical sophistication?
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