I’m not sure if they knew it at the time, but composers such as Johann Sebastian Bach and Ludwig van Beethoven composed music notes that obeyed the principals of fractals, and in particular, a power law function. Hsü et al (1) reported in 1990 that log-log linear plots can be used to determine if a particular piece of music obeys the power law, and suggesting a fractal nature to music composition. When discussing the properties of fractals, it is the dimension of the object that reveals the existence of a fractal. In general, fractals are characterized by three concepts:
- Self similarity
- Response to measure of scale
- Recursive subdivision of space
Self Similarity: A self similar object is exactly or similar to part of itself, essentially the whole object has the same shape as its smaller parts.
Response to measure of scale: One concept of self similarity is scale invariance, which states that at any magnification there is a smaller piece of the object that is the same as the whole object. For example, the Thue-Morse sequence exhibits scale invariance.
Recursive subdivision of space: Known as a finite subdivision rule, which is a recursive way (i.e. a process of repeating items in a self similar way) of dividing a polygon or a two-dimensional shape into smaller and smaller pieces. Subdivision rules can be applied to the study of hyperbolic surfaces, this is important to me because I think of the layout of notes in a composition as a hyperbolic surface. The patterns exhibited by quasicrystals represent another example of a hyperbolic structure. Also, the Fibonacci sequence, which is defined by the following recursive relationship:
Fn = Fn-1 + Fn-2
which translate to the following infinite sequence:
0, 1, 1, 2, 3, 5, 8, 13, …
has fractal properties. It turns out that the Fibonacci sequence is related to many other interesting quantities, such as the golden ratio, Pascal’s triangle, and right triangles, just to name a few.
Now, onward to our discussion of fractal dimension. Fractal dimension is defined as:
D = -log10N/log10(1/R)
N ≡ number of segments created when dividing an object
R ≡ length of each segment
this is called the Hausdorff dimension. The Hausdorff dimension is related to the power law:
N = R-D
As a fundamental starting point to appreciate the dimension of fractals, lets calculate the dimensions of other shapes in Euclidean space. We will use examples from one, two, and three dimensional space. To start with, lets calculate the dimension (D) of a line. All that is necessary is to figure out how many parts to divide (N) the line into and the scale (R) of the line. The following digram explains how the line is segmented up and the dimension calculated:
It is no surprise that the dimension of a line is equal to 1. Next, lets calculate the dimension of a square and a cube. The flowing diagrams provide for a visual representation of how these two dimensions are calculated:
Now, we will show you why fractals are called fractals. The following diagram explains how the dimension of a Koch snowflake is calculated:
As you can see, the dimension is a fraction, hence, fractal. Finally, lets create a piece of music and use the power law to calculate if the structure of the composition can potentially be called fractal. We will use the note duration as a criteria for producing fractal music. To do this we will create 1/16, 1/8, 1/4, and 1/2 notes, then design a pattern of ratios of the notes that obey the power law. The following table explains how many notes are used for each duration:
To see if these values do indeed obey the power law, a log-log plot is created from these values, and if this plot shows a straight line with a fractional slope then the music created has the criteria met for obeying the power law of duration. The following is a representation of the log-log plot:
and as can be seen, the points map out a straight line. Therefore, with a straight line as a result we assume that this particular piece of music satisfies a power law distribution. However, the idea that this pieces of music, based solely on the note duration, has a fractal structure could be premature. This is due to the length of the piece and how evenly distributed the notes are. By inspection, this piece of music may not contain the correct distribution of notes where any given subregion would contain the same distribution as the whole (i.e. self similarity). The power law property can also be applied to other aspects of music, such as pitch, but I will leave this for another post.
Here is the piece of music called, Power V, which satisfies the power law for note duration:
Please enjoy listening to this piece.
1. Hsü & Hsü (1990) Fractal Geometry of Music (Physics of Melody) Proc. Natl. Acad. Sci. Vol. 87 pp. 938-941.