Fractal Geometry and Music Signals

A fractal is generally “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,” a property called self-similarity.
The term was coined by Benoit Mandelbrot in 1975 and was derived from the Latin fractus meaning “broken” or “fractured.” A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals - for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics; for instance, it is regular enough to be described in Euclidean terms.

Musical research over the last century has become increasingly entwined with the scientific areas of acoustics, psychoacoustics, and electro acoustics, among others. During the last half century, the computer has become the central site of this research, including sound synthesis, digital signal processing and computer-assisted composition.

The application of fractal geometry to musical signals with respect to Indian musical instruments has not been widely experimented.

Tanpura is a multi-stringed instrument extensively used as a drone instrument and is an integral part of classical music in India. The instrument is plucked by finger and is used as an accompaniment with the vocal music. The special form of the bridge has a remarkable influence on the tone quality. When the adjustment of the position of contact of string on the oval shaped bridge is made carefully by trial, by using a cotton thread, the instrument is highly sonorous, giving a tone of fine musical quality. This is known as “jwari”. The average fundamental frequency is extremely steady and consistent. The amplitude however shows a regular long-term variation, a sort of waxing and waning, three to four in number during the course of a single plucking. It has been noticed that the harmonics of the tanpura strings’ sound exhibit a periodic waxing and waning. This period is shorter as one goes up along the higher harmonics. This is connected with the jwari of the tuning. These are, however, regular predictable nature of complexity variation, which may have functional values other than providing a rich melodious sound. The source of origin must be related to some sort of non-linearity associated with the strings, the slightly convex form of the lower bridge and the mode of attachment of strings and/or some sort of feed back mechanism which uses the total acoustic environment including the global resonance structure of the instrument.

The best known example of this process (non-linear dynamic systems) in the field of Indian musical instruments is Tanpura. The analysis of the Tanpura sound signals defy the conventional assumption based in linear models that complex behaviour results from complex factors. It is also characteristic of these models that their component elements act as cells or quanta, and that the global behaviour emerges over large numbers of iterations, usually such that a computer is required for the intensive calculations involved.

Fractal dimensions of time series data generally reveal the presence of non-linearity in the production mechanism. Tanpura signal is considered as repetitive quasi-stable geometric forms. Time series data is a quantitative record of variations of a particular quality over a period of time. One way of analysing it is to look for the geometric features for categorising data in terms of concept. The signals emitted by a Tanpura is characterised by varying complexity with undulations of intensity of different harmonics with different frequencies as well as multiple decay. All these suggest interplay of source at various point of time like attack time, quasi-static state and end decay. Study of fractal dimensions might be a technique to analyse this bevaviour. Non-linear dynamical modeling for source clearly indicates the relevance of non-deterministic approaches in understanding these signals.