The Ubiquitous $1/f$ Noise

A work of art has to be complete and at the same time it should be devoid of any extra part; meaning that a complete piece needs no part added to it while nothing can be taken out of it. In the case of music composed with traditional notational systems, no notes can be taken out, and there is no room for any new notes to be added. This means that every single note should have a meaning and a function. Every note contributes not only to the instantaneous color of sound (i.e. creating its own individuality and meaning), but also it satisfies a context built by the previous note and sets up a new context for the notes which proceed it (i.e. satisfying its function). Satisfying immediate functions means that successive notes have to be ``correlated'' with each other. A complete correlation in the time scale of notes dictates very boring melodies. It is important to note that in a longer scale of time the phrase ``it satisfies a context built by previous notes'', does not mean that there has to be a conformation to the immediate context. It may be that a conscious breakdown of context is needed to satisfy a higher level goal (context) in a higher time scale, and that might be what creates the element of surprise. This breakdown of lower level context can also be controlled by higher level organized chance operations.

Now we can simply replace the word ``note'' with ``melody'' in the previous paragraph, and move to a higher plane with the same type of requirements. When we apply this idea to all levels of time in music we reach a rather obvious fact: that a piece of music has to have structures on all levels of our perception. However, these structures themselves have to be related in some way to each other. Again the same rule which we described for the successive elements (e.g., notes, melodies, etc.) applies to the entities which these correlations create. If we visualize music laid out in the conventional time-frequency plane of spectrograms, then the relationship among successive events is a relationship along the horizontal axes, while the relationship between correlations in different time scales of perception is in the vertical direction. In other words a piece of music has to have some ``correlation'' in all its time scales while the values of the correlations are in turn correlated within themselves. Having this correlation and at the same time not being boring, a piece of music creates a plexus for every note (event) which has to strive for its individuality while conforming to its context.

Different techniques in signal processing provide us with ways to become more concrete about qualities such as ``correlation'', as long as we are precise about what comprises our signal. For example, there are different algorithms for pitch detection using fast Fourier transform or analysis of the auto-correlation function by using the sound pressure level as the signal. Such analyses take a physical signal (e.g. sound pressure level) and try to come up with a perceptual value (pitch). It seems plausible that applying the same type of analysis, which finds some type of correlation in the physical signal, to the newly found perceptual values would result in some tangible understanding of a higher level entity. There are two questions which we have to keep in mind. (1) Is there any clear-cut boundary between perceptual and physical events? (2) Are the physical and (many) perceptual levels of our mind governed by the same principles, and therefore can they be analyzed in a similar fashion?

In this section we will briefly touch upon these two questions by analyzing ``pitch signals''. A pitch signal is composed of a single line melody which is extracted from a piece for the duration of the whole piece. Please note, we make no claim to the fact that the extracted is ``the'' melody of the piece; we define a procedure and extract ``a'' melody from the piece. Voss and Clarke[50] conducted some such studies and concluded that what they assumed to be a pitch signal of almost all music behaves like $1/f$ noise. Before explaining their results, we will try to achieve an intuition about how a $1/f$ noise behaves and what are its properties. The text is written in a way that, with the help of graphs, the formulas may be ignored. One of the goals of this section is to show how musical signals such as pitch can be analyzed in the same way that we analyze sound.