Scalings of Pitches in Music

We investigate correlations among pitches in several songs and pieces of piano music by mapping them to one-dimensional walks. Two kinds of correlations are studied, one is related to the real values of frequencies while they are treated only as different symbols for another. Long-range power law behavior is found in both kinds. The first is more meaningful. The structure of music, such as beat, measure and stanza, are reflected in the change of scaling exponents. Some interesting features are observed. Our results demonstrate the viewpoint that the fundamental principle of music is the balance between repetition and contrast. The aesthetic sense and perception remains a mystery in scientific level. One may expect that its mechanism should be dependent on the structure, especially a kind of collective effect of information possessed by the artistic or natural beauty and then the sensory processing of this information. The quantitative nature of most elements of music make analysis using methods of statistical physics feasible. 1/f noise was discovered in an earlier measurement of loudness fluctuations in music. Motivated by what was done recently on DNA sequences and human writings, we study here correlations among pitches according to musical scores. 25 songs and several pieces of piano music are studied. In equally tempered scale, the octave is divided into 12 semitones. A semitone is the interval between two tones whose frequency ratio is the twelfth root of 2. We consider two kinds of correlations, referred to as “frequency-dependent” (FD) one and “frequency-independent” one respectively. In the study of FD correlation, we map the real values of tones to positions within a one-dimensional walk, with a semitone corresponding to the unit of distance. The least of the duration of tones in the piece of music studied is selected as the duration unit. The rests are neglected. Hence the step of the walk, which corresponds to the difference between adjacent pitches, is not a constant. This walk reflects directly the change in the logarithm of frequency. Consequently, notes on the staff just make up an approximate landscape of this so-called “FD musical walk”. The correlation can be quantified by studying the displacement ∆X(t, T ) of the walker after T steps, which is the difference between the positions X(t+ T ) and X(t), ∆X(t, T ) = X(t+ T )−X(t). (1) ∆X(t, T ) is just the sum of changes in each step, i.e.,