Correlations in a Mozart's music score (K-73x) with palindromic and upside-down structure

In this work, we study long-range correlations in a “Scherzo-Duetto di Mozart” score (K-73x) for two violins. This is a fascinating piece, as the second violin part is upside down on the same sheet below the first violin, and some parts are like a palindrome. Given such ingenious structure, it is expected the existence of long-range correlations in the score structure. In order to quantify long-range correlations, we considered the music score as a sequence of integer numbers, each of them corresponding to last common denominator units of note. By using detrended fluctuation analysis (DFA), correlations are quantified by means of the scaling exponent that reflects the type of correlations for a given distance between neighbors note. The following conclusions can be drawn from the analysis: (a) For about 10-25 neighbor note distances, correlations are similar to 1/f-noise. This is an interesting finding since it has been shown that pleasant sounds for humans display a behavior similar to 1/f noise. (b) As the neighbor note distance increases, the long-range correlations decays continuously. For some score sections, the music score behaves like non-correlated (i.e., purely random) noise. Summing up, the results show that the studied Mozart's score contains a certain degree of correlation for relatively small note distances, and becomes close to non-correlated behavior for long note distances. We considered also the sequence constructed by considering the distance between the simultaneously played notes of the two violins. Interestingly, for relatively small neighbor note distances, a scaling behavior similar to that found for individual violins is also displayed. In some sense, this is an expression of the specific structure (palindromes plus upside down construction) used by Mozart in the composition of this music score. Although we focused on a particular high-art music score, our results suggest that modern methods borrowed from statistical physics can be useful for the systematic study of music composition techniques.