Preprint del Dipartimento di Matematica

Preprint n. 2/2003

A mathematical model for a metric index of melodic similarity

PIETRO DI LORENZO

What is a melody? When is there similarity between two melodies? Musicians cannot operatively ask for this question. Always, they doubt whether there is some objective definition. Furthermore, it is very difficult to estimate musical and melodic similarity. A solution is provided by signal analysis method. In this paper each melody is encoded into a numerical sequence of integer number that rappresents the pitchs of the notes. Starting from a multidisciplinary point of view, I propose to build up a metric index in the space of all the simple melodies, based on the cross-correlation function. This is a well defined and known function in statistics, used in a several natural sciences (electric circuits, geophysics etc.). Cross-correlation is applied to two melodies; its value for lag time t = 0 is used to define the index. This index (real-valued) is like the Mercalli-Cancani-Sieberg empirical seismicity scale. A metric index of melodic similarity turns out to be a very practical and helpful instrument to catalogue Gregorian chant musical variants among same canticulae and in folk melodies. I have applied this method to investigate for several melodies the distance between original melody and its variants. The results show a good agreement with the perceptive point of wiev. Between a melody and its picth trasposed variant (up or down) there is not distance. If I select and change only a note (without a particular role as tonic or dominant) I obtain a little distance. The distance increases as much as the pich of the note aspected in variants is different from original one.