  The concept of time series irreversibility -- the degree by which the
statistics of signals are not invariant under time reversal -- naturally
appears in non-equilibrium physics in stationary systems which operate away
from equilibrium and produce entropy. This concept has not been explored to
date in the realm of musical scores as these are typically short sequences
whose time reversibility estimation could suffer from strong finite size
effects which preclude interpretability. Here we show that the so-called
Horizontal Visibility Graph method -- which recently was shown to quantify such
statistical property even in non-stationary signals -- is a method that can
estimate time reversibility of short symbolic sequences, thus unlocking the
possibility of exploring such properties in the context of musical
compositions. Accordingly, we analyse over 8000 musical pieces ranging from the
Renaissance to the early Modern period and certify that, indeed, most of them
display clear signatures of time irreversibility. Since by construction
stochastic processes with a linear correlation structure (such as 1/f noise)
are time reversible, we conclude that musical compositions have a considerably
richer structure, that goes beyond the traditional properties retrieved by the
power spectrum or similar approaches. We also show that musical compositions
display strong signs of nonlinear correlations, that nonlinearity is correlated
to irreversibility, and that these are also related to asymmetries in the
abundance of musical intervals, which we associate to the narrative
underpinning a musical composition. These findings provide tools for the study
of musical periods and composers, as well as criteria related to music
appreciation and cognition.
