High $n$-gram occurrence probability in baroque, classical and romantic melodies

An $n$-gram in music is defined as an ordered sequence of $n$ notes of a melodic sequence $m$. $P_m(n)$ is calculated as the average of the occurrence probability without self-matches of all $n$-grams in $m$. Then, $P_m(n)$ is compared to the averages Shuff${}_m(n)$ and Equip${}_m(n)$, calculated from random sequences constructed with the same length and set of symbols in $m$ either by shuffling a given sequence or by distributing the set of symbols equiprobably. For all $n$, both $P_m(n)-{\mathrm{\text{Shuff}}}_m(n)$, $P_m(n)-{\mathrm{\text{Equip}}}_m(n)\ge 0$, and this differences increases with $n$ and the number of notes, which proves that notes in musical melodic sequences are chosen and arranged in very repetitive ways, in contrast to random music. For instance, for $n\le 5$ and for all analyzed genres we found that $1.6<-log(P_m(n))<8.6$, while $1.6<-log({\mathrm{\text{Shuff}}}_m(n))<14.5$ and $1.9<-log({\mathrm{\text{Equip}}}_m(n))<18.3$. $P_m(n)$ of baroque and classical genres are larger than the romantic genre one. $P_m(n)$ vs $n$ is very well fitted to stretched exponentials for all songs. This simple method can be applied to any musical genre and generalized to polyphonic sequences.