As an aid for musical analysis, in computational musicology mathematical andinformatics tools have been developed to characterise quantitatively some aspectsof musical compositions. A musical composition can be attributed by ear a certainamount of memory. These results are associated with repetitions and similarities ofthe patterns in musical scores. To higher variations, a lower amount of memory isperceived. However, the musical memory of a score has never been quantitativelydefined. Here we aim to give such a measure following an approach similar tothat used in physics to quantify the memory (non-Markovianity) of open quantumsystems. We apply this measure to some existing musical compositions, showingthat the results obtained via this quantifier agree with what one expects by ear.The musical non-Markovianity quantifier can thus be used as a new tool that canaid quantitative musical analysis. It can also lead to future quantum computingcontrollers to manipulate structures in the framework of generative music.