The mathematical possibilities of the music concept of mode from Ptolemy to Messiaen

In the 5th century A.D. Boethius translates the Greek terms tonos and tropos into the Latin modus. The theory of Greek tonoi is thus connected to the practice of church modes, producing what has been defined “l’imbroglio des modes”. The main difficulties connected to this concept arise from the fact that both Greek and Medieval theorists presented the modes sometimes as different sequences obtained by linearizing the same cycle of music intervals, sometimes as pitch translations of the same sequence. After the “tonal” period, when the modes were reduced to two (our Major and Minor), in the 20th century the search for new modal possibilities was resumed, and led to the encounter with what Messiaen called “mathematical impossibilities”. In this paper, we try to get out of the “imbroglio” by using an extensional method, which consists (a) of building a model that contains all the mathematical possibilities related to the concept of mode and (b) identifying the subsets of the model that correspond to the various forms that this concept has historically taken on.