Pourquoi la gamme a sept notes? Une math{é}matique des notes et des gammes

We present an algebraic construction of music notes and show how to associate them inseveral ways to construct music ranges. Then a family of ranges emerge with a fixed number of notes: two, three, five, seven, twelve, seventeen, etc. A classification of these scales is simple with the concept of multiplicative structure. The action of such a multiplicative structure on a music note introduces the definition of tonality. Multiplicative structure classification is then straightforward with the notions of type and mode. We detail the case of music scales with five and seven notes and propose musical examples for a wide range of modes.