David Lewin and maximally even sets

David Lewin originated an impressive number of new ideas in musical formalized analysis. This paper formally proves and expands one of the numerous innovative ideas published by Ian Quinn in his dissertation, to the import that Lewin might have invented the much later notion of Maximally Even Sets with but a small extension of his very first published idea, where he made use of Discrete Fourier Transform (DFT) to investigate the intervallic differences between two pc-sets. Many aspects of Maximally Even Sets (ME sets) and, more generally, of generated scales, appear obvious from this original starting point, which deserves, in our opinion, to become standard. In order to vindicate this opinion, we develop a complete classification of ME sets starting from this new definition. As a pleasant by-product we mention a neat proof of the hexachord theorem, which might have been the motivation for Lewin's use of DFT in pc-sets in the first place. The nice inclusion property between a ME set and its complement (up to translation) is also developed, as occurs in actual music.