A Label-Free Development for 12-Pitch-Class Systems
David I Lewin · Journal of Music Theory · 1977
For some time, it has been customary in much of the important theoretical literature to label the twelve pitch-classes' by the numbers 0 through 11. For instance, Babbitt,2 Perle,3 and Forte4 all do so. The numbers involved should not strictly be called integers: the appropriate mathematical term is integers modulo or better, because it avoids all possible confusion, modulo An to a mathematician is an ordinary counting number: after counting to 11, one continues on to 12, 13, etc. In contrast, the residues modulo 12 (or 12) have a circular structure: if one continues past 11, one returns to 0. The residues mod 12 have a natural algebraic structure. It would be out of place to develop the structure rigorously here, but I shall outline the basic elements of its manipulation. One can add the residues m and n. Their residue sum is that residue corresponding to their integer sum, if the latter is less than 12. In case the integer sum is 12 or more, the residue sum is that residue corresponding to the integer sum minus 12. Under these rules, 3 + 4 = 7, 8 + 5 = 1, 6 + 6 = 0, etc. By analogy, three hours clockwise and four more hours