Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

automorphism $~ .7 : Z12--> Z12$ , i.e., we consider the synonymous form

FiPiM od12-@-->.7 Syn(PiM od12)

which means that pitch denotators are now thought in terms of multiples of fifths, a common point of view in harmony. On this pitch space, two extensions are necessary: extension to intervals and extension to chords. The first one will be realized by a new form space

IntM od12---> Simple(Z12[e])

with the module $Z12[e]$ of dual numbers over the pitch module $Z12$ . We have the evident form embedding

ox 1 : F iP iM od12 >-> IntM od12 : x '--> x ox 1

of this extension, where we should pay attention to the interpretation of a zero-addressed interval denotator

D : 0~>IntM od12(a + e.b).

It means that $D$ has cantus firmus pitch $a$ and interval quantity $b$ in terms of multiples of fifths. For example, the interval coordinate $1+ e.5$ denotes the pitch of fifth from the basic pitch (say »g« if zero corresponds to pitch »c«), together with the interval of $7.5 = 11$ , i.e., the major seventh (»b« in our setup). The set $Ke$ of consonant intervals in counterpoint are then given by the zero-addressed denotators with coordinate $a+ e.k,k (- K = {0,1,3,4,8,9,}$ . The set $De$ of dissonant intervals are the remaining denotators $a+ e.d,d (- D = Z12- K$ .

The counterpoint model of mathematical music theory (Mazzola, 1990a) which yields an excellent coincidence of counterpoint rules between this model and Fux’ traditional rules (Fux, 1742) is deduced from a unique affine automorphism, the autocomplementary involution $2 AC = e .5$ on the pitch space: we have $AC(K) = D,AC(D) = K$ . It can be shown (Mazzola, 1990a; Noll, 1995) that this unique involution and the fact that $K$ is a multiplicative monoid uniquely characterize the consonance-dissonance dichotomy among all 924 mathematically possible 6-6-dichotomies. This model’s involution has also been recognized by neurophysiological investigations in human depth EEG (Mazzola, 1995b). Consider the consonance stabilizer $T rans(Ke, Ke) < Z12[e]@Z12[e]$ . This one is canonically related to Riemann harmony in the following sense.

In his PhD thesis (Noll, 1995) succeeded in reconstructing Riemann harmony on the basis of »self-addressed chords«. This means that pitch denotators

y D : Z12~>F iPiM od12(e .x)

are considered instead of usual zero-address pitch denotators which here appear as those which factor through the zero address change $a : Z12 --> 0$ , i.e., the constant pitches. A self-addressed chord is defined to be a local composition with ambient space $F iP iM od12$ , and Noll’s point was to replace zero-addressed chords by self-addressed ones.