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

One may easily observe, that two tones in a pythagorean strip $P[n]$ have the same fifth coordinate if and only if their generic step numbers are congruent modulo $n$ . According to this observation one may study two cycles of length $n$ : a cycle of steps $Pn$ and a cycle of fifths $Qn$ . Both cycles can be modeled by the additive group $Zn$ of residue classes mod $n$ . They may be converted into one another by virtue of the formulas:

step-to-fifth(k) = pq(s[-n1](k)) mod n fifth-to-step(k) = s[n] ((k .q,0 .o)) mod n

Here, $pq$ denotes the projection of a pythagorean tone $(a.q,b.o)$ onto its fifth coordinate $a$ .

The idea of Norman Carey and David Clampitt is the following: If both modes of counting (steps and fifths) are musically relevant, then one should also study the conversion maps between the two. How can one easily balance between the two modes of counting? The answer to this question is the property of well-formedness. A pythagorean strip $P[n]$ realizes a well-formed tone system if the step-to-fifth conversion is an algebraic map, i.e. a multiplication

step- to-fifth(k) = m .k mod n.

In other words: A system is well-formed, if each fifth has the same step size $m$ (and vice versa).

The contrafactual experiment yields a surprising result: The well-formed tone systems correspond to the numbers n = 1, 2, 3, 5, 7, 12, 17, ... . In the pentatonic case each fifth consists of 3 steps, in the diatonic case each fifth consists of 4 steps and in the twelve-tone system each fifth consists of seven steps. The reader is invited to meditate upon the possible interpretations of this finding.

2.2 Structure Theory of Consonance and Dissonance

Guerino Mazzolas investigation (c.f. Mazzola, 1989) into phenomena of two-part counterpoint aims at formulating the idea of contrast between consonance and dissonance on the grounds of purely structural criteria. Consequently, it is not the properties of a single interval that matter, but rather the properties of the entire interval system. The contrafactual assumption of this investigation is that any ordered pair $(A,B)$ of complementary six-element interval sets out of the 924 possibilities is taken into consideration as a possible model for such a contrast. It turns out the the pair

(K,D) = ({0,1,3,4,8,9},{2,5,6,7,10,11})

(intervals are numbered in a circle of fifth order) has several interesting properties which distinguish this particular pair (it is called the »Fux-Dichotomy«) from all other 923. The idea of contrast is expressed in terms of an »autocomplementary function« $f(x) = 5x + 2 (mod 12)$ mutually exchanging the two sets $K$ and $D$ . Furthermore, the two sets exemplify an abstract property of distance: The intervals of each half are located close to one another, while the average distance