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

An even more refined filtering of all chords in their partially ordered roles as ”fuzzy” representatives of a given chord $X$ is suitably described in terms of the global morphem $(int(X)s >,{Ext(M )|M (- int(X)s>}).$ In section 4 we give an example of this kind (but see also Noll, 1997, 1998).

Remark 8 Another situation where global morphemes can be considered is the study of saturated monoids $M (- MON s$ without constant tone perspectives. We give only a sketch of this procedure. For any monoid $M$ of tone perspectives one has the relation

GM = {(s,t) (- T × T| E f (- M withf(s) = t}

$M0 = Ø$ implies that the equivalence closure of $GM$ splits into more than one chord as equivalence classes. The stable images $ImM (Xi)$ of these chords $Xi$ ( $i = 1,,,,,n$ ) under iterated application of the elements of $M$ yield the minimal local placeholders $ImM(Xi) 0$ for the missing global constant tone perspectives within $M$ . The whole variety of candidates is given by collections of chords ${Yi (- Ext(M )|ImM (Xi) (_ Yi (_ Xi}$ , where i runs at least over two indices between 1 and n.

4 Bridges to Hugo Riemann

The present morphological approach yields some interesting relations to ideas in Hugo Riemann’s approach to harmony (cf. Riemann, 1877, 1887). We divide them into two subsections.

4.1 Consonant and dissonant Tone Perspectives

First we recall that Riemann’s understanding of the consonance/dissonance dichotomy differs from the traditional one in several ways. He applies a notion of relative consonance to the relation of the dominant/subdominant triads to the tonic and he applies a notion of relative dissonance to the relations of »Terzwechsel« and »Leittonwechsel«, i.e. to the relations usually denoted by $R$ and $L$ in more recent writings.⁴

⁴: Riemann actually speaks of »Scheinkonsonanz« in order to point out that--for example--in with respect to a tonic C the D-minor triad is an apparently consonant triad as auch, but appears to be dissonant relative to the subdominant triad

A first obstacle for a direct link to Riemann is our choice of the 12-tone system as a point of departure. Although Riemann confirms its theoretical importance he nevertheless insists to investigate tone relations, chord relations and key relations on the basis of a free tone net, generated by fifth and (major) third. This abstacle is shared by other Neo-Riemannian approaches as well. In another --quite different--investigation (Noll and Nestke, 2001) cover this gap on a theoretical level of tone apperception.

A mathematical bridge from the counterpuntal consonance/dissonance dichotomy of intervals to Riemann’s concept we can buid as built as follows by : We encode the (directed) intervals in $T$ in fifth-circle order, e.g.

prime = 0, fifth = 1, major second = 2, ... fourth = 11.