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

IV/Eb ~ I/Eb ~ IV/Gb ~ I/Gb ~ IV/A ~

|_ vi/A ~ vi/E ~ I/E ~ vi/B _| vi/A ~ vi/E ~ I/E ~ ii/Gb vi/A ~ vi/E ~ I/E ~ iii/E vi/A ~ vi/E ~ IV/B ~ vi/B vi/A ~ vi/E ~ IV/B ~ ii/Gb vi/A ~ ii/B ~ IV/B ~ vi/B vi/A ~ ii/B ~ IV/B ~ ii/Gb vi/A ~ iii/A ~ I/E ~ vi/B vi/A ~ iii/A ~ I/E ~ ii/Gb vi/A ~ iii/A ~ I/E ~ iii/E -vi/A---~---iii/A--~---V/A----~---vi/B-- -vi/A---~---iii/A--~---V/A----~---ii/G-- vi/A ~ iii/A ~ V/A ~ iii/Eb vi/A ~ iii/A ~ IV/B ~ vi/B vi/A ~ iii/A ~ IV/B ~ ii/G ii/E ~ vi/E ~ I/E ~ vi/Bb ii/E ~ vi/E ~ I/E ~ ii/G ii/E ~ vi/E ~ I/E ~ iii/Eb ii/E ~ vi/E ~ IV/B ~ vi/B ii/E ~ vi/E ~ IV/B ~ ii/G |_ ii/E ~ ii/B ~ IV/B ~ vi/Bb _| ii/E ~ ii/B ~ IV/B ~ ii/G b

~ viio/Gb ~ III/Gb ~ vi/Gb

All of the 22 pathways start with the same 5 loci and end with the same 3 loci and differ only in the choice of the 4 loci inbetween. The authors were surprised by the observation, that Cohn (2003) presented exactly one of those shortest analyses as a >Lerdahl-Style-Analysis<, namely No. 11 (enclosed between horizontal lines) as a point of departure for his further analysis. The shortest analyses support a plagal interpretation of the opening rising fifths. Lerdahl’s analysis is in better accordance with David Lewin’s observation, that the regional centers of this passage form a Klingsor motive (c.f. Lewin, 1987).

5.6 Diatonic Triad Stratifications

In this last subsection we recapitulate Lerdahl’s definition of the elementary intra-regional, inter-regional and local pivot-distances. We introduce our own terminology and eliminate a technical problem.

A diatonic locus $(t (k),Dia(n)) type$ is a triad with(in) a diatonic collection, where $type (- {M aj,min,dim}$ and $n,k (- F$ such that $|(t (k)| < |Dia(n)| type$ ( $|S |$ denotes the set of the elements occouring in a sequence $S$ ). As an example consider $I = (t (0),Dia(0)) C Maj$ , which denotes the C major triad within the ’C-major’ diatonic collection.⁹

⁹: The label ’C-major’ should not include a modal connotation when it is applied to a diatonic collection.