system. Before entering into the details we recall other authors’ suggestions and observations which motivate the main direction of the present study.

1.3 Anchors in American >Set Theory<

The treatment of chords as sets-of-tones refers to the american music-theoretical tradition named »set theory« which uses the term »pitch class« for elements of the twelve-tone-system and »pitch class sets« for unordered collections of these. While this theory has been originally devoted to the study of atonal music, there are numerous ramifications and synergies with other approaches which are dedicated to tonal music, such as the studies of diatonic set theory. A suitable anchor for our investigations is David Lewin’s generalized set theory (c.f. Lewin, 1987, chapter 6). For readers being familiar with this approach we mention that our investigations concentrate on the study of chords (pitch class sets) with respect to fully internal transformations, i.e. chords under study having maximal injection numbers with respect to some of the affine endomorphisms of . For readers being familiar with Morris (1987) we mention that our study systematically studies the transformations for all multiplication factors . Usually scholars spend most attention to the invertible factors . In our approach especially the non-invertible transformations for appear to be of particular importance for a morphological reconstruction of Riemannian harmony.

1.4 Anchors in >Neo-Riemannian Theory<

Neo-Riemannian analyses (c.f. Cohn, 1996, 1997) consider the 24 major and minor triads as basic objects, i.e. as elements or >points< of a harmonic configuration space. Pathways trough this space are studied with respect to the Schritt/Wechsel-group or Riemann-group transitively acting on this space. The practical interest in such studies is connected with prototypical chord-sequences in romantic and late-romantic pieces. Besides, there is also a theoretical insight which had significant impact on the Neo-Riemannian paradigm, namely Richard Cohn’s finding on the solidarity between voice-leading parsimony and -transformations for major and minor triads, pentatonic scales (and their compliments). With regard to this result Cohn characterizes the major and minor triads as >overdetermined< in the sense that this property is not related to the traditional >overtone< explanations, but rather distinguishes these triads as sets from all other tone sets according to minimal voice-leading connections. With regard to our discussion on chords as sets in subsection 1.2 we mention that Richard Cohn also switches between the -group and the Riemann group , i.e. between operations on chords-as-tone-sets and operations on chords-as-points in a Neo-Riemann space. In other words, this switch is a very interesting music-theoretical construction in connection with the >either/or<-problem. Strongly speaking, both kinds of triads are not the same music-theoretical objects. A possible mathematical bridge is offered by Julian Hook’s duality between transformations and -transformations within his theory of (Pseudo-)Uniform Triadic Transformations (c.f. Hook, 2003). In the present article we will discuss morphological properties -- different from the voice-leading