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

Figure 4:

Snapshots of the PianolaScore (left) and the SliceScore-windows displaying the presence of the morphemes $MR$ (middle) and $ML$ (right).

Figure 4 simultaneously displays the presence of two morphemes $MR$ and $ML$ in the d-Minor passage (bars 173 - 203) from the first movement of Schubert’s $Bb$ -Major Sonata (D 960). This passage is immediately followed by the >official< recaptitulation and--at the same time--has some characteristics of an already >pretended< recapitulation (cf. Cohn, 1999). The double-interpretation of the d-minor triad as a relative-minor-triad ( $R$ ) of the dominant and the leading-tone-triad ( $L$ ) of the tonic supports this ambiguity. The Monoid $M11,5$ with $|M11,5| = {2,5,9}$ is the intension of the bigeneric morpheme $M11,5$ . With respect to semitone-encoding this is associated with the d-minor triad. The corresponding left ideals

2 5 9 5 9 0 2 5 9 11 8 0 R = { 0, 0,0, 8, 4,1} and L = { 0,0, 0, 3, 9, 1}

represent the truth values of the relative and the leading-tone functions within the internal logics of $M11,5$ (c.f. subsection 4.2). Considering them as intensions of new morphemes would yield extensions solely consisting of super-chords of the d-minor triad. Therefore we try to get rid of at least one constant tone in both monoids. There is only possibility to do so (otherwise we do not get monoids):

MR = {50,90,58,94,01} and ML = {20,50,113,89,01}.