Analysis by Key: Another Look at Modulation

only serves to make musical time a more powerful symbolic representation of
human temporal experience. And further, a piece that struggles to achieve its
tonic presents a world of sound and feeling very different from one where the
tonic is asserted as a given from the outset.
Ex. 2 Chopin: Prelude, Op.28, No. 18
Allegro molto
+Aye J: : ,> ag, '/ : |
augmented 4th
t) tb e 5 b > _ * t
The Tonic as Matnx
X. *
ANALYSIS BY KEY
During the first nine or so bars of another Chopin Prelude, No. 24 in D minor,
there is no diminished fifth C# -G to specify D as the tonic pitch. That the key of
D minor is never in doubt during these bars results from the ever-present D
minor triad arpeggiated in both the left-hand ostinato and the melodic line; that
the few non-harmonic notes ali belong to the D minor scale adds further
confirmation. If we look a bit further - say up to b.21 - we shall see that the
tonicized harmonies are the opening D minor, the F major of b. 15 and the A
minor of b. 19, forming a huge expansion of the same arpeggio.
Despite its complex and unusual - perhaps unique - tonal plan, the Prelude
sustains its key with a minimum of help from the diminished fifth on the leading
note. Indeed in seventy-seven bars of highly chromatic music, C," occurs only
twice: before the reprise in b. 50 (introduced enharmonically as DS), and in the
final cadential dominant, b.64 (not counting a few rapid passing notes of no
harmonic significance in bs 55-6). D minor is securely established as a key
because the D minor chord so clearly forms the matrix that generates the
ostinato figure, the melodic line and the large-scale harmonic structure of the
Prelude's extended opening phase (bs 1-20).
Central to Schenker's work is the notion that the tonic triad, an image of the
overtone series generated by the tonic note, functions as a matrix- the source of
the Fundamental Structure that governs large-scale harmony (through the bass
arpeggio) and melody (through the Fundamental Line) as well as the ultimate
source of the middleground structures and foreground details that grow out of
the Fundamental Structure. As matrix, the tonic triad has rhythmic properties:
it defines the beginning and end of complete and self-contained harmonic and
melodic progressions; it also provides the foundation for form and design, since
motivic and thematic elements always connect (usually quite closely) to tonal
structure. As Fred Lerdahl and Ray Jackendoff write, 'the tonic is in some sense
MUSIC ANALYSIS 6: 3, 1987 291