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<font face="Calibri">In so far as Schenkerian prolongation can be
represented by letters, I'd answer something like this:<br>
<br>
If the prolongation of a chord <b>Y</b> produces a chord <b>x</b>
(not all prolongations produce new chords), then, in specific
cases, the result may be of the form <b>y–x–y</b>.</font> It
would however be mistaken to believe that the resulting <b>y</b>'s
are the same thing as the original <b>Y</b>: they do not belong to
the same level.<br>
<br>
Going on with alphabetic labels, the model representing this
situation might be formulated as follows:<br>
<u><b>y – x – y</b></u><br>
<b>Y</b><br>
<br>
In what sense does Schenkerian prolongation "let Y grow"? I don't
suppose it necessary to repeat here Schenker's organic metaphors.
From a more technical point of view, Schenker views the initial
chord as delimiting a "tonal space", intervals between disjunct
degrees, open to be filled by conjunct voice leading. This
particularly is the case of triads, the degrees of which remain
disjunct in any inversion. The conjunct movements therefore
necessarily involve notes foreign to the initial chord. This process
of filling the tonal space by conjunct movements is the main process
of prolongation: in lets the prolonged chord extend on a
considerable span of time. [Calling this extension a "growth" is
part of the organic metaphor, which one may or may not like; yet,
even "extension" already is a metaphor: any description of music has
to be metaphoric in a way or another.]<br>
Schenker himself gives the example of the first two bars of
Chopin's Prelude op. 28 n. 6; one could easily extend the example to
the first four bars: they form a four-bar prolongation of the chord
of b minor, by arpeggiation, neighbor notes and passing notes,
without any intervening new chord being created. One could similarly
quote the first four bars of Chopin's Nocturne op. 27 n. 2,
prolonging a Db major chord.<br>
In the first four bars of Bach's C major Prelude of WTC I, the
chord of C major is prolonged by neighbour notes exclusively. Here,
however, these filling notes meet to produce new chords, new tonal
spaces, which might have been further prolonged if Bach had wanted
to produce a longer work. Here, <b>Y</b> (C major) grows, is
prolonged, into <b>y–z–x–y</b>; each of these is turn is shortly
prolonged by arpeggios.<br>
<br>
This brings us back to Dmitri's initial opposition between the
descriptive usage of Roman numerals and the "intentional" idea of
prolongation. In the case of Bach's Prelude, <b>y–z–x–y </b>could
be replaced by I–ii–V–I. But that would say nothing of the origin of
these four chords, of the reasons why they follow each other in that
sense, etc. [In this particular case, the description by the voice
leading does not say much either about the grammaticality of the
phrase and, in particular, does not say why Bach could not have
written I–V–ii–I; but that is another discussion.]<br>
<br>
Yours,<br>
<br>
Nicolas Meeùs<br>
Université Paris-Sorbonne<br>
<br>
<br>
<br>
Le 17/07/2011 10:22, Eytan Agmon a écrit :
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<p class="MsoNormal"><span style="font-size: 11pt; font-family:
"Calibri","sans-serif"; color: rgb(31,
73, 125);">Nicolas wrote:<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:
"Calibri","sans-serif"; color: rgb(31,
73, 125);"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size: 11pt; font-family:
"Calibri","sans-serif"; color: rgb(31,
73, 125);">“As to Y–X–Y as a ‘model of prolongation’, I can
only repeat that this would form an excessively superficial
view of the process. There is only one ‘Y’, that which is
prolonged; the prolongation does not consist in splitting Y
in two in order to insert X between the two parts, but in
letting Y grow. ‘X’, if any, fully belongs to Y, of which it
is an organic outgrowth. Schenker's thought is not an easy
one; it certainly never is ‘naïve’.”</span><span
style="font-family:
"Calibri","sans-serif";"><br>
<br>
</span><span style="font-size: 11pt; font-family:
"Calibri","sans-serif"; color: rgb(31,
73, 125);">Suppose that the “Y-X-Y model” includes a
relationship between Y and Y-X-Y. Then certainly Nicolas’s
first objection is overcome (“there is only one ‘Y’, that
which is prolonged”). As to his second objection, I think we
need to have a clearer sense of what it means to “let Y
grow” (or “let X grow organically out of Y”) before we can
judge the “Y-X-Y model” in this sense as a “superficial” or
“naïve” interpretation of Schenkerian Prolongation.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 11pt; font-family:
"Calibri","sans-serif"; color: rgb(31,
73, 125);"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size: 11pt; font-family:
"Calibri","sans-serif"; color: rgb(31,
73, 125);">Eytan Agmon<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size: 11pt; font-family:
"Calibri","sans-serif"; color: rgb(31,
73, 125);">Bar-Ilan University <o:p></o:p></span></p>
<p class="MsoNormal"><span style="color: windowtext;"><o:p> </o:p></span></p>
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