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<font size="-1"><font face="Georgia">Dear Thomas,<br>
<br>
I consider two types of progressions only, dominant and subdominant
vectors, and I have no specific need of a "tonic vector".<br>
<br>
The case that you raise (I–vi) is that of the "progression" from a
given chord to one of its substitutes (T–Tp in this case). Similar
questions may arise about other substitutions, for instance S–S<
(IV–vi; "S<" is a Lettonwechsel subdominant). I wouldn't consider
that such successions form progressions properly speaking. They merely
materialize substitutions that in other cases may remain implicit.<br>
<br>
Such successions, however, usually are ambiguous because they involve
another relation which, in turn, can be expressed as a vector. This is
because any chord can be (and often is) a substitute for two other
chords: vi, in your example, can be either Tp or S<; T–S< clearly
is a dominant vector. In other words, I–vi expresses both an absence of
progressions and a (weak) dominant progression.<br>
<br>
In my MTO paper
(<a class="moz-txt-link-freetext" href="http://mto.societymusictheory.org/issues/mto.00.6.1/mto.00.6.1.meeus_frames.html">http://mto.societymusictheory.org/issues/mto.00.6.1/mto.00.6.1.meeus_frames.html</a>),
I had described the double emploi as follows:<br>
<small><br>
</small></font></font><font face="Courier"><small><font face="Georgia">I
––> IV –> V ––> I = I ––>
IV(=II) ––> V ––> I <br>
I –> II ––> V ––> I = I ––> (IV=)II ––> V ––> I</font><br>
</small><small><font face="Georgia"><br>
In these cases, the implicit substitution is expressed in notation by
IV(=II) or (IV=)II; if the substitution had been fully spelled, I would
have written, for instance, I––>IV=ii––>V––>I. As you can see,
the sign = stands here for the substitution, which I will not call a
"tonic vector". Also, –> in the notation above stands for ––>= (a
dominant vector followed by a substitution) or =––> (a substitution
followed by a dominant vector).<br>
<br>
To turn back to your I-vi example, my old harmony teachers gave us
rules (mainly, a formal ban on vi-V) that I interpret now, after many
years, as meaning that I-vi can only be T––>S<.<br>
<br>
I hope that this answers your question.<br>
<br>
Cordialement,<br>
<br>
Nicolas Meeùs<br>
<a class="moz-txt-link-abbreviated" href="mailto:nicolas.meeus@paris-sorbonne.fr">nicolas.meeus@paris-sorbonne.fr</a><br>
<a class="moz-txt-link-freetext" href="http://www.plm.paris-sorbonne.fr">http://www.plm.paris-sorbonne.fr</a><br>
</font></small><br>
<br>
<br>
</font><br>
Thomas Noll a écrit :
<blockquote
cite="mid:6470E9A8-FAEA-4B93-9B52-0A0DC487990F@cs.tu-berlin.de"
type="cite">Dear Nicolas,
<div>If you could pardon my misunderstanding, I would like to ask
about the secondary tonics. If I understand it better now, it is the
concept of transitivity which turns chord-progressions into the focus
of the theory rather than simply chords in succession. The progression
as such can be compared to a transitive verb whose two valences are
saturated by two chords. From this view point I may imagine a strategy
to consider higher order dominants as trivalent progressions, (or some
theoretical derivative). With secondary tonics it is harder to me to
follow. When a tonic is considered as not being intransitive, what are
its constituents then? E.g. Is it a progression or a paradigmatic
relation between two "tonic" chords (without the necessity to appear in
sequence)? Is I - vi a manifestation of a transitive tonic? </div>
<div>Sincerely</div>
<div>Thomas <br>
<div><br class="Apple-interchange-newline">
<blockquote type="cite"><span class="Apple-style-span"
style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">But
I do realize that the term "transitivity", as I use it, may lead to
misunderstandings and I will think of abandoning it altogether.</span></blockquote>
</div>
<br>
<div> <span class="Apple-style-span"
style="border-collapse: separate; border-spacing: 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-indent: 0px; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px;"><span
class="Apple-style-span"
style="border-collapse: separate; border-spacing: 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-indent: 0px; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px;">
<div>*********************************************************</div>
<div>Thomas Noll</div>
<div><a moz-do-not-send="true"
href="http://flp.cs.tu-berlin.de/%7Enoll">http://flp.cs.tu-berlin.de/~noll</a></div>
<div><a moz-do-not-send="true" href="mailto:noll@cs.tu-berlin.de">noll@cs.tu-berlin.de</a></div>
<div>Escola Superior de Musica de Catalunya, Barcelona </div>
<div>Departament de Teoria i Composició </div>
<div>Tel (priv.): +34 93 268 75 19</div>
<div>Tel (mobil): +34 66 368 12 02</div>
<div><br class="khtml-block-placeholder">
</div>
<div>*********************************************************</div>
<div><br class="khtml-block-placeholder">
</div>
<div><br>
<br class="khtml-block-placeholder">
</div>
<br class="Apple-interchange-newline">
</span></span> </div>
<br>
</div>
</blockquote>
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