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<div class="moz-text-html" lang="x-western"> <big><font face="Georgia"
size="-1"><big>Thomas,<br>
<br>
I am somewhat at loss to answer you because I utterly lack the
background to which you refer. Be indulgent. I will try to provide
simple answers to questions that struck me as highly complex.</big></font><font
face="Georgia"><br>
</font></big>
<blockquote
cite="mid:D151E109-C85A-40B2-8C86-1237AB7EC4EC@cs.tu-berlin.de"
type="cite">
<div>
<div><font face="Georgia">(1) The concrete treatment of "dominant-of"
as a binary relation on chords, as Nicolas proposed, is a good reason
to address the connection between two concepts: <i>iteration</i> and
(primitive) <i>recursion</i>. Both are exemplifications of
transitivity but they have opposed meanings in functional programming,
from which we may benefit in our discussion. For the scope of this
posting, let as assume the "dominant-of"-relation on chords to be a
graph of a fifth-transposition-function (i.e., the Cayley-graph of a
group of transpositions acting on these chords with the group being
generated by the fifth transposition).<i> </i>This may be too narrow in
comparison to Nicolas' function-relations on chords, but it helps me to
connect some ideas.</font></div>
<div><font face="Georgia"><i>Iteration</i> starts with an initial
argument Y and applies a certain function D to give an output X= D(Y)
which is used again as an input argument to which the function is
applied again to yield D(X) = D(D(Y)). Iteration leads from given
objects to new objects. We read and understand the expression
D(D(D(Y))) from inside outwards. </font></div>
<div><font face="Georgia">In recursion we read and understand the
expression D(D(D(Y))) from outside inwards. We apply the function D to
something we don't know yet, so we need to understand what this is. It
is the value of another application of D to something we don't know
yet, and so fourth until we recognize the inner argument Y as something
known or explicitly given. I wonder wether in the study of reasoning,
iteration could be a concatenation of deductions along the same rule,
while (primitive) recursion could be a concatenation of abductions
along the same rule. The more I wonder wether this is a relevant
distinction for the interpretation of secondary, tertiary etc.
dominants. I'm particularly puzzled by the typical syntagmatic order in
fifth-fall sequences for chords. Do they revert raising paradigmatic
order?</font></div>
</div>
</blockquote>
<font face="Georgia">Don't you fail here to fully acknowledge my
transitive description of the function? My claim is that the "dominant
function" is a relation, say X–>Y, which I call "dominant vector".
This is not identical with your idea of "dominant-of" – say, D(X) –
because your presentation retains both a hierarchy and a direction. In
your presentation, D(X) somehow implies that X is given a priori (as
the tonic) and that D(X) is a possible way to relate to X. I want to
imagine that in X–>Y, both X and Y acquire their reciprocal function
because of the –> that links them: X is D(Y), for sure, but that
implies the reciprocal function where Y is T(X) – without hierarchy,
temporal or otherwise, between these two functions : they imply each
other.<br>
[Let me stress in passing that this "vectorial" view is mainly
methodological: I do not claim that it represents a truth, merely that
it is methodologically worth considering because it helps renewing and
reviewing common conception of tonal functions.]<br>
Things being so, my present view of the secundary-dominant function
certainly cannot be notated as D(D(Y)), because that would imply
precisely the type of hierarchy (or direction) that I try to avoid. The
secundary-dominant relation can be represented either as
(W–>[X–>Y]), or as ([W–>X]–>Y) without any possibility to
choose between the two. Or else, a secondary dominant really must be
symbolized by [–>]–>, or by –>[–>], of better still by
–> –> –> because the function is in the relation, in the
"vector", not in the related. The whole idea of harmonic vectors is
that no tonality needs be presupposed: in a chain of dominants, I
cannot predict a priori which one will resolve on the tonic.<br>
[Allow me a second comment in passing, namely that I believe this
theory to be rather stronger than may seem at first, even if not
entirely novel. It may seem simple or naive, and as such negligible,
but for those who put some confidence in it, it proved unexpectedly
fruitful.]<br>
<br>
In short, my "harmonic vectors" are more truly transitive than you seem
prepared to admit. And I think that their transitivity somewhat rejects
the possibility of iteration as you describe it, because iteration has
a direction.<br>
</font>
<blockquote
cite="mid:D151E109-C85A-40B2-8C86-1237AB7EC4EC@cs.tu-berlin.de"
type="cite">
<div><font face="Georgia"><span class="Apple-style-span"
style="font-family: Georgia; font-size: 10px;"><span
class="Apple-style-span"
style="font-family: Helvetica; font-size: 12px;">F<big>urther you
mention an obstacle for the acceptance of tonnetz-recursion in the
music (rather than only in the theory). This obstacle is of syntagmatic
nature, as I understand: You argue that a sequence I-P-V-I which is
typical for chord progressions is less typical for regional
progressions. What exactly is the role of this interesting argument?</big></span></span></font></div>
<div><big><big><big><big><big><font face="Georgia"><big><span
class="Apple-style-span" style="font-family: Georgia; font-size: 10px;"><span
class="Apple-style-span"
style="font-family: Helvetica; font-size: 12px;"><big>- Firstly, we
can reject the argument as far as the acceptance of recursion alone is
concerned. I recall a comparison by Michael Leyton which comes into
play again later in (3): Suppose we would like to analyze the control
of the limb motions as a relative motion system. Every limb (from the
finger tip to the shoulder and further on) moves relative to the motion
of another limb. Let as (silently) assume that (some) researchers in
robotics and cognitive science interpret this capacity as an instance
of recursion (I'm not sure though :-). Should we then expect - e.g. for
the gestures of a pianist - that the trajectories of the elbow or the
wrist should resemble the trajectory of the fingertip? Similar
objections (against the lack of self-similarity as an obstacle for the
assumption of recursion) have been raised before in this thread.</big> </span></span></big></font></big></big></big></big></big></div>
<div><big><big><big><big><big><big><font face="Georgia"><big><span
class="Apple-style-span" style="font-family: Georgia; font-size: 10px;"><span
class="Apple-style-span"
style="font-family: Helvetica; font-size: 12px;"><big>- But: Secondly,
with respect to fifth progressions alone, there seems to be a
resemblance of both levels (chordal and regional). How we should bring
this into the overall picture? Shall we therefore(!) be more strict
with our theoretical expectations about recursion in harmony in order
to keep this self-similarity aspect on the table?</big> </span></span></big></font></big></big></big></big></big></big></div>
<div><big><big><big><big><font face="Georgia"><big><span
class="Apple-style-span" style="font-family: Georgia; font-size: 10px;"><span
class="Apple-style-span"
style="font-family: Helvetica; font-size: 12px;"><big>- Thirdly, what
is the role of secondary dominants in this picture? Do secondary
dominants imply secondary regions? </big><br>
</span></span></big></font></big></big></big></big></div>
</blockquote>
<font face="Georgia">I do believe that the two cases behave quite
differently. <br>
The I-P-V-I succession must be read in terms of the <i>double
emploi</i> : either I-P or P-V relation is a "dominant vector": either
I–>P [i.e. I=D(P)], or P–>V [i.e. P=D(V)]. In either case, the
other progression (that which is not truly a dominant vector) is a
substitute dominant vector (this can be demonstrated in various ways).
The progression as a whole, therefore, can be represented either as
I–>P-->V–I, or as I-->P–>V–I (or, better still, either as
–> --> –> or as --> –> –>; see my MTO communication
of 2000, which used be available at
<a class="moz-txt-link-freetext"
href="http://societymusictheory.org/mto/issues/mto.00.6.1/mto.00.6.1.meeus_frames.html">http://societymusictheory.org/mto/issues/mto.00.6.1/mto.00.6.1.meeus_frames.html</a>
– but the link does not work tonight).<br>
The T–D–T regional progression, on the other hand, is a mere
"pendular" vectorial progression, <– –>, lacking the teleological
quality of the chordal one described above. (Such pendular progressions
do appear at the chordal level as well, but at that level they do not
form complete harmonic/tonal phrases).<br>
The progressions are superficially similar, but logically quite
different. Whether the second can be accepted as a recursion of the
first seems to me at least questionable.<br>
</font>
<blockquote
cite="mid:D151E109-C85A-40B2-8C86-1237AB7EC4EC@cs.tu-berlin.de"
type="cite">
<div><big><big><big><big><font face="Georgia"><big><span
class="Apple-style-span" style="font-family: Georgia; font-size: 10px;"><span
class="Apple-style-span"
style="font-family: Helvetica; font-size: 12px;"> </span></span></big></font></big></big></big><font
face="Georgia"><span class="Apple-style-span"
style="font-family: Georgia; font-size: 10px;"><span
class="Apple-style-span"
style="font-family: Helvetica; font-size: 12px;"><big>I realize, that
you take an <i>embedded region</i> into account in connection with the
example </big><span class="Apple-style-span"
style="font-family: Georgia; font-size: 10px;"><big> I–[V/V–I/V]=V–I, </big><span
class="Apple-style-span"
style="font-family: Helvetica; font-size: 12px;"><big> But I don't
fully understand the theoretical and argumentative connection of this
decision with the other two arguments in your posting, (i.e. that "X is
dominant of Y" is a relation between chords (or between regions) and
that the Tonnetz-recursion involves the danger of being a theoretical
artifact.) With these two arguments alone it is more plausible to also
dismiss the application of mutual embedding of regions. Do you assume a
Schoenbergian concept of mono-tonality on the paradigmatic(!) side?
Otherwise a <i>region </i>could be simply the set of those chords
which are in functional relation to a fixed chord, which is in
identical (tonic)<i> </i>relation to itself. Although the definition of
regions in Fred Lerdahl's "Tonal Pitch Space" is still different, it is
nevertheless non-recursive in this sense. In TPS mono-tonality is not a
paradigmatic concept, it is manifest in the prolongational reductions
on the syntagmatic side.</big></span></span></span></span></font></big></div>
</blockquote>
<big>I would consider this in relation with Lerdahl's <i>Tonal Pitch
Space</i> network representation. Superficially, it would seem that
passing from one tonal cluster to another in the TPS is similar to
passing from one degree to another within a given cluster. But this is
not so, the two cases obey distinct rules, as indicated above (passing
from one region to another obeys a logic of the pendular type, while
passing from a chord to another usually obeys the rule of dominant
vectors.) The idea that secundary dominants are borrowed regions may
not be entirely true, and if so may not be accurately represented in
the TPS. In other words, I-II-V-I (instead of I-ii-V-I) may not be
adequately notated as I-V/V-V-I.</big><br>
<blockquote
cite="mid:D151E109-C85A-40B2-8C86-1237AB7EC4EC@cs.tu-berlin.de"
type="cite">
<div><big><font face="Georgia"><span class="Apple-style-span"
style="font-family: Georgia; font-size: 10px;"><span
class="Apple-style-span"
style="font-family: Helvetica; font-size: 12px;"><span
class="Apple-style-span" style="font-family: Georgia; font-size: 10px;"><span
class="Apple-style-span"
style="font-family: Helvetica; font-size: 12px;"><big> </big></span></span></span></span></font></big><font
face="Georgia">(3) I argued that a distinction between the types of
X and Y in "X is dominant of Y" would imply obstacles for a treatment
of secondary dominants as instances of recursion. But in fact, there is
a paper by Michael Leyton, entitled <big>"</big></font><big><font
style="font-family: Helvetica; font-style: normal; font-variant: normal; font-weight: normal; font-size: 12px; line-height: normal; font-size-adjust: none; font-stretch: normal;"
face="Georgia" size="3"><big>Musical Works are Maximal Memory Stores"
(In Mazzola et al (eds.): "Perspectives of Mathematical and
Computational Music Theory", Osnabrück), which actually addresses this
problem in a challenging way. In (1) above we consider the iteration of
a single map on chords or roots (fifth transposition). In Leyton's
approach (see also his book "A generative theory of shape", Springer,
2001) the iteration occurs on a level of group actions. The operation
which is iterated is the wreath product between copies of an underlying
group, such as Z12, for example. Leyton compares musical modulation
with relative motion systems and connects his treatment of the latter
with the treatment of the former. He also challenges the traditional
interpretation of the group theory involved in robotics and relates
this to various domains in cognition. The lack of a convincing
non-trivial example left an ambiguous feeling when I studied the paper.
But in the context of our discussion I believe it offers a promising
line of thought. In particular it would be good to inspect connections
to Jay Hook's investigation of uniform triadic transformations.</big></font></big></div>
</blockquote>
<big>Does the above somehow answer this as well? If not, I may need
more precisions on Leyton's claims.<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>
<br>
[PS. Note, as a comment to other messages in this thread, that my
theory of harmonic vectors precludes the existence of tonal tensions a
priori. Tonal tension, in my opinion, always is retrospective. In
X–>Y, the tension arises only after ther resolution, because X led
to Y (and Y liquidated X). X as such remains free to go anywhere and
the sensation of obligato movement appears only retrospectively. It
goes without saying that memory, in the case of such a stable system as
the tonal one, suggests tension; but this always is based on the
assumption that the usual progression will happen. The exceptional
resolutions of V<sup>7</sup> (i.e. those resolutions that take any note
of the chord as the leading tone to the following one) can be shown to
retrospectively create tensions in unexpected directions).]<br>
</big>
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