<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
<meta content="text/html;charset=ISO-8859-1" http-equiv="Content-Type">
</head>
<body bgcolor="#ffffff" text="#000000">
<font size="-1"><font face="Georgia">This is not what I meant. To me,
"X is Dominant of Y" means either that chord X is Dominant of chord Y,
or that region X is Dominant of region Y. To me, harmonic or tonal
functions are exemplifications of transitivity (this is also meant as
an answer to Idar Khannanov's "absolute" view of the harmonic
functions). A chord (or a region) is dominant of another chord (or
region) merely because the two are placed in this relation. A dominant
necessarily is Dominant OF something; the same for a tonic. It is the
relation (say V–I) that defines the functions: there is no V without I,
no I without V. There is no Dominant "by definition". (Similarly, 1+1=2
is not true only because of properties of 1 and 2, but also by virtue
of properties of + and =).<br>
<br>
[On this point, allow me to refer to my paper, "Transitivité, rection,
fonctions tonales",
<a class="moz-txt-link-freetext" href="http://www.plm.paris-sorbonne.fr/Textes/NMTransitivite.pdf">http://www.plm.paris-sorbonne.fr/Textes/NMTransitivite.pdf</a>. I'm really
frightened ascertaining the extent to which what we publish in French
is not read on the other side of the Ocean. I may not be right in this
paper, but it was first published more than 15 years ago... <br>
(I know that some of you do read French ;-).]<br>
<br>
The recursion that I see in the Tonnetz consists in the fact that a
device that had originally be conceived (by Euler) to describe just
intonation (i. e. individual pitches) later was used (1) for harmonic
relations; (2) for tonal relations. In other words, music theory
recognized (or at least assumed) that the functionings at the level of
tonalities (or regions) were similar to those at the level of chords.
In other words, if one may consider that a tonal region minimally is
defined by chords in I–V–I relation, then one might consider that a
tonal scheme of the type T-D-T (by which I mean Schoenberg's
designations of regions) actually is recursive as follows:<br>
<br>
T --> I–V–I<br>
D --> I–V–I [i.e. I/V–V/V–I/V]<br>
T --> I–V–I<br>
<br>
The limit of such a model soon becomes obvious, however, because the
true minimal tonal phrase would include a predominant (I–P–V–I), which
hardly could be found at the level of regions. This is why I wondered
whether recursion exists in the theory only, or in the musical works
themselves.<br>
<br>
Secundary dominants (which, in my transitive conception of functions,
necessarily involve a secundary tonic) do represent cases of regions
recursively embedded in regions. But not in the form of
chords-as-regions, I think. I–V/V–V–I must be understood as
I–[V/V–I/V]=V–I, where the embedded region is not merely II (i.e. V/V)
as a "chord-as-region", but rather V as a chord with dual function,
being both I/V (or I/D) in the embedded D region, and V (V/T) in the T
region. The same could be said, of course, of IV/T=I/SD.<br>
<br>
To sum up: Schoenberg's concept of monotonality as a land of regions
probably is a good example of a conception of recursivity. On the other
hand, the usage of the Tonnetz both as chart of the monotonal land and
as a descriptor of harmonic functions, assumes a level of recursion
that may not be substantiated. This question must be raised also about
Lerdahl's charting of tonal pitch spaces, which similarly may assume an
unsubstantiated level of recursion. (The paths within each of the tonal
nuclei in Lerdahl's space do not obey the same rules as between the
nuclei.)<br>
<br>
Yours,<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>
</font></font><br>
Thomas Noll a écrit :
<blockquote
cite="mid:0FAEE425-0222-4A5F-956B-451F063917F1@cs.tu-berlin.de"
type="cite">What does "X is Dominant of Y" mean (in the paradigmatic
sense)? Is X a chord and Y a region? Under such an assumption it would
be impossible to speak of secondary dominants without an additional
operation of "typecasting" wherein chords are turned into regions. What
is the basis for such a "typecasting" operation: Something like the
recursivity of the Tonnetz - as mentioned by Nicolas Meeùs? It seems
that the acceptance of secondary dominants as instances of recursion
has strong consequences for the entire theory.
<div>A typical syntagmatic trace for the chord-as-region-casting is a
ii - V/V progression. If we assume recursion we assume ths at behind
the scene the "V" in "V/I" is casted as the "V" in "I/V". </div>
<div>Analogously we have the reverse direction, the V - ii/IV
progression (think of Chopin's E-minor prelude). Here the "IV" in
"IV/I" is casted as "IV" in "I/IV" behind the scene. </div>
<div>What are other typical syntagmatic traces for
such chord-as-region-castings? </div>
<div>
<div><br class="webkit-block-placeholder">
</div>
<div>Sincerely </div>
<div>Thomas Noll <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;">The<span
class="Apple-converted-space"> </span><i>Tonnetz</i><span
class="Apple-converted-space"> </span>would appear to me as an example
of recursion, in that it can be taken to represent pitches (as in
Euler), or chords (as at times in Riemann), or tonalities (Schoenberg's
regions). This exemplifies the assumption that functions are the same
or similar at these three embedded levels. Some may remember that at
one of the early OxMac conferences (in the late '80s, I think), Leonard
Meyer forcefully questioned this assumption.</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>
</div>
</blockquote>
</body>
</html>