[Smt-talk] Classical Form and Recursion

Ildar Khannanov solfeggio7 at yahoo.com
Sun Mar 29 17:33:06 PDT 2009

Dear Nicolas,
I apologize for being a bit straighforward with my definition of D, S, and T. Mine is not an absolute view. Well, of course, if it mirrors the absolute view of a voice leading model, in which everything is determined by adjacency, then, may be. The point is, however, that if 1 and 2 are determined by + and =, and + and = are determined by 1 and 2, then we are in le cercle vicieux. Harmony is determined by voice leading, voice leading is determined by harmony. Not that we have not been there, or, hermeneutic tram is unfamiliar to music theorists. However, there must be the way out of this cercle. Ths is why I suggested that Tonic is Tonic because it is Tonic. Dominant is Dominant, even if in many cases it is left unresolved (Wagner/Scriabin).  I side with Dmitri on the  exigency of multiple sources. He calls it compatibilism, I use heterogeneite and multiplicite. And the history of music shows us that the origins of our concepts are multiple. Some
 see counterpoint as diminution, others as primarily the  rules of intervals. A millenium was long enough to accomodate both.  There is a Pythagorean beginning, but also a khalifat influence. Plied nature, in other words.
In this sense, I see no problems identifying l'accord parfait not necessarly by its placement in the environment of other chords, but as a sort of religious-philosophical construct, pure perfection. This is how French textbooks define it: it is a combination of a perfect fifth and a third. Yes, it has numeric origin, but that math was different from our trivial attempts to measure and calculate. Then, in the 5th century B.C., One was a male, Two--a female, Three--perfection, Four--unity of life and death, Heavens and Earth, the fourfold. Later, in book 6 chapter 6 of Plotinus' Enneads (which has been ommitted from many translations as obscure and impertinent), we read that the Many is the One.  This is not to say, that l'accord parfait has nothing to do with dominant-tonique or/and the sous-dominant.. No, it may come into contact with these sonorities, but their interaction is not the ontological condition. 
As for tonal spaces, regions and placement of chords, do we realize that we are dealing with poetic metaphors here? Yes, we want to talk about music in a precise and rigorous way, like mathematicians talk about their topics, but what can one do with temporal and essentially a-spatial character of music? When I listen to music, there is indeed, some sensory experience which is related to space. That is, the source of sound can be far or close to me. I can hear something in the back or in front of me. But this is not what you mean by musical space and it does not matter that much. Sound can make an impression of a figure and the background, but that is not what you mean by musical space. Notes  can be high or low? But here we stumble upon a poblem. The last statement is very problematic. In ancient Greek barus and oxun do not mean low and high. Rather, dark and wet against squeaky and unpleasant. In Tatarian and Bashkirian they are ache and k'olun,
 acidic and thick. High note is a metaphor of Christian eschatology, a relatively new thing. What you mean by tonal space is technical description of a playing surface (again very different on many instruments and non-existent in voice) and the paper and ink plane. As if Heidegger did not spent his life struggling with visual metaphors as a substitution in the question of Being and Husserl did not fight with transplanting terms from one scientific jargon to another. 
One can listen to music with eyes closed, in complete darkness. This will shut down the visual metaphorization process, together with spatial determination. It will be dark there  like in the Mutterleib. It will be the real Lebenswelt of harmony, using the terms of Edmund Husserl. Tension, tone.. Audible space has been nicely described by Husserl in his Ideen. Maurice Merleau-Ponty went a step further in Le visible et l'invisible. Music can present the world without the Other. 
Let us continue using visual metaphors, but let us also be cautious of their poetic-metaphoric character (I nod to Michael Spitzer).
ildar Khannanov
Peabody Conservatory
solfeggio7 at yahoo.com

--- On Sat, 3/28/09, Nicolas Meeùs <nicolas.meeus at paris-sorbonne.fr> wrote:

From: Nicolas Meeùs <nicolas.meeus at paris-sorbonne.fr>
Subject: Re: [Smt-talk] Classical Form and Recursion
To: "smt-talk Talk" <smt-talk at societymusictheory.org>
Date: Saturday, March 28, 2009, 3:13 PM

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 =).

[On this point, allow me to refer to my paper, "Transitivité, rection, fonctions tonales", http://www.plm.paris-sorbonne.fr/Textes/NMTransitivite..pdf. 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... 
(I know that some of you do read French ;-).]

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:

T --> I–V–I
D --> I–V–I [i.e. I/V–V/V–I/V]
T --> I–V–I

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.

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.

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.)


Nicolas Meeùs
nicolas.meeus at paris-sorbonne.fr

Thomas Noll a écrit : 
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. 
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". 
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. 
What are other typical syntagmatic traces for such chord-as-region-castings? 

Thomas Noll 

The Tonnetz 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.

Thomas Noll
noll at cs.tu-berlin.de
Escola Superior de Musica de Catalunya, Barcelona 
Departament de Teoria i Composició 
Tel (priv.):   +34 93 268 75 19
Tel (mobil): +34 66 368 12 02


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