[Smt-talk] Classical Form and Recursion

Thomas Noll noll at cs.tu-berlin.de
Fri Apr 10 12:11:11 PDT 2009

Dear Nicolas, dear Colleagues,
I would like to go again into secondary dominants and continue along  
and across the line of argument in Nicolas' posting from march, 28.

(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: iteration and (primitive) recursion.  
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). This may be too narrow in comparison to  
Nicolas' function-relations on chords, but it helps me to connect  
some ideas.
Iteration 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.
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?

(1') In extension of deduction/abduction-idea: The concept of higher  
order maximally even sets (Jack Douthett's "Stroboscope) involves an  
iteration (or recursion?) of Clough/Douthett's J-functions (mapping  
generic to specific pitches) with varying parameters from application  
to application. Eytan Agmon argued - from a perceptual/cognitive  
point of view - that the J-function has the wrong direction.  
Therefore he introduced a "generic function" which turns specific  
pitches into of generic ones. But in objection to his objection one  
can replace the "wrong direction"-argument by an abduction argument.  
Specific  pitches on one level could be abductively interpreted as  
the result of some J-specification (of a generic pitch).


(2) My question was about the type of arguments on both sides of the  
predicate "X is dominant of Y". It was dangerous to use the term  
region as a possible type for "Y". A more cautious formulation of my  
original question might have been: "Shall we make a distinction  
between the types of X and Y" ?

Nicolas, you convincingly dismiss this option in favor of an argument  
which features the productive role of transitivity. By transitivity  
we may extend a given functional relation on chords.
Further 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?
- 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.
- 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?
- Thirdly, what is the role of secondary dominants in this picture?  
Do secondary dominants imply secondary regions?

I realize, that you take an embedded region into account in  
connection with the example  I–[V/V–I/V]=V–I,   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 region could be simply the set  
of those chords which are in functional relation to a fixed chord,  
which is in identical (tonic) 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.

(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 "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.

Thomas Noll

> 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.)
> Yours,
> Nicolas Meeùs
> nicolas.meeus at paris-sorbonne.fr
> http://www.plm.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?
>> Sincerely
>> 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
>> http://flp.cs.tu-berlin.de/~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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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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