[Smt-talk] Classical Form and Recursion

[Thomas Noll]

Dear Nicolas,
many thanks for you reply. I think I did too many things at the same  
time in my posting and I got several point wrong. It would be good to  
disentangle some points.
With my clumsy paraphrase of a binary "dominant-of" relation I tried  
to silently circumnavigate a possible source of misunderstanding [in  
particular for mathematicians like me] in the connection of the terms  
"relation" and "transitive". Obviously you did not(!) mean  
"transitive relation" in the mathematical sense: If X -> Y and Y ->  
Z, then also X -> Z (with -> denoting the dominant vector as a  
relation). But I still thought that you mean something in this  
direction. That was the condition I (again wrongly) had in mind::
To every chord Y which is in dominant relation X -> Y to a chord X,  
there exists a chord Z which is in dominant relation Y-> Z to Y.
In that case the mathematicians would say that the underlying  
directed graph is connected. This is very close, however, to another  
meaning of "transitive" in mathematics, namely to "transitive group  
action". David Lewin's GMIT is mainly about musical applications of  
such actions. In other words: I believed that you aim to explain  
secondary dominants as concatenations of dominant vectors.
[i.e. something like this: If we treat the dominant vector as a  
"generalized musical interval" we may construct higher order vectors,  
such as a secondary dominant X ->-> Y, tertiary dominant X ->->-> Y  
and so fourth. With these vectors we may calculate concatenations ->  
+ -> = ->->. -> + <- = ø. So we obtain a group of intervals, ..., <-<- 
<-<-, <-<-<-, <-<-, <-, ø, ->, ->->, ->->->, ->->->->, ...
The concept of transitivity, which I try to project onto your  
approach can be expressed by this condition: To every pair of roots X  
and Y there is exactly one vector v = ->->...-> such that  X v Y.]

But reading again more closely what you said, I understand that in  
your understanding secondary dominants involve secondary tonics.   
Thus the above construction is insufficient for this and I will need  
to think about that again. However I'm left with this intermediate  
question: From your analytical applications I tend to infer that the  
dominant vector is a relation between roots rather than chords. Is  
this correct? Your explanation of the term transitivity involves  
aspects of voice leading and therefore I'm not sure.
With best wishes
Thomas

> Thomas,
>
> 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.
>> (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?
> 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.
>     [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.]
>     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.
>     [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.]
>
> 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.
>> 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 do believe that the two cases behave quite differently.
>     The I-P-V-I succession must be read in terms of the double  
> emploi : 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 http://societymusictheory.org/mto/issues/mto. 
> 00.6.1/mto.00.6.1.meeus_frames.html – but the link does not work  
> tonight).
>     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).
>     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.
>>  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.
> I would consider this in relation with Lerdahl's Tonal Pitch Space  
> 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.
>>  (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.
> Does the above somehow answer this as well? If not, I may need more  
> precisions on Leyton's claims.
>
> Cordialement,
>
> Nicolas Meeùs
> nicolas.meeus at paris-sorbonne.fr
> http://www.plm.paris-sorbonne.fr
>
> [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 V7 (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).]

*********************************************************
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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