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<div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">Dear Nicolas,</font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">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.</font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">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::</font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">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.</font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">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.</font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">[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, ->, ->->, ->->->, ->->->->, ... </div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: normal normal normal 12px/normal Helvetica; ">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.]</font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><br class="webkit-block-placeholder"></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">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. </div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">With best wishes</font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font face="Helvetica" size="3" style="font: 12.0px Helvetica">Thomas</font></div><div><br class="Apple-interchange-newline"><blockquote type="cite"> <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> </blockquote></div><br><div> <span class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 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-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; 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 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-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><div>*********************************************************</div><div>Thomas Noll</div><div><a href="http://flp.cs.tu-berlin.de/~noll">http://flp.cs.tu-berlin.de/~noll</a></div><div><a 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></body></html>