Music Theory Nerd Fight II: Why Michael Buchler is Wrong About Klumpenhouwer Networks

Note: second in a series of fairly esoteric articles on issues in the music theory discipline. 

First, let me say that I really like Michael Buchler personally and I’m quite sorry to do this. However, since his 2007 Music Theory Online article “Reconsidering Klumpenhouwer Networks” I have been more than once dinged by peer review for failing to take into account Buchler’s article when using Klumpenhouwer Networks (hereafter K-nets) myself, and I cannot tell you how annoying that is.  (Once I was so dinged by a peer-reviewer who was so passionate on the subject that I deeply suspected that the reviewer was Buchler himself.  Who else cares about this as much as he does?)

Like Straus’s article “The Problem of Prolongation in Post-Tonal Music,” Buchler’s “Reconsidering K-Nets” has achieved almost the force of a Supreme Court decision in Music Theoryland, and it has definitely put a crimp in the style of what could otherwise be some very interesting, freewheeling and progressive K-net-based analyses.  I am going to repeat a theme that I suggested in Music Theory Nerd Fight I, which is that I am perennially puzzled by the very common phenomenon of politically liberal professors who are not at all liberal in their academic bailiwicks.  Indeed, it often seems the more politically progressive the academic, the more likely that academic is to cling to orthodoxies in his or her chosen field.  So it is the case with Buchler, who, I don’t think it is any kind of great outing to say, is quite politically progressive judging from my encounters with him through social media.

Regarding K-nets, I question Buchler on two fronts: what’s the harm?  And what’s the alternative?  If Buchler can demonstrate harm, then, as far as I’m concerned, he wins the day.  However, if there’s no harm, then his complaints are entirely misplaced.  As far as an alternative model goes, Buchler proposes one, which is to his credit, but is it really a superior model?

1. What’s the harm?

Unlike Straus’s claim in 1997 that post-tonal prolongational analysis was dead, Buchler in 2007 observed that K-nets were alive and well: “Since David Lewin’s introductory article in 1990, K-nets have been among the most frequently discussed and analytically utilized tools for post-tonal transformational analysis” (“Reconsidering,” par. 1). One of the immediate harms Buchler identifies is that K-nets entail “a Pandora’s Box of relational permissiveness” (par. 2). He further elaborates, “Clearly, the more ways that it is possible to draw equivalent relations, the less significant those relations become” (par. 2). Buchler describes “problems” (par. 3) occurring because of the overabundance of relations that K-nets identify.

Buchler finds an ally in Straus, who finds harm in K-net recursion which he says “is only a problem when our desire for it leads us to emphasize musical features that might otherwise be of relatively little interest” (emphasis mine, par. 4). Straus goes on to criticize the dual-inversion aspect of K-nets as “hav[ing] no intrinsic interest [emphasis added], they correspond to no musical intuitions, they provide an answer to a question that no one has cared to ask” (par. 4). Again, as before, there is more than a hint of solipsism in Straus’s comments. Interest for whom? Just because Straus might find an observation uninteresting does not make it inherently uninteresting. Straus’s insistence upon the “correspondence to musical intuitions” is also puzzling, since he was so critical of the intuition-based analyses of Travis and Salzer in “The Problem of Prolongation in Post-Tonal Music.” (His term for intuition-based there was “ad hoc,” which is no kind epithet.  But here he insists on “correspondence” to “intuitions.”  So which is it, Prof. Straus?  Are intuitions good or bad?)  As for “providing an answer to a question that no one has cared to ask,” is he certain? Is it really a problem that observations about music may come to the fore without investigative antecedents? Is this the harm? Is this harm at all?

Buchler goes on to say that his alternative to K-nets “convey[s] clearer and more meaningful musical connections,” echoing Straus’s call for “more meaningful” relationships in “The Problem.” Again, clear and meaningful for whom? Is a lack of clarity really the problem with K-nets? If anything, I would say K-nets represent an immediately apprehendable entrée into the world of transformational theory, which only becomes more impenetrable as one goes, to which many who found Lewin’s Generalized Musical Intervals and Transformations difficult can probably attest. Buchler complains that K-nets are really dual transformations in disguise [par. 20-26], the harm of which eludes me. It strikes me as comparable to the competing set-theoretic taxonomies of Forte and Perle: both equally valid, but a preference for one as more elegant and comprehensive than the other emerging in consensus. The harm is obvious if one is a Perle partisan, but one is still able to use Perle’s nomenclature rather than Forte’s if one wishes (scholars such as Elliott Antokoletz who do precisely that come to mind).

Buchler critiques K-nets as leading to counter-intuitive results in analyzing a short passage from Lutoslawski’s Symphony No. 4. I would remind again that one of the great values of any kind of analysis— far from being a harm— is its capacity to lead the analyst to observations that could not be had by intuition alone. Counter-intuitive observations are valuable. Buchler (and Straus), however, tend(s) to find them “uninteresting” or “indefensible” [27-31].  On the other hand, I find much music analysis that serves only to reinforce the intuitive to be uninteresting to say the least, however defensible such analysis may be.

Buchler refers to the overabundance of K-net relationships as “promiscuity,” using quite a loaded term. He says that this is a harm because the more relationships a model can show between musical artifact A and musical artifact B, the less meaningful those relationships are. Let me interrupt the argument about harm here and point out that one of the primary problems with Buchler’s article is that he has essentially misapprehended the K-net model. K-nets come out of Henry Klumpenhouwer’s 1991 Harvard dissertation A Generalized Model of Voice Leading for Atonal Music (emphasis mine). Putting their recursive capabilities aside (and it must be pointed out that the recursive possibilities of K-nets were not promoted at first by Klumpenhouwer but rather by his mentor David Lewin), K-nets were originally conceived as voice-leading apparatuses.

Given K-net A and K-net B, every corresponding node describes a voice-leading motion from Node A to Node B. Voice-leading motions are indeed quite promiscuous. Between tetrachord A and tetrachord B one has sixteen potential voice-leading motions; between pentachord A and pentachord B, twenty-five potential voice-leading motions, and so on. Buchler confuses a K-net with a pcset.[1]  He believes that two K-nets are static things that show “relationships” rather than motions, like pcsets. If K-nets were intended to show pcset-like “relationships,” then there certainly would be too many “relationships” to be meaningful, the thrust of Buchler’s argument. However, K-nets describe motions from single notes to other single notes, not pcset-like relationships. It is of no moment, then, that there are many possible transformational motions that can be described between any musical artifact A and a musical artifact B of the same cardinality, just as it is of no moment that there are many possible voice-leading relationships that can be described between any two musical artifacts of the same cardinality.

However, let us suppose that we agreed with Buchler that the possible relationships are too promiscuous. What is the harm? Relational abundance is “problematic” (par. 32). He points out: “Since the most promiscuous trichord classes include many of the most common and familiar melodic and harmonic structures found in a wide range of repertoire, trichordal isography generally comes easily to those who seek it. When the standard for pcset relatedness [emphasis added][2] is this low, analysts ought to exercise particular diligence and discretion in making a strong case for the uniqueness and musicality of their readings.” So what is the issue? Let us continue to use K-nets, and let the analyst exercise particular diligence and discretion in making a strong case for the uniqueness and musicality of their readings, just as Buchler suggests. Problem solved.

Buchler devotes an entire section to the “problem” of multiple interpretations (par. 42-52). One is either in the business of analysis to find “the” definitive interpretation of a piece, or simply “an” interpretation of the piece. I prefer the latter mission, as I am skeptical of the possibility of the former mission. Suffice to say, I think the potential for multiple interpretations of music is hardly a harm.

Buchler criticizes K-nets on phenomenological grounds: “It would be difficult to imagine a situation in which dual transformation did not provide a more straightforward phenomenological account than K-nets” (par. 58). Straightforwardness is fairly subjective, however. Some people find one model straightforward (e.g., Forte) while others find a competing model straightforward (e.g., Perle). This too is barely a harm.  Hooray for alternatives!  Vive la difference!

Buchler devotes a section to the problems of K-net recursion. He finds a more troubling harm than that of Straus’s mundane “but can we hear it”-type critque. He says: “Recursive analysis requires us to locate positive and negative surface-level isographies in the same quantity[3] as shown in any one local K-net. This often entails skewing surface readings into representations that simply provide the right type of graph to fit the situation” (64). I find this to be his best argument: that the abstract attractiveness of K-net recursion entices the analyst to fit the music into a Procrustean bed. But then, to remedy this, I think one simply has to call on analysts to “exercise particular diligence and discretion in making a strong case for the uniqueness and musicality of their readings” when creating recursive K-net analyses.  Plus, the dangers of Procrustian beds are everywhere in music theory; they are certainly a danger of Schenkerian analysis (as Eugene Narmour has forcefully and repeatedly pointed out).  These are remedied by care, due diligence and keen judgment.

Buchler never overtly calls for the abolition of K-nets in his article, and, to be sure, he proposes some improvements to the model (such as the suggestion that more explicit numerical arguments could be used to describe transformations). However, when he says “We all have different goals for analysis, but surely one central purpose is to clarify and explain. There may not be any inherently easy ways to model difficult music; I just want to be certain that my analytical tools help me elucidate more complexities[4] than they introduce. That might be the simplest and best reason to reconsider Klumpenhouwer networks,” what does he mean by “reconsider Klumpenhouwer networks”? The only conclusion that makes any sense is that he means we should reconsider using them at all. He does not title his article “Taking Greater Care with Klumpenhouwer Networks” or “Some Suggested Improvements for Klumpenhouwer Networks.”

Just as Straus is reluctant to admit to technologies of certain degrees of complexity in addressing post-tonal music (e.g., prolongational schemas like those of Olli Vaisala which he says are “too complex” to hope to achieve widespread adoption), so too is Buchler, and it is just as puzzling.  Did I miss a memo?  I thought we were all on board with the proposition that post-tonal music is really, really complex and as such, requires analytical techniques to address this really, really complex music that are themselves really, really complex, commensurate with the complexity of the music the analyst hopes to address.  I don’t think that K-nets introduce more complexities than they elucidate; I think instead they are complex to the same degree as the music that they describe, which is fine.

2. What’s the alternative?

Buchler’s alternative is to recast K-nets as dual transformations. However, precisely his point is that it is much more difficult to locate recursive possibilities in dual transformations than it is in K-nets. Recursion is obviously a great harm to Buchler, since it too heaps on more potential “relations” that are possibly meaningless, and because such recursive relationships are simply harder to hear.

Phenomenology is such a great bugaboo with both Straus and Buchler, but it is not as though their preferred models do not entail great challenges on the front of audibility as well. Buchler compares K-nets (par. 5) to a “host of other tools” such as “similarity relations, split or near transformations, and topographical distance metrics,” but does not observe that these tools have also entailed perennial phenomenological questions of audibility. Both Buchler and Straus are practitioners of Schenkerian analysis, but they do not observe that Schenkerian analysis too has been long questioned on phenomenological/audibility grounds (paging Eugene Narmour again).  I would furthermore remind that just because a recursive K-net analysis might lead to something counter-intuitive (which is what I think Buchler really means when he talks about phenomenology, that analysis should match his own intuitions of what he believes would be audible) does not mean that the analysis is not valuable.

This again gets at fundamental questions about the mission of music theory and analysis.  If the enterprise is supposed to be about finding empirical justification for what we intuit about music, then, sorry, but I’m out.  I would rather discover something delightfully counter-intuitive that challenges my predispositions.  I find that K-nets are amazing tools to this end.

-Robert Gross


[1] Revealingly, at one point Buchler says “I find myself forced to think of [K-nets, K-classes and K-families] abstractly, in the same basic way as I think about set classes” (par. 32). In the same paragraph he also criticizes K-nets because of the propensity for analysts to ask “can these two pcsets [emphasis added] be diagrammed in such a way that they appear isographic?” I suspect Buchler is so steeped in the pcset model that he does not truly appreciate the salient differences between the pcset model and the K-net model.

[2] Not wanting to beat the proverbial dead horse, but this is another revealing phrase that shows Buchler thinks that K-nets are pcsets, and that K-net “relatedness” is exactly like pcset relatedness.

[3] He further complains (par. 65): “A more pragmatic problem that arises when constructing K-net superstructures is that K-nets of size q must be grouped into q-sized hyper-networks for recursion to be drawn.” However, this is not actually true, as I demonstrate with my K-Tonnetz model (Gross, “Post-Tonal Hierarchization in Wozzeck,” Journal of Schenkerian Studies, Fall 2014).

[4] I would point out that “complexity” is another troublesomely subjective term. Not all people find the same things to be complex.


I would like to share two gigantic K-nets I made to show a possible link between the digraphic qualities of the K-net and the mechanisms of genetics.  The first graph shows all sixteen possible Punnett Squares realized as a mod-2 K-net (the only values being 0 and 1); the second graph shows the entire genetic code (DNA codon version) as a mod-4 K-net (the values being 0, 1, 2 and 3).

Robert Gross - Analogy Networks 10-6-13 P20

Clearer downloadable version

Robert Gross - Analogy Networks 10-6-13 P23

Clearer Downloadable Version

And click here if you are interested in a really adventurous exploration of how the K-net model could help explain, in part, the origin of the universe.

Klumpenhouwer Networks Surreal Numbers Strange Loops and Cosmology the Case for Somethingness from Nothingness 1-25-15

One thought on “Music Theory Nerd Fight II: Why Michael Buchler is Wrong About Klumpenhouwer Networks

  1. First of all, I like you, too! And I like Henry Klumpenhouwer. And I’ve been very good friends with Shaugn O’Donnell for more than a couple of decades. It’s a friendly discipline (I like to think) and there’s absolutely no reason to let epistemological or methodological or whatever differences get in the way of that. Besides, when engage in polemics, you’ve gotta be prepared for polemics to be directed back at you. (Unless, you’re Donald Trump, I guess.) All that’s to say that I’m glad that you wrote this and that you shouldn’t feel the slightest bit sorry or painful about doing so; indeed, I rather hope you’ll consider submitting for publication.

    I think that most of our disagreements are philosophical and I believe that we’ve got considerably more common ground than you might be imagining. You don’t disagree with me that K-nets can be reconfigured into dual transformations; I don’t disagree with you that the choice to draw K-nets instead of dual transformations is a choice that once should be allowed to make.

    About choices: on reflection, I probably shouldn’t have used “promiscuity” as a descriptor (though it was Shaugn O’Donnell’s who, to my knowledge, originally applied that term to K-nets). I still find it an apt descriptor, but I fear that it distracted people into thinking that I was making a moral judgment (Henry certainly reflected that in his MTO response to me). Nothing could be farther from the truth! My most recent pair of papers on the subject (on transformational analysis writ large and whether there are inherently good or bad applications and on ornamentation in atonal music) have both been attempts to clarify my position. Briefly, though: I’m all for interpretative freedom: lots of it! However, my problem with K-nets is that because they can be musically opaque (the design of K-nets can be rather distant from the score) and because they can be promiscuous (there’s that word again), they can be easily and not always intentionally used to bolster some very strange (and dare I say unmusical) analyses.

    Yes, discovery is important, but it’s also important to be clear about what you’re saying when you are presenting an analysis and my experience has been that quite a number of presented and published K-net analyses have been less than clear. Indeed, I was prompted to write that article after seeing a young scholar give a conference paper in which he showed that every musical object in the examined piece was relatable via isographic K-nets to every other one. His parameters were more abstract than mod-12 pc space and, as was made obvious in the Q & A session, he himself didn’t realize that every object in his space could, willy nilly, be related to every other object in his space using K-net isography. This is a case of going too far. When everything is related to everything else, you shouldn’t only rely on K-nets. As I showed, even in our usual mod-12 pc space trichords are exceptionally easy to relate. So, then, I would hope that K-net analyses should justify the particular isographies that have been drawn.

    On this, I think we agree more than we disagree: we’re both sheepish about jumping into Procrustean beds; we both want musically sensitive analyses; we both want our tools to be used with care. My fear is that a lot people (not you, Robert!) learn to use these devises and don’t honestly understand their limitations and their extensions. On the continuum of meaningful to trivial statements, where does each drawn K-net isography fall? This aspect of K-net structure and entailment was not explored by either Lewin or Klumpenhouwer, and I think it’s important to understand.

    So, I do want us to “take greater care with Klumpenhouwer Networks” and indeed I wasn’t looking to abolish them. (But that’s not as good an article title as mine!) Could I honestly wield that kind of power? I hope not! I guess I should note at this point that if my goal were to abolish these things, I have done a pretty lousy job of it: even one of my own doctoral students is presenting a talk on K-nets (of a sort—a very creative sort) at SMT this year. And it’s a project that he first drafted in one of my classes.

    I guess I should also respond to your comment that I don’t seem to understand the difference between pcsets and voice-leadings. On this account, you might be right… or you might be drawing too great a difference. In any case, it’s here that your claims of my methodological conservatism are right on (nowhere else, though, I think). I have long been a critic of Straus’s pc voice leading (and he certainly knows that and we’ve had very good discussions about this point). I admit that I want the default meaning of “voice leading” to be stepwise motions (m2, M2) that mostly unfold in p-space (with some allowances for octave couplings and transfers, just as in tonal music). Now, there are some things that could sway me to be more expansive in what I think of as a “voice” (e.g., orchestration), but I find a lot of the supposed “voice leadings” that are expressed by either pc voice leading or K-nets are not truly voice leadings (in my admittedly conservative definition), but rather another way of expressing pcset transformation (e.g., when the “root” of set A maps to the “root,” of set B, etc.). When I see things like K-classes (Lambert), I become more convinced that my reading of K-nets as equivalent to broadly-defined pcset classes is a common (mis?)understanding.

    Anyway, there we are. I’m deeply sorry if my work has been used to block your work from seeing print (rest assured, that wasn’t my doing), but I still agree with the arguments I made way back when. I’ve got nothing against abstraction or against complexity, though where an equivalent simple solution exists, I prefer it (again, I’m with Occam!) unless there’s a very good reason not to prefer it. You and others have provided some good reasons and more power to you.


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