Networks for Fixed Media

Or, My Big Ideas, Vol. II.

The genesis of Networks comes out of a desire to prove the efficacy of composition with Klumpenhouwer Networks (K-nets). It is my belief that for the post-tonal composer looking for a happy medium between strict formal procedures such as serialism (which is, apparently, outmoded in this day and age) and completely free post-tonality loosely governed by an intuitive set-theoretic approach, K-nets offer exactly this happy medium.

Rather than focusing on sets as fixed objects, K-nets focus the composer’s attention on transformations. Composition with K-nets means a post-tonal music that is governed, regulated and unified on the one hand, but which places its emphasis on motion and transformation rather than on the stasis of fixed sets or unchanging row forms.

The specific networks used for the composition of Networks come from an article I wrote for the Journal of Schenkerian Studies entitled “Post-Tonal Hierarchization in Wozzeck.” In Wozzeck I discovered a Tonnetz of K-nets at work, unifying together on one grid the 37 sets defined as salient to Wozzeck by Allen Forte. On this Tonnetz, which I call a K-Tonnetz, every four-square box is superisographic with every other foursquare box (for a definition of K-net superisography, one should consult the article); furthermore, all the pentachords, hexachords, septachords and octachords described by Forte as salient are findable as adjacencies on the K-Tonnetz.

So I use this Wozzeck K-Tonnetz as my pre-compositional harmonic landscape for Networks. Any foursquare K-net on the K-Tonnetz is available to me, in any transposition, as is every identified pentachordal, hexachordal, septachordal or octachordal adjacency, in any transposition, that is superisographic to the parent hexachord, which is a 6-31 omnibus governing sonority of Wozzeck. (It should be noted that Perle would never have called it a 6-31 sonority, but they did agree that this sonority was of great importance, and if those two agreed on anything, there was probably something to it.) Transposition away from the parent K-Tonnetz denotes a hierarchically inferior set, giving the composer a means of post-tonal hierarchization. (For more details on why this is so, again, consult the article.)

The K-Tonnetze identified as governing Wozzeck and also used as a pre-compositional device for Networks are given below.

Next, it would be remiss not to address the issue of the medium for which Networks is composed. Like Diogenes looking for an honest man, I had been looking for years for an elder statesperson to guide me in my career goals which are frankly more concerned with being a composer/thinker and a composer-theorist hyphenate than they are with having the Big Composition Career (capital letters intended). I lament what seems to be the paucity of ideas in today’s concert music establishment landscape. Where are the heirs apparent to Babbitt and to Perle? Where are the essayist composers who are having the Big Careers (capital letters intended again)? Where are the Arthur Bergers and (whether one agrees with his assessments or not) the George Rochbergs?

So it has been my pleasure to have become reacquainted recently with the work of Benjamin Boretz. Realizing that I had been tardy for some time in my intention to compose a self-conscious K-net piece, I came across Boretz’s fixed media piece Group Variations (or, more properly, Group Variations II). This piece is based on an acoustic version for large chamber ensemble (Group Variations I). The piece is very dense and complex, and probably defies human performance realization. The turn to the electronic medium was understandable for Boretz.

I got to thinking, whatever happened to pitch-determinate pieces for fixed media? This too seems to be a lost art. Where are today’s Group Variations, today’s Philomel? This criticism is not to be taken with anything less than a heaping tablespoon of salt: the current state of affairs in electronic music is marvelous. So much is possible. But when so much is possible, it seems as though some of the fundamentals have gotten lost. Contemporary fixed media pieces strike me as approaching an all-texture-all-the-time sort of affair.

So I thought it might be refreshing to try to compose a pitch-determinate piece for fixed media, as a successor to Group Variations and to Philomel. I conceived of the piece as an ensemble piece: an ensemble comprised of ten pitch-determinate sounds that have analogues to acoustic music instrumentation. I chose ten sounds in my sound bank that are abstract enough not to directly imitate the intended analogue instrument (most direct samples of instruments are dreadful) but which behave in some analogous way to the intended analogue. (The one exception is the contrabass sample which I used, intact, which I think is quite good.) The analogue (analog?) instruments are wind quintet and string quintet. The piece could conceivably be performed by this standard ensemble, if the ensemble can achieve some of the more complex polyrhythms and subdivisions that exist therein.

That said, the piece is a fixed media piece in its own right for determinately composed pitches and rhythms. It owes greatly both to Berg and to Boretz, and it is hoped here that it is a worthy tribute to both of them.

-Robert Gross

October, 2015

For the K-Tonnetz formations and score:

Networks SCORE 10-26-15 v2

The piece itself:

My Big Ideas, Vol. I

Okay, hotshot, I can hear someone out there thinking.  You’ve complained about  the disappearance from widespread prominence of the composer/thinker in the American concert music scene.  You recently complained on facebook about the lack of any heirs apparent to Babbitt and Perle, to Boretz and Morris, who occupy the same sort of presence on the concert music stage as did they in days gone by.  Fine, but what are your big ideas?  What do you actually stand for in composition?

Nobody’s actually asked this, yet, but I think if I keep complaining loudly enough and persistently enough, the question would become inevitable.  So here are a few ideas that I’ve developed in my theoretical writings that I also apply to my composing, and where possible, an example piece.

1. Post-Tonal Prolongation, alive and well.

My view of course is that post-tonal prolongation exists, and I vary wildly from anyone who says otherwise.  I won’t totally rehash my views as expressed in the link just given, but I will say that the fact that post-tonal music is vastly more complex than tonal music means that the odds are greater, not lesser, that some kind of prolongational syntax is at work in order to make this music coherent.  And don’t say that it’s not coherent; everybody loves post-tonal music and understands it just fine when it is used in film and television.

Prolongational graphs of post-tonal pieces probably are more complex than orthodox Schenker graphs of tonal works, if done well.  That does not mean that these complexities are insurmountable; I certainly don’t believe that the prospect of vastly complex prolongational relationships in post-tonal music means we should throw our hands up, give up, and simply pretend that the relationships don’t exist.  That’s certainly not the can-do attitude that made America great.

Moreover, I think post-tonal prolongation can be reverse-engineered to be a compositional device.  (For that matter, I think functionally tonal composers can sketch in the form of an orthodox Schenker graph and realize the graph in the form of a fully composed piece.  Indeed, I would urge any composer insisting on writing functional tonality to do this because there is great risk that one’s functionally tonal music can verge on superficial surface mimicry without a deeper understanding of tonal architectonics.)

My post-tonal prolongational thinking comes in two flavors: vertical and horizontal, and I do not believe these approaches are in any way mutually exclusive.  I call my vertical post-tonal prolongational analytical schema projection-constructive analysis.  My horizontal schema I would more traditionally call post-tonal linear-reductive analysis (no newly coined term).

Projection-constructive analysis posits that lurking in the background of most post-tonal pieces is a pitch field that represents the structural culmination of the work.  Suppose that pitch field is C#2/A4/D5/E5 with the numerals representing fixed register.  Projection-constructive analysis is predicated on the idea that fixed register assertion of pitches replaces functionality, which is missing in post-tonal music, as a means of asserting structural supremacy of those pitches.  The C#2/A4/D5/E5 pitch field might occur toward the end of the post-tonal piece, and I would call that the Structural Tetrad (sort of like the “Structural ^4” we might encounter in a Schenker graph).

How do we know it’s the Structural Tetrad?  We have to look for non-pitch parameters to determine structurality of the event.  We do this in orthodox Schenkerian analysis too: in determining whether a pitch is a potential head-tone, or should be flagged, or whatever, we often ask the same contextual questions, like, does it fall on a downbeat?  Is the event strongly placed metrically?  Is the event agogically accented?  Is the event dynamically extreme?  Is it colored by orchestration in a particularly marked way?  In the case of the post-tonal work, we can ask this additional question: is the set salient on the surface as well as in the deep background?  Notice that our C#2/A4/D5/E5 Structural Tetrad is a [0237] set.  Do we see plenty of [0237] sets in operation on the surface of the piece?  Projection-constructive analysis shares with Schenkerian analysis the idea that there is a significant nexus between the surface events of a piece and its background, and that the two mutually reinforce one another.

Projection-constructive analysis then looks for events in the piece, one event at a time, in which the final structural x-ad assembles itself vertically.  So if C#2/A4/D5/E5 is our Structural Tetrad, then we might find, for example, a prominently placed C#2 at the beginning of the work.  We can label this C#2 the Structural Monad (the post-tonal equivalent of a Schenkerian head tone).

Next, we would look for a Structural Dyad.  We could expect something like this: somewhere along the way, we might find a C#2 and a D5 in close proximity to one another (either vertically simultaneous or very near to one another), with both pitches contextually made prominent.  Once we’ve found this, we can label the dyad our Structural Dyad.  Then we would look for a Structural Triad (the term “triad” here is not used to denote tertian harmony but rather any literal three-note collection).  Suppose we find at some prominent juncture of the piece, occurring after the Structural Dyad, a C#2/D5/A4 collection.  So long as the three notes are contextually prominent, this would make a fine Structural Triad.

The arrival of E5 working in some prominent conjunction with C#2/A4/D5 completes the Structural Tetrad, the culminating structural point of the post-tonal work.  Of course, unlike with Schenkerian analysis where we have three prescribed cardinalities of structural events possible (the 3-prong descent from ^3, the 5-prong descent from ^5 and the 8-prong descent from ^8), projection-constructive analysis posits up to twelve possible structural events (if there are twelve, then the piece culminates in a Structural Dodecad).  I have not found a Structural Dodecad in a piece yet, but I have found a Structural Octad, lurking in the background of The Rite of Spring.

(The Structural Monad of The Rite of Spring is D#6 at Rehearsal 8+2; the Structural Dyad is C6/Eb6 at Rehearsal 9+4; the Structural Triad is G5/C6/Eb6 at Rehearsal 11; the Structural Tetrad is A4/G5/C6/Eb6 at Rehearsal 34 through 34+1; the Structural Pentad is E4/A4/G5/C6/Eb6 at Rehearsal 36+3 through 37; the Structural Hexad is C#4/E4/A4/G5/C6/Eb6 at Rehearsal 39+1; the Structural Septad is A#3/C#4/E4/A4/G5/C6/Eb6 at Rehearsal 70+5 and the Structural Octad is F#2/Bb3/C#4/E4/A4/G5/C6/Eb6 at Rehearsal 80+3.  Note that the Structural Octad is fully octatonic, giving support to those who have maintained that The Rite is essentially an octatonic work.  Once the Structural Octad is in place, those eight pitches in fixed register continue to assert themselves prominently in various combinations throughout the rest of the work, in what I call an Epilogical Dissipation.)

You would be amazed at how often one can find a projection-construction lurking in the background of any post-tonal work.  I believe that projection-constructions might be tropes of post-tonal pieces, because they are the results of the way post-tonal composers hear music: they go back to the same fixed-register pitches in various combinations again and again in order to tether together the work.  Webern does this very clearly.  Stravinsky does this.  Schoenberg does this.  Samuel Adler, whose music I’ve analyzed extensively, does this.  Birds do it.  Bees do it.  Post-tonal composers make projection-constructions— not intentionally, but by virtue of the way that post-tonal composers organize their pieces when they are listening closely to their materials.

A piece of mine that is structured in such a way is called Twelve Structures for Piano and Cello or Twelve Charming Little Pieces for Cello and Piano.  Each movement is a one-minute miniature that projects one of the twelve possible trichords across the background of the movement.  A performance of the piece by Viktor Valkov, piano, and Lachezar Kostov, cello, can be heard here:

The other post-tonal prolongational idea I work with is essentially linear.  The big idea here is that post-tonal prolongation is predicated on post-tonal counterpoint.  I have found that many pieces idiostructurally assert their own rules of post-tonal counterpoint on a piece-by-piece basis.  Certainly, it takes sleuthing and analysis to find the rules of counterpoint lurking in a post-tonal piece, but again, what are we analysts to do but analyze?

Stefan Wolpe composed a short passage called Modulation as Process which aesthetically modulates from a tonal harmonic landscape to a post-tonal harmonic landscape with remarkable fluidity.  Analyzing the short work, I determined these contrapuntal principles at play:

For the tonal section:

Traditional rules of counterpoint apply for the section described by diatonic intervals, with the exception of the following:

1. 7ths are allowed on strong beats if they are approached and left by step.

2. Tritones are allowed on strong beats if they are approached and left by step.

For the post-tonal section:

1. No tritones on any part of the downbeat.

2. Every beat must entail at least four distinct interval classes.

3. Interval classes 3 and 6 never appear together in a purely 4-voice sonority, unless on an upbeat.

4. Wide leaps greater than an octave must be a registral displacement of a linear motion (i.e., the only leap greater than an octave that is allowed is a leap of some kind of ic1 or ic2).

5. Tritone leaps must be followed by step in either direction.

6. Leaps from 6 to 12 semitones must be followed by another leap in the opposite direction.

Taking these idiostructures as rules for background contrapuntal skeletons, I am now composing a set of Wolpe Variations for pianist Viktor Valkov.  Wolpe’s Modulation as Process provides the framework of events that I prolong; the contrapuntal principles provide the rules by which I can compose-out the prolongations.

The purposes of the Wolpe Variations are two-fold: first, to show that post-tonal music can be prolonged through contrapuntal principles; second, to show that post-tonal music and tonal music can coexist peaceably and without superficial juxtapostion as long as there are large-scale architectonic forces in governance of both.

The Wolpe Variations are not yet complete; I’m shooting for 45 minutes of music.  The piece, I’m told by Viktor, will require significant amounts of editing to make it more idiomatic for piano.  Nonetheless, here is a MIDI realization of what I have so far.  I think piano samples are reasonably okay to listen to in order to get an idea of the work; they’re certainly better than any string samples.  There is about a half hour of music here; about two-thirds of the way.

To be continued:

Vol. II: Klumpenhouwer Networks as Compositional Devices

Vol. III: Geometric Formal Proportions other than Golden Sections

-Robert Gross

Schenker, Earlier

Here is a blog post making the argument that we need to integrate pop and jazz music theory into mainstream music department and conservatory undergraduate curricula.

I completely agree.  The thing I would add is that we need to teach Schenker a lot earlier, and apply it to jazz and pop as well. It’s absurd that we make vertical everything in music theory, and save the linear dimension for grad school specialists.  I’m not saying that everyone has to become a Carl Schachter-level Schenkerian; I’m just saying that we need to introduce linear concepts in all music theory at all levels as we go so that as composers and performers we are thinking in both the vertical and the horizontal dimensions, and making those connections accordingly.

I’m sure that there are those who would object on the grounds that traditional music theory, as it is structured, does not have the time to accommodate these additional demands.  Theory is traditionally taught in four units in the freshman and sophomore years: semester one, devoted to first principles which often coincide with Baroque principles of four-part chorale theory (also known as part writing); semester two, devoted to expansion and refinement of these ideas, with some form and structure thrown in, which nicely coincides with the formal structures introduced in the Classical era; semester three, we add advanced chromatic— but still functionally tonal— ideas to our plate, which just so happens to coincide with what Romantic-era composers did; and then in the fourth semester we cover 20th and 21st century techniques, which, by way of a really amazing coincidence, coincides with what 20th and 21st century composers actually did.

Then a student is sometimes given a one- or two-semester elective.  Some programs take this basic model and stretch it out to five semesters; some compress it into three; but what is amazing is the invariance of this model across the board.  I would propose adding two more semesters: junior year, a seminar in jazz music, team taught by a theorist and a musicologist; and then a seminar in pop music, team taught by a theorist and a musicologist.  I would make the course team taught because if theorists get an extra two semesters to do their thing, musicologists are inevitably going to want two more semesters to do their thing too.

More central to the argument of what this essay is about, though, is the need for linear reductive analysis from the beginning.  When we make everything vertical, we are making the argument that music happens from event to event to event.  This idea calcifies in the minds of young, impressionable musicians, and disadvantages them musically perhaps for their entire lives.  The idea that there are broad-scale architectonic ideas at play in musical works should not be Masonic wisdom reserved only for an elite, secret order of initiates.  We should teach this idea from day one: the vertical and horizontal dimensions in music are coequal.

I am not saying that inordinate amounts of time on graphing technique should be taught.  I am saying that when an instructor gives a chorale part-writing exercise, Schenk it when he or she is done.  In the first semester, look at real Bach chorales; they are often some of the most interesting literature to read from a Schenkerian perspective since this is the corpus of work most likely to reveal the rare descent from ^8.

In the second semester, when we’re teaching sonata form, the instructor might want to talk about why it is that one sees a descent from ^3 in major mode more often, and why one sees descent from ^5 in minor mode more often.  (Here’s why, if you’re wondering: pieces tend to descend from ^3 as a norm.  But in minor mode, I pushes to III just before the development section.  III can support ^5 but it is just as likely to support ^3, with ^4 supported by V/III inevitably along the way.  And it is very interesting to look at the salient differences between a development section in a minor-mode piece governed by ^3, which anticipates the arrival of ^2 before the interruption, and a development section in a major-mode piece governed by ^2, which maintains ^2 just before the interruption.)

In the third semester, it can be very instructive to look at chromatic voice leading from a linear perspective.  Honestly, it is sometimes the only way to make heads or tails out of densely chromatic Romantic-era music.  I remember as an undergrad studying chromatic harmony and thinking that the Roman numeral system was becoming extremely contorted to the point of meaninglessness, even though I was arriving at the received wisdom of acceptably correct “answers” on the assignments (going to remote key areas by a series of common-tone pivots and such).  Why is this music the way it is?  Is it really because of a string of improbable key areas forever modulating into one another with myriad pivot chords and common tone pivots?  Or does one elegant linear progression really explain what’s going on?

Finally, when one gets to post-tonal music, linear progressions can really be one’s friend in reassuring the novice that comprehensive relationships can indeed be teased out of this seemingly abstruse stuff.  One does not even have to get into the many controversies about post-tonal prolongation; suffice to say that the general idea of linear progressions are still at play (whether they are truly prolongational or merely associative).  But you cannot do this if the scaffolding has not been put previously into place.

Many musicians will graduate from their undergraduate institutions and never look back at academia.  We want these musicians to be as literate as possible, and to give performances or compose pieces in which there is some broad-scale concept of the architectonics involved in music-making.  We need to start introducing Schenkerian/linear/prolongational concepts much earlier in our mainstream music theory tuition.

-Robert Gross

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