The Deserving Police

In the United Kingdom, there have been some serious questions about whether musical composition constitutes bona fide academic research.

It seems to me another call being made to the Deserving Police.  The job of the Deserving Police is to make sure that nobody on God’s green earth gets a goody from a government or other goody-dispensing authority (like, say, a public university department) that they did not rightly Deserve (because, of course, what people Deserve is always self-evident and easy to calibrate).

In the U.S., the Deserving Police go around snooping into the grocery lists of shoppers using food stamps and EBT cards, and sound the alarm if the shopper is purchasing something nice for themselves, like a candy bar, or ketchup.  (Since there have been calls in some states to ban the purchase of ketchup with food stamps, this is not an exaggeration.)

In the U.K., apparently, the Deserving Police are more sophisticated and lofty in their aspirations.  They investigate whether or not composers are receiving undeserved doctorates that are institutionally held to be on a par with the doctorates of the musicological colleagues.

I also can’t help but notice that this argument strikes as an invitation to a back-door re-litigation of the argument about whether or not pop music is a bona fide subject for academic study.  (The Deserving Police are very interested in this question too: do musicologists and music theorists concerned with pop music Deserve, capital-D intended, the same professional status as musicologists and music theorists concerned with self-evidently prestigious and worthy classical music?)  It seems interesting to me that the Deserving Police became interested in the research status of composition more intently when composers themselves became less interested in obligatory fugues and more interested in appropriating hip-hop samples as is often done by electroacoustic composers.

I’m going to sidestep the arguments pro and con on all fronts just to say, generally speaking, that life on the Deserving Police Force strikes me as a very petty, very unhappy waste of time.

-Robert Gross

Compose a Piece That Goes Against Everything You Believe In

Apparently one of Morton Feldman’s favorite assignments for his composition students was “write a piece that goes against everything you believe.” I’ve never quite done this, but I had a similar experience a few years ago when I was teaching a course on postmodernism. I felt obliged to include a short unit on John Zorn, whom I back then despised. I felt that he stood for everything I was against— his juxtapositions were superficial, and that everything he did was superficial whereas the “true artist” unifies things in the deep background in some subtle way, etc.

But I didn’t want to be a drag and a Debbie Downer, so I tried to find positive things to say about John Zorn. And the more I really listened to what he was doing, the more I came to really appreciate what he was doing.

And because of John Zorn, Ken Downey and I composed our first serious piece together as Blind Labyrinth, “Dyschordia.” And it’s been a fruitful partnership that has been going on for about three years now. All because I was willing to teach a composer whom I thought “went against everything I believed.”

Oddly enough, I’m not sure what I would do if I had that assignment today. I think I would write an orchestra piece full of octatonicism and tinkly percussion effects (especially crotales), because I feel like this particular kind of piece is so overdone by everybody, and for very crass reasons (partly to be “just accessible enough” for the lay audience and “just smart enough” to impress the academic colleagues). I don’t know if Feldman’s gambit would pay off in my case if I wrote this piece (the article says that invariably Feldman’s student would write his or her very best piece as a result of the assignment). I think it might certainly result in my most *marketable* piece, but best? I’m not so sure.

-Robert Gross

Most People Like Post-Tonal Music

I had the misfortune to have lunch with an actual cognitive scientist from Canada— I don’t remember his name— who trotted out the same solipsistic arguments.  He hated post-tonal music and tried to argue the point that post-tonal music is completely invalid and cognitive science shows us so.  That tonal music is inherently biological and hard-wired into our brains.  That the human brain is incapable of understanding post-tonal music because it does not conform to the inherent biological hard-wired and proven-by-cognitive-science syntax that we need to truly understand and appreciate music.

Me: Okay, then, why do *I* like post-tonal music?

Him: I don’t know.  Most people don’t.

Me: Sure they do.

Him: Really?

Me: Sure.  Any time you watch a horror movie or science fiction film and post-tonal music appears, people understand it and love it.

Him: Oh, well, then, it goes to show that post-tonal music can’t communicate a full range of human emotions, certainly not any positive ones, like love.

Me: But that’s not what you said.  You said it wasn’t *understandable*.  That cognitive science proves it’s incomprehensible.  Now you’re backtracking, conceding the point that it is comprehensible, and now saying that it’s merely incapable of showing a full range of emotions.  Well, so is tonality.  You can’t express terror or alien invasions in D major, can you?

Him: Awkward change of subject.

Completely frustrating and impassive conversation for the rest of lunch.  We were never in touch again.

I find it amazing how many legitimate so-called scientists want to throw actual science, logic and reason out the window and instead use their authority as scientists to make an argument-by-authority to validate their prejudices about music.  Many of them love tonal music dearly; it’s so sacred to them that they are willing to absolutely and utterly upend what they know they should be saying and doing as scientists on the subject— to confirm that music is complex and even complex, post-tonal music probably has a logic and syntax that is innately comprehensible to the brain, or at least acknowledge that the jury is still out on the question— and instead go into full-on worship mode at the altars of Beethoven and Brahms busts.  Many of them won’t begin to concede to others the same kind of faith-based tunnel visions when it comes to other people’s worship of gods and practices of religions that they reserve for themselves when it comes to the worship of tonal classical music.

I know I’m out of lock step with the vast majority of people in western culture, but I find it very difficult to listen to 90% major-minor, 7% diminished and 3% augmented as my harmonic palette for forty-five minutes, or an hour, or three hours. I just can’t. Post-tonal music drives other people nuts with its complexity. Tonal music drives me nuts with its paucity— its paucity of harmonic choices. I get so bored with functional tonality. I really, really, truly do. I know in some circles in concert music culture it’s basically farting in church to acknowledge that common practice period music doesn’t really do it for you. But there are a lot of us out here, more than you might think, who became attracted to concert music because of all that stuff that happened *after* 1900, not before, and who just sort of nod along and pay lip service when everyone else drones on and on and on about the greatness of Brahms, or even Mahler, whom we’re supposed to get excited about because he ended movements in different keys than he began them in.

So is there something wrong with my brain then, because I prefer music that gives me a richness of harmonic experience? Because the harmonic dimension is that important to me? I know that even the great 20th and 21st century composers say, dutifully, that we need to respect, know and even love the common practice period masters. I’ll concede the respect and the knowing. But I cannot love music that is ubiquitously triadic in its harmonic constructions and never anything else for any great length of time.

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

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

Music Theory Nerd Fight I: Why Joseph Straus is Wrong About Post-Tonal Prolongation

Note: this article is pretty technical. If arcane issues of music theory are not your cup of tea, this may not be the article for you.

It never ceases to amaze me how good, progressive-thinking professors and scholars can be as progressive as they are in what I would call, for lack of a better term, “real life,” while being unbending stalwarts for purist orthodoxy in their own scholarly fields.

Take Joseph Straus, for instance. I don’t know for a fact that Straus is politically liberal per se, but he has done an awful lot of good for people with disabilities. He almost singlehandedly created the field of disability studies in music, and that ain’t all bad. Straus’s book Sounding Off has become the cornerstone of this important emerging field, and his efforts to this end should be applauded.

Unfortunately, though, Straus’s pronouncements in a different arena have made life very difficult for those of us who care deeply about post-tonal music. In 1987, Straus wrote a vastly influential article for the Journal of Music Theory called “The Problem of Prolongation in Post-Tonal Music.”  In this article, he says that post-tonal prolongation is only possible if post-tonal music can fulfill four conditions— four conditions that it is impossible to fulfill. He then concludes that post-tonal prolongation is not possible.

Prolongation, to put it very generally, is the glue that holds music together. We don’t hear music note-after-note-after-note. We hear music as events in a hierarchy— more important events, and less-important events. Prolongation is the technique composers use to make the less-important events function so that we retain the more-important events in our memory until the next important event comes along. The music theorist who discovered this and wrote about it extensively was Heinrich Schenker. He is the guy Schenkerian Analysis is named after.

The controversy in the music theory world that persists to this day is whether or not prolongation exists in post-tonal music (music that is not in any particular key) or whether prolongation only exists in tonal music (music that is in a key). Schenker said that prolongation only exists in tonal music, and a lot of adherents to Schenker insist that he was correct. Straus is one of them.

There are many problems with Straus’s argument, and here are some of them.

1) His reasoning is circular. I am not the first to point this out (that distinction goes to the late composer-theorist Roy Travis, who gave us some sensational post-tonal prolongational analyses of the music of Bartok, which Straus dismisses in his 1987 article without ever really saying why, other than to call these analyses “ad hoc” and without really taking a look at the actual purely musical insights those marvelous graphs give us). But I am continually amazed at how Straus’s view continues to be the “law of the land” with all the force in the music theory world of a Supreme Court decision, even though it has been pointed out time and time again that his reasoning is completely tautological. You can’t get anything into peer-reviewed ink directly challenging Straus on this— believe me, I have tried. But it has been indirectly suggested on several occasions by scholars like Olli Vaisala and the late Steve Larson that there is circular reasoning going on, and they’re right.

Straus says that any kind of music that has prolongation has to fulfill four conditions that just so happen to be four conditions that contra-indicate post-tonal music. (Those four conditions are, in case anyone is wondering: the music has to have consonance and dissonance; the music has to have scale degrees; the music has to have ornamentation; and the music has to have a distinction between harmony and melody.) So the conclusion— post-tonal music does not have the features required to create prolongation— is assumed the in the premise— music, to have prolongation, has to have features that contra-indicate post-tonal music. See how that works? Travis called Straus’s four conditions ex posteriori (that’s Latin for “pulled out of his ass”) and he was absolutely right. Exactly where and how Straus divined his four conditions, and based on what scholarly precedents, he never says.

2) Post-tonal prolongation has to exist somehow. To say that post-tonal prolongation does not exist means then that we hear post-tonal music from note to note to note and no other way. I do not accept that. To be fair, there is a kind of post-tonal music called “moment form” which deliberately attempts to do exactly this; its premise is that we hear notes as a never-ending stream flowing by, and no moment ever has any obligation to reflect the past in the piece, nor does it have any obligation to predict the future. Boulez and Stockhausen are/were exponents of this idea. The only trouble is, cognitive science says that we are going to hear patterns in music whether they are there or not. It seems, then, in my view, that the post-tonal composer may as well avail upon this fact of our cerebral hardwiring and use it to his or her advantage.

Anecdotal though the evidence may be, I have to say that it has never been my experience of post-tonal music to hear it from note to note to note. I hear it hierarchically. I hear moments that are clearly more important, and moments that are clearly less important. Furthermore, I have never known a post-tonal composer— and I know a lot of post-tonal composers— who thinks that he or she is simply stringing together notes, one after another, without thinking about a bigger picture. I’ve watched post-tonal composers sketch. They think about long arcs. They think about getting from Point A to Point B, with Point B perhaps being five or ten minutes later. To suggest that post-tonal prolongation does not exist is to suggest that post-tonal composers are deluded in terms of what they think they are doing.

I am certain there are anti-post-tonalists who are giddy and eager to say, “Yes! Yes! Post-tonal composers are deluded, because post-tonal music is completely invalid!” But Joseph Straus cares deeply about post-tonal music. He wrote one of the most widely used textbooks about post-tonal music. That is why I think his argument is so counter-productive: he is just playing into the hands of those who would say that post-tonal music is inferior to tonal music, if not completely invalid altogether.

Straus proposes that an “associational” model gets us from Point A to Point B. That Point A and Point B may have something in common, and that’s what links them, but— and here is the important point— it doesn’t matter what comes in between. If that’s true, then post-tonal music becomes pastiche. As long as we invest Point A and Point B with some common factor, we can literally do anything we want in between Points A and B. Point A and Point B may be unified by their both being orchestral tutti on a 4Z-29 tetrachord; but in between Points A and Point B we have a bebop tenor sax solo for five minutes.

Unless your name is John Zorn, this is not how post-tonal music works. To say that post-tonal music is this incoherent is an insult to most post-tonal composers. But that is precisely what Straus’s associational model does. It posits that nothing matters on the way to Point B from Point A in any post-tonal work.

Post-tonal composers care a lot more deeply about their materials than this. I think that there are truly prolongational devices that get us from Point A to Point B in post-tonal music. We have yet to uncover them all. But to throw up our hands and say that because it’s a difficult and thorny question to consider, that means we should just give up and declare that post-tonal prolongation is impossible: I ask you, is that the can-do attitude that made America great, Prof. Straus?

If we don’t have prolongation, then we have pastiche at best and moment-form at worst. I don’t know any post-tonal composers, though, who would be satisfied to say that they are merely writing whatever between the signposts of large structural events in their music. And, yes, death-of-the-author notwithstanding, the viewpoints of composers really matters. Because, as I like to say, without composers, music theory would be just… well, theory.


One area of investigation I am beginning to explore are the tropes of post-tonal music. Straus says that post-tonal music may just be too “multivalent” (to use his exact term) to understand from a prolongational standpoint. Granted, post-tonal prolongation may be a lot more complex than prolongation in tonal music. (Post-tonal music, by and large, is typically more complex than tonal music, so that follows.) But that does not mean post-tonal prolongation is impossible to obtain, just difficult to obtain. I believe that there really may be tropes that are common to most, or even all, post-tonal pieces that are worth pursuing. A composer friend once joked to me, “you know, the major seventh is the octave of post-tonal music.” But I don’t think that’s a joke at all. I think there is something to it. The ic2 (interval class 2, i.e., a semitone) spaced eleven semitones apart (or in compound) really does have a quality of being a stable reference point in a lot of post-tonal music. So if this is a trope, perhaps there are other tropes. And in those tropes, maybe we find prolongational devices. What is an Ursatz but a trope of tonal music?

I am not guaranteeing this line of investigation will bear fruit. But it is worth investigating, enough so that I think it is premature to declare the non-existence of post-tonal prolongation until lines of investigation such as these have borne themselves out.


3) Post-tonal counterpoint exists.

Straus concedes in the 1987 article that if a post-tonal counterpoint could be shown to exist commensurate with a tonal counterpoint as underpinning post-tonal music, then he would concede the existence of post-tonal prolongation. Again, he is assuming in the premise his wanted conclusion— of course, it must be that there is no such thing as post-tonal counterpoint. That’s his assumption.

Not so. Post-tonal counterpoint does exist. The only difference between post-tonal counterpoint and tonal counterpoint is that post-tonal counterpoint tends to exist on an idiostructural (piece-by-piece) basis. Post-tonal pieces find their own contrapuntal rules. But that does not mean that contrapuntal rules cannot be divined.

Post-tonal pieces create their own structures for what is consonant and what is dissonant. (It is absolutely a myth that all twelve tones are completely egalitarian and hierarchies do not exist; that myth is unfortunately promulgated by freshman-year textbooks that include chapters about Arnold Schoenberg’s so-called “liberation of the dissonance”.) But these structures can be divined. It takes analysis.

Here is a sneak preview of a chapter I have written that I am contributing to a book Timothy L. Jackson and I are writing on post-tonal prolongation. It is a rough draft, but it shows a post-tonal contrapuntal analysis of pieces by Samuel Adler, Carl Ruggles, Shulamit Ran among others.

Contrapuntal Principles of Post-Tonal Prolongation 5-14-14

Straus is careless in that he never says that post-tonal counterpoint has to entail universal rules in order for us to obtain post-tonal counterpoint. But this is no mere loophole I am exploiting; I think this is precisely how post-tonal counterpoint works. If each piece creates its own consonance and dissonance conditions (along with the other three conditions), then each piece creates its own unique universe in which idiostructural post-tonal prolongation obtains. To me, that is an incredibly exciting possibility. To Straus, it is a reason to throw up hands and say, oh well, can’t be done. I like my way— the way that is actually rife with possibility rather than defeatism— a little better.

4) Separate but equal is inherently unequal. The associational model is the domestic partnership of music theory. It is the segregated school of music theory. Straus asserts that post-tonal music has a separate model governing its large-scale architectonics, and it’s the associational model.

The only trouble is, the associational model is an inferior model. It is capable of creating only one level of hierarchy deeper than the foreground, and that’s the background of associations themselves, and nothing else. By contrast, tonal prolongation (and remember, as an adherent of orthodox Schenkerian analysis, Straus signs on to this premise) creates rich nests of foreground, many middlegrounds, and a background. The associational model is too weak to do this.

As long as we consign to post-tonal music an inferior model, we are saying that post-tonal music is itself inferior. Straus would like us to believe that the associational model is a separate, but equal, model in which to find hierarchy in post-tonal music. But it simply is not. One layer of associations does not suffice to explain the intricate and complex dynamics of post-tonal music.

Therein lies Straus’s ultimate failure. His model assumes that post-tonal music is simpler than tonal music, and therefore a simpler model can suffice. No, no, no. Post-tonal music is vastly more complex than tonal music. Straus told me in a private e-mail that Olli Vaisala’s complex schema for finding post-tonal prolongation in intervallic-registral analysis was correct, but would probably not find widespread use because it is so complex.

That does not make it wrong, Prof. Straus! Okay, so you’re going to find fewer post-tonal analysts than tonal analysts using fewer post-tonal prolongational schemas for analysis. So what else is new? More people listen to tonal music than post-tonal music anyway. More people analyze tonal music than post-tonal music. More people teach and perform tonal music than post-tonal music. To hold up the correctness of an analytical approach to a popularity contest is absurd. Post-tonal music will always be a tiny niche province. What Straus does not want to admit is that Vaisala found a way to locate post-tonal prolongation that fulfills Straus’s four conditions, thus proving that Straus is wrong. “Widespread use” is now one of the conditions for post-tonal prolongation? That’s not what he said in 1987. Not once did he say that a post-tonal prolongational schema had to achieve widespread use in order to be correct. Straus is moving the goalposts here.

In conclusion, the time has come for Straus’s edict to fall by the wayside, and for exciting, freewheeling, progressive investigations into the nature of post-tonal prolongation to see the light of day. Otherwise, we are relegating post-tonal music to the status of inferiority, and that would be a shame. I do not think this is Straus’s intent— I think he cares very much about post-tonal music— but he has to realize that his pronouncements have had lasting consequences in shaping attitudes toward post-tonal music. Separate but equal is inherently unequal. The time to recognize the equality of post-tonal music is now.

-Robert Gross

Coming soon in this series:

Music Theory Nerd-Fight II: Why Michael Buchler Is Wrong About Klumpenhouwer Networks (With Apologies, Because I Really, Really Like Michael Personally)

Music Theory Nerd-Fight III: Why Hepokoski and Darcy Are Wrong About Pretty Much Everything