Chasing My Past with Harpoon and Row Matrix

The semester is over, and so is my 12-tone analysis class, which made me work harder than any class I’ve ever taught. I added about 18 works to my analytical repertoire, including behemoths like Mantra, Sinfonia, Le Marteau, and Threni. Even having analyzed most of the music over the summer, I still spent most weekends checking rows and poring over dense JSTOR articles. And aside from me having wanted to learn all that stuff anyway, it was a continually rewarding class. I especially enjoyed showing the row matrix from Ben Johnston’s String Quartet No. 6, with a row consisting of six harmonics of D- plus six undertones of D#, comprising, if I counted right, 69 63 61 different pitches in his Just-Intonation notation:

That 11th pitch in the third row, by the way, is called F-double-sharp-down-arrow-upside-down-seven-plus. It’s the 77th subharmonic of the perfect fifth above D#. But you knew that.
Babbitt was really fun to teach (which explains, I guess, why so many theory professors teach him). I demonstrated how there are 16 ways to make a rhythmic pattern within a half-note using only eighth-notes, and then showed how Babbitt assembled those 16 possibilities into a rhythmic row that covers the first eight measures of his jazz band piece All Set and then reappears elsewhere in the work, now augmented, now in the percussion – and I heard a voice major, who’d had no prior interest in 12-tone music and was only taking the class to get a theory credit, whisper under her breath, “That’s incredible!” She ended up doing a final paper on Babbitt’s Du, which I took as one of those rare personal triumphs a professor gets only every few years. Still, overall the students remained a little dubious about the whole 12-tone thing, which is good – interested, curious, but only intermittently convinced. The last day I played, following the scores, some pieces I love without analyzing them, including Maderna’s Aura, Zimmermann’s Monologe, Ligeti’s Monument-Selbstportrait-Bewegung, and Xenakis’s Mists, to show them where 12-tone music had led in Europe. The most recent work I played was Mikel Rouse’s Quick Thrust (1983) for rock quartet which uses only one form of the row amid elegantly serialized rhythms. In playing Le Marteau I noted that my birth was historically closer to Rhapsody in Blue than the students’ was to Le Marteau. The 12-tone era is now just another historical period, to which we could bring a historical perspective, and I taught it that way. The music was too old and too ensconced to engender the slightest controversy, and too distant to embody any mandate for the present. It is what it was, only now immune to partisanship in either direction.
The biggest problem was finding good examples of 12-tone analysis to serve as models. Most of the books and articles are written as though to exclude outsiders from a secret club. If you don’t already understand, you can’t read them. Especially irritating are the digressions into meta-analytical issues, meant to create some kind of general 12-tone theory rather than to address the piece at hand. For instance, is it ever necessary to launch into a discussion of first-, second-, third-, and fourth-order combinatoriality? Sure it determines what rows are available to combine polyphonically, but who gives a shit? The best article I found by far was Richard Toop’s analysis of Mantra in his “Lectures on Stockhausen” – perhaps because they were lectures rather than articles, he was the only writer who seemed to really care that his readers got drawn into the analysis, and truly understood. As I’ve said before, I used the Osgood-Smith book on Sinfonia, which was thorough if indifferently lucid, and Wayne Wentzel’s “Dynamic and Attack Associations in Boulez’s Le marteau” (Perspectives) went a long way toward clarifying Lev Koblyakov’s impenetrable Boulez book, possibly the worst-written music book in history. I regretted throwing in the towel on Sessions’s Third Sonata, but I asked George Tsontakis, a Sessions protégé, and he said, “Oh, don’t analyze that piece, it’s like two pieces happening at once”; and the published analyses were little help.
Most of all, the class meant to me – and this conditioned what it meant to them – a chance to go back through a repertoire that had seemed numinous when I was a teenager. That’s the music I loved before minimalism came along and seduced me away, seeming fresher and more full of possibility. I remember clearly what it sounded like in 1971, and I needed to find out how I’d react to it now. I was bringing up demons from my youth to exorcise, and I hope I didn’t often sound like Captain Ahab chasing his personal white whale. But I was told that some appreciated learning that repertoire from someone who didn’t insist that they pledge allegiance to it. Now that I’ve gone through all that analysis and kept records of it, I may well teach it again someday.


  1. Sarah says

    Have you looked at Sofia Gubaidulina’s 1965 Piano Sonata? It only uses one row over all three movements, but she juxtaposes the row with a bunch of other musical palettes: white-note scales, chromatic scales, stacks of perfect fourths, major triads, and even Cage-inspired inside-the-piano timbral cadenzas. Check it out, it’s catchy and a bit spunky—probably my favorite twelve-tone piece ever.
    I caught a live performance of Stockhausen’s Mantra last month at UNC Chapel Hill and I have to say that it was just too long. All those enjoyable theatrical moments were just sort of swallowed up by the sheer length. Maybe it would be better performed outdoors at a summer music festival.
    KG replies: Thanks, it turns out I have a PDF of the Gubaidulina (friends give me these things), and I’ll check it out. I’ve never heard Mantra live, but I can imagine I might have the same reaction you did. It may be one of those pieces that works better on record. Come to think of it, my students didn’t register the same level of enthusiasm with it that I did in 1972.

  2. Rodney Lister says

    I had a similar situation teaching a course on post-tonal analytic techniques, which was an enormous amount of work and from which I learned a lot. Not being as systematic or as thorough as you are, I covered less ground. Although the purpose of the course was somewhat different–thinking about analytic techniques–what the toolbox was and which of the tools were best in approaching which music.
    I wanted to include thinking about Britten as well as Babbitt. For some reason the piece (music)that seemed to meet with the most resistance was Lutoslawsi (Parole Tisees).
    Anyway, I’m sure you know Words About Music by Babbitt, which seems to me to be the clearest and most convincing discussion about why one might want to write twelve-tone music and what it might do for you. I get lost with partitions (which I suppose I shouldn’t admit), but I like the rest of it.
    KG replies: Words About Music is indeed an oasis of plain-talking about 12-tone issues. I should have looked through it to see whether any excerpts could have applied for my class, but didn’t think of it, and I don’t recall him mentioning All Set. I relied heavily on the Andrew Meade book, which is a godsend but you really have to plow through most of it to get all the benefit. I found it difficult to excerpt for students working on individual pieces, because (and also true with Joseph Straus’s late Stravinsky book) the concepts are all cumulative. So it was easier to absorb them myself and feed students the ones they needed for the individual pieces. I should add that the Babbitt piece I used to always use was Post-Partitions, which they always hated. I had much better success analyzing All Set and playing them Philomel – though I also have to say that the ones with good ears picked up how miserably inaccurate the Nonesuch recording of All Set is.

  3. says

    ‘Twas a great class, Kyle. My strong desire for just the type of mathematical problem solving which goes into creating and using and altering a row was greatly appeased, and I now feel that I could talk for a long and excited time about many of these pieces, without feeling any necessity to like them. That said, “All Set”, and Tenney’s “Chromatic Canon” and of course “Sinfonia” will remain in my itunes playlist until further notice.
    Moreover, on the topic of the Ben Johnston 6th SQ, this is just about the richest set of codebreaking material I’ve ever encountered, they should have used this piece in WWII to communicate military secrets.
    Not to give too much away (before you read my paper), but when you graph the rhythmic modulations as a function (after converting them to ratios, with which they make a simple just scale with 9 notes to the octave), you can take the derivative and get a perfect parabola towards the exact center of the piece (a complete palindrome).
    Cool stuff, thanks for the fun times and the analytical tools, I wouldn’t want to be taught 12tone by anyone else!

  4. peter says

    Kyle — I would love to have sat in on your class. Have you thought about writing up your lectures as a book?
    KG replies: [Sigh] No. That may require a whole other blog entry.

  5. Ernest Ambrus says

    Merry Christmas, Mr. Gann.
    Included in the Scala file folder is a 61-tone 11 limit tuning for “String Quartet No. 6”. If it is 69 tones, could you tell us which are missing?
    KG replies: You could have figured this out yourself, but you knew that if you asked I’d get so curious I wouldn’t be able to do another thing till I solved it. On recount I get 63 pitches, but I get completely different fractions than you’ve got, which must mean I’m starting from a different reference point. I used D- as 1/1 because it’s the 1st harmonic of the 1st hexachord, but if you (or Manuel) used A as 1/1 (center of the matrix), or any other pitch, you’d get different numbers. So I can’t compare with what I’ve got. I’ll have to work this out another time, unless my genius student Dylan Mattingly does it for us. This piece has always been a puzzle, and I can’t find my analysis from 26 years ago, oddly enough.
    I have to add I can’t figure out what the reference pitch is in your list. Since you have a 160/81, that means if A is 1/1, there should be an A-, and there isn’t; and if D- is 1/1, there should be a D–, and there isn’t. So I don’t see any logical 1/1 that would give a 160/81. [To all you non-microtonalists: see how much fun we have?]
    UPDATE: I’ve confirmed 61 pitches. The A7b- above should be A7b, and the Db– should have an up-arrow on it. And I think your ratios are calculated on C. I also think your 77/81 should be 154/81. (Can’t believe I’m spending all this time with a Christmas tree waiting to be put up.)

  6. Ernest Ambrus says

    Thank you very much for that, Kyle. I appreciate it.
    KG replies: And thanks for making me straighten that out.

  7. mclaren says

    Many people think that sophisticated mathematical transformations only recently came into use in music. But the mensuration canons of the Gothic period can be regarded as tensor transformations of rhythmic fields. Since the longer notes weren’t lengthened as much as the shorter notes in mensuration canons, the transformation involved is nonlinear, and can be viewed as a vector field.
    Dylan Mattingly might also find provocative Larry Polansky’s work on the transformations of differentiable manifolds representing musical parameters. Most common musical parameters are multidimensional, and so qualify as manifolds (n-dimensional surfaces); loudness, for example, obviously has two dimensions, since the Fletcher-Munson curve tells us that the loudness of a sound depends on its frequency. Recording engineers have to take this into consideration when mixing down music, because bass parts must be inordinately boosted in order to sound as loud as instruments with a higher tessitura.
    In the same way, pitch has two dimensions (louder musical notes sound higher in pitch — up to a minor third higher, depending on the loudness and the range, a fact noted by S.S. Stevens in Proceedings of the National Academy of the Sciences as long ago as 1934), duration has three dimensions (see the section “Rhythm and Timing in Music” by Eric Clarke in The Psychology of Music, ed. Diana Deutsch, 2nd ed., 1999): meter, rubato/accelerando, and expressive deviation within the overall rubato/accelerando; and, as John Gray discovered in 1977, timbre has three perceptual dimensions. See “Multidmesional Perceptual Scaling of Musical Timbres,” Gray, John, 1977.
    More recently Larry Polansky has done breakthrough work on characterizing transformations of these multidimensional manifolds. This allows musicians to reliably control the transformation of one set of rhythmic values into another, one timbre into another, one set of pitches into another, or one set of loudness values into another. See “Morphological Metrics,” Polansky, Larry, Journal of New Music Research,, 1996.
    As heard, music involves patterns, not math. So the useful results of these sorts of conjurations arise from the intersection twixt psychoacoustics and mathematics. Talented composers have shown a gift for intuiting structures later confirmed by cognitive research. Consider Ben Johnston’s use of 9 rhythmic values: he could not have known that Geoge Miller’s paper “The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information,” Psychological Review, 1956, Vol. 63, pp. 81-97, specified nine items as the outermost limit of human short-term memory. Was Ben Johnston’s use of nine rhythmic values (as opposed to the twelve-value or larger rhythmic rows used by Babbitt) mere coincidence?
    Some of us prefer to call these kinds of convenient coincidences “musical talent,” rather than happenstance. When composers like Johnston get “lucky” and anticipate the results of research in cognitive science, at a certain point you have to stop ascribing this sort of result to “luck” and start considering it evidence of deep musical intuition.
    Indeed, it would prove intriguing to re-analyze many classic minimalist classics as examples of intuitive foreshadowing of the later results of cognitive research. Much of the structure of Tom Johnson’s “An Hour For Piano,” for example, can be explained by Bruno Repp’s paper “Perceiving the Numerosity of Rapidly Occurring Auditory Events in Metrical and Nonmetrical Contexts,” Perception & Psychophysics, Vol. 69, No. 4, pp. 529-543, 2007.
    While the Darmstadt crowd claimed to be doing musical research, subsequent cognitive science experiments have not borne out the limits they imposed on pitch, rhythm, timbre, et al. The Darmstadt crowd consistently overran the human channel capacity for processing information. Minimalist composers, however, quietly and thoroughly did a series of pieces whose grouping limits tend to be confirmed by subsequent cognitive science experiments. At a certain point you have to wonder who was really doing the “musical research” here — guys like Babbitt, or Tom Johnson?

  8. says

    Re books on serialism, have you read Reginald Smith-Brindle’s Serial Composition?
    It’s the most demystifying book on serial composition I’ve found. It doesn’t feature any in depth analysis but uses examples in its various chapters designed more for practical composition instruction (writing melody, polyphonic writing etc etc). It’s practical and easy to read, same goes for his other books, they’re designed to be understood.
    Also I thought Rahn’s analysis of Webern’s op21 in Basic Atonal Theory was worthwhile and pretty clear, it’s one I remember off the top of my head anyway.
    It’s been a while since I read any of that stuff but the above examples stayed with me more than Perle’s work or Forte’s atonal analysis for instance (or Lewin or Ockelford style analysis for that matter, dense indeed).
    Also I only search these things out to further my own efforts in composition, not to discover some ‘truth’ about a piece/s via analysis (I’ll leave that to the theorists/musicologists). Re Forte for instance, I think his taxonomy/system is really useful for composition in basic terms, as a form of analysis I’m not so sure (again, not my thing).
    KG replies: I have the Smith-Brindle, I should look at it again, thanks. A couple of students found Rahn’s Babbitt analysis “How Do You Du” laughably pedantic, but I hadn’t seen his Webern.

  9. Eric Bruskin says

    I have to agree with your student, and I didn’t even take the course. As a veteran of teachers, books and articles, I can confidently say that I wouldn’t want to be taught this stuff by anybody else. Between your scholarship and your attitude is a very sweet spot.
    The great physicist Enrico Fermi published his Notes on Quantum Mechanics – simply images of his raw lecture notes. People who already know some of the basics can enjoy watching a teacher teach himself and others. (Check it out on Amazon.) Is there any chance you could offer us a peek at yours?
    — Eric
    KG replies: Thanks very much, Eric. But I almost never use lecture notes. I just walk into class with a score and start talking.

  10. says

    “KG replies: I have the Smith-Brindle, I should look at it again, thanks. A couple of students found Rahn’s Babbitt analysis “How Do You Du” laughably pedantic, but I hadn’t seen his Webern.”
    Would be interested to know what you think of the Smith-Brindle. It certainly doesn’t fit into the “If you don’t already understand, you can’t read them” category. On the other hand it might be a bit basic for your purposes.