Saturday, August 14, 2010

Master of the Infinite Series - Nørgård’s Second Symphony

Per Nørgård (1932-) has been Denmark’s leading modernist composer since the 1960s. His fertile musical imagination has led to the creation of seven symphonies over the course of fifty years, the most recent premiering as recently as 2006. Nørgård's music is rigorously constructed but surprisingly approachable, in some cases even ecstatically enjoyable. 

Among the deeds Nørgård (pronounced "Ner-gore") is known for is the planning of large scale compositions around the same principles that would eventually be formalized in the idea of the “fractal” was even coined. The key to his prescient anticipation of fractals is his use of a specific music-composition device that he (and he alone) invented. Since 1959, a great deal of Nørgård's music has been based on what he called the “infinite series.” (alternatively "infinity series"). His Second Symphony, a one movement work lasting about one half hour, is among his first and most rigorous applications of this tool. Here’s how it works:

Nørgård’s infinite series is actually an integer sequence produced by a relatively simple algorithm that “unpacks” a single musical interval. A single interval is all you need to generate an unstoppable Nørgård sequence. Say you want to begin a piece with the melody G - A. Let’s assume just white notes (diatonic) are in use. From this melody, Nørgård would extract an essential piece of info, the ascending +1 “go up by one” interval between G and A. Then, he composes out that interval, using its inverted form as instructions on what the next pitch shall be: go -1 away from G. Thus, the 3rd note in the sequence is 1 below G, or F. He does the same for A, only with the original interval, so that the 4th note in the series is +1 from A, or B. We’ve got a nice little tune, already fanning away from G! The instructions we’re following are essentially: take each new interval that appears in the sequence starting at the front, and go that far (in inversion) from the second to last note in sequence, and then go that far (uninverted) from the last note in the sequence.
Infinity Series Algorithm for Initial Interval +1 (white-note) step. Click to expand.
This diagram shows the process for two iterations, the first (+1 interval) and the second (+2), producing six notes from the original two. As you get further into the sequence, the new intervals get increasingly far away from the pitches they are “producing” at the other end. This is not a barrier to understanding, however; no one expects you to hear how specific notes are being chosen, because the effect is one of carefully tuned chaos – totally dependent on the initial condition (the +1 interval), seemingly random but thoroughly determined.1

Continue generating the melody,2 and you’ll begin seeing notes slightly further away from the starting point. But not by much – a fairly (statistically) tight grip around the starting range is always retained, and big leaps tend to be followed by leaps in the opposite direction. The resulting succession of pitches is what mathematicians call non-monotonic. No, they’re not referring to its lack of a clearly defined central pitch! Rather, it’s the tendency to avoid continuous motion in the same direction; the iterative process Nørgård uses produces unstable melodies, constantly flopping up and down, locally unpredictable but globally secure.

Pick any sample slice of this pitch sequence and it’ll likely look pretty similar to any other given slice. But in order to take advantage of the more rarefied property of self-similarity, these resemblances must show up on multiple levels. Cue Nørgård’s truly recursive compositional process in the Second Symphony.

The clearest fractal property at work in this piece is the use of a single G-to-A-flat based infinite series at several time-scales. The strict sequential orderings of pitches can be difficult to discern aurally, but you can easily tell that there are multiple orchestral strata doing different but related things. At measure 60, the orchestra splits into three streams. Woodwinds trade sprightly runs in constant eighth notes, buzzing in the vicinity of G. Brass operate at a more leisurely pace, generally 4x slower (half-notes), while the entire string section explores pyramidal figures at a *much* slower rate – roughly one change every 30 measures, or 1/120th the speed of the metronomic winds. Around halfway through the piece, these three main temporal roles suddenly begin alternating, shifting between players.

Sounds good in theory, but how does it all come out sounding? At times, Nørgård’s procedures produce truly dazzling passages. Here’s a clip (and, for the curious/masochistic, a score excerpt) of the initiation of the woodwind stream. The recording is from Segerstam's glittering performance with the Danish National Symphony orchestra.

Woodwind Stream:
[Score Example: Measure 60]

A great deal of the symphony sounds like this, with some stratum chugging away at their 8th-note pitch sequence while the rest of the orchestra slowly shifts in hue. Because pitch, as determined by the sequence, is usually so tightly wound around a certain range, our attention drifts to other matters, especially tone color, and Nørgård’s imagination for orchestral combinations is impressive. There are no catchy themes, but little shards of melodies do phase in and out of focus, and various ideas do come back. One is the throbbing unison pulses from brass at several form defining moments, celebrating the arrival at an important member of the infinite series with bizarre fanfare.

Brass Fanfare:

This is the kind of piece you can only write once, and Nørgård’s subsequent output, while equally ingenuous, tends to treat his infinite series less as the structuring principle as here, and more as a jumping off point. Which is not to say his Second Symphony isn’t successful. There is a hypnotic quality to this music quite unlike anything from the minimalists. And a sense of yawning expanse that pushes beyond much of the “sonorist” work from the 60s. Whether he beat chaos-theorists to the punch with his unpredictable, recursive music or not, Per Nørgård certainly created the bar and then raised it ridiculously high for anyone wishing to write a “fractal symphony.”
— Frank Lehman

1. This pretty extraordinary website has tons info on (and can play back!) any integer sequence you can dream up, including several from and inspired by Nørgård. For example the following functions specify the "infinite series" sequence beginning with 0-1: [pitch(starting place) = 0 ; pitch (2n places) = - pitch(n places ; pitch (2n + 1 places) = pitch(n places) + 1]

2. A “fun” exercise, if you’d like to try yourself. Check with the website above to see if you’re right, or consult Kullberg, “Beyond Infinity” in The Music of Per Nørgård in Fourteen Interpretive Essays.


  1. I had no idea music could sound like fractals. I also think Jeff Goldblum in Jurassic Park would enjoy this piece.

  2. So, so, so, so, so awesome. This is a fantastic blog, great for someone (such as myself) who's just getting ready to start at a university school of music - such a great educational tool, and the writing/analysis is in-depth without being too dry, and very interesting.

    This blog is my new favourite website :-D

  3. @Anonymous - thanks for your kind words, and best of luck at your school of music!

    @Mugshot - maybe we should put a link to this site on the front page:

    1. Hi Frank,

      Where did you get that score excerpt? I am a composer applying to conservatories next year and would really like to read Norgard's scores.

      Thank you.

  4. I just started getting into Norgard a couple of months ago, and I never really understood the infinity series before seeing this article. Thanks for the wonderful (and finally SIMPLE) explanation of its workings!