Permutation
http://jan.ucc.nau.edu/~tas3/perm.html
This lesson ©1997 Arizona Board of Regents and Timothy A. Smith

| Bach | Baroque | Beyond | Assignment | Links | Notes |


In our first lesson we considered how a simple musical idea could be varied by interval direction and size. Such techniques for achieving motivic variation were exploited in the baroque as a means of perpetuating a single Affekt for the duration of a composition. While Affekt as an aesthetic paradigm did not survive the baroque, the ideal of motivic variation continued to be very important in the music "beyond."

In our second lesson we explored techniques composers have used to develop the motive and its variants by stating them on different scale degrees, in different keys or modes, and in different textures. Recall that this latter process involved an exchange of registers in which two motives, one in the high voice and one in the low, switched places. This technique for developing musical textures is called "double counterpoint," or, in instances where three voices are involved, "triple counterpoint." While "Inversion" is often used as a synonym for double counterpoint, it ought to be specified as "contrapuntal inversion" in order to distinguish double and triple counterpoint from "melodic inversion." Whereas melodic inversion may be performed upon a single motive by simply reversing the direction in which intervals move, contrapuntal inversion requires two or more motives in counterpoint with each other. So, in terms of the two lessons thus far, melodic inversion is a technique for motivic variation (lesson one), while contrapuntal inversion (double counterpoint) is a technique for textural or motivic development (lesson two).

In this unit we continue to "develop" the preceding concepts by studying how composers use permutation to insure that maximum variation and development of the motive are taking place. In the mathematical sense, "permutation" is a process in which an array of numbers is arranged in all of its possible subsets. Integers of the set [1, 2, 3], for example, could be paired as follows: [1,2], [2,1], [1,3], [3,1], [2,3], [3,2]. Observe that, if we were to consider only the various combinations of integers (as opposed to orders), there would be but three permutations. But, when the order of integers is considered, there are six: e.g. [1,2] and [2,1] having different orders, are distinct permutations.

What use, one might ask, might a composer have for techniques of permutation? Insofar as such techniques enable composers to imagine every possible way that three motives, for example, might be combined in triple counterpoint, they are useful indeed. Permutation is, in short, a sure way to achieve the goal of saturating a composition with motive without textural redundancy. But, beyond any purpose the composer may have, the ability to demonstrate permutational operations can be an important analytical tool showing structural cohesion, design and process. Thus, whether the composer used logic or intuition to create them, the mere existence of permutations justify themselves and are of great interest to music theorists.


Extending back to the fourteenth-century Ars Nova, permutation is one of the oldest techniques in western music. It is also one of the primary components of the classical music of India and the folk music of Africa. The following simple exercise illustrates how all three of the aforementioned cultures might have invoked permutation as a process to create larger musical structures. Find a friend who can keep a steady beat and ask him or her to clap a repetitive pattern of TWO beats followed by one beat of rest. Then, at some point, you join in but clapping THREE beats followed by one beat of rest. You will soon notice that every twelve beats you will have a rest that coincides with your friend. All of the intervening "stuff," however you want to mark them off--in measures of three, or measures of four, beats--represent the gamut of rhythmic permutations replicable from this process.

The isorhythmic motet of the Franco-Flemish Ars Nova represents the first overt use of permutation technique in western music. Composers of this period--notably de Vitry, Machaut, Dunstable, Dufay and Binchois--conceived of the rhythm and pitch content of a motet as two separate events which they called the talea and color. If the talea and color were not of the same length, then one or the other had to be repeated until their endings coincided (as in yours and your friend's rests eventually coincided in the preceding paragraph). Thus, each time the pitches of the motet were repeated to a different rhythm....sort of like writing out the "Star-Spangled Banner" with the 1st pitch to the 2nd pitch's duration, the 2nd pitch to the 3rd duration, etc.

Donald Grout observes that, ordinarily, relationships of permutations to each other as a consequence of asynchronous talea and color are not intelligible to the ear. At the very least, the mechanism creating them is transparent...it would be very difficult to discern, by listening alone, the process whereby each permutation was created. While one might not be able to detect the process, one ought to perceive unity or coherence as a consequence of process. Grout writes: "...the very fact of the structure's being concealed--of its existing, as it were, at least partially in the realm of abstraction and contemplation rather than as something capable of being fully grasped by the sense of hearing--would have pleased the medieval musician." Later Grout describes this dichotomy between perceptible coherence growing of imperceptible process as "mystic" and "supra-sensory." Grout writes that this "delight in concealed meanings, sometimes extending to deliberate, capricious, almost perverse obscuring of the composer's thought, runs like a thread through late medieval and early Renaissance music. We may think of this as a typically medieval propensity, but it is present also in other historical periods: one finds it in Bach...."

We conclude this section with the observation that twentieth-century composers, too, have found obvious delight in the use of abstract permutational processes. For most of them, however, permutation has been a means to an end rather than a metaphysical symbol. The total serializations of Webern (1883-1945) and Babbitt (b. 1916), or the stochastic formulations of Boulez (b. 1925), Xenakis (b. 1922) and Stockhausen (b. 1928) employ permutation in order to avoid repetition. But for Arvo Pärt, whose Fratres shall be the subject of the last half of this unit, permutation may represent a sacramental reality through which the invisible world breaks through, or becomes flesh, so to speak, so that we listeners may apprehend and perceive spiritual realities in physical ways--i.e. sounding music. This sacramental conception of artistic form and process is, I believe, the reason why Pärt's music is often described as "spiritual"...but more about that later. For now the point should be pressed that Pärt's mature style of composition is strongly influenced by the choral polyphony of the fourteenth and fifteenth centuries: i.e. the isorhythmic motet about which you have been reading.


Fugue in C-Minor (from Passacaglia & Fugue BWV 582)


Please continue this lesson by listening again to the fugue this time following the score. You may also wish to preface this section by listening to the Passacaglia and studying Tom Parsons' analysis.

While Bach was certainly influenced by Renaissance choral polyphony, there is no evidence that he was attracted to isorhythm. But there is plenty of evidence that he employed permutational logic in order to insure that his double- and triple-contrapuntal textures would not duplicate themselves in precisely the same manner. When it comes to evidence that the composer thought consciously in terms of permutation, his C-Minor Fugue (BWV 582) is the "smoking gun."

This fugue is special because it has two countersubjects, each of them sounding every time the subject itself sounds. In addition to the subject and its two satellites, a fourth voice either rests or occupies itself in free counterpoint against the other three.


Subject & Countersubjects of the Cm Fugue (BWV 582)

As you can imagine, the many ways to layer these ideas presented the composer with an opportunity as well as a challenge. The opportunity was to create as many contrapuntal textures as possible. By way of analogy, if you were hired to bake a four-layered wedding cake of chocolate, vanilla, banana, and angel food, and the couple didn't specify the order of each layer, you'd be left to invent one. Hmm, shall we put the chocolate above the vanilla, or below, or maybe the banana sandwiched between? Lots of possibilities, eh! If you couldn't make up your mind, you could always bake twenty-four--one for each possible combination--then let the couple decide. That's almost what Bach does in this fugue. He reassembles the cake eleven times, each time in a different order. The challenge...do it without repeating a texture.

The first question the composer must have asked is: "Exactly how many textures are possible?" Well, if each idea could appear in the soprano, we start with four permutations. And, if each of the four sopranos could run counter to one of the three remaining motives in the alto...twelve. Finally, if the twelve could be followed by the remaining two ideas in the tenor with the leftover motive in the bass, it should be possible to create a grand total of twenty-four textures.

So, it is neither difficult to calculate the number of permutations, nor to determine what they are. But if one were to think theoretically, one would realize that this inchoate mass of possibilities can be organized into three groups, each member of each group generated from one progenitor. Think, by way of analogy, about the seasons of the year, and you'll see what I mean. There is not just one way to organize them, but four:

  1. Calendar year: winter/spring/summer/fall
  2. Liturgical year: spring/summer/fall/winter
  3. Fiscal year: summer/fall/winter/spring
  4. Academic year: fall/winter/spring/summer
All of the foregoing "years" may be conceived as progeny of one cosmological generator fueled, as it were, by Newtonian physics. Our culture thinks a priori of the calendar year as the "prime form," but other cultures fix the new year in the spring, and still others, in the fall. The point: regardless of when the year begins, it is possible to rotate it, seasonally, to create four "other" years...going clockwise, that is.

If, for some weird reason, you decide to organize your time counterclockwise, you could produce four more years. These would be like getting into a time machine and traveling backward. Imagine that, doing your school year backward: 1st semester begins in the fall with final exams in the summer; 2nd semester begins in the summer with finals in the spring; and, if you don't need to get a job, you may take a session or two of "winter school." There is a term for this strange and backward world...it is called "inversion." If the calendar year is the prime form (p-form), then the calendar year backward is its inversion (i-form). And, if there are four clockwise permutations of the p-form of the year, then there are, likewise, four COUNTERCLOCKWISE rotations of the i-form of the year, for a total of eight permutations, all more-or-less dictated by which direction the earth orbits the sun...forward or backward.

But wait! Suppose you were to get into a spaceship and travel to a parallel earth with the following orbit: winter/summer/spring/fall! Ah, that would represent a new "primal order" (prime form). It, too, would have four prime rotations and four inverted rotations which, added to the eight terrestrial "years" would total sixteen. Then you might hop into your spaceship again and travel to hyper space where the seasons were: winter/summer/fall/spring. Back to terra firma--and I think you're beginning to get the point--in spite of what your astronomy teacher may have taught, there are three primitive ways to order the seasons, each way having four prime-rotations (clockwise) and four inverted rotations (counterclockwise), for a total of twenty-four distinct types of "years." Of these, only the calendar, liturgical, fiscal and academic years are familiar to us.



Now take a moment to explore these twenty-four possibilities as Bach might have thought about them while composing his Cm fugue BWV 582 (click on the Permutation Primer to the right).

Whew! That was something else! You see, anybody can calculate ways to state four musical ideas simultaneously in four voices, but only a master would restrict his use to half of the possible variations, employing them in such a way as to demonstrate a remarkable understanding of the theory behind it all:


Arvo Pärt: Fratres


I have discovered that it is enough when a single note is beautifully played. This one note, or a silent beat, or a moment of silence, comforts me. I work with very few elements--with one voice, with two voices, I build with the most primitive materials--with the triad, with one specific tonality. The three notes of a triad are like bells. And that is why I call it "tintinnabulation." --Arvo Pärt
One of three contemporary composers whose works breathe themes of Eastern Orthodox Christianity and Roman Catholicism, Arvo Pärt began his career as a composer by experimenting with post-romantic serialism. Pärt's recent works are characterized by minimalist and Gothic influences, especially music of the fourteenth and fifteenth centuries. Thus Pärt's music looks "beyond" by looking back and within. In this sense Pärt's compositional journey may be likened to J. S. Bach's which drew inspiration from the stile antico of Palestrina. While Pärt's music sounds not to be overtly indebted to the baroque, his Collage on B.A.C.H. does contribute to an important body of literature attributing some level of admiration for the 18th-century master. Please refer to David Pinkerton for an excellent introduction to Pärt's mature style.

Partly for its brevity (ten minutes) but mostly for beauty and clarity of design, I have chosen Pärt's Fratres (1977) as a contemporary example of permutation technique in music. Notwithstanding the composer's nod to uncomplicated materials (see quotation above), Pärt here employs a simple idea (a scale), transforming it by means of development (melodic inversion and retrograde) into an asymmetrical pattern which he then permutes into an elegant series of nine variations. While, in terms of organic relationships, the structural elements of Fratres are simple, the completed work is anything but, and will provide grist for additional discussion to follow. For now, please acquaint yourself with the fundamental materials and processes in Fratres by studying the compositional notebook that follows:

Arvo Pärt Fratres for Strings and Percussion


Graphics and analysis ©1997 Tim Smith with kind permission of Universal Edition A.G., Wien

It seems clear to me that, like Bach, who did not specify instrumentation for his Art of Fugue, Arvo Pärt may have conceived of Fratres as existing independently of sound. By this I do not mean to imply that sound is unimportant...exactly the opposite. Before there can be meaningful sound there must be intellect, thoughtfulness, and design. The most meaningful sounds emanate of contemplation...in Pärt's case, an attempt to fathom the Cloud of Unknowing (to quote a medieval English text). Such thought is evident in Pärt's permutation technique which makes the various "incarnations" of Fratres possible. As if to say the same thing as Bach, though entirely opposite in manner, Pärt specifies MANY instrumentations. Notice that I do not say that he has "arranged" the piece for different combinations, because that would imply that one of the combinations was primary, and the others adaptations. No, it seems clear that this composer thinks of his music as existing first in the timeless and soundless dimension of thought, out of which, and only after which, sounds emanate. In that sense Fratres is sacramental--a physical manifestation of spiritual signification--a materialization of that which is spirit, transcendent, and therefore unfathomable. In Pärt's own words:

That is my goal. Time and timelessness are connected This instant and eternity are struggling within us. And this is the cause of all of our contradictions, our obstinacy, our narrow-mindedness, our faith and our grief."

Tintinnabulation is an area I sometimes wander into when I am searching for answers--in my life, my music, my work. In my dark hours, I have the certain feeling that everything outside this one thing has no meaning. The complex and many-faceted only confuses me, and I must search for unity. What is it, this one thing, and how do I find my way to it? Traces of this perfect thing appear in many guises--and everything that is unimportant falls away.

In the Soviet Union once, I spoke with a monk and asked him how, as a composer, one can improve oneself. He answered me by saying that he knew of no solution. I told him that I also wrote prayers, and set prayers and the texts of psalms to music, and that perhaps this would be of help to me as a composer. To this he said, "No, you are wrong. All the prayers have already been written. You don't need to write any more. Everything has been prepared. Now you have to prepare yourself." I believe there's truth in that. We must count on the fact that our music will come to an end one day. Perhaps there will come a moment, even for the greatest artist, when he will no longer want to or have to make art. And perhaps at that very moment we will value his creation even more--because in this instant he will have transcended his work.

Assignment

Questions pertaining to the Cm fugue (BWV 582)

Begin by listening to the Cm Fugue (BWV 582) while following the score. When you've become thoroughly familiar with the piece, listen again while following the analysis of each page.

  1. Normally a fugue's exposition is concluded after all the voices have stated the subject. Most writers recognize, however, that some fugues are better analyzed with one additional statement of the subject. I have analyzed the Cm fugue in this manner, the exposition of this four-voice fugue includes five statements of the subject, beginning and ending with the subject stated in the alto voice. Suggest two reasons why this would be the preferred analysis. Hint: for one of your reasons think of what the word "exposition" means and what exactly Bach is exposing in this fugue. Appeal to permutation theory for your second reason.

  2. The Cm fugue contains twelve statements of the subject. List the keys, in order, in which each subject begins. Write a paragraph discussing the relationship of each key to that which precedes it (e.g. closely related, foreign, relative M/m, parallel, enharmonic). Note especially any episodes in which the subject is mutated to the parallel mode.

  3. Of the Cm Fugue (BWV 582) Christoph Wolff writes: "The first section...complies in its regulated sequence of the different subjects with the principle of the permutation fugue that served Bach in his vocal works, particularly those of the early period. Bach departs from a systematically applied permutation principle after what might be considered the first exposition, which moves back and forth between tonic and dominant harmonies, eschewing wider modulation." Write three or four paragraphs discussing the implications of Wolff's assertion. When Bach departs from what Wolff calls "a systematically applied permutation principle" (after the exposition), what exactly is Bach departing from and why was it necessary for him to do this? In that light, was Bach actually departing from permutation principle, or was he just getting started? What evidence might you cite to support the proposition that the remainder of the fugue also manifests systematic permutation technique?

Questions pertaining to the Pärt Fratres

  1. Each of the nine variations in Pärt's Fratres are made of the same stuff...the same A/E drone in the bass, the same d-minor scale in the high and low voices, and the same a-minor pitches (A/C/E) in the middle voice. Yet, in spite of its sameness, Pärt achieves an unusual tonal development in which the ear begins to perceive two tonal centers at the same time. This sense of centeredness in two pitches is called "bi-tonality." The prevailing tonality is, of course, a-minor--reinforced by the drone as well as the pitches of the middle voice. While the high and low voices begin and end the piece with an A-Major chord, toward the middle, the permutation process compels the ear to acknowledge two more tonal centers. Listen to each variation, then identify which are bi-tonal and which are firmly centered in A. Of the bi-tonal variations, what other key do you hear in each (specify mode as well). One of the bi-tonal variations is particularly difficult to pin down and you might even hear it as hinting of three keys simultaneously! Which variation might that be? Theorize as to why some of the middle variations sound bi-tonal. If they are all made of the same pitches shouldn't they sound in the same key? (Hint: observe the clock diagrams as the piece moves through each variation.)

  2. At the top of the string and percussion score of Fratres are these words: "in memoriam eduard tubin." Tubin was a composer and conductor from Pärt's native Estonia. Letting your imagination run freely--stream of consciousness--listen to another "incarnation" of Fratres (violin, strings and percussion) then write a short paper describing your thoughts. Relate the sounds to Pärt's words above. Note well the agitation of variation 6 versus the tranquillity of variation 9, and relate these to your "theory" of question 1 above. Speculate as to the meaning of the drone, if any, and the symbolism of bitonal tension against it followed by repose. Do you see permutation technique as having anything to do with the multiple guises of Fratres? How do you think Pärt would respond to the complexity of textural permutation in Bach's Cm fugue? In doing this exercise you might find it helpful, as well, to listen to all the various instrumentations of Fratres.

    • Fratres for strings and percussion Fratresfor violin, strings and percussion> Fratresfor wind octet and percussion Fratresfor eight 'cellos Fratres for string quartet
    • Fratres for 'cello and piano

  3. For extra credit: describe the permutation pattern in the tabla part at the beginning of Pärt's When Sarah Was Ninety.
Make sure that your response is organized as pertaining to the Cm fugue (answers numbered 1-3) and the Pärt Fratres (answers numbered 1-3). Briefly restate the question before you answer it using complete sentences, correct grammatical construction and spelling.

Links

  1. David Pinkerton: Pärt Information Archive and Bibliography

  2. ClassicalNet: Pärt Biography, CD Discography, Works List, and Recommended Recordings by Doug Maskew

  3. ECM Records: Program Notes on Pärt Litany, Psalom, and Trisagion

  4. The official Stockhausen homepage and notes on Stockhausen's search for "a unified underlying order beyond the dimensions of pitch and time independent of themselves."

Notes

  1. From the liner notes to ECM New Series disk 817-764-2 Cantus in memory of Benjamin Britten, Tabula rasa, and Fratres.

  2. The other two composers are: John Kenneth Tavener (not to be confused with Tudor period composer John Taverner), and Henryk-Mikolaj Görecki.

  3. Donald Jay Grout, A History of Western Music third edition (New York: W. W. Norton, 1980), p. 121-122.

  4. Christoph Wolff, Bach: Essays on His Life and Music (Cambridge: Harvard University Press, 1991), p. 314.

  5. As quoted in the liner notes to ECM New Series disk 817-764-2 Cantus in memory of Benjamin Britten, Tabula rasa, and Fratres.
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