tonal, schmonal
stephen dembski

I first met Stephen Dembski in late 1991 when I was assigned by the University of Wisconsin-Madison to write this article about his work. Words of warning: it was The Man who hired me to write this story—which was intended for a lay audience—and so it is bland and corny. Dembski is neither.

a system of circles
To the untrained ear, some twentieth-century classical music can seem like a stream of unrelated notes forming unfamiliar patterns, much like abstract art. Modern compositional techniques, such as“serialism” and“limited sets,” disregard the structures of keys, blurring the distinction between melodies and chords that for centuries provided the framework within which people interpreted music.

Stephen Dembski, a University of Wisconsin-Madison music professor, composed most of his early works within these“post-tonal” systems. But in the early 1980s, convinced that the extensive mental processing required by such music overwhelmed most listeners, he started to search for new compositional techniques.“I thought people would have a better shot at understanding new music if it was more like tonal music,” he says.“And at the same time I wondered how I could make writing atonal music feel more like writing tonal music, which had always felt good to me, without being limited to the structures of tonal music?’”

In his quest to enhance the music systems of his predecessors, Dembski is like other composers through the centuries. The classical music that we take for granted evolved as composers experimented, building on the musical discoveries of their forerunners. Just as Einstein relied on the physical laws Newton first explained, Beethoven built on musical ground broken by Mozart and Haydn.

Early classical music concentrated on seven-tone major and minor scales and simple three-tone chords, called triads, built from those scales. In“tonal” music--the compositional system traditionally used in eighteenth- and nineteenth-century classical music and in popular music such as rock n’ roll—chords are built from alternating pitches, or“tones,” within a scale. Melodies, which weave the chords together, are usually drawn from the scale, but can include any pitch. Some composers, including Dembski, think of melodies as combinations of“lines of pitches”—which are two or more successive pitches—and other pitches that don’t form lines.

Hearing the difference between a line and a chord, can, however, be difficult. Although many people think of chords as groups of notes that sound simultaneously, a chord’s notes can also sound separately, like the rippling of harp strings. Such a series of chord notes is called an arpeggio. And although lines usually contain some notes that are found in the chords they connect, they also can include notes from“outside” the chords.

To picture the relationship between chords and lines, think of the pitches in a tonal scale as numbers from 1 to 7 or as do, re, mi, fa, sol, la, ti. Some of the chords—which are constructed from alternating tones—could be the pitches 1, 3, and 5 (do, mi, sol), 6, 4, and 2 (la, fa, re), or 5, 3, and 7 (sol, mi, ti). Some lines—which are made of successive tones—could be 2, 3, 4 (re, mi, fa), 6, 5, 5, 4 (la, so, sol, fa), or 1, 2, 3, 2 (do, re, mi, re). And a melody—which can include lines, arpeggios, and single notes—might be an arpeggio 5, 3, 7 (sol, mi, ti), followed by a line 5, 6, 7 (sol, la, ti), which leaps to another line 5, 5 ,4 ,3, 2, 1 (sol, sol, sol, fa, mi, re, do).

Or imagine a singer accompanying herself on guitar. As she sings a ballad’s distinctive melody, she strums a series of chords. With each chord change, you can feel the song move forward. This shifting from chord to chord provides the song’s“structure.” It’s the singer’s voice, however, that provides the melody. Often the melody’s pitches are“chord tones”: tones that match one of the notes in the chord she is playing and sound in perfect harmony with the chord. Other times the pitches are found in the scale for the song’s key, but are not“chord tones.”

Starting from these simple principles, early composers soon found systematic ways to move between scales and chords and to“modulate” from one key—or set of scales and chords—to another. Composers also learned to build more intricate chords. But no matter how elaborate the chords and movement between keys became, for each section of a composition, a single tone, the“tonic”—which was usually the first note in the scale being used—was treated as a reference, a sort of focal point. A section of a string quartet that is in the key of D major, for example, would treat D as the reference tone, and, most often, the first and last chords would be D major chords.

In the early twentieth-century, composers started looking for ways to incorporate musical materials that crossed tonal music’s boundaries of keys, scales, and triads. The new concept of“atonality” was hinted at in the works of Claude Debussy, as the French composer explored tone combinations not found in major and minor scales. Soon the influential German composer Arnold Schoenberg further defined and developed the idea into a 12-tone music system, which he and other composers used instead of traditional harmony.

Rather than pivot on a single, reference tone and its related chords, as tonal music does, 12-tone music draws from series of 12 notes, which is why the system is sometimes called serialism. These series of tones—or“tone rows”—contain, in the order the composer determines, all twelve tones used in western music. One tone row might be C, D, F#, C#, B, G, G#, A#, E, D#, F, A. Unlike tonal music, serial music has no reference tone. Each tone refers to all of the others. It is the order of the set, or“tone row,” of selected 12 tones—rather than a single reference tone, conventional scale, or set of triads—to which the composer refers when creating lines and chords.

Strict rules, however, have not applied to 12-tone music. Composers could make chords from sequential tones, every other tone, blocks of six tones, any other regular or irregular pattern applied to the 12-tone row, or no pattern at all. Sometimes the tone row was varied so that combinations of rows were used simultaneously or one after another. Lines could be parts of the row passed from instrument to instrument, perhaps the first three tones sounding on an oboe, the next five on a bassoon. But regardless of what followed, the composer’s original 12-tone row was always the reference.

During the past 80 years, modern composers have thoroughly explored and elaborated upon this system. Milton Babbit, for example, Dembski’s teacher at Princeton, applied serial techniques not just to pitch, but to every aspect of composition, including such things as rhythm and orchestration.

Dembski, a Romnes Faculty Fellow, wrote his early works according to the sophisticated refinements that this 12-tone system has undergone. And it was this system Dembski wanted to transcend.

Moving Along
Although most people recognize tonal music by remembering melodies—whether the tune from“Misty” or the theme of Beethoven’s Fifth Symphony—underlying those melodies are groups, or“progressions,” of chords. Tonal compositions“move forward” by progressing from chord to chord.

But 12-tone music doesn’t move from chord to chord. Instead, as the composer uses the 12-note tone row as a source, it moves a“12-tone aggregate” at a time.“And unless you use all 12 tones, you lose many of the structural capabilities of the system,” Dembski says.“The unit of harmonic progression is the complete aggregate of all 12 tones. You can’t interpret any one tone—such as C or A—unless you understand its relationship to each of the other 11. Unless you hear all 12, everything is left hanging and you can’t move the composition forward.” But, he worried, many listeners would find it difficult to keep track of the relationships between 12 pitches as they sound in different combinations or move from instrument to instrument.

Dembski also wanted his non-tonal compositions to include a useful characteristic of tonal music. In the tonal system, manipulating a note here or there can alter the interrelationships between sets of tones, shifting one tonal key to another. For example, by placing the pitch F# wherever“F” would otherwise have appeared, a composer can modulate from the key of C to the key of G. This distinction wasn’t possible with a 12-tone reference set.

Limited Sets
Dembski thought his audience might be better able to understand his chords and melodies if the collections of tones he used were smaller than 12, even if they weren’t based on major and minor scales. For compositions such as“Digit,” a duet for clarinet and prerecorded computer tape, he constructed an eight-tone“limited set” that had particular pitch interrelationships he wanted to explore. Then he transposed the eight-tone set into other eight-tone sets that shared the same structure. As the composition progressed, Dembski used each of these sets to eventually move through all 12 tones in a tone row.

But Dembski wasn’t satisfied with such limited sets. They didn’t offer the structural depth of tonal or atonal music. Without tonal music’s scale system, they didn’t offer an innate way to distinguish between chords, which in tonal music were built from alternating pitches in a scale, and lines, which were built from successive pitches. They didn’t have a well-defined way to move forward. With such limited musical resources, Dembski found it impossible to write compositions that were more than a few minutes long.“I became totally obsessed with this problem,” Dembski says.“If it was a line it was moving, if it was a chord it was just hanging there—it had no harmonic rhythm.”

In 1981, still using limited sets, Dembski began writing“Alta,” a piece for solo piano that Alan Feinberg, a primary interpreter of Dembski’s work, was to perform and later record on the CRI label. One of Dembski’s goals in writing this piece was to find a way to make lines and chords in a non-tonal framework.

“That’s why the piece goes on so long with just one pitch sounding at a time,” he explains,“It’s to see whether you can sense the difference between a line and an arpeggiation—a broken chord.”

In that sense, Dembski says,“Alta” is similar to some of the compositions of Johann Sebastian Bach—especially the partitas for violin. Long sections of the partitias gradually progress from chord to chord by altering one pitch at a time within arpeggios—chords that are played a note-at-a-time rather than simultaneously. Although—in“Alta” or a Bach partita—the repeating strings of notes may seem like melodies, the listener doesn’t sense that the composition has moved forward until a crucial note is altered, changing one chord into the next.

As he developed“Alta,” Dembski asked himself:“What is the basic underlying structure that makes tonal music so flexible and easy to understand?” The answer, he decided, was that the fundamental structures of tonal music could be extracted from two unique 12-tone sets: the circle of fifths and the chromatic scale.

Fifths, notes that are five pitches apart in a major or minor scale, and chromatic steps, or“half-steps,” have an attribute shared by no other intervals: they can be represented as regular“circles” that include all 12 pitches, evenly spaced in terms of steps.

Starting at any note and moving by either fifths or half-steps uses all 12 pitches before repeating the original pitch. The notes within these circles are always in the same order. C, for example, in the circle of fifths will always follow F (because C is the fifth of F) and will be followed by G (which is the fifth of C). In the chromatic circle, C is followed by C#, which is a half-step higher, which is in turn followed by D, another half-step higher.

Regularly spaced circles can also be constructed from other pitch intervals. But these circles will always return to the original pitch before all other 11 pitches are used. A circle of thirds starting on C, for example, uses just three notes—C, E, and G#—before returning to C. So except for the circle of fifths and the chromatic circle, any complete 12-tone circle would have to include at least some irregular spacing: a half step here, two-and-a-half steps there.

These two regularly spaced 12-tone sets, Dembski found, form the structural basis of tonal music. By selecting any continuous section of seven adjacent pitches from the circle of fifths—for example G, D, A, E, B, F#, and C#—and then reordering them according to order of pitches in the chromatic circle—D, E, F#, G, A, B, C#—he could construct tonal music’s fundamental scales, in this case D major.

Dembski’s Circles
“What if,“ Dembski thought,“I made my own circles from irregularly spaced 12-tone sets I construct, and then drew my seven-tone scales from those circles?“ One irregular circle could serve the same function for his new system as the circle of fifths served for tonal music: providing seven-tone sets from which scales are constructed. The other irregular circle could serve the same function as the chromatic circle: putting the tones in order.

Because the 12-tone sets were irregular, he found that every rotation of the first circle produced a change in the internal structures of the seven-tone set. That created 12 scales. Dembski could then“transpose“—or shift—each of these scales 12 times, once for each of the 12 pitch classes western music uses; 12 times 12 equaled 144 scales. Then he could reorder each of these 144 scales 12 ways—once for each transposition of the second circle. That’s 1728 possible pitch combinations, each of which he could treat as a scale. And unlike the limited sets he used for earlier compositions, all of these scales were drawn from two interacting 12-tone circles, just like tonal scales.

“It seemed like tonal music in a deep way, without being restricted to the regular scales and triads of tonal music,“ Dembski explains.“So it seemed more reasonable to expect people listening to such a piece to be able to understand it, even eventually to guess the next note, like many people can do when listening to Bach.“

As with the tonal scales built from regular circles, these new scales built from irregular circles gave Dembski a framework in which some combinations of pitches formed lines and others formed chords. Chords were built from groups of alternating pitches in the scales. In, for example, the tonal scale of D major—which is D, E, F#, G, A, B, and C#—the first chord is built from D , F#, and A. The chord skips over E and G.

In one of Dembski’s new scales comprised of F, F#, D#, E, A#, B, and C#, for example, the first chord is F, D#, and A#. The chord skips over F# and E. This system made possible note combinations—such as F# and E as two notes of a triad—that weren’t found in tonal triads.

Each of the 1728 scales that was generated from a set of two irregular circles had seven triads. That’s more than 12,000 three-pitch chords—although many were repeated—for each set of two circles.

“All of a sudden a whole universe of possibilities, within a vigorous structural framework, opened up for me,“ Dembski says. It was in“Alba,“ a five-movement piece for flute, clarinet, percussion, violin, and cello, that he first began applying the new technique. Dembski shaped the first two movements for“Alba,“ which he had started before inventing his 12-tone circle system, from limited sets built from chunks of 12-tone rows. The last three movements, and everything he has written since, grew out of his circles.

It Takes a Computer to Explore the Universe
Although Dembski felt liberated by the prospect of exploring the universe of scales and chords his new system offered, the very immensity that excited him also presented obstacles.

“It was laborious to work out just a few of the possibilities from this enormous constellation of harmonic structures and scales,“ Dembski says.“It was like writing a piece of tonal music except that if you wanted to make a simple modulation you’d have to go through a six-hour process to figure out the chords. By the time I figured out the next set of pitch material, I had forgotten much of the idea I wanted to express.“

Thousands of scales and chords were too much for a human mind to master. But, Dembski decided, a computer could easily map his new universe. Drawing on skills he had developed writing electronic compositions on mainframe computers, Dembski designed programs for an Apple II personal computer to generate circles with pitches represented as numbers from zero to 11. As he acquired faster computers and more sophisticated programs, he found that in just a few minutes he could chart all the relationships between two 12-tone circles.

“In half an hour these programs generated all the pitch material related the way I wanted,“ Dembski says.“So if I was looking for a particular harmonic or linear construct, it could tell me if it was right next door in the next circle, or way over there by Alpha Centauri. It felt like an enormous expanse of scales and chords within which structures were well-defined, as in tonal music.“

The European Tradition
Within these new universes of tone rows, scales, and chords, Dembski develops the raw material for compositions whose stylistic influences range from the lush romanticism of late nineteenth-century classical music to the improvisations of free jazz. Often, Dembski says, the ideas for his works originate in the physical way he wants the instrumentalists to play.

His experience at Antioch College in the late 1960s, playing flute in radical jazz pianist Cecil Taylor’s big band, for example, inspired Dembski to write“Pterodactyl,“ a piece filled with dense clusters of notes. Taylor, who taught at the UW-Madison for one controversy-filled year, is best known for almost chaotic, but highly disciplined, improvisation.

“I thought, ’I can’t play piano like that, but what would it be like if I tried to write it down, and then based a piece around that way of playing?’“ Dembski says.“That was one of the ways I heard music, and I wanted to see if I could write music that way.“

There is, however, an important distinction between imitation, which post-modern composers have often been accused of, and drawing on playing techniques, Dembski cautions. And unlike many definably American artists, such as John Cage or Terry Riley, many of whom draw heavily from pop culture and Third World folk music, the roots of Dembski’s work extend back to Gregorian chants of the Middle Ages.

“I think of myself very much in the long-term European tradition,“ he explains. Between 1920 and 1950, powerful composers with dominating personalities—Europeans such as Schoenburg and Stravinsky—brought the traditional disciplines of harmony, polyphony, and counterpoint to the United States. Their musical descendants—such as the Princeton professors Milton Babbitt and Roger Sessions—in turn shifted the center of the European tradition to the United States. Now Dembski, a student of Babbitt, is finding new structures in which to explore those traditions.