Patterns and Time-Scales
by Anthony Brandt
We are in the process of investigating musical concepts that transcend eras and styles, and which remain as relevant for the music of today as they are for the music of the past. In my last column, we discussed the idea of "similar patterns on multiple time-scales," and I would like to delve into this a little further.
A strong illustration of this principle in classical music centers on the harmonic basis for tonality. In a tonal piece, one piece, more than any other, represents maximum rest, order, stasis, repose; this pitch is called the tonic, and its chord the tonic triad. When we speak of Beethoven's "Eroica" Symphony being in the key of E-flat Major, we mean that E-flat is the tonic. When a piece returns to the tonic with maximum emphasis, it has achieved closure, peace, rest.
Against the tonic is poised another pitch, supported by its chord: this pitch, which is always the fifth degree of the scale, is called the dominant. The dominant--and, more specifically, the dominant chord--represents motion, disorder, instability, chaos. Thus, the stability of the tonic is juxtaposed against the unresolved nature of the dominant.
Moving from chord to chord in a piece of tonal music, we will find an abundance of tonic chords moving to dominant ones, and dominants resolving to tonics, as well as other chords which mediate between these two poles. But tonal music is also perceived in larger units, called phrases, which embrace an entire progression of chords. Phrases are punctuated by harmonic arrival points, called cadences. It so happens that many of these cadences will also be to the tonic or to the dominant. Thus, the tonic-dominant opposition is already expressing itself on two temporal levels.
Now phrases may be gathered into sections. It is very characteristic for the first section of a piece to be in the tonic, and the second to be in the dominant; the final section will then return to the tonic. An opposition which expresses itself very immediately--from chord to chord--now spans the entire composition. If our ear searches among the time-scales, it discovers essentially the same relationships being expressed.
The above discussion applies to a large number of tonal pieces. But as we examine a specific composition, we find the great composers engraving this same principle of "similar patterns on multiple time-scales" in all sorts of contextual ways with imaginative variety. Let me offer one brief illustration. Beethoven's Bagatelle, opus 126, No. 1 for piano opens with a lovely, lyrical tune in the upper register. This tune spins itself out elegantly, and eventually leads to a contrasting middle section. Then, at a critical moment, the tune reappears. But it does so in an unexpected way: instead of returning in its original register, as might be expected, it comes back way down low. Other transformations heighten the sense of displacement, of disguise. The question is: why does the return happen in this way?
If one investigates the opening tune, one discovers that it, in fact, also passes from the upper register to the lower register. This is not immediately apparent to the ear, because the upper register continues onwards with a very captivating line; but, meanwhile, down below, the low register is continuing the rhythm and direction of the upper register's first few measures.
Thus, the surprising shift to the low register at the return echoes, on a larger time-scale, the shift that occurs right at the beginning of the piece. These are not even the only examples within this concise composition to express this idea of registral shifts: indeed, it is happening all the time. It is crucial to their musical meaning that these melodic transfers are happening at different structural rates-- that is, that is, that the same relationship is being expressed on different time-scales.
Another example from Beethoven: his Grosse Fuge for string quartet, one of his most elaborate and original compositions, begins with a stark tune played together by the four strings. The tune begins with very slow values, then gradually speeds up. This idea of accelerating will be repeated in many ways: as the main theme reappears in various guides, there are various passages which speed up.
But, even more interesting, the piece is divided into a sequence of clearly delineated sections. The first of these is extremely long. As the piece progresses, by and large, the sections get shorter and shorter, one passage giving way to the next more and more rapidly, so that the rhythm of the form seems to accelerated as well! Thus, the rhythmic profile of the theme also stretches across the landscape of the entire work.
In tonality, the opposition of tonic and dominant forms the back- bone of this principle of multiple time-scales. To this is added the contextual working out of the material specific to a given piece. This contextual working out becomes even more important in 20th-century music, when the influence of tonality became more complex and unsettled. In non- tonal 20th century music, there are fewer "givens;" instead, it is more than ever up to the composer to create the correspondences which we are describing.
The twelve-tone composer Anton Webern, for instance, is a master of carrying out this principle; indeed, the more one looks at his music, the more one realizes how obsessed he was about doing so. In his Symphony, opus 21, for instance, one of the patterns he focuses on is the interval of the tritone.
In music, an interval is no more than a measure of distance. In conventional Western music, the semi-tone, or "half-step," is the basic unit of measure, much like the inch or centimeter. An octave is made up of twelve semi-tones: start on pitch "A," count up twelve semi-tones, and you return to A, an octave higher. Divide the octave in half, and you get the tritone; the tritone is therefore the octave's center of symmetry.
In Webern's music, the tritone opposition does not represent stasis versus motion, as the tonic and dominant do in tonal music. Rather, the two notes that form the tritone are in balance with each other; they represent symmetry, equilibrium.
How is the tritone expressed in the Symphony? The central tune of Webern's composition starts on a note and ends, eleven notes later, a tritone higher. Now, if you look carefully at the tune, you realize that is constructed as a sort of mirror: the tune progresses towards the middle, then continues with a backwards version of itself. The midpoint of the tune, the "mirror-glass," so to speak--is also a tritone. So, already, just within the tune, the tritone is expressed on two levels: from end to end, but also from note to note in the middle. As the first movement progresses, there is a change from one version of the tune to a second version. In this second version, the tritone that was in the middle goes to the ends, and the one at the ends goes to the middle. The first movement as a whole is divided into three sections: lo and behold, the harmonies in the first section center around the first note of the main tune; the middle section is more complex; and the harmonies of the last section center around the last note of the tune. Thus, the interval of the tritone is now reaching structurally across the entire time-span of the movement. What Webern has done is grasped a principle first articulated in tonal music, and expressed it in a totally original way. His compositional thinking is therefore utterly conservative, and utterly radical at the same time.
It is well established that arhythmia, the absence of a steady pulse, is symptomatic of an unhealthy heart. But recent research by physician Ary Goldberger theorizes that the opposite extreme, a pulse that is too regular, is also unhealthy. The healthy human heart's beat exhibits a zone of fluctuations; fractal-like, the pattern of these fluctuations are reproduced on multiple time-scales. Goldberger theorizes that these variations on multiple time-scales heighten the heart's capacity to adapt to different environments. He also acknowledges the difficulty of building a medical instrument that can keep track of multiple time- scales simultaneously. We, of course, are such an instrument; we keep track of multiple time-scales all the time. A steady pulse is usually described as the "heartbeat" of music; music with an insistently steady beat is usually felt to be more visceral, more human. But it is possible that music's connection to our heart may actually lie more in the domain of what we have been discussing.
As a final remark, I would like to make clear that music is not literally "fractal." Music and fractal geometry do share this central principle of "similar patterns on multiple time-scales." But, as we will see, composition is about the integration of a whole variety of necessities. Indeed, music is deeply involved with time but it is also deeply involved with sound. In my next column, we turn our attention to the nature of sound, and to the concept of "resonance."
©1995 Anthony Brandt, all rights reserved.