Intuition and Algorithm in Einstein on the Beach
[Ed. Note: The following paper was presented at the Einstein on the Beach conference of the University of Amsterdam, January 6, 2013. The present version is slightly expanded based on questions and comments from that session. All score excerpts from Einstein on the Beach by Philip Glass ©1976 Dunvagen Music Publishers. Used by Permission.]
Some of us are old enough to remember the public impression that minimalist music made before the premiere of Einstein on the Beach. Minimalism in its first manifestation was a strict, objectivist style. We thought of it, pretty much, as gradual-process music. Philip Glass’s Music in Fifths and Music in Contrary Motion offered us perceptual exercises in additive process, and taught us to hear the gradual expansion of a time frame. Steve Reich’s out-of-phase tape loops amplified microscopic phenomena of the human voice. The similarly phasing 12-note pattern of Piano Phase created its own objective geometry, as did the change-ringing patterns of Jon Gibson. La Monte Young’s sine-tone installations were comfortingly based in mathematics. Charlemagne Palestine’s piano improvisations conjured up overtones. Only Terry Riley seemed a little loopy, pardon the pun, but even in In C we heard the echoes of melodies as a strict canonic process.
It is odd to remember at this distance how important this criterion of objectivity seemed at the time. We had come out of a period of serialism and strict chance processes, glorifying mathematics and the natural world. In the conditioned milieu of 1960s avant-garde music, mere expression of emotion seemed at the time unworthy of serious study; subjectivity was distrusted. The worship of science was rampant, and music had become scientific. For many musicians involved in the avant-garde it accelerated the acceptance of minimalism, I think, that it seemed to be about natural and or logical phenomena. Some of us weren’t yet ready to return to intuition, art, shading, eccentricity, and the unsteady foundation of personal preference.
There were enough hints of gradual process in Einstein on the Beach, I think, that it was accepted as fitting this objectivist paradigm at the time. The use of numbers and solfège syllables as text facilitated this impression. So did Glass’s liner notes to the original Tomato recording of the work. Of the lightning fast patterns in the Building scene, Glass states that “the repeated figures form simple arithmetic progressions,” and he refers to a figure in the Trial scene which, he writes, “slowly expands and contracts […] through an additive process.” While this is arguably true of the Trial scene, I will show that the progressions of the Building scene are far from arithmetically simple, and also that many of the other scenes are devoid of predictable algorithmic thinking. Looking back in retrospect, Einstein seems a far more intuitively written work than we thought at the time. For several decades the score wasn’t available, and to transcribe these lightning-fast patterns would have taken a lot of patience; we thought we had a pretty good idea what we were hearing. Eventually a score became available via links on Glass’s website, and I bought it in 2008. Upon opening it I was immediately struck by how much more unpredictable the music was than I had remembered it, how circuitous its forms were, how difficult it often was to pinpoint the musical logic. I was struck by how compositionally playful the piece is. In particular it offers some striking examples of recomposition, of writing through the same parallel succession of motives and harmonies several times within one piece and doing it a little differently each time. It is this playful, intuitive technique of recomposition in the composed musical scenes of Einstein on the Beach that I plan to focus on here.
Let me begin, almost pro forma, by reviewing the recombinant elements that make up Einstein, since I’ll be referring to them later. First of all are three recurring chord progressions, one with three chords, one with four, and one with five.
(Glass’s notes also refer to ideas of two chords and one chord, but these don’t appear as frequently.) The famous five-chord progression modulates from F minor to E major, and is the basis of the Spaceship scene, appears in the Train and Building scenes, and is the basis of the internal Knee Plays. Glass refers to it as “cadential,” but the three-chord progression heard in the first and last Knee Plays seems cadential as well: a simple vi-V-I. The four-chord progression appears in the Trial and Bed scenes, always in the kind of slowly arpeggiated motion seen here. In addition, there are a few other features found from scene to scene. The upward A-minor triad with a variety of continuations serves as a prelude to the four-chord progression. A la-fa-la-si-do-si motto in A-flat appears as a kind of section marker in the Train and Night Train scenes, and the following figure of four modules appears in both those scenes and constitutes almost the entire notated material of the Building scene.
Many movements of Einstein seem to be written in a kind of stanzaic form, wherein a movement is divided into stanzas which are parallel in their function and similar (though varied) in their progress through the same harmonies and motifs. Sometimes the beginnings are marked by introductory motives which I will call incipits: such as the three perfect fourths in the saxophones at each new section of the Train scene, the two four-note patterns of Night Train, and the reduction of the Building scene to a 6/8 pattern:
Likewise, some of the stanzas end in signalling characteristic figures that I will call envois, after the medieval poetry term. These include a quick F-minor triad in the Train scene, the solo violin playing A minor scales in Trial 1, and, again, the la-fa-la-si-do-si motive in Night Train. Not all stanzas in the various scenes are marked by these devices, but in those that are the incipits and envois are quite clear in their framing intent.
There are moments in Einstein at which a more stereotypically linear minimalist logic prevails. The most obvious is the bulk of the Bed scene, where the soprano sings over the four-chord progression. As this chart of the rhythms for each chord shows, the length of the rhythmic cycle expands in a fairly predictable manner with each iteration.
The voice part, however, is a more intuitive element, drawing lines that use only notes from the four triads. Out of 81 possibilities (not counting octave displacements), Glass chooses only seven of the available such lines and repeats three of them, creating a mild climax by using sevenths in the 7th and 8th cycles. (Each voice line is repeated once before proceeding to the next.)
Likewise, the violin part of the Trial 1 scene does, as Glass says, go through a process of expansion and subtraction, though not in a completely linear manner.
The chorus doubles the low C and A in the first part of each pattern; thus, once the music begins leaving this part of the phrase out of each repetition, the chorus disappears and only the violin continues.
At the other extreme is the Building movement, in which only the two organs are notated, as the voices and other instruments drone and improvise. As notated, the piece is entirely in eighth-notes in contrary motion, much like several of Glass’s minimalist works of the late 1960s. (It should be kept in mind that all of these patterns repeat 2, 4, or 8 times before proceeding to the next one.)
The move from one pattern to another, however, is not at all predictable. The following chart of the right-hand Organ 1 part shows all the pitches for the first 30 (out of 37) cycles, lined up vertically to show easily what notes are added or subtracted to get from one repetition the next (here the lower-case “e” denotes the E-flat in the upper octave, and there is a key signature of three flats).
What making this chart clarifies is that the movement is made up of only four modules whose changing combinations make up the form:
The movement’s seeming micro-complexity is due to the fact that module A is contained in module B, and likewise module C in D. The following chart shows the deployment of these four modules throughout the entire section:
At each step one can see a kind of additive or subtractive logic: the 3/8 modules start out with a 6/8 feel, and then the A module is added to give kind of a quarter-note bump to the repetition of BD, then another A is added, then steps 2 and 3 are added together, and so on. After expanding via additions of the A module, the music strips back down to just B and D, after which module C starts to be added in. Towards the end the music begins to emphasize the ten-note pattern ABCD, and finally resolves to the opening BD with which it began. The musical continuity here is not illogical, but neither is there any place where one could look at two or three successive phrases and guess (with any confidence of accuracy) what the next one will be. Glass’s comment about “simple arithmetic progressions” notwithstanding, this is a very unpredictable sequence. And, it must be said, this is all background structure anyway, since in this scene the pentatonic drones and improvisations tend to greatly override the subtlety of the organ patterns.
The Train scene, one of the most musically complex movements, is made up of three recurring sections, structured in the form ABABCABC. The B sections, which seem to serve a connective function, are instrumental (without voices), using the same four modules from the Building scene that were just identified. The C sections are based on the five-chord cadential progression that is the basis of the Spaceship scene. The A sections are unique to this scene. The first two B sections are identical, the third one considerably expanded. The two C sections are identical except that the chorus is added in the second one; the rhythmic patterns here follow those at the beginning of the Spaceship scene. What is illustrative of Glass’s approach to form in Einstein, though, are the three A sections.
The diagram here outlines the three A sections in a kind of shorthand that isolates an abbreviated set of features, namely the length of the repeated phrases and the notes of the soprano solo (once again in a key signature of three flats).
A three-note drone ostinato in the saxophones runs throughout all the A sections, imposing an underlying 3/4 meter. The voices sing repeated patterns of various lengths, entirely in quarter-notes except for a recurring refrain which I will identify in a moment. When the number of quarter-notes in the voice pattern is not divisible by three, the voices and saxophones run through a brief out-of-phase pattern, and the number of their repetitions must be divisible by three to make the phrases come out evenly at the end of the measure. The numbers on each left-hand column indicate the number of quarter-notes in each voice phrase. A number given as 4×3, 5×3, and so on, indicates that the phrase goes out of phase with the saxophone ostinato. A number given as 6+4+2 or 5+4+3+2+1 points to a subsidiary rhythm within the phrase suggesting an additive or subtractive rhythmic process; note that these occur only in the second A section. In each right-hand column are given the pitches in the solo soprano voice part, and since there are three flats in the key signature, A should be read as A-flat, B as B-flat, and so on. As in earlier examples, all these melodic fragments are stated in quarter-notes except for a recurring refrain in 8th-notes on Ab-F-Ab-Bb-C-Bb, sung on the solfège syllables la-fa-la-si-do-si, which recurs both here and in the Night Train scene as a kind of motto. One can see that here it appears twice at the beginning, and then at the end of each A section.
Other points that could be made here are even clearer in the following example. This comparison of the three soprano parts (and the tenor is always either a perfect fourth or major third below, in parallel) shows that the soprano begins each section alternating between Ab and Bb, gradually making her way up to Eb and then F, then descending back to Bb before concluding with the la-fa-la-si-do-si motto.
Notice, however, that the melodic and rhythmic character of the route is quite different in each A section. The first section reaches the high F relatively quickly, the second takes a long time to get there, and the third stays on F and Eb for a long time as a kind of climax. The second section, as previously noted, contains more patterns which contain an internal subtractive or additive process, and the third section contains the only additive process among phrases, and one additional subtractive process. At the end, each of them finally goes into a subtractive rhythmic process of 5-4-3-2 before lapsing into the la-fa-la-si-do-si motto. What is evident, then, is how many rhythmic and melodic options were open to Glass to get him from the Ab up to the F and back to the closing refrain, and how carefully he recomposed this process for three parallel sections achieving the same function through different routes. This possibility of intuitively recomposing a section is far removed from the typical concept of early minimalism as being something logically predetermined. We’re not just listening to nature here; we’re listening to variations in a large-scale melody conceptualized as rhythm.
Dances 1 and 2
One of the most fascinating views of Glass’s compositional process in Einstein is the subtle contrast between Dances 1 and 2. Many of the materials are identical from one dance to another, but the second is somewhat transformed in consequence of its use of the solo violin representing Einstein. Both dances contain a trio of drone notes that sound throughout: the pitches A, D, and E. These pitches are heard in the solo voices, the saxophones, and the left hand of Organ 1 in every measure. In addition, a fourth pitch, F, appears in the culminating repetitions of each large section. These drone pitches are recontextualized by the changing harmonies around them. In Dance 1, the arpeggios in the organs and piccolo move among chords of F major, A major, Bb major, G major, and C major, the constant D, E, and A being reinterpreted in each new harmony.
(Actually, Glass describes it in his program notes as always returning to D, which is justifiable if you consider the F major as part of a D minor 7th chord; but which isn’t the way I hear it.) The singers and saxophones use only the pitches D, E, F, and A, voiced either as quarter notes, dotted quarter notes, half notes, or, at one arguably climactic point, dotted half notes. A listing of the chord progressions and the rhythms in each repetition throughout Dance 1 reveals a clear division into three parallel parts, though these are not marked by clear incipits and refrains as in the Train scene:
Though there is no key signature, the general tonality of Dance 1 sounds to me to be in F major; the piece begins and ends on an F major chord (with an added sixth D), and the phrase rhythm frequently makes F major sound like a resolution, though it can also sound like a flatted-seventh adjunct to G major. Listing the harmonic progressions of each repetition, we see a clear parallelism among phrases 1 to 17, 18 to 33, and 34 to 50. That is, there are 50 phrases in the dance, divided into three stanzas with lengths 17 + 16 + 17. (By the way, the Nonesuch recording of Einstein omits the second stanza, as did the recent production of the opera in Amsterdam.) As is clear from the diagram, each stanza starts by alternating F major and A major, then adds in Bb major, and finally moves to an alternation of G and F, inserting between them a C major chord to make a kind of II-V-I cadence. Although this pattern is clear, there is some variety in the tonal emphasis: stanza 3 spends less time on Bb than stanzas 1 and 2 do, and stanza 2 has a long middle passage on F major lacking in stanzas 1 or 3. Likewise, in an overview of the rhythms of each repeated phrase one can note rough parallels, but no clear isomorphism. In the first half of each stanza there is one repeated pattern longer than the ones around it; in stanza 1 it’s the fourth, in stanza 2 the third, and in stanza 3 the fifth. In stanza 1, half-notes appear in phrases 5 to 7, in stanza 2 in phrases 1 to 4, and stanza 3 contains no half-note rhythms, but offers dotted half-notes in phrase five.
The last four or five phrases of each stanza are rather climactic. While elsewhere the soprano and soprano saxophone use only the pitches D and E, in these final phrases they use a repeating DEFE, as marked in the diagram. Note that the rhythms here are entirely in quarter-notes, and that in stanzas 1 and 2 there occurs a nine-note pattern of DEFE-DEF-DE, a kind of small 4+3+2 subtractive motif. (By the way, we’ll see Glass using this 4+3+2 pattern 15 years later in his Columbus opera The Voyage, and many of these other patterns as well.) These DEFE motives appear only over the harmonic progressions G-F and G-C-F. All of these features point to an overall form divided into three parallel parts, but the musical continuity is so static, with its endlessly sustained A, D, and E, that the listener does not distinctly experience the piece as sectional, but as a smooth continuum with variations in the symmetry of the rhythm.
Turning to Dance 2, we find many of these same characteristics. Again the pitches D, E, and A are sustained throughout, the DEFE motive appears as a kind of relative intensification, and the rhythm moves among quarter-note, dotted-quarter, and half-note beats. But now the saxophones and piccolo are replaced by Einstein’s violin, and the necessity of writing a playable if still extremely virtuosic part for that instrument seems to suggest quite a few alterations.
Most noticeably, the tonality of F that dominated Dance 1 is replaced with a feeling of A minor, or at least an A natural minor scale (denoted here by a lower-case “a”), a tonality that marks much of the violin solo’s music throughout the opera. The tonality of Bb no longer appears. For the first ten phrases the music merely alternates between an A natural minor scale and an A major triad. The DEFE motive appears only twice in the piece, about a third of the way through and for a long time at the very end. Replacing Dance 1’s middle stanza is a long section (phrases 15 to 25) in which the violin articulates a strict process that is both additive (in terms of adding phrases together) and subtractive (in terms of each new phrase addition being fewer notes than its predecessor): it plays a scale starting on A and going up to G, then another phrase going up to F, then up to E, D, C, and B. After reaching maximum length it then begins subtracting the opening phrases one at a time, and I tried to spell out in the diagram the strict process of addition and subtraction. Later, starting in phrase 37, Glass reinserts this scalar process into larger phrases on the chords of G, C, and A, resulting in repetitive sections far longer than anything in Dance 1. At last the chorus and organ suddenly drop out, and the violin is left to play a solo transition to Knee Play 4.
One of the realizations one draws from such a reading of Einstein, I think, is that it does not much matter whether the progression of patterns follows a strict algorithm, or whether the music moves from pattern to pattern more arbitrarily. An algorithm that is sufficiently complex will prevent the listener from gaining a firm sense of what the patterning is; and, conversely, given a severe restriction of material, a series of similar but nonlinear patterns can be interpreted as probably following some pattern too complex to tease out by ear. The effect can be much the same. The two Dance movements are rather different in this respect without the effect being noticeably formally contrasting. However, I think there is one exception to this in Einstein, and it is the Bed scene with which we started, the opera’s penultimate major scene. Here the rhythmic progression is not only linear and predictable, but proceeds so slowly that it is quite easy to count, and the listener will be tempted to do so. In this case I find something nostalgic about the linearity, as it is the one part of the opera that exposes the underlying process in an audibly discernible way, inviting the listener to step behind the curtain, as it were, and find out how the music works. Given Einstein’s historical appearance just after the repertoire of strict-process minimalism, this mood may even be felt as a nostalgia for the musical period that had just passed. As a kind of retrospective adagio, the Bed scene virtually invites us to remember the minimalist works of the late 1960s and hear how far the rest of the opera has moved away from them.
Aside from the special case of sonata form, in which a composer rewrites the exposition as a recapitulation in order to transpose all themes into the tonic, this idea of using several recomposings of the same passage within one piece does not come up often in the history of music. The composer who wants to get from point A to point B typically figures out the best way to do so, and then proceeds to point C. To find three different and functionally interchangeable ways to get from point A to point B and then use all of them in the same piece, as Glass does here in the Train scene, is rather rare, I think. (After all, had Glass merely repeated three identical A sections, how many listeners would have noticed, how many analysts would have found that anomalous in this context?) And one would have to go to the music of Erik Satie, such as the Gymnopedies or the Pieces Froids, to find a composer writing two large movements of a piece with such similar content as Glass does between these two dance movements. And yet the Dance 2 is quite different in feeling than Dance 1, with its focus on the violin soloist, its incessant running up and down the scale, its greater reliance on additive and subtractive process, and its lack or the comforting F major into which Dance 1 tended to resolve. All the usual jokes about minimalism aside, I find it remarkable that Glass could generate 45 minutes of his opera with so little material, shaping each internal stanza so intuitively, and differentiating the two dances into such different purposes and moods. It is a real piece of compositional virtuosity, and not at all the kind of predetermined logic that we tend to associate with early minimalism.
As my predecessor at The Village Voice Tom Johnson wrote in 1981 about Glass’s Music with Changing Parts, “Yet as I listened once again to those additions and subtractions I realized that they are actually rather whimsical. Composers like Frederic Rzewski, Louis Andriessen, and William Hellermann have written such sequences with much greater rigor. By comparison, Glass is not a reductionist at all but a romantic.”  Romantic is not quite the word I would have used—I would be loathe to think that the mere absence of a generating algorithm suffices as evidence of passion or individualism. (In fact, that Tom would use the word on such minor grounds in 1981 is indicative of the atmosphere I began this essay by describing.) But I do think that there is a kind of inherent mystery in Glass’s circuitous, unpredictable paths through extremely circumscribed material, and that Einstein on the Beach would be a less compelling work than it is had he been more content with mere concept and less generous with his subtly-shaping artistry.