In most of Cage’s Number Pieces, a series of works that occupied the composer’s attention between 1987 and his death in 1992, pitch—even harmony—takes pride of place. This fact poses a paradox, even a contradiction, because harmony as such had hardly ever been central to his concerns. In fact, while Cage’s early studies of harmony and counterpoint with Adolph Weiss and Arnold Schoenberg between 1933 and 1936 proceeded satisfactorily enough, there came a time when his development as a composer reached a point of crisis. One of his anecdotes about his work with Schoenberg stands, probably, as his most famous expression of that crisis:
After I had been studying with him for two years, Schoenberg said, “In order to write music, you must have a feeling for harmony.” I explained to him that I had no feeling for harmony. He then said that I would always encounter an obstacle, that it would be as though I came to a wall through which I could not pass. I said, “In that case I shall devote my life to beating my head against that wall.
To understand what harmony meant to Cage in his final years, one needs to understand what it meant to him earlier. I have spent some time locating all of the remarks Cage made about harmony in his published writings and some of his interviews. Taken in toto, these remarks show a richer—and what at first seems contradictory—viewpoint than has been generally assumed. First and foremost, Cage saw himself in opposition to tonal harmony, with its predictable (and arbitrary) rules of voice-leading and harmonic progressions. By learning these rules, listeners condition themselves to expect the harmonic progressions that establish tonality; any deviations from those progressions—for instance, an unexpected modulation or a deceptive cadence—would be understood in terms of the “expected” norms. To Cage, listeners so conditioned would never hear sounds as sufficient in themselves—would not, in other words, be able to hear with the kind of open mind that he felt was essential. What’s more, as Cage wrote, the rules of tonal harmony foster “old-style” European thinking which stresses understanding all single events as the causes or consequences of other single events: Cage called the succession of musical events in a composition a “continuity,” and this old-fashioned way of thinking simply “making that particular continuity that excludes all others.”
In the 1940s, in particular, Cage began to criticize harmony on other counts too, especially for its capacity to move music away from the ideal of tranquility to which he thought music should aspire: “I now saw harmony, for which I had never had any natural feeling, as a device to make music impressive, loud and big, in order to enlarge audiences and increase box-office returns.” For Cage, harmony was a device that composers used to manipulate the audience, to make them feel certain emotions instead of others; for that reason it served the personality of the composer more than it did the sounds themselves.
Cage proposed replacing this type of thinking with the Oriental concept of interpenetration. He explained the idea in his lecture “Composition as Process” by recalling one of Daisetz Suzuki’s references to it in his lectures at Columbia, which he attended:
Interpenetration means that each [idea] is moving out in all directions penetrating and being penetrated by every other one no matter what the time or what the space. So that when one says that there is no cause and effect, what is meant is that there are an incalculable infinity of causes and effects, that in fact each and every thing in all of time and space is related to each and every other thing in all of time and space.
Each sound, pitched or otherwise, carries with it the same amount of significance as all other sounds. When we hear them, we are overwhelmed by the infinite complexity of their interrelatedness; our awareness might focus on a fractional part of this complexity but never apprehend it entirely. In another hearing, we might choose to focus our attention on something else yet again. Cage pursued these ideas with far-reaching implications in the late 1950s and ’60s—for example, by separating instruments spatially so as to help ensure their sounds would be perceived as “sounds in themselves” and, in several works from the Variations series, by leaving undefined the types of sound chosen for a given performance.
Too often, I feel, we have been admonished that paying attention is beyond the point in Cage’s music—that because all events cause all other events, it’s unnecessary to focus our attention too specifically. But if we don’t pay attention, surely we run the risk of coming to the superficial conclusion that all of Cage’s chance music sounds alike. Furthermore, it’s worth recalling that Cage occasionally admitted the possibility of finding relationships or continuity in his music, an activity that can only take place when we pay attention:
I would assume that relations would exist between sounds as they would exist between people and these relationships are more complex than any I would be able to prescribe. So by simply dropping that responsibility of making relationships I don’t lose the relationship. I keep the situation in what you might call a natural complexity that can be observed in one way or another.
The awesome complexity of interpenetrating sounds set in motion can never be comprehended in its entirety. But by observing that complexity “in one way or another,” we experience what we can; we might even share that experience with others, without suggesting that our experience is superior to anyone else’s. What pleased Cage was the way in which his chance operations prevented him from imposing a single continuity of his own design in the music, and thus opened up the possibility for any number of equally valid continuities. A given performance would reveal different patterns to different listeners; indeed, the same listener might make different connections in a subsequent performance—further approaching, but never totally comprehending, “the incalculable infinity of causes and effects.”
In later years, Cage extended his interest in all the sounds around him to tonal harmony itself—for instance, he tried to liberate such harmony from its rigid syntax in works like Hymns and Variations (1979). In this and similar works, Cage took traditional pieces by William Billings and other eighteenth-century American composers and subjected them to chance operations so that certain tones were prolonged and others removed altogether, replaced with silence. In this way Cage retained the “flavor” of the original music but made it possible for listeners to hear it in a new way—to hear the individual sounds and silences as equally important, relating to each other in countless ways that could not be explained or predicted by rules and habit.
Finally, in the Number Pieces—particularly those involving the piano—Cage returned to harmonies entirely produced by chance operations and unrelated to past music. More important, Cage discovered a way to use harmonies in such a way that they could not suggest any single pattern of coherence. In most of the Number Pieces, all the performers have some freedom through Cage’s use of time brackets, flexible measures that show a range of possible starting and ending times. The time bracket system of notation used in these works allows a certain amount of flexibility in the performance: individual notes or chords may always occur in the same general time frame, but their specific order and duration varies slightly and unpredictably from performance to performance. In this way Cage could create a new kind of harmony, a harmony that, as he described in a 1991 interview, simply “means that there are several sounds . . . being noticed at the same time.”
Though Cage had pursued the idea of multiple unusual and flexible connections between elements in many ways during his career, his literary works of the 1980s supply some elegant examples of the principle—examples that I believe are particularly relevant to the Number Pieces. One form that he referred to over and over again was renga, a Japanese poetic design of five, seven, five, seven, and seven syllables expressed at least thirty-six times. In the introduction to Themes and Variations (1978), Cage’s description of renga shows the powerful affinity that it shares with his own work:
Traditionally renga is written by a group of poets finding themselves of an evening together and having nothing better to do. Successive lines are written by different poets. Each poet tries to make his line as distant in possible meanings from the preceding line as he can take it. This is no doubt an attempt to open the minds of the poets and listeners or readers to other relationships than those ordinarily perceived.
While Cage did not adhere to the syllabic structure of renga, he clearly embraced the idea that sequences of words which did not make conventional sense could, nevertheless, be meaningful. To help create the illusion of the many poets characteristic to renga, he began his work by writing a “library” of source mesostic poems. (Cage’s mesostics have a central vertical string of letters, all capitalized, which spell out a name or idea that may or may not be connected with the horizontal lines of poetry.) Once he had written a large number of poems, Cage selected individual lines through chance operations to form new mesostics that make a fuzzy kind of sense. His method also allowed certain lines of poetry to reappear; recurring lines from the first and second sections of Themes and Variations are underlined in the excerpt below:
whAt did you say
clearing the Mind of music
if i gave up my Sense of accomplishment
of anYthing else
we are in yuCatan
out thE window
if i gave up my Sense of accomplishment
what’d you Just
anything in frOnt
without anY waiting
we are in yuCatan
and Every unpredicted thing
Cage’s late poetry exploits an ineffable and arresting alchemy of the noble and the common. Particularly fascinating in this passage is the juxtaposition of such phrases as “we are in yuCatan / and Every unpredicted thing.”
In Two2 (1989), written for the pianists Edmund Niemann and Nurit Tilles performing together as the piano duo Double Edge, Cage unites these two themes—a new kind of harmony and renga—in a marvelous way. One finds all manner of sonorities in this work: various triads and seventh chords, triads with other added notes, dissonances with two, three, or more notes, even single tones. More interesting, however, is the fact that so many chords—even identical successions of chords—recur in the piece. The manner of their overall succession, of course, is determined by chance procedures and performance, which guarantee a nebulous ordering that offers a modicum of coherence without any predictability whatsoever. And so the logic of recurrence in Two2 resembles the way that our different friends and acquaintances pass in and out of our lives—a continuity that doesn’t make conventional sense but is meaningful nevertheless.
Cage expresses the renga idea in Two2 in several ways: first, each line of music is divided into five measures, just like the five lines of the poetry. The first measure contains five separate musical events—chords or single tones, usually shared between the two pianists—which correspond to the five syllables of the first line; the second measure has seven events, and so on. And there are a total of thirty-six such five-measure sections in the piece. The pianists proceed one measure at a time. They must read the music in their respective measures from left to right; and while they can take any amount of time to perform each measure, each pianist must wait until both have finished the same measure before proceeding to the next. In this way, the order of the “syllables” composing one “line” almost always occurs in an unpredictable order. Cage also respects the analogy with renga by keeping one line of poetry—expressed as one measure—separate from another.
Of course, one can also hear the music itself as renga: dissonant chords cohabitate amicably with simple triads and ambiguous, neutral single tones: some sonorities are sublime, while others seem the product of harsh, even unformed technique. Every time we think we settle into the mode of Cage’s discourse, some unexpected sound—a simple seventh chord or even a seemingly banal augmented triad—suddenly intrudes, throwing us off balance and disorienting our reactions. And yet that mingling of “incompatible” elements is fundamental to Cage’s aesthetic and the very source of its magic. This mixture of the generic and the exquisite evokes what Cage admired in Duchamp’s work, its ability to resist becoming a mere “art object,” though Cage was rarely able to achieve a Duchampian ideal.
Two2 stands apart in the sequence of Number Pieces because it has no time brackets; the two pianists are directed simply to play the music in their respective measures at their own speed, taking care (as noted above) not to continue to the next measure until both have finished playing the music for that particular measure. As Cage explained in his performance notes for the work, his decision not to incorporate time brackets owed itself to a remark made by the soviet composer Sofia Gubaidulina, whom he had met in 1988 at the Third International Festival of Contemporary Music in Leningrad: “There is an inner clock.” This gives the pianists the luxury of playing the piece at the speed that suits them best; in the finest performances, the freedom also allows them to discover surprising new relationships between the sounds—relationships unexpected even to them, no matter how much rehearsal time they spend preparing.
There is one final topic to discuss in Two2, and that is the relationship of the two performers. I have always sensed something intimate—even erotic—in all of Cage’s Number Pieces for two musicians. More than other works in the series, I think, they act as a metaphor for a healthy, loving relationship between two people. In connection with this theme, we might turn to Cage’s own (few) words on the subject to deduce the nature of that relationship. As early as 1966, Cage wrote, “The most, the best, we can do, we believe (wanting to give evidence of love), is to get out of the way, leave space around whomever or whatever it is.” He later simplified this formulation to “love = space around loved one.” The composer discussed intimate human relationships rarely, but when he did, it was clear that he had considered such relationships deeply all of his life. In a conversation with Joan Retallack from October 1991, he described friendship using similar thoughts: “It isn’t a fixed thing that you come to and keep. It’s something which is not dependable. Even if you think it is, it isn’t, hmm? And it gets richer as it encounters obstacles and surmounts them.”
To leave space around loved ones, we must learn to let people live their own lives wherever that life may take them—that having or possessing the person is beside the point. Similarly in the Number Pieces for two individuals, and perhaps Two2 above all others, our performance is a time in which we agree to come together and be in each other’s company—a time which we expect to be happy but which holds no guarantees.
 John Cage, “Indeterminacy,” in Silence: Lectures and Writings (Middletown: Wesleyan University Press, 1961), 261. For more on Cage’s early studies, see Robert Stevenson, “John Cage on his 70th Birthday: West Coast Background,” Inter–American Music Review 5, no. 1 (Fall 1982): 3–17; Michael Hicks, “Cage’s Studies with Schoenberg,” American Music 8 (1990): 125–40; and David W. Bernstein, “John Cage, Arnold Schoenberg, and the Musical Idea,” in John Cage; Music, Philosophy, and Intention, 1933–1950, ed. David W. Patterson (New York: Routledge, 2002), 15–45.
 See, for instance, Cage, “Lecture on Nothing,” in Silence, 116.
 Cage, “Lecture on Something,” in Silence, 132.
 Cage, “A Composer’s Confessions,” in John Cage: Writer, ed. Richard Kostelanetz (New York: Limelight Editions, 1993), 29.
 Cage, “Composition as Process,” in Silence, 46–47. David Patterson discusses historical problems in Cage’s recollections of Suzuki and his influence in “Cage and Asia: History and Sources,” in The Cambridge Companion to John Cage, ed. David Nicholls (Cambridge: Cambridge University Press, 2002), 53–55.
 Cage quoted in Michael Nyman, Experimental Music: Cage and Beyond (New York: Schirmer Books, 1974), 25. Nyman can no longer remember the source for this quotation, but believed he cited a printed source (e-mail to the author, 21 November 2002).
 See, for example, “Where Are We Going? And What Are We Doing,” in Silence, 250–52.
 John Cage and Joan Retallack, Musicage: Cage Muses on Words, Art, Music. John Cage in Conversation with Joan Retallack, ed. Joan Retallack (Middletown: Wesleyan University Press, 1996), 108.
 John Cage, “Introduction to Themes and Variations,” in Composition in Retrospect (Cambridge: Exact Change, 1993), 64. My emphasis.
 Cage, Themes and Variations, in Composition in Retrospect, 74 and 100.Composition in Retrospect, 133. The mesostic string spells the name of James Joyce.
 For a description of the compositional process of Two2, see Rob Haskins, “‘An Anarchic Society of Sounds’: The Number Pieces of John Cage” (Ph. D. dissertation, University of Rochester, Eastman School of Music, 2004), 119–21 and 214–19.
 There are a few occasions when, probably as the result of the chance operations Cage used, one pianist has a silent measure and all the “syllables” for that “line” are performed by a single pianist.
 For more on Cage’s thoughts on this aspect of Duchamp’s work, see Cage and Retallack, Musicage, 103–5.
 John Cage, performance notes for Two2 (New York: Henmar Press [C. F. Peters], 1989), n.p.
 In the “Diary: How To Improve the World (You Will Only Make Matters Worse) Continued 1966” in Cage, A Year from Monday: New Lectures and Writings (Middletown: Wesleyan University Press, 1967), 52.
 As part of the introduction to Themes and Variations; see Composition in Retrospect, 61.
 Cage and Retallack, Musicage, 151.