Nancarrow's Canons: Projections of Temporal and Formal Structures.
A number of twentieth-century composers incorporated canonic procedures into their music, of course. At one level, canon continued its historical role as a pedagogical tool elemental to the study of both counterpoint and performance. Those contemporary composers who have written a substantial number of canonic works often have done so in the context of collections having a pedagogical intent, carrying forward the tradition of such composers as Antonio Caldara and J. S. Bach. (4) Bela Bartok's Mikrokosmos (1926-39), particularly vol. 1, (5) and Forty-four Duos for two violins (1931), and Paul Hindemith's Sing- und Spielmusiken fiir Liebhaber und Musikfreunde (1928) and Schutwerk fur Instrumental-Zusammenspiel (1927) (6) are important examples. In these types of works the canons are usually quite simple and straightforward: melodic contour and rhythm are imitated strictly, with minor adjustments made for synchronized endings. But there are other contemporary composers who, like Nancarrow, have utilized canon in more "serious" works and have expanded its compositional possibilities, sometimes to make an overt historical connection to the centuries-old technique, and sometimes to infuse canon with a more modern spirit. Canons appear in works by such composers as Elliott Carter, Luigi Dallapiccola, Peter Maxwell Davies, Gyorgy Ligeti, Steve Reich, Igor Stravinsky, and Anton Webern. (7)
While Nancarrow was not alone in writing canons he seems to have done so out of his own impetus, not in response to canons he heard elsewhere. Nancarrow, though American-born, composed in near-isolation in Mexico City for most of his career. His formal compositional training was minimal. He did study counterpoint briefly with Roger Sessions (1934-35), and composition with Walter Piston and Nicolas Slonimsky (during the same time period). Nancarrow claimed that the more significant influences on his compositional development, however, were Henry Cowell's treatise New Musical Resources, (8) which he read in 1939-1940, and jazz, which he grew up playing and listening to. (9) The connections to Nancarrow's music are quite apparent. Cowell advocated the use of complex polyrhythms, polymeters, and polytempos, all of which effects are common in Nancarrow's studies. Cowell even suggested the player piano as a useful instrument on which to produce complex rhythmic structures. The polyrhythms common in jazz certainly f ind direct parallels in Nancarrow's music, and many of his works clearly evoke jazz and popular styles. Neither Cowell nor jazz seems to have provoked Nancarrow's interest in canons, however; if anything, that interest might be linked to his studies with Sessions. Three factors in particular characterize Nancarrow's unique use of canons: (1) the prominence of canon in his output, appearing as it does in the majority of his works; (2) his primary emphasis on tempo-proportion canons, utilizing proportions that range from 2:3:4 to 60:61 to e:[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII] and (3) the performance medium of the player piano, which enables the accurate realization of such tempo proportions, coupled with tremendous rhythmic complexities.
Nancarrow's heavy usage of such a strict compositional procedure as canon might make these works seem somehow formulaic and self-evident: to say, for example, that the two or three or more voices of a work are in a tempo-proportion canon at 3:4 or 3:4:5 or the like is to explain the work fully. But the technique of canon is much richer for Nancarrow than that, yielding a wide variety of textures. As discussed by Kyle Gann, Nancarrow's canonic technique follows an approximate progression from the early, fairly straightforward tempo-proportion canons to the later canons he calls the "sound-mass" canons, in which the canon itself is not the focus of the work, but, rather, the means to achieve a highly complex, multidimensional texture. (10) The paradoxical fertility and attraction of the canonic technique for Nancarrow is precisely this ability to project not only perceptually clear tempo and textural structures, but also obscured and otherwise rich textures, which accounts for his longstanding use of the techni que over the evolution of his style. This paper focuses on canons in three studies, Numbers 14, 24, and 20.11 These works exhibit in miniature the wide stylistic, formal, and textural variety found in Nancarrow's canons, and serve as a good introduction to his music.
Before turning to the studies, an overview of Nancarrow's canonic techniques will help clarify the perception issues fundamental to the canons. In keeping with his stated compositional emphasis on rhythm and tempo over pitch, Nancarrow's canonic pitch imitation is always direct, never by inversion or retrograde. Given this constancy of the melodic dimension, Nancarrow addresses temporal issues by using four primary canonic procedures that accommodate layers with different speeds; I term the four canon types converging, diverging, converging-diverging, and diverging-converging. These are illustrated in hypothetical two-voice settings in Example 1, along with a conventional canon. The horizontal lines in the example represent the individual voices, while the vertical and angled lines that connect them show the placement of equivalent points in their melodic material. In a converging canon (Example 1b) the entrances are staggered--the slowest voice enters first, followed by ever faster voices--and the voices eve ntually meet up at a simultaneous final articulation. In a diverging canon (Example 1c) the voices enter simultaneously and gradually move apart, the fastest voice completing the material first and dropping out of the texture, followed by the next fastest, and so on. In the converging-diverging canon (Example 1d) the voices enter as if in a simple converging canon. They do not stop when they reach their point of synchrony, however, but continue beyond it, taking that point as the beginning of a diverging canon. As a result, both the entrances and endings of the voices are staggered. In the diverging-converging canon (Example 1e) the entrances and endings are simultaneous. The voices diverge at the start and then exchange speeds at some point so that they come back together, arriving eventually at a simultaneous final articulation. In the tempo-proportion canons Nancarrow often states the ratios of the tempos as a subtitle to the work; if not, metronome markings are usually provided so that the ratios can be c alculated. The ratios range from simple rational ratios (3:4 in Number 15) to irrational ([square root of (2 2:2)] in Number 33). There are also some canons that do not follow one of these four plans, but rather incorporate a technique of gradually changing speeds, so that the relationship between the voices constantly changes. (12)
Several perception issues arise with regard to the canons: Can the individual voices be followed? Can the tempo proportions be heard? Can a particular canon actually be heard as a canon? The answer, of course, depends on both the listener and the piece. As stated above, Nancarrow's canonic studies constitute a widely diverse group of pieces. In some works canon is deployed straightforwardly and the canonic process is quite perceptible. But in other works the canonic process may be obscured by the complexity of the canonic line, the number of voices, the fast speed at which it proceeds, or the specific canon type used.
There are two special properties of the tempo-proportion canon that seem to have drawn Nancarrow to the technique. First, Nancarrow's primary aesthetic goal is to project different tempos, and he cites canon as an aid in this: "One reason [to use canons] was my interest in temporally dissonant relationships. ... When you use canon, you are repeating the same thing melodically, so you don't have to think about it, and you can concern yourself more with temporal aspects." (13) In some of his studies canon does seem to help articulate the tempos. And even if the canonic voices or process are obscured, some of the component gestures of the voices are still distinguishable, and their respective speeds thereby evident. Canonic organization ensures that the voices share melodic gestures. The second special property of the tempo-proportion canon is its impact on a larger scale, whereby distinctive formal processes are produced, whether or not the canon itself is perceptible: each of Nancarrow's four basic canon types (converging, diverging, converging-diverging, and diverging-converging) creates a unique formal shape. The points of synchrony--whether at the beginning, ending, or middle of a piece or passage--operate as significant structural moments and thus help to articulate the design of the studies. As a result, the works achieve a directed flow toward and from these focal points. Not only can the technique of canon therefore help to clarify tempo relationships, temporal dissonance, and (sometimes) texture, but it is also responsible in large part for the well-formed, process-driven structure of many of the studies.
Study Number 14 is one of several early studies, Numbers 13-19, that were originally grouped together as a set entitled Seven Canonic Studies, and whose rhythms are all derived from cyclic presentations of the following duration series: <n, n+1, n+2, n+1>, where n is a duration of three, four, five, or six eighth notes. Nancarrow later gave the studies individual numbers, and, indeed, there is enough variety in them, in their canonic techniques, to warrant individual status. Study Number 14 is a fairly simple converging-diverging canon in which the technique of canon, and the stratified texture created by its two compound voices, is quite clear. The tempos of the voices, at [MUSICAL NOTES NOT REPRODUCIBLE IN ASCII] = 88 and 110, are related at 4:5, and their entrances are staggered, with the faster voice entering later than the slower voice. They reach a point of synchrony with a simultaneous downbeat at page 3.2, (14) the midpoint of the piece, when the faster voice catches up to the position of the slower v oice in the canonic material. After this they diverge, so that their endings are also staggered, with the faster voice ending first.
The faster voice, as is characteristic of Nancarrow's canons, is higher in register, imitating the first voice two octaves plus a fifth above. Their registral and temporal identities ensure that the voices are distinct, as does their canonic relationship. The voices themselves are polyphonic (see the opening page of the study in Example 2, where voice 1 is notated on two staves, both in bass clef, and voice 2, entering on the second system, is notated in two staves, both in treble clef). The polyphony within both voices is highly coordinated, however, both rhythmically and harmonically. Thus the full texture remains quite clearly that of only two compound voices. This textural clarity helps to keep the canon and the processes of imitative convergence and divergence audible. There is a palpable sense of the voices being in different places at the same time, of gradually moving closer together, of a brief moment of coordination, and then a departure. In essence, the canon creates an overriding sense of asynchro ny between the two voices, superimposed on a formal structure that is mirror symmetrical about the central point of synchrony.
But Study Number 14 sounds much less simple than this description implies. While the differentiation of the voices is always made clear by their registral separation, canonic relationship, and perceptibly distinct tempos, the specific ratio of those tempos is never quite clear. Nancarrow metrically and rhythmically undermines any potential predictability a 4:5 ratio might have. If we now return to the source duration series of the study, <n, n+l, n+2, n+l>, we can better understand its rhythmic construction. (15) The canonic line is a resultant of four ongoing separate lines whose attacks are determined by the duration series, with n = 3, 4, 5, and 6 eighth notes. The generation of the resultant rhythm of page 1, voice 1 of the score is illustrated in Example 3 (compare Example 2). Following their simultaneous beginning at the initial downbeat, each series undergoes relentless cyclic presentations for the duration of the study. We do not hear the four components individually at all; Nancarrow deploys them mel odically and gesturally so as to create the effect of a compound canonic voice consisting of a melody and bass line. But the significance of the four duration series is more than trivial. The lengths of the four series, 16, 20, 24, and 28 eighth notes, have as their least common multiple 1680 eighth notes; in other words, it would take that long for the four series to be realigned as they are at the beginning. The length of Study Number 14, however, is only 336 eighth notes. As a result, there is no exact rhythmic repetition during the piece but, rather, a continually changing rhythmic surface. This is important to the complexity of the canon. While regular, repetitive rhythms and an unchanging meter could simplify the individual canonic voices and facilitate our discriminating their tempo relationship, the study's irregular rhythms and near-continual meter changes (among 2/8, 3/8, 4/8, 5/8, and 6/8) do the opposite. The canonic line does not express a regular (metrical) pulse larger than the eighth-note micr opulse, and this keeps the 4:5 tempo relationship of the canonic voices from being obvious.
Before discussing issues of meter and tempo in the study more fully it will be useful to consider three ways in which a 4:5 tempo proportion could be deployed, as illustrated in Example 4, where the eighth-note speeds are given as 88 and 110 so as to resemble the speeds in Study Number 14. In each of the three scenarios a regular eighth-note rhythm is assumed, and accent marks in the examples represent the accentual weight given downbeats in a metrical setting. Vertical dotted lines indicate simultaneous articulations, which occur every four/five eighth notes in each of the examples. What differs among the three scenarios is their number of metrically significant simultaneous articulations, that is, simultaneities that coincide with mutual downbeats. This has an impact on the level of coordination in a passage, and the perceptibility of its tempo ratio. In 4a the complementary meters of 5/8 and 4/8 create identical measure lengths for the two voices. As a result, each simultaneous articulation coincides with a downbeat in both voices. The effect is actually polymetric rather than polytemporal in the metrical coordination of the voices. The sense of coordination is reduced somewhat in Example 4b. The only difference in 4b relative to 4a is that the voices now share the same meter, 4/8. Although the frequency of simultaneous attacks is therefore the same, the frequency of metrically significant simultaneities is reduced to every four (in the slower layer) or five (in the faster layer) downbeats rather than eighth notes. Example 4c introduces the possibility that the two voices do not begin with coordinated downbeats: the first downbeat of the faster voice coincides with beat two of the slower voice. The simultaneous articulations in this scenario are thus never metrically significant, and, as a result, the voices sound quite uncoordinated.
These three examples of a 4:5 proportion are simple, to be sure, and do not emulate a real musical context, but they will help to clarify the temporal relationship of the voices in Study Number 14. There are additional issues, moreover, important to an understanding of the rhythmic differentiation of voices in a tempo-proportion work. Notice that in each model of Example 4 the eighth-note pulsations of the two layers are coordinated on some basic level, so that there are relatively frequent simultaneous articulations, whether or not they are metrically significant. This certainly does not have to be the case--one can envision a deployment of the tempos in which one of the layers is slightly shifted, by a sixteenth note, for instance, so as to eliminate the possibility of eighth-note simultaneities--yet this fundamental, low-level coordination is almost always found in Nancarrow's studies that use ratios. Rather than shifting the alignment in this way, Nancarrow commonly uses two other techniques, both found i n Study Number 14, for reducing the simultaneities found in the straightforward rhythmic schemes of Example 4: rhythmic variety and meter changes.
We have already seen how Nancarrow achieves irregular rhythms with the four generative duration series. Since not every eighth note is articulated, there are innumerable points where a possible simultaneity is avoided because one or both voices is either sustaining a note or resting. Notice, in Example 2, that there are only three simultaneities between the two canonic voices on the first page of the score, which I have indicated with arrows below the systems. (16) The issue of meter, and thus the metrical significance of simultaneities, is an interesting one in Nancarrow's works. As noted above, Study Number 14 features meters that change nearly every measure; because downbeats shift continuously the possibility of significant simultaneities is further reduced. But what does meter really mean in a player-piano work?
Meter is often thought of as an aid to performance: it guides a performer in placing accentual weight and creating a large-scale pulse organization. The studies do not involve a performer, however; the score is merely a convenience for those examining the works. Once a study was written, Nancarrow punched its player-piano roll and it was a finished product. Nancarrow could have used dynamic or even durational accents to express downbeats, but he did not. Meter would therefore seem to be completely irrelevant to his music, and, indeed, some of his works are notated without meters. But Study Number 14 is set with meters, as are most of Nancarrow's works. The study's changing meters reflect the perceptible suggestion that certain articulations function as downbeats. These downbeats can be viewed as arising from some combination of four principles (see Examples 2 and 3): (1) the second note of semitone melodic motion, either ascending or descending, is usually placed on a downbeat; (2) measures are often preceded by an eighth-note anacrusis; (3) the coincidence of two or more rhythmic strands on an eighth note is marked by a chordal articulation with as many pitches as there are coinciding strands (see boxes in Example 3), and these chords are usually placed on downbeats since they sound accented relative to single-note articulations; and (4) the harmonies across barlines often present dominant-tonic motion, with the tonic sonority falling on a downbeat (see horizontal bracket overlays on Example 2). These four principles relate to traditional tonal and rhythmic practices, upon which Nancarrow freely draws for many of his studies. The familiar gestures help to project the metrical structure of Study Number 14, whose frequent changes of meter shift the downbeats continuously. The possibility of metrically significant simultaneities is reduced as a result. Of the simultaneous articulations in Example 2, for example, only one occurs on a shared downbeat, that near the end of the second system (at the large arrow). This scarcity of metrically significant simultaneous attacks is typical of the entire study. The rhythmic and metric irregularity is built into the canonic material, and its effect is to supplement the asynchrony of the 4:5 tempo proportion by decreasing any sense of coordination between the two canonic lines. The process of canon, however, remains clear throughout the study.
The perceptible prominence of the canonic process is one of the areas in which Nancarrow's studies vary most dramatically. The variance can result from many factors, including the canonic type itself, the canon material, the number of voices, and the tempo proportions. Some of the canons in the next study under consideration here, for example, the highly sectionalized Study Number 24, are not audible. Each of the study's twelve sections is comprised of a three-voice diverging-converging canon (see Example le), featuring the tempo proportion 14:15:16. (17) The voices are synchronized at the beginning and ending of each section, although that synchronization is sometimes masked by a rest occupying one or both points of synchrony, and the fastest and slowest voices exchange speeds midway while the middle voice retains the same speed. For the most part the sections are relatively short, featuring an alternation of dynamic level and (usually) articulation type, as summarized in Example 5. (18)
Because the ratios of the tempos at 14:15:16 prevent the speeds being dramatically different, the voices in the shortest sections of Study Number 24 (4, 6, 8, 9, and 12) do not tend to diverge very far before their tempo exchange and subsequent convergence. The imitative gap is thus never very big; consequently, the process of canon is not perceptually prominent. Section 4, pages 9.2-10.1, for example, is a case in point, lasting just 7.5 seconds (see score in Example 6). The fastest and slowest voices at their most divergent (page 9.3) are separated in their placement in the material by approximately just one-half second. We hear not so much an imitative process as three voices presenting the same material at approximately the same time, moving out of and back into phase with one another, an effect not unlike Reich's phase pieces. Section 4 contains a series of roughly coordinated gestures, separated by brief rests, that can be described as octave tremolos followed by semitonal trills. The synchronization at both ends of the divergence-convergence (shown by arrows) is compromised, thereby masking the processes of divergence and convergence: the point of synchrony at the beginning, and thus the true start of the divergence, is placed at the downbeat of the final measure of the preceding section, and the point of synchrony at the end occurs at the downbeat of the next section. Thus the voices are slightly staggered at the beginning and ending of the section proper, as defined by the distinctive material of the section, and, of course, throughout the main body of it. The section is so short, moreover, and the activity within it so fast, that the canonic process is further blurred. The extreme slightness of the misalignment of the three voices, along with the lack of clear synchronous reference points, detracts from our perception of a canon. Another factor is the interval of pitch imitation, which is at the double octave in this section: the voices have registral but not pitch-class integrity, and so are difficult to track as contrapuntal entities.
Notice in Example 5 that the intervals of imitation in most of the sections create triadic vertical sonorities, while a few employ octave duplication (sections 6, part of 10, and 12, in addition to 4), and one, a stacked-fourths sonority (8). These intervals seem to be related to the canonic process. Octave duplication occurs in short sections like section 4, where the canonic divergence is minimal, and the effect is more one of heterophony than canon. The exception to this is the middle portion of the lengthy section 10, which is discussed below. In the sections with triadic pitch imitation the processes of divergence and convergence are enhanced by the stable tonal connotations of the triadic sonorities: at the boundaries of such sections, near the points of synchrony, the voices form triads vertically, but as they diverge the notes of the triadic imitation no longer align, and the ensuing verticalities are generally dissonant and nontonal. As the voices subsequently converge the imitative gap eventually be comes small enough that they once again form triadic verticalities in support of their temporal realignment. This effect is illustrated in Example 7, which contains the first and last systems of section 1 of the study; triads whose notes actually sound as verticalities, regardless of simultaneous articulation, are encircled. Notice that the initial starting notes, [D.sub.1]-F[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII]-[A.sub.5], have little large-scale impact (a fact that is true for all the sections of the study). The section does not express D as a tonality, nor does the canonic line itself suggest a "tonic," which could create a polytonal texture. In a typical fashion, Nancarrow incorporates tonal elements nonfunctionally into a pitch language that is freely nontonal yet not deliberately dissonant. The one significant reassertion of D in the study occurs at the very end, where, in octaves, the voices state A-D as the last two notes, rounding off the work.
The length of a section controls how far the voices can diverge canonically. This becomes an important feature in the longest section, section 10 (pages 13.1-23.3), which accounts for nearly half the length of the entire study. Because of its length the voices have the time to diverge more dramatically than in the shorter sections. The changing relationship of the voices, that is, the process of canon, thus becomes perceptually more prominent. Throughout Study Number 24 adjacent sections present canonic material that contrasts in dynamics and characteristic gestures (see Example 5), but within a section the material tends to be consistent in its activity rate and articulation type. Section 10 is the exception to this pattern of internal consistency, largely due to its length and increased divergence. The section contains changes in dynamics and material type to include isolated staccato chords, motion by eighth note, dotted eighth note, or sixteenth note, and lengthy rests. The placement of different gestures generally reflects the changing imitative gap of the voices. The beginning of the section, page 13.1-13.2, is given in Example 8. Articulations are separated by rests, and are placed so as to minimize overlapping motion among the voices; due to the canonic divergence this requires in general progressively longer rests between articulations. By page 15 a new, repeated eighth-note figure is introduced (Example 9). It is preceded by almost five measures of rest, so that its first statement (top voice, page 15.1) is unaccompanied. In other words, the rest is just long enough that the slower voices complete the preceding material before the new figure begins. The length of the figure, moreover, is such that the middle voice begins it immediately after the top voice completes it. The figure seemingly passes from the top to middle voice (page 15.1), a motion supported by contour and register, and then on to the lowest voice (page 15.2).
This interconnectedness of material and canonic process peaks at page 18.1-18.2, the midpoint of the section, where all the voices proceed, for the only time in the entire study, at the same speed ([MUSICAL NOTES NOT REPRODUCIBLE IN ASCII] = 240, the speed of the middle voice), with the same rhythms, and with the same octave tremolo motion for eight measures. This excerpt is provided in Example 10, in which the eight unison measures are bracketed. These coordinated measures have a unique and profound effect: just where we least expect it, at the moment of greatest divergence among the voices (at the middle of the longest diverging-converging canon in the work), there is a section, not just a moment, of synchronized motion, after which the fastest and slowest voices exchange speeds and the convergence begins. This focal point is possible because the canonic material contains the presentation of the same eight-measure rhythmic pattern with octave tremolo motion three times consecutively, and because the diverge nce of the section is plotted so that the fastest voice begins its third presentation of the pattern just when the middle voice begins its second presentation, and the slowest voice, its first presentation. Nancarrow exploits and highlights this coincidence by momentarily using the same tempo in all three voices. The passage effectively functions as the climactic focus of the study, positioned, appropriately enough, approximately three-quarters of the way through it.
The integrity of the voices in Study Number 24 is never in doubt: the work consistently and clearly presents a three-voice texture. Unlike Number 14, however, whose canon is relatively easy to follow, thereby assisting the listener in comprehending its two-voice structure, most of the canons in Number 24 do not contribute much to the differentiation of the voices. Register is a more important factor: the voices are widely separated throughout, by at least two octaves each. Canon type is important, as well. In general, diverging and diverging-converging canons tend to be more difficult to track perceptually than simple converging canons, and the primary reason is quite basic: the listener can easily grasp the canonic melody of a converging canon since it begins with the spare texture of a single voice. With the subsequent gradual introduction of additional layer(s) the process of canon and the degree of textural multidimensionality attained is usually also easily discernible. With a diverging canon there is no such blatant introduction of the single-voice melody, or of individual voices.
Not all diverging canons are alike, of course, and a number of factors can affect the textural character, beyond the registral integrity of the voices. Both the ratios of the speeds, which control the rate of divergence, and the length of the canon, which controls the degree of divergence, can be crucial. In the short sections of Study Number 24, for example, the voices have very little time to diverge and to assert their independence, and therefore, the result is a complex, roughly coordinated texture rather than a contrapuntally polyphonic texture. The construction of the melodic material also plays a big role. When the material has a clear identity--usually a line of single pitches with a distinct character--it is easier to identify as a single voice, and thus the individual canonic voices are easier to perceive. Portions of Number 24 do contain lines of such melodic distinction, but others, such as those focusing on octave tremolos, do not. Nancarrow's canonic voices almost invariably imitate contour stri ctly, but because of disjunct lines and the like, canonic imitation is not always the most conspicuous feature of a work. But, as we have seen, canon can have a tremendous impact on gesture length and placement, and formal structure.
Study Number 20 goes even further than Number 24 toward exploiting the technique of canon to create a wonderfully rich, but not perceptibly canonic, textural effect. The focus here will be the canon of pages 11.3-24; the opening of the passage is given in Example 11. This six-voice canon, beginning with the last note of the top voice at page 11.3, is conventional in type, which is unusual for Nancarrow: the voices have staggered entrances and all proceed at the same pace, written in proportional notation, (19) so that they remain staggered with an unchanging imitative gap. It is a rhythmic canon in that the rhythms, or rather, durations, are subjected to imitation, as can be determined by measuring the distance between the notes in the score. But it is difficult to hear as a canon for a number of reasons. First, because the rhythms are irregular, and not relatable to an underlying equable pulse, they defy tracking. (20) Second, and more important, the voices do not have distinct registral or melodic identitie s, nor do they participate in a true pitch canon. The canonic entrances of the six voices are bracketed in Example 11.
The material of each layer can be characterized as the twelve statements of a single pitch, using irregular durations, followed by successive changes to four other pitches, each of which, except for the last, is also stated twelve times. Each set of twelve articulations, however, follows a different pattern of durations. The final pitch of each voice is stated fewer than twelve times because the canon breaks off just as the last voice reaches its fifth note (page 23). Although the pattern of twelve durations changes with each pitch, the sum of those twelve durations remains constant, measuring approximately 610 millimeters in the score. The metronome marking of the piece is given as 13 millimeters equals 120. Each pitch occupies the same amount of time in the piece, therefore, approximately 23.5 seconds.
While the six voices participate in a strict rhythmic canon, they do not share the same pattern of pitch changes, even in transposition. The pitch succession of each layer for pages 11.3-23 (where the canon breaks off) is given in Example 12a; pitch imitation occurs only in pairs of layers, as shown in Example 12b. The layers are numbered according to whether they are in the upper three (I) or lower three (II) notational layers, and their letters represent whether they are the first (a), second (b), or third (c) layer to enter within that set. The pitch imitation of the lettered pairs does not begin with the first note of each layer, but is rotated so as to match the first pitch of one layer with the second pitch of a layer that enters later, after the entrances of two intervening voices. The imitation is thus so widely spaced in the music that it is imperceptible: the gap from [B.sub.3] in voice II-a to the corresponding [D.sub.4] in voice I-a spans nearly six systems, or around thirty-seven seconds, for exa mple. Clearly, the rhythmic canon and these imitative pairings are not the central perceptual factors of this passage, but they combine to produce its most apparent features: a continuously changing rhythmic surface, a sequential composite melodic line, and characteristic trichordal and hexachordal sonorities.
As seen in the individual pitch successions of Example 12, the voices of Study Number 20 do not have distinct registral spans: they all fall within the [G[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII].sub.3] to [E[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII].sub.5] span. Moreover, pitch change is slow and staggered among the voices because of the repetitions and canon. As a consequence, a resultant melodic line emerges from the composite of the six layers that contains a great deal of stepwise motion. The gradual succession of new pitches in the passage, formed first by the canonic entrances of the voices and later by their subsequent pitch changes, comprise this melodic line, which is presented in Example 13a. It contains a sequential pattern as a result of the paired imitation of the individual voices; Example 13b shows how the line is imitatively deployed between the upper and lower three notational layers.
The registral proximity of the adjacent notes helps us to hear them as forming this single line, as does the temporal spacing of the notes, which is relatively regular: starting from the first note of the first voice, page 11.3, the successive distances to the entrance of each following voice (and thus the next new pitch) are approximately 114, 96, 108, 70, and 101 millimeters. The distance from the entrance of the sixth voice to the first change of pitch in the first voice is 116 millimeters. Twenty-six millimeters in the score represent one second, so these distances correspond approximately to 4.4, 3.7, 4.2, 2.7, 3.9, and 4.5 seconds. This pattern recycles throughout the passage as a result of the canon, and it supports the perception of a single melodic line through its fairly regular introduction of the pitches of that line, which are spaced widely enough so that there is plenty of time to perceive each new member of the line.
In addition to this sequential, mostly stepwise resultant line, there is another important factor guiding the pitches of this canon: the upper and lower three notational layers form successive 3-2  trichords (a favorite of Nancarrow), closely spaced, as shown in Example 13c. Each pair of trichords combines to form 6-Z25 . Both sonorities follow transpositional patterns that are linked to the melodic sequence:  moves by [T.sub.5], and  by [T.sub.10]. Indeed, these sonorities, with their transpositional chains and the rhythmically free pulsations of their components, characterize this passage and the entire study. The canon and imitative pairs are crucial to the passage, but not perceptually so. They facilitate the formation of the sonorities and melodic line that we hear prominently, while remaining imperceptible themselves. The process of canon is further obscured by additional, low-register material that enters at page 16.1, and by the break-up of the canon with the beautiful cadenc e on E[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII] at page 24. But in its unique and repetitive material, its irregular durations, and the registral closeness of its layers, the canon is not intended from the start to have perceptual clarity. The characteristic trichords and hexachords, and the overriding melodic line of the passage, claim our attention instead, to the near exclusion of the canonic process responsible for their formation. This points to how fertile a concept canon was for Nancarrow, and how varied its application can be.
In fact, this survey merely touches the surface of the wealth of canonic effects in Nancarrow's music. There are canons whose tempo proportions and/or canonic material are considerably more complex than those seen here, canons with up to twelve voices, and canons that feature changing tempos rather than fixed. What I hope to have shown is that canon is a remarkable and seemingly inexhaustible tool, one that provided Nancarrow with a mechanism to achieve multifarious textures, rhythmically and texturally elaborate at time, to be sure, but always grounded in a process of systematic imitation. The resulting variety seems to belie Nancarrow's typically self-effacing account of his reasons for using canons to such a degree. When asked by Charles Amirkhanian in 1977 how he had gotten interested in canons, Nancarrow credited the shared melody of canonic voices with making their temporal relationship easier to hear, and then added: "Also, I forgot who pointed this out to me--I don't have much of a what-do-you-call-it , melodic imagination--so if it's the same melody going it's easier for me. I just have to do it once!" (21) Of course, as we have seen in Studies Numbers 14, 24, and 20, Nancarrow had an impressive imagination, melodic and otherwise, and was clearly a master of canonic writing. The canonic process and resulting rhythmic/temporal complex may well be the point of some of his works, while in others they may be instead the crucial means to distinct textural, harmonic, and formal designs.
MARGARET THOMAS is Assitant Professor and Coordinator of Music Theory at Connecticut College.
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(1.) See, for example, Studies Numbers 3c, portions of 7, and 26.
(2.) Nancarrow, quoted in Peter Garland, "Conlon Nancarrow: Chronicle of a Friendship," in Americas: Essays on American Music and Culture, 1973-80 (Santa Fe: Soundings Press, 1982): 185.
(3.) Nancarrow, quoted in Charles Amirkhanian, "Interview with Composer Conlon Nancarrow," in Conlon Nancarrow: Selected "Studies for Player Piano," Soundings 4 (Spring/Summer 1977): 13.
(4.) See Caldara's Divertimenti musicali percampagna (1729), and Bach's Musical Offering (1747) and Art of Fugue (1749).
(5.) See, especially, nos. 10, 23, 24, 28, 29, 31, and 36 in vol. 1 of the Mikrokosmos.
(6.) See, especially, no. 2, Acht Kanons fur Zwei Singstimmen mit Instrumenten, op. 45, no. 2, and no. 7/II, Acht Kanons in der ersten Lage fur wenig Fortgeschrittene fur zwei Geigen oder zweistimmigen Geigenchor mit begleitender 3. Geige oder Bratsche.
(7.) Representative canonic works by these composers include the following: Carter, Canon for 3--In Memoriam Igor Stravinsky (1971) and Canon for 4--Homage to William (1984); Dallapiccola, Quaderno musicale di Annalibera (1952) and Sonatina Canonica (1989); Davies, Alma redemptoris mater (1957) and Vesalii icones (1969); Ligeti, Lux aeterna (1966), Lontano (1967), and Melodien (1971); Reich, Piano Phase (1967), Violin Phase (1967), and Phase Patterns (1970); Stravinsky, In Memoriam Dylan Thomas (1954) and Double Canon for string quartet (1959); and Webern, Five Canons on Latin Texts, op. 16 (1923-24), Symphony, op. 21 (1928), and Piano Variations, op. 27 (1935-36).
(8.) Henry Cowell, New Musical Resources (New York and London: Alfred A. Knopf, 1930). See especially "Part II: Rhythm," 49-108.
(9.) For an excellent biography of Nancarrow see Kyle Gann, The Music of Conlon Nancarrow (Cambridge: Cambridge University Press, 1995), chapter 2, 36-50.
(10.) Gann, especially chapters 6 ("Canon: Phase I"): 109--45, and 8 ("Beyond Counterpoint: The Sound-mass Canons"): 173-239.
(11.) The scores to these works can be found in Conlon Nancarrow: Collected Studies for Player Piano, vol. 5 (Santa Fe: Soundings Press, 1984).
(12.) For examples of converging canons, see Studies Numbers 18, 19, 31, 32, and portions of 33 and 37. The diverging type may be found in Numbers 9, 33, and 37. Converging-diverging canons appear in Numbers 14 and 36, and in portions of 25 and 37. See Numbers 15, 17, 24, and portions of 33 for diverging-converging canons. Canons that exhibit changing speeds occur in Numbers 8, 21, and 27.
(13.) Nancarrow, quoted in Roger Reynolds, "Conlon Nancarrow: Interviews in Mexico City and San Francisco," American Music 2, no. 2 (Summer 1984): 5. For a discussion of the aesthetic and compositional importance of Nancarrow's concept of "temporal dissonance" see my dissertation, "Conlon Nancarrow's 'Temporal Dissonance': Rhythmic and Textural Stratification in the Studies for Player Piano" (Ph.D. diss., Yale University, 1996), especially chapter 4, "Temporal Dissonance": 93-141.
(14.) I provide musical references to the studies as follows: page 1.1 and page 1.2 refer to page 1, system 1, and page 1, system 2, respectively, and so on. The page numbers are those hand-written by Nancarrow at the top of each page. This reference method is clearer than reference to measure numbers for the studies, since at any given moment each voice is likely to be in a different measure because of their staggered entrances and different tempos. Furthermore, the scores to some studies do not contain barlines.
(15.) Gann discusses the use of the duration series in each of the "Seven Canonic Studies," 114-27.
(16.) Most of the published scores to the studies are reproductions of Nancarrow's handwritten scores, in which he took painstaking care to represent the complex rhythmic relationships of voices neatly and accurately.
(17.) My analysis focuses on the fourth and tenth canons of Study Number 24. See Gann, 176-82, for an overview of each section.
(18.) Section lengths for Example 5 are calculated between points of synchrony, which do not always coincide precisely with the beginning of new canonic material, but which are never offset by more than one measure. There is one omission in the score that confuses the eighth-note count: the measure that begins with the last articulation on page 20, middle voice, is in 14/16, not 12/16.
(19.) Nancarrow's proportional notation is a nonmetric kind of notation. The placement of notes on the page reflects their occurrence in time, and traditional durational symbols acquire different meanings: usually an "eighth note" represents a staccato note, while a "quarter note" represents a sustained note whose precise duration is represented by a horizontal line following the notehead. Nancarrow uses proportional notation in approximately one-third of the studies, including Numbers 8, 20,21, 27, and 41.
(20.) Gann notes that the distance given in the printed score, 1 1/3 centimeters, as the basic beat for a tempo of 120 indicates that the piece could have been written conventionally with that distance represented by a quarter note. He goes on to demonstrate, however, that such notation would have "resulted in needless complexity," and would not have provided a clear "picture of the piece's intentions and aural effect" (106), precisely because that duration does not significantly underlie the rhythms of the piece.
(21.) Nancarrow, quoted in Amirkhanian, 13.
EXAMPLE 5: STUDY NUMBER 24, SECTIONAL DIVISIONS Section Tempos Pages 1 [MUSICAL NOTES NOT REPRODUCIBLE 1.1-4.3 IN ASCII] = 149 1/3/160/170 2/3 2 [MUSICAL NOTES NOT REPRODUCIBLE 4.3-6.3 IN ASCII] = 224/240/256 3 " 6.3-9.2 4 " 9.2-10.1 5 " 10.1-11.2 6 " 11.2-11.3 7 " 11.3-12.2 8 " 12.2-12.3 9 " 12.3-13.1 10 " 13.1-23.3 11 [MUSICAL NOTES NOT REPRODUCIBLE IN 23.3-25.1 ASCII] = 149 1/3/160/170 2/3 12 [MUSICAL NOTES NOT REPRODUCIBLE IN 25.1-25.2 ASCII] = 224/240/256 Section Starting notes Length, in number for the three voices of eighth notes 1 [D.sub.1]-[F[MUSICAL NOTES NOT 270 REPRODUCIBLE IN ASCII].sub.3] -[A.sub.5] 2 [B.sub.0]-[D.sub.3]-[F[MUSICAL NOTE 180 NOT REPRODUCIBLE IN ASCII].sub.5] 3 [G.sub.1]-[B.sub.2]-[E.sub.6] 270 4 [C.sub.1]-[C.sub.3]-[C.sub.5] 90 5 [C.sub.1]-[D[MUSICAL NOTES NOT 135 REPRODUCIBLE IN ASCII].sub.3]-[G [MUSICAL NOTES NOT REPRODUCIBLE IN ASCII].sub.5] 6 [F[MUSICAL NOTES NOT REPRODUCIBLE 45 IN ASCII].sub.3]-[F[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII].sub.5]-[F [MUSICAL NOTES NOT REPRODUCIBLE IN ASCII].sub.7] 7 [B.sub.1]-[D[MUSICAL NOTES NOT 60 REPRODUCIBLE IN ASCII].sub.4]-[F [MUSICAL NOTES NOT REPRODUCIBLE IN ASCII].sub.6] 8 [C[MUSICAL NOTES NOT REPRODUCIBLE 15 IN ASCII].sub.2]-[F[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII].sub.4]- [B.sub.5] 9 [E[MUSICAL NOTES NOT REPRODUCIBLE 30 IN ASCII].sub.1]-[G.sub.3]- [C.sub.6] 10 [G.sub.1]-[B.sub.3]-[D.sub.6] 1110 [right arrow][B.sub.2]-[B.sub.4]- [B.sub.6] [right arrow][B.sub.1]- [E[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII].sub.3]-[A[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII].sub.4] 11 [C[MUSICAL NOTES NOT REPRODUCIBLE 90 IN ASCII].sub.1]-[E.sub.3]-[A.sub.] 12 [A[MUSICAL NOTES NOT REPRODUCIBLE 30 IN ASCII].sub.2]-[A[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII].sub.4]- [A[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII].sub.6] Section Duration, Dynamic and character in seconds 1 33.75 pp; legato 2 15 ff; staccato, repeated notes 3 22.5 pp; sustained notes 4 7.5 ff; trills and octave tremolos in sixteenth notes 5 11.25 pp; staccato 6 3.75 ff; arpeggios and octave tremolos in sixteenth notes 7 5 pp; legato 8 1.25 ff; trills in sixteenth notes 9 2.5 pp; sustained chords 10 92.5 ff-p-ff; staccato, repeated notes, turns, trills, octave tremolos 11 11.25 pp; legato, as reprise of opening 12 2.5 ff; staccato, repeated notes
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|Title Annotation:||Conlon Nancarrow|
|Author:||Thomas, Margaret E.|
|Publication:||Perspectives of New Music|
|Article Type:||Critical Essay|
|Date:||Jun 22, 2000|
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