Thesis

What is Tempo




A major structural feature of Conlon Nancarrow’s tempo canons, where canonic material moves at different speeds in different voices, is the places where the musical lines converge. These convergence points can vary in importance and function, and, depending on the type of canon being used in the piece, there may be only one or two convergence points per piece, or many. This paper will examine the features of convergence points in Nancarrow’s tempo canons and illustrate the many ways they are used in his Studies for Player Piano.

[1] The most characteristic structural feature of Conlon Nancarrow’s tempo canons is the convergence point, or “the infinitesimal moment at which all lines have reached identical points in the material they are playing” (Gann 1995, 21). Convergence points (CPs) serve varying purposes in Nancarrow’s Studies: sometimes they seem to be the raison d’être for an entire piece (particularly in his earlier brief tempo canons with just one CP, such as Nos. 14, 18, and 19), sometimes they vary considerably in significance within the same piece (such as the thirteen CPs in No. 24), or they may serve to mark the most structurally significant points in a piece (such as the three CPs in No. 27). 

[2] Unlike conventional canons in which the voices proceed at the same tempo, tempo canons present the possibility for one or more CPs between the canonic voices because the voices are moving at different speeds and either converging or diverging. Margaret Thomas classified tempo canons into four basic types: (1) the converging canon, with one CP at the end; (2) the diverging canon, which begins with a CP; (3) the converging-diverging (arch) canon, in which a single CP is somewhere in the middle; and (4) the diverging-converging canon, which begins and ends with a CP. Nancarrow wrote examples of each type (although he did not write any complete diverging canons—there is, after all, not much aural interest in a piece that diverges immediately away from a beginning point of convergence, never to return).  (1)


[3] This paper will examine the presence and placement of CPs in Nancarrow’s tempo canons and their significance to the overall structure. CPs will be examined in their more conventional uses in the canons, and other devices examined will include tempo switches and overlaps (which can be used to create additional CPs) and techniques for emphasizing and de-emphasizing CPs. Several of Nancarrow’s more complex works feature many CPs, and further complexity is possible through the addition of more voices to a canon, providing opportunities for interior CPs among smaller groups of voices. A special sub-type of Nancarrow’s tempo canons is the “acceleration canon,” in which voices are accelerating or decelerating. 

[4] The effect of convergence in Nancarrow’s tempo canons is perceptually heightened because it involves not only temporal convergence, but convergence of canonic material. The choice of canon as a means of portraying temporal proportions is quite natural. Thomas notes the aptness of this technique when she says “One of the clearest compositional strategies in which to hear proportions is canon” (Thomas 1996, 97). She adds further that in Nancarrow’s tempo canons “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” (p. 66). This “brief moment of coordination” is the CP, or Gann’s “infinitesimal moment” at which temporally divergent voices converge. 

[5] A CP is an excellent example of what Jonathan Kramer refers to as a “timepoint,” or “an instant, analogous to a geometrical point in space” (Kramer 1988, 454). A time-point has no dimension, as explained by Kramer: 
 

But what is a timepoint? Whereas a timespan is a specific duration (whether of a note, chord, silence, motive, or whatever), a timepoint really has no duration. We hear events that start or stop at timepoints, but we cannot hear the timepoints themselves [emphasis mine]. A timepoint is thus analogous to a point in geometric space. By definition, a point has no size: It is not a dot on the page, although a dot may be used to represent a point. Similarly, a staccato note or the attack of a longer note necessarily falls on and thus may represent a timepoint, but a timepoint in music is as inaudible as a geometric point is invisible. (Kramer 1988, 82–83) 

 

 

[6] Like one-dimensional geometric lines in space, Nancarrow’s canonic lines converge at this dimensionless point in time, the timepoint, and the CP is an audible event that takes place at this point. Eytan Agmon, in his discussion of musical durations as mathematical intervals, noted that “every common musical duration is uniquely associated with a single moment in time, namely its attack” (Agmon 1997, 48). Indeed, most of Nancarrow’s CPs do occur on a coincident beat attack; in a number of cases, however, CPs occur on rests; and, in several cases (the end of Studies No. 32 and 37), a CP is at the end of a long held note. But, whether articulated or not, a CP always represents a timepoint. 

 

Tempo Canon Terminology and Features

[7] In addition to convergence point, Gann defined several terms relevant to tempo canons. An illustration directly from Nancarrow’s scores will be given here to show these features and introduce some additional concepts. Figure 1 shows a portion of Study No. 14 (“Canon 4/5”) at the point where the CP occurs, and this example will illustrate the relationships among the terms convergence point, convergence period, echo distance, and potential points of simultaneity. Study No. 14 is a two-voice arch (converging-diverging) canon in which the CP occurs exactly in the middle of the piece. Echo distances (“the temporal gap between an event in one voice and its corresponding recurrence in another”; Gann 1995, 21) are shown in Figure 1 by arrows. Gann pointed out that the echo distance “will grow shorter and shorter as the convergence point is approached, reach zero at the convergence point, then grow progressively longer as it moves away” (ibid., 21), and this is clearly seen in this example. Thomas alludes to the aural effect of the echo distance when she discusses the relative degree of temporal dissonance near what she calls the “point of synchrony”: 

The notable and increasingly proportionately significant changes in the gaps, and the attendant modifications in temporal dissonance, are responsible for the intense perceptual focus a point of synchrony achieves in a tempo canon, whether that synchrony occupies the beginning of a diverging canon, the ending of a converging canon, or the middle of a converging-diverging canon. Although it is clear that voices at different tempos are gradually and continuously drawing near to or away from one another, their degree of dissonance can seem nearly uniform for a good portion of a canon. Only near the point of synchrony does the fast approaching/departing [temporal] consonance claim perceptual prominence. (Thomas 1996, 138) 


[8] Gann generalizes that as a CP approaches in a piece and the echo distance decreases, motives tend to splinter and become briefer, and that the opposite occurs as a CP recedes and the echo distance increases: 

Typically, in the late canons, the following motion occurs: immediately following any convergence point, a quick echoing of brief figures creates excitement signalling the entrance to a new section of the piece. Usually voices lose their distinguishability in a bristling texture, then slowly separate. Short figures are gradually displaced by longer and longer motives which sound calmer (by virtue of their temporal stability) but also more complex, even developmental. As a new convergence point is approached, the echo-tempo picks up again, and figures splinter into ever briefer motives, creating a deliciously gradual feeling of cumulative climax. (Gann 1995, 173–74) 


[9] Gann defines the convergence period as the distance between potential simultaneities, and points out that in canons involving superparticular ratios the echo distance “will approximate n beats at a point n convergence periods from a convergence point” (Gann 1995, 21). In Fig. 1, this is obscured somewhat by rests but can be most clearly seen at the points marked ⊗, where the echo distance is equal to the convergence period of five beats in the top voice and four beats in the bottom. At these points it is easiest to see that the respective lengths of the echo distance (in beats) in each voice are proportional to the operational tempo ratio at that point; for example, at a point in this 4:5 canon where the echo distance is 2 beats in the top voice, it will be 1.6 beats in the bottom voice. 

[10] In some of Nancarrow’s canons where the meter does not change, the location of some convergence periods is clearly marked by shared barlines. Also, for as long as the metric pulse does not change, the length of the convergence period remains the same. Shared barlines in Nancarrow’s scores are important markers of convergence and sometimes the only way one can accurately measure the location of events elsewhere in the score (such as the number of beats elapsed to the entrance of a later voice).

[11] Two additional terms, timespan and point of simultaneity, are important in making determinations about the placement of CPs and the amount of delay that is needed in later-entering voices to make a CP occur at a certain point. Kramer defines the timespan as the “interval between two timepoints” (Kramer 1988, 454). For determining the timespan of Nancarrow’s canonic subjects, the critical question becomes the location of the first and last timepoints. The first timepoint is clearly the first beat attack. There are generally two possibilities, however, for the location of the final timepoint: coinciding with either the attack of the final note, or the release of the final note (which can be said to coincide with the attack of the hypothetical beat that would follow the end of the final note). The most important criteria for making the determination of which should be the final timepoint are usually how the canon ends metrically and its prevailing metric pulse.

[12] Study No. 14 will serve as an example of a canonic subject in which the final timepoint coincides with the final attack. Gann notes that there are 337 total eighth-note beats in the subject of Study No. 14 (Gann 1995, 117). I concur with this assessment because of the concluding nature of the final attack (with the remainder of the final measure filled with rests) and because the final note matches the prevailing eighth-note metric pulse (see Fig. 2). By measuring from the attack of beat 1 to the attack of beat 337, the distance between these two timepoints, or the timespan, is 337 – 1 = 336 beats. From that measurement we can determine that the midpoint of the canon, where the CP occurs, is the attack of beat 169, with 168 beats occurring prior to this point and 168 beats after. Beat 169 is, indeed, the location of the CP (see Fig. 3).

[13] Measuring the timespan is straightforward when the CP occurs at the end of the canon (i.e., converging and diverging-converging canons) because, in these pieces, the final CP defines the concluding timepoint. Most of the time (e.g., Studies No. 18, 19, and 24) the CP occurs on the final attack and the timespan would be measured from the first attack to the final attack. Occasionally, however, the CP occurs after the final attack (e.g., Study No. 32) and the timespan must be measured to a timepoint that follows the CP (usually a mentally-supplied terminal beat). In other cases (e.g., Study No. 36), the final timepoint is the release of the final note.

[14] Potential points of simultaneity (i.e., the beginning of a convergence period) have tremendous importance in the structure of tempo canons. Obviously, a CP can occur only at a point of simultaneity in all sounding voices. A piece may have a large number of potential points of simultaneity, if the tempo ratio involves fairly small numbers, or it may theoretically have only one point of simultaneity (especially in those tempo canons whose proportions involve irrational numbers). At the same time, many potential points of simultaneity are de-emphasized through the use of rests or held notes. The importance of the location of potential points of simultaneity to the structure of tempo canons can be illustrated as shown in Fig. 4. This example compares two Studies already mentioned, Nos. 14 and 36, both arch canons with a single CP in the middle. Nancarrow composed Study No. 14 so that its CP would fall exactly in the middle of its 337-beat theme—that is, on beat 169. In Study No. 36, the midpoint of the canon’s 843-beat timespan falls between beats 421 and 422, and is obviously not an appropriate spot for an audible CP. Nancarrow thus moves the CP slightly later to beat 427; this slight shift means that 426 half-note beats occur before the CP, and 417 occur after (see Fig. 8).

[15] If it does not matter to the composer whether or not the entrances of later-entering voices coincide with beat attacks in voices that are already sounding (i.e., with the beginning of a convergence period)—and for Nancarrow it often did not—then the CP can essentially be placed wherever the composer chooses. Convergence periods will then proceed both backward and forward from the CP, depending on its location. In the two Studies shown in Fig. 4, the entrance of the second voice in No. 14 is between beats 33 and 34 of the first voice, and none of the later voices in No. 36 enters on a beat attack in an earlier voice. In fact, in most of Nancarrow’s canons’ later entrances do not coincide with beat attacks.(2)


[16] Finally, two tempo change devices are used with some frequency in Nancarrow’s tempo canons: the tempo switch and the tempo overlap. The tempo switch is defined by Gann:

Another major, but less audible, event is the tempo switch, a device in which Nancarrow switches the fastest line to the slowest tempo and vice versa, so that the line that has been lagging catches up with the one that has been pulling ahead. By mathematical necessity, the tempo switch always occurs halfway between two convergence points. (Gann 1995, 21) 


Tempo switches are useful in creating additional CPs. They are a significant part of Study No. 24 (“Canon 14/15/16”), where they are used to create thirteen CPs which delineate twelve sections.

[17] Occasionally a tempo changes in only one voice to create a tempo overlap, where the same tempo is occurring in more than one voice. Fig. 5shows how two tempos can form a tempo overlap in a diverging-converging canon. This is, in fact, exactly how Study No. 15 (“Canon 3/4”) is configured. The tempo overlap section in a two-voice canon will consist of a portion of the entire canon that is equal to  , where m and n are the two elements of the tempo ratio. For instance, in Fig. 5 the two tempos (and the sectional durations) are related by the ratio 3:2, so the tempo overlap section occupies one-fifth of the entire canon.

[18] Another Study with tempo overlaps is No. 17 (“Canon 12/15/20”). Tempo overlaps occur between section A in the top and bottom voices, and between section B in the top and bottom voices (see Fig. 6). Because of the inverse relationship between time and duration, the “duration ratio” of this 12:15:20 tempo canon is 3:4:5, which is how the durations of sections C, A, and B, respectively, relate to each other. In general, in a three-voice tempo canon where the elements of the duration ratio (not the tempo ratio) are x:y:z (with x being the smallest element and z the largest), the difference between the durations of x and y related to the length of the entire piece is, between y and z it is, and between x and z it is. Thus, in No. 17 the sum of the duration ratio’s elements, x:y:z, is 12, and the durational relationship between contiguous elements in the ratio (3:4 and 4:5) is one-twelfth of the whole piece while between the outer elements (3:5) it is one-sixth. As shown in Fig. 6, the tempo overlap involving section A occupies one-twelfth of the piece, and that involving section B occupies one-sixth of the piece. The section B overlap is long enough to be bisected by a tempo change in the middle voice between sections C and A.
Convergence Points in Nancarrow’s Tempo Canons
Tempo Canons With One (or Almost One) CP

[19] Two of the basic types of tempo canon classified by Thomas have just one CP: the converging-diverging (arch) canon, and the converging canon.  In the arch canon the CP falls somewhere in the middle, and in the converging canon it falls at the end. Fig. 7 shows line diagrams of Nancarrow’s tempo canons with just one CP.


 

Convergence Points in Nancarrow’s Tempo Canons

Tempo Canons With One (or Almost One) CP

[19] Two of the basic types of tempo canon classified by Thomas have just one CP: the converging-diverging (arch) canon, and the converging canon.(3)
In the arch canon the CP falls somewhere in the middle, and in the converging canon it falls at the end. Fig. 7 shows line diagrams of Nancarrow’s tempo canons with just one CP.

 

 

Arch Canons

[20] Nancarrow’s arch canons vary widely in ambition and scope. Study No. 14 has its CP exactly at its canonic midpoint, and the realization of this structure seems to be the primary goal for the piece. Study No. 36 has one of the most spectacular CPs in all of Nancarrow’s oeuvre in terms of its preparation and realization. Study No. 21 has one true canonic CP in addition to an area of “crossing tempos” and a loosely canonic CP on the last note. Study No. 49B returns to a simpler texture but also has a rather arresting CP.

[21] Study No. 14 (the first that Nancarrow subtitled “Canon”) exhibits the basic arch canon form with the CP exactly in the middle of its 337-beat theme (see Fig. 3). With a tempo ratio of 4:5, the faster voice occupies 80% of the duration of the slower voice, with the remaining 20% of the timespan split so that 33.6 beats elapse prior to that voice’s entrance, and 33.6 beats after it concludes; thus, the second voice does not begin at the beginning of a convergence period with the lower voice, but rather three-fifths of the way between the lower voice’s beats 33 and 34.

[22] Another important feature of arch canons, and any tempo canons with interior CPs, is the changing leader-follower relationship between the voices that takes place at interior CPs (see the CP in Fig. 1). In fact, in Study No. 14 there is so little else calling attention to the CP that the changing leader-follower relationship is thrust into relief and is what most immediately captures the listener’s attention. Gann points out (p. 118) that Nancarrow exerted so much rhythmic control in the creation of his canonic subject in No. 14 that melodic and harmonic coordination between the voices is virtually absent, and this lack of coordination does little to prepare and support the CP. There is also a fairly low incidence, in general, of rhythmic coordination between the voices due to the numerous rests; both Gann and Thomas note that the fairly small 4:5 ratio between the voices would potentially allow for convergence every four to five beats, but actual simultaneous attacks are avoided more often than realized. Thus, in Study No. 14 the CP’s placement at the exact midpoint defines the structure, but Nancarrow’s highly contrived rhythmic scheme precludes a high incidence of convergence at the metric level.

[23] The formal sophistication of Study No. 36 (“Canon 17/18/19/20”) makes it a stark contrast to the somewhat humble beginnings of Study No. 14. It is an arch canon in which the canonic midpoint falls between beats because the timespan is an odd number: 843 half-note beats, placing the canonic midpoint between beats 421 and 422. In this canon, then, it is not possible for the CP and midpoint to coincide on a beat attack as they did in Study No. 14 (Fig. 4). Nancarrow instead delays the CP to beat 427, thus creating two sections consisting of 426 and 417 beats (see Fig. 8). The final timepoint in this canon falls silently on the final release.

[24] Philip Carlsen notes that the placement of the CP on beat 427 creates a particular relationship of the elapsed time between statements of a recurring theme (which he calls a rondo theme) within the canon and the CP (see Fig. 8, bottom diagram):

 

The rondo theme’s second appearance [at m. 107] is twice as fast as the first; it occurs at a point (system 36 in the bass voice)(4)
exactly halfway between the beginning and the canonic midpoint, providing an obvious parallel with the well-known acoustical fact that the halves of a string vibrate twice as fast as the whole. (Carlsen 1988, 30)

The halfway point that Carlsen refers to is actually with the CP and not the canonic midpoint, as shown in Fig. 8. Carlsen’s observation concerning the acoustical parallel can be carried further by noting that the first three statements of the theme occur on C2, F#2, and C3 so that the second statement “bisects” not only the elapsed time to the CP but the octave between the first and third statements. Such connections are quite common in Nancarrow’s Studies.

[25] The table of elapsed beats in Fig. 8 describes the number of beats elapsed to the entrance and from the drop-out point of the top three voices, in relation to the first (or lowest) voice. The first column shows the beat elapse needed to place the CP exactly at the canonic midpoint; the second and third columns show the beat elapses that actually occur prior to and after, respectively, the statements in the top three voices.

[26] None of the voices enters at the beginning of a convergence period, as shown in Fig. 9. In fact, even the first voice begins on the last beat of a convergence period, as half-note beat 2 marks the beginning of the first complete convergence period. The second voice’s 23.7-beat elapse places its entrance within the second convergence period. Fig. 9 also shows how the echo distance decreases by one beat with the passing of each convergence period; at the beginning of each period, the number of beats between the voices is equal to the number of convergence periods remaining to the CP.

[27] The stunning CP in Study No. 36 (Fig. 10) is one of Nancarrow’s finest moments. Even the composer was surprised to hear the effect of what Reynolds refers to as “low-register difference tones” (Reynolds 1984, 16) generated by the extremely rapid iteration of notes in all the voices. Carlsen describes the passage this way:
 

 

In this passage, the four voices sweep up to their point of convergence with rapid chromatic glissandos, arriving simultaneously on their highest notes—B3, D5, F6, and A7.(5)
Each high note is then rapidly repeated at the speed of a quarter-note in its own voice’s particular tempo. . . . Even in the slowest voice, the reiterated high notes fly by at a rate of more than 340 per minute (5 per second). But that is not all: the spaces between high notes are filled in with ascending thirty-second-note glissandos. At such speeds, the perception of individual pitches is impossible. The thirty-second-notes (which, in the bass voice, move at over 3000 per minute or 50 per second) are fast enough to theoretically start generating additional low pitches with frequencies in the range of approximately 50–60 Hz. (Carlsen 1988, 25)


[28] The convergence point is preceded by what Gann identifies as a “mega-glissando” that begins 80 measures before the CP, creating a fine example of a “collective effect” that is an extraordinarily long preparation to the CP. Fig. 11 shows a section in the middle of this passage. Exactly halfway between the beginning of the mega-glissando and the CP, the C1 half note in the lowest voice begins a restatement of the opening transposition levels, which spell out a widely spaced major seventh chord. The last voice reaches its restatement at the beginning of the next convergence period; two measures prior to this, the lowest voice starts another restatement, this time an octave higher.

[29] This section of rising glissandos creates a tremendous build-up to the CP. The glissandos emphasize to the listener the leader-follower relationship of the four voices, and Nancarrow very cleverly manipulates the decreasing echo distance, by using steadily decreasing gaps between the glissandos, to eventually create long, continuous glissandos through the four voices. After the CP, the glissandos descend through the voices and the listener is once again acutely aware of how the leader-follower relationship has changed at the CP.

[30] An unusual and unique canon is Study No. 21 (“Canon X”), an acceleration canon whose subtitle is derived from the manner in which the tempos of the two voices “cross” in the course of the piece. Study No. 21 is unlike any of the other tempo canons, with its gradually changing tempos (accelerating in one voice while the other voice decelerates) that almost imperceptibly “cross” somewhere in the middle of the piece. In No. 21’s proportionally-notated score,(6)
there is no meter and no shared downbeats are notated, but it appears that there are three significant convergences. This Study thus fits somewhat uneasily into this section on arch canons with one interior CP; however, only one of the convergences is a true canonic CP, and it does appear in the interior of the piece. The first convergence, just over one-third of the way through the piece where the two voices cross tempos, is not a canonic convergence point but more of a convergence area (see Fig. 12). Thomas (1996, pp. 126, 281) identifies the bracketed area in the figure as a passage of “nearly simultaneous motion,” which occurs just before the tempos of the two lines actually coincide somewhere near the beginning of the next system (at a point where the note articulations are nearly evenly staggered between the voices). The crossing of the tempos is thus not marked by a convergence point.

[31] At just over two-thirds of the way through the piece, an actual canonic CP does occur (see Fig. 13). Its significance is enhanced by several factors: the 54-note row (B–C–Aβ. . .) that comprises the canon begins again in each voice; there is a registral shift upward in the lower voice; and the texture of the upper voice changes from triple octaves to quadruple octaves. Thus, this CP is both the conclusion of a converging canon and the beginning of a diverging canon. At this point in the piece there are about 16 notes in the bottom voice between successive articulations in the upper voice. Study No. 21 concludes with a non-canonic CP on the final note; at this point there are about 48 notes in the lower voice between articulations in the upper voice. The canon is altered at the end so the piece ends on a V–I cadence with a quintuple C octave.

Converging Canons


[32] The next group of canons to be examined is the converging canons shown in Figure 7, which include Nos. 18, 19, 31, 32, 34, 48, 49A, and 49C. One of the distinguishing characteristics of converging canons is that the leader-follower relationship of voices is maintained to the final CP. This is easily seen in a structural diagram for Study No. 18 (“Canon 3/4”), which was mentioned earlier as an example of a canon where the entrance of a later voice coincides with the beginning of a convergence period (see Fig. 14). This is made possible because the canon’s timespan (1,680 eighth-note beats) is divisible by both 3 and 4 and the first voice begins at the beginning of a convergence period.

[33] In Study No. 19 (“Canon 12/15/20”), like most of Nancarrow’s tempo canons, later-entering voices do not enter at the beginning of a convergence period (Fig. 15). The voices are related to each other by a duration ratio of 3:4:5,(7)
and the bottom voice relates to the middle voice by the ratio 4:5 and to the top voice by the ratio 3:5. The timespan of 336 eighth-note beats is divisible by 3 and 4, but not by 5; thus, since both the middle and top voices relate to the bottom voice in ratios containing the factor 5, neither of these voices enters at the beginning of a convergence period with the bottom voice. The CP at the end of Study No. 19 is noteworthy because Nancarrow somewhat uncharacteristically alters the interval of imitation between the voices (from elevenths to double octaves) to create a unison V–I cadence (see Fig. 16). I surmise that he likely did this to convey an additional sense of finality because this canon concludes the six-canon set of Nos. 13–19, all of which are based on the rhythmic series {n – 1, n, n + 1, n}.

[34] Study No. 32 (“Canon 5/6/7/8”) is a straightforward four-voice converging canon whose structure is shown in Fig. 17. No. 32’s final convergence point (see Fig. 18) is unusual in that it occurs at the final cut-off—that is, at the hypothetical downbeat that follows the final beat. There are 431θ. beats in the canon, and the timespan is the same number because the CP actually occurs on hypothetical beat 432.
[35] In Study No. 34 (“Canon  ”), measuring entrances to later-entering voices is complicated by the almost complete lack of noted shared downbeats in the score, the multiple sections of the canon, and the unmetered rests that occur between many of the sections (making it impossible to establish a length for the canon’s timespan). The CP (Fig. 19) occurs on the attack of the final beat (this is, in fact, the only downbeat in the entire score shared by all three voices).

[36] Study No. 48 (“Canon 60/61”) is a remarkable achievement in convergence. It consists of three movements: movements A and B are each 60:61 canons, which are then played together to create movement C (with approximately a 60:61 ratio between A and B). The performance of C involves two carefully coordinated player pianos. Movement A has a major tenth level of imitation, with an echo distance that begins at about eight seconds (the score is proportionally notated and includes additive acceleration in all four voices, thus measurements are inexact), dwindling to zero over the course of some 127 pages. There is a minimal level of convergence between the two voices built into the canon; the most pervasive is regularly-recurring pairs of arpeggios in which the second of the pair in the bottom voice aligns with the first in the top voice (see Fig. 20a).

[37] Movement B is slightly shorter than movement A, as, during the performance of C, it should enter after movement A’s 61-voice (see Fig. 20b). Movement B’s interval of imitation is the perfect fifth; this allows the final chord of movement C to spell out a Bβ major seventh chord at the extraordinary moment of coordination where all four voices converge. Dynamics play a critical role in the preparation for this convergence; Nancarrow brilliantly creates an almost complete coordination of rapidly changing and highly contrasting dynamics (mostly between ΦΦ and ) for the final 28 pages of the piece, creating a convergence of dynamics long before the temporal convergence.

[38] Study No. 31 (“Canon 21/24/25”) is an unusual case. It is a converging canon with no CP, as the canon is truncated before the CP is reached. As shown in Fig. 21, the canon’s actual length is 669 eighth-note beats, which is 36 beats shy of the 705 needed to reach convergence (the timespan is 704 beats) at a ratio of 21:24:25. The lack of a temporal convergence is seemingly underscored by the three key areas a fifth apart to create what Thomas (1996, p. 67) refers to as an “unresolved asynchronicity” (see the canon’s conclusion in Fig. 22). The canon is in three distinct sections, each separated by an eight-measure rest, and the third section is both registrally enlarged (the level of imitation increases from a fifth between voices to an octave plus a fifth) and the canonic line reinforced by octaves and full of melodic leaps. Based on these features, Thomas offers a very perceptive and plausible reason for the lack of convergence at the end:

 

 

It is in the final section of the study that the canon becomes most difficult to perceive, and in this lies what may have been a potential reason for ending the study before it can converge on a point of synchrony. Each single voice is difficult to follow in and of itself in this section because of its extreme registral leaps and because the articulation types are limited exclusively to staccato. . . . Since it is difficult to follow a single voice, it is nearly impossible to follow the canonic relationship among voices. The staggered ending thus reorients us to the canonic nature of the passage: as each voice drops from the texture in quick succession we are reminded retrospectively of them as linear entities, and we hear in their concluding gestures their shared material and imitative relationship. (Thomas 1996, 69–70)

 

 

Tempo Canons With More Than One CP


[39] Canons with more than one CP are of the following types: diverging-converging canons (which begin and end with a CP), canons with interior tempo switches, and special cases such as acceleration canons. Fig. 23 shows the simpler canonic structures that contain more than one CP. One of the basic types of tempo canon, the diverging-converging canon, has two CPs (one at the beginning and one at the end). Nancarrow wrote two examples of this type of canon: Studies No. 15 and 17. Canons of the diverging-converging type have at least one tempo overlap (the number depends on the number of voices and tempos in the canon). Study No. 15 contains a simple example of a tempo overlap. This two-voice Study has a tempo and duration ratio of 3:4 and a canonic subject of 336 eighth-note beats; the canon is stated twice in each voice, once at the slower tempo and once at the faster tempo. When the faster voice repeats the canon at the slower tempo, both voices are stating the slower tempo for a period of 84 eighth notes (one-fourth of the canon’s length), and this section comprises the middle one-seventh of the entire length of the piece (see Fig. 24). An interesting by-product of this structure is that the top voice, which initially states the faster tempo, remains the leader in stating the canon throughout the piece. 

[40] Another important structural feature of diverging-converging canons is that each voice successively states every tempo, with each voice stating the tempos in a different order and each tempo uniquely associated with a canonic section that is the same length in each voice. This can be clearly seen in the structure of Study No. 17 (“Canon 12/15/20”), which was shown in Fig. 6. The three sections are stated once in each voice, with the A section associated with the tempo η = 172.5 and a duration of 4, the B section associated with the tempo η= 138 and a duration of 5, and the C section associated with the tempo η= 230 and a duration of 3. As shown in Fig. 6, there are no tempo overlaps involving C, the shortest section. The longest section, B, has the longest overlap, with the A section having an overlap half that length.

[41] Study No. 17 does not exhibit as much harmonic convergence at section changes as No. 15 does, and, with almost continuous eighth-note motion in three different (and quite fast) tempos, it would be difficult to hear such coordination anyway. The CP at the beginning of No. 17 coordinates on a F major triad while the final CP features a cadence on A (see Fig. 25). Throughout the piece, the A section always begins on the note C, the B section on A, and the C section on F. Nancarrow keeps to a limited number of frequently changing key areas (mostly F, A minor, C, G, and D, with their affiliated accidentals Bβ, F#, C#, and G#), which allows some brief and random areas of harmonic convergence to occur. 

[42] Several Studies (Nos. 24, 33, and 43) use internal tempo switches to create additional CPs. The structures of Nos. 33 and 43 are shown in Fig. 23; No. 24’s structure is shown in Fig. 28. Study No. 43 (“Canon 24/25”), with its single tempo switch and two CPs, provides a simple structural framework for looking at the tempo switch; its two CPs and the tempo switch are shown in Fig. 26. The two voices of Study No. 43 are not only very close in tempo ratio (24:25) but also in their interval of imitation, a major third. Gann describes the proportions of the Study in relation to the CPs as follows (p. 219):

 

 

 


Thus, the tempo switch occurs exactly halfway between the two CPs, after twelve convergence periods at a point where the echo distance is twelve quarter-note beats (this is shown by the arrow connecting the bars in the middle example of Fig. 26). 

[43] As he did in Study No. 36, Nancarrow precedes the first CP in No. 43 with ascending movement. The second CP, which is actually a rest, is followed by descending movement in sixteenth notes (the most rhythmically active section of the piece). The CPs are used to set off a middle section in the palindromic structure of the piece; Gann describes that structure in this way:

 

 

This 24/25 canon is more nearly palindromic than No. 36; no extravagant post-CP event appears to throw the symmetry off balance. Instead, the textures and motives of the first half of the piece return in almost exact reverse order in the second half, though varied, some of them more elaborately developed, others shortened.
    . . . No. 43 is palindromic in spirit, although there are no actual retrograde passages. (Gann 1995, 218)


The “palindromic spirit” and arch shape of the piece are also reinforced by increases and then decreases in note density and dynamics before and after the CP section. Gann’s generalization about the splintering of motives that become briefer as a CP approaches somewhat holds true here, as some of the motives prior to the first CP are subjected to rhythmic compression, and in the section leading up to the CP there is a countdown extending from 12 quarter notes down to 1.

[44] Study No. 33, with its irrational tempo ratio of √2:2, presents special challenges in measuring and calculating distances and durations. The structure of the Study is shown in Fig. 23. There are four distinct canons (which Gann labels A through D in the diagram, but refers to as Canons 1 through 4 in his narrative): Canon 1 is a diverging-converging canon with a tempo switch in the middle; Canon 2 is an arch canon with a CP in the middle; Canon 3 is another diverging-converging canon; and Canon 4 is also a diverging-converging canon, but it begins with a short interlude prior to the canon’s first CP. All told, there are seven CPs. Only the first one takes place on a note attack in both voices; most of the remainder occur either on held notes or rests. Even the final CP does not take place on a coincident beat attack. See Table 1 for a description of the canons and convergence points in Study No. 33.

 

 



Table 1
Description of Canons and Convergence Points in Study No. 33
 


[45] Fig. 27 shows three of the unarticulated CPs in Study No. 33. CP 4 marks the beginning of Canon 3, and begins on a rest. CP 6 is technically the beginning of the diverging-converging canon in Canon 4, but it also occurs at a rest. CP 7 is an enigma—the final beat attacks in both voices are almost coincident, but not quite, as indicated by the lack of a shared barline at the final measure. The effect, then, is that the top voice finishes slightly ahead of the bottom voice to create a sort of V–I cadence in Dβ. Unfortunately, it is impossible to calculate exactly where the final CP falls in the last measure. Due to a shared barline, the exact location of CP 6 is discernible, but because the exact measurement from CP 6 to the tempo switch between CP 6 and CP 7 is not known or calculable, it is not possible to use the measurement of that half of Canon 4 to determine the full canon’s length and project the exact ending point of the canon. If it were possible to calculate the exact measurement of the first half of the canon to the tempo switch, the exact location of CP 7 would be calculable. Thus its location can only be approximated.

[46] Study No. 24 (“Canon 14/15/16”) has twelve canonic sections demarcated by thirteen CPs (see Fig. 28). Each of the twelve canons is of the diverging-converging type, with a tempo switch between the outer voices (i.e., the fastest and slowest voices) in the middle of each canon; the middle voice stays at the middle tempo throughout the piece. Canon 10, however, has a tempo overlap section in the middle before the tempos switch. Also, in addition to the tempo switches in each of the other eleven canons, an additional tempo switch is inserted at the beginning of Canon 9. Thus, there are twelve tempo switches and one tempo overlap in this Study.

[47] Table 2 lists and describes the canons and CPs that occur in Study No. 24. As the table shows, the majority of the CPs elide the end of one canon with the beginning of another; the exceptions are CPs 4, 7, and 10. The table also shows that CPs 1, 4, 5, 7, 8, 11, and 13 are articulated in all voices while the others are unarticulated (i.e., take place on rests or held notes).

[48] CPs and tempo switches sometimes occur with startling rapidity in this Study, particularly in the area of Canons 6–9 where (as shown in Fig. 28) CPs occur every 3, 4, 1, and 2 convergence periods, respectively. Fig. 29 shows Canon 8 (the briefest canon, consisting of only one convergence period of 15 sixteenth notes) and the beginning of Canon 9. As mentioned earlier, an additional tempo switch is inserted at the beginning of Canon 9. Normally the leader-follower relationship between the top and bottom voices reverses at each CP; at Canon 9, however, the additional tempo switch allows the bottom voice to remain the leader for both canons.

[49] At Canon 10 (by far the longest canon, comprising 74 convergence periods), Nancarrow inserts a tempo overlap on the middle tempo (θ. = 240) that lasts for 60 sixteenth notes (four convergence periods). The tempo overlap allows for a rare rhythmic convergence to be created in all three parts by repeating the same rhythmic pattern three times, as shown in Fig. 30. After this rhythmic convergence, the top and bottom voices switch to the tempo opposite the one they had before the convergence.

 

 

 


[50] Study No. 37, whose structure is shown in Fig. 31, is another canon with multiple CPs. Its twelve voices state the tempo ratio of the justly-tuned scale, which seems imposing but allows Nancarrow the ability to create widely-ranging textures and to combine the tempos in different groupings (comparable to chordal structures) that are no longer rigidly bound by his practice of assigning the slowest tempo to the lowest voice, etc. Gann identifies twelve canons and five CPs in this Study, and although the structure in Fig. 31 does not show this, different tempos are consistently applied to different voices. In Canon 1, the tempos are in Nancarrow’s usual order of fastest tempo in the top voice to slowest in the bottom voice and expressing a rhythmic analogue of just intonation. In Canon 2, the order of tempos is reversed. In later sections of the Study, Nancarrow creates “chordal” groupings of tempos, such as in Canon 6 where the tempo order from bottom to top is 150187   240160   200250168   210262    180225281   (see Fig. 32, left edge of score); as shown by the bold and italic type, the tempos are arranged into three interlocking “diminished seventh chord” groupings.

[51] Convergences between voices are clearly important here as in many places Nancarrow has marked with vertical broken lines where entrances of new voices coincide with beats in other voices (something he could have marked in many other scores, but did not), even when the voices are widely separated in the score and the convergence is with an interior beat rather than a downbeat. These convergences between voices are more common than might be supposed from the unwieldy tempo ratios. In several of the later canon sections (e.g., canons 8, 9, and 10) where tempos are in “minor third” groupings, simpler tempo ratios such as 6:5 and 7:6 are common between voices. These middle and later canonic sections, then, are more temporally consonant, which is a virtual necessity for perceptual purposes because the melodic material here is much more active than in the earlier sections.

[52] Gann says about the convergence points in No. 37, “Convergence points are brilliantly de-emphasized in this study, for this is the work in which Nancarrow learned how to create beautiful effects with convergence points by omitting them” (p. 195). The CPs are not really omitted, of course, but unarticulated, as not one of the five CPs occurs at a note attack. CPs 1 and 2 occur on a rest, with the voices entering in a staggered fashion just after the CP. CP3 also takes place at a rest, just after the conclusion of canon 3. The most climactic CP is unquestionably CP4 at the beginning of canon 7 (Fig. 32). This CP marks the most rhythmically active section of the piece and comes at a place where the texture thickens to all twelve voices after an extended section in canons 4, 5, and 6 where the voice saturation is considerably thinner. In comparison, the concluding CP5 is notably anti-climactic, occurring at a whole note in all twelve voices that is held over from twelve tied half notes and marked with a fermata.

[53] The processes of acceleration and deceleration create special challenges (and opportunities) for planning and placement of convergence points, and for this we look at Studies No. 8 and 22.(8)
Nancarrow used two varieties of acceleration: arithmetical and geometric, where arithmetical tempo changes involve adding or subtracting the same invariable rhythmic unit to the previous value, and geometric changes involve changing note values by an invariable percentage. Arithmetical acceleration creates a constantly increasing or decreasing rate of change, but can be notated in conventional notation; geometric acceleration creates a smooth continuum of changing speed that must be notated in proportional notation.

[54] Although it predates the first tempo canons, Study No. 8, an early example of Nancarrow’s acceleration technique, makes considerable use of convergence points. Its first section, called by Carlsen the “trio” because it is in three voices, consists of canonic lines that alternately accelerate and decelerate within a series of nineteen durations, each of which consists of a pair of notes in a 2:1 rhythmic relationship (notated as an eighth note followed by a quarter note with a sustain line); the 2:1 rhythmic pattern creates a loping effect that particularly elucidates the processes of acceleration and deceleration. Although the notation is proportional, the process of tempo change is arithmetical as an invariable background unit is continually added or subtracted to determine the next note values.

[55] Nancarrow imposes strict order on the form of this section by bringing in each new voice at a convergence point as shown in Fig. 33. As Carlsen notes, the main pitch in this section is G, and all CPs take place on G octaves or triadic intervals involving G. No two voices are ever stating the same rate of change at the same time, even if they are using the same tempo process. For instance, in the fourth phrase of this section, where both the first and second voices are in accelerating-decelerating patterns, the patterns begin and end at different places in the nineteen-duration series and also change speed direction at different places. One of the interesting effects of this procedure is that the echo distance between voices is constantly different, and not steadily increasing or decreasing as would normally be the case. Despite this variety in rate of change within the voices, Nancarrow is able to ensure convergences at regular time intervals by keeping the total number of background units the same in each voice within each phrase. There are also occasional near-convergences that pop out of the texture due to the use of the same limited number of durations within each voice; also, the short note in each durational pair casts such nearly-aligned attacks, when they do occur, in stark relief.

[56] In contrast, Study No. 22 (“Canon 1%/1½%/2¼%”(9)
), with its geometric acceleration, has only three widely-spaced convergence points (see Fig. 23): the first concludes Canon A, a converging canon; the second is in the middle of canon B, an arch canon; and the third begins Canon A’, a diverging canon. Study No. 22 is a strict palindrome in both rhythm and pitch; CP2, which takes place at a rest, is thus the rhythmic line of symmetry in the piece. In Canon A, the first note of each voice is the same length (75 millimeters in the score’s proportional notation), but by the CP the different rates of acceleration (with the 2¼% line accelerating 125% as fast as the 1% line) have resulted in very different lengths for the last note before the CP: 7.5 mm. in the fastest voice, or a duration 10% as long as at the beginning; 17 mm. in the middle voice, or about 23% as long as at the beginning; and 30 mm. in the slowest voice, or about 40% as long as at the beginning. Thus, at the CP the fastest voice has sped up to four times the speed of the slowest voice. As a result, as the CP approaches one hears the exponential change in the echo distance between the voices.


 

Conclusion

[57] By exploring so many techniques through which he varied the tempo relationships between voices in his tempo canons, Nancarrow developed a variety of musical structures in which convergence points between voices play a primary role. Besides their structural importance, they are often the most arresting aural features of each piece. One knows they are there, even if they are not articulated or otherwise de-emphasized, and Nancarrow became a master at using CPs to create a palpable sense of anticipation for the listener. The great variety of effects he was able to achieve with CPs is a fascinating and fruitful area of study for music analysts.



REFERENCES

Agmon, Eytan. “Musical Durations as Mathematical Intervals: Some Implications for the Theory and Analysis of Rhythm.” Music Analysis 16/i (1997): 45–75.

Carlsen, Philip. The Player Piano Music of Conlon Nancarrow: An Analysis of Selected Studies. I.S.A.M. Monographs, no. 26. Brooklyn, N.Y.: Institute for Studies in American Music, 1988.

Gann, Kyle. The Music of Conlon Nancarrow. Cambridge: Cambridge University Press, 1995.

Kramer, Jonathan D. The Time of Music: New Meanings, New Temporalities, New Listening Strategies. New York: Schirmer Books, 1988.

Reynolds, Roger. “Conlon Nancarrow: Interviews in Mexico City and San Francisco.” American Music 2/1 (1984): 1–24.

Scrivener, Julie A. [Julie A. Nemire]. Representations of Time and Space in the Player Piano Studies of Conlon Nancarrow [Ph.D. dissertation, Michigan State University, 2002].

Thomas, Margaret Elida. Conlon Nancarrow’s ‘Temporal Dissonance’: Rhythmic and Textural Stratification in the Studies for Player Piano [Ph.D. dissertation, Yale University]. Ann Arbor, Mich.: University Microfilms, 1996.


 Footnotes


(1) Perhaps a good real-life analogy to the perception of the diverging-converging process is airplane travel. A passenger on an airplane has a far more accurate perception of when the plane is going to meet the ground during the landing process (i.e., convergence) than of when the plane is going to reach cruising altitude after takeoff (i.e., divergence). Also, the passenger’s perception of exactly when plane and ground will meet becomes more acute as the plane approaches the point of “convergence” with the ground and is much more acute than the perception of when cruising altitude will be reached
(2)  One clear exception is Study No. 18 (“Canon 3/4”), a two-voice converging canon with the CP on the final attack. Since the 1,680 half-note-beat timespan of this canon is divisible evenly by both factors in the canon’s ratio (3 and 4), the faster voice in this canon enters after 420 half-note beats have elapsed, coincident with the 421st beat attack of the slower voice. Thus, in this Study the second voice enters at the beginning of a convergence period, and this is confirmed in the score by a shared barline at the entrance of the second voice.
(3) The diverging canon type also has a single CP, but Nancarrow did not write any complete examples of these.
(4)  Since there are no measure numbers in Nancarrow’s scores, analysts often number the systems and refer to these.
(5) The levels of imitation in Study No. 36 spell out a major seventh chord.
(6) Note values in Nancarrow’s scores do not always have a relative time value but instead, in his proportional notation, the length of the note is approximated by the distance between notes on the page.
(7)  The inverse relationship between tempo ratios and duration ratios is discussed in Ch. 2 of my 2002 dissertation, Representations of Time and Space in the Player Piano Studies of Conlon Nancarrow.
(8)   Study No. 27, “Canon 5%/6%/8%/11,” is Nancarrow’s tour-de-force in the use of geometric acceleration. This piece is analyzed separately in a companion paper to this symposium.
(9) Each contiguous pair of percentages relates to the same ratio—3:2.

 

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