Online Symposium: Conlon Nancarrow, Life and Music

© 2012 Julie Nemire


STRUCTURAL SIGNIFICANCE OF CONVERGENCE POINTS IN NANCARROW’S STUDY NO. 27 (“CANON 5% / 6% / 8% / 11%”)


Julie A. Nemire


Abstract

Study No. 27 (“Canon 5%/6%/8%/11%”) is one of two “acceleration canons” written by Conlon Nancarrow. The Study is Nancarrow’s tour-de-force in the use of geometric acceleration and features four canonic voices that use acceleration and deceleration at the four rates given in the title. A middle line of unvarying tempo, the “clock line,” is a constant throughout the piece and provides a temporal reference point against the constant acceleration and deceleration of the canonic voices. Study No. 27 contains three canonic convergence points, and these are somewhat structurally significant but not as much so as in many of Nancarrow’s tempo canons.

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[1] Study No. 27 (“Canon 5%/6%/8%/11%”) is one of two “acceleration canons” written by Conlon Nancarrow (the other being No. 22). Like Study No. 22, No. 27 has three widely-spaced convergence points. However, whereas No. 22 consists of three of the basic canon types (converging, converging-diverging, and diverging), Margaret Thomas notes that No. 27 is unlike any of the four basic canon types and instead “incorporate[s] the technique of gradually changing speeds, so that the relationship between the voices is constantly changing” (Thomas 1996, p. 13). This gave Nancarrow considerable freedom in arranging relationships among the four canonic voices and in placing convergence points in the piece.

General Observations

[2] The canonic material is primarily in four voices (sometimes doubled) representing the four acceleration/deceleration percentages of the subtitle. Due to the geometric acceleration technique, the Study is notated entirely in proportional notation. In the middle of the canonic texture is a line of non-varying tempo that is commonly referred to as the “clock line” (Figs. 1 and 2), representing what Nancarrow referred to as the ticking of an “ontological clock” (Reynolds 1984, 9), or a time-keeping device against which one can detect the constantly changing speeds of the other voices. According to Jan Jarvlepp (1983/84, 219), Nancarrow considered the clock line to be an ostinato even though it seems to have no recurring pattern of pitch or rhythm. Thomas’s description of the clock line emphasizes its function as an ostinato:
It is presented in staccato articulations at intervals equivalent to either quarter or eighth notes. . . . The basic rhythmic and pitch material of this layer is limited, to be sure, but it is subjected to perpetual, random reorderings. As a result, it has no recurring patterns nor metrical implications. Nonetheless, the layer functions contextually as an ostinato because of its constant presence and its relatively steady character: although its articulations are not regular, its tempo is unchanging, while all the other layers in the study are continually subjected to accelerandi and ritardandi. (Thomas 1996, 14)


[3] Fig. 3 shows Gann’s formal diagram of Study No. 27. It identifies the relative positions of the clock line and the four canonic voices, with the percentage of acceleration or deceleration indicated for each voice. It also identifies Gann’s eight sections; these eight sections correspond to the changing textures of the clock line as shown in Fig. 1. Section 1 contains four different canons (identified separately by Gann because the levels of imitation do not remain the same), and I have modified Fig. 3 to identify these four canons; each of the other sections has one canon, for a total of eleven. CPs (at the end of Canons 2, 5, and 11) are also identified, and I have modified the diagram to more accurately reflect the relative relationships between canons and where voices enter and drop out. As shown in the diagram, within each canon each percentage is uniquely affiliated with a specific voice that does not change within that canon; in some canons both acceleration and deceleration are used, and in some only one process is used.

Overall Structure


[4] The importance of convergence points in this Study seems a matter for some disagreement. Gann asserts (p. 163) that “convergence point is not an important issue in this study.” Thomas, on the other hand, points to the first CP as a place where “the process of convergence is very plainly the point of this particular canon” (p. 135), and, indeed, it seems that with the kind of mathematical manipulations that were required to create this and the other CPs it could hardly be otherwise. If convergence is not the main point of this Study, it certainly is at least a significant one. The first two CPs involve the canonic voices but not the clock line, while the CP that concludes the piece involves all the voices, increasing its emphasis. (1)

[5] Thomas characterizes this Study as “decidedly non-tonal” (p. 17), and, although like many of Nancarrow’s pieces it contains numerous tonal elements such as scales and triadic structures, there is little evidence of functional tonality or long-range tonal motion. These tonal elements tend to be used together in a manner suggestive of what Tenney refers to as “aggregates”—for instance, the seventh chords in the clock line in Canon 9 (Fig. 1, sixth section).

[6] Carlsen points to this Study (along with Nos. 19 and 36) as one in which “It is significant that the exact midpoint on Nancarrow’s instruments is the note E4. . . . This has a direct bearing on the symmetrical structures Nancarrow sets up using every note on the instrument” (p. 5). The clock line itself, consisting of the pitches D#4, E4, F4, and Gb4, is almost evenly deployed about the piano’s central note, and thus it not only continuously marks time in the piece, but also constantly articulates the vertical axis about the center of the player piano keyboard. Symmetries also exist in the levels of imitation within some canons, as shown in Table 1. Fig. 4 shows the levels of imitation in each of the canons; these intervals form vertically symmetrical structures in Canons 1, 2, 3, 4, 6, 8, 9, and 11, while Canons 5, 7, and 10 are not quite symmetrical.
table 1

[7] The first three lines of the ubiquitous clock line are shown in Fig. 2. The clock line consists of two rhythmic values related 2:1, with the longer value articulating the stated 220 tempo. Thomas points out that the clock line never approaches temporal consonance with the other voices (p. 134), and since the rhythmic values in the canonic voices are always changing there are only random, non-consecutive, cases where these values match the clock line’s. All references to the clock line in the main analytical sources comment on its apparent total randomness of both pitch and rhythm throughout the piece, and any rhythmic or melodic patterns or non-musical associations that Nancarrow may have incorporated into this line are so well hidden as to be indecipherable. I even entertained the possibility that Nancarrow might have created the rhythm by using a system such as Morse code to encode a text in this line; however, in Morse code the duration of a dash relates to a dot by the ratio 3:1 and the system includes breaks equal to a dot between elements of a character, the length of a dash between characters, and the length of seven dots between words.(2) Since the only relative durations in Nancarrow’s clock lines are lengths of 2 and 1, this system could not have been used. Short repeated pitches appear fairly frequently in the clock line, with the first set of three repeated notes articulating E4, the piano’s central note. The longest series of short notes in the line is nine, and of long notes it is five.

[8] Despite the overall appearance of randomness in both pitch and rhythm in this ever-present line, it still seems likely that some type of patterning must have been involved in its creation. For one thing, as Pickover (1995) has noted, it is quite difficult for humans to generate truly random sequences:
Sit down at a computer keyboard . . . and randomly press the 1 and 0 keys. Try to make the string of numbers as “patternless” as possible. In other words, try to generate random numbers. [Pickover then gives a 6-line example of 1’s and 0’s that he generated.] . . I have tested dozens of supposedly random sequences typed by colleagues, and found out that it is curiously difficult for humans to type patternless sequences. To start with, we would expect about a 50 percent occurrence of each digit, and I did amazingly well. The [example I typed] had 49.3 percent 1’s and 50.7 percent 0’s. Next, we would expect 25 percent occurrences of the following pairs: 00, 11, 01, 10. In fact, doing my best to make a random sequence, I produced 15 percent, 13 percent, 36 percent, and 36 percent occurrences, respectively. Apparently my fingers preferred to oscillate rather than producing doublets such as 00 and 11. Perhaps I was trying to avoid clumpiness of digits, when in fact strings of identical digits should exist in a truly random sequence. (Pickover 1995, 233)
In creating the clock line, then, Nancarrow would not only have been faced with this difficulty in creating random rhythmic values, but randomness among the four pitches. Perhaps he created a 4 x 2 matrix (four pitches by two rhythmic values) and threw darts at it!

[9] The percentages in the Study’s subtitle are used to create both accelerating and decelerating series of rhythmic values. Table 2 shows the multiplication factors needed to derive successive values of the series. The factors can be applied over a series of accelerating or decelerating values by using successive multiplication; for example, an accelerating series of twenty values (i.e., a series of nineteen value changes) at 6% that begins with a rhythmic value of 100 millimeters will end with a value of 100 • 0.943419, or 33.1 millimeters, while a decelerating series of twenty values at 6% that begins with a rhythmic value of 20 millimeters will end with a value of 20 • 1.0619, or 60.5 millimeters.

table 2

 

Sections of the Study


Canons 1 and 2


[10] Study No. 27’s first CP concludes the section consisting of these two canons. Canon 1 is one of only a few canons in the piece in which more than one basic rhythmic value is used, and in this way it forms a rhythmic parallel to the clock line. Canon 2 forms a pitch parallel to the clock line in that each of the four canonic lines, like the clock line, consists of four contiguous chromatic pitches. Both Canons 1 and 2 (as well as 3 and 4—see Table 1 and Fig. 4) have symmetrical patterns of levels of imitation between the voices. In Canon 1, the first two voices (A6%/R6% and A11%/R11%) finish stating the canon before the other two voices enter; in Canon 2, the four voices enter in succession so that the canon is eventually being stated in four voices at once.

[11] Thomas asserts that “the canonic and ostinato layers seem completely unrelated and unrelatable” (p. 133); in the first canon, however, rhythmic values are sometimes halved and doubled to create a 4:2:1 rhythmic duration relationship, and this partially parallels the 2:1 relationship of the values in the clock line. Because both accelerative and decelerative processes are used in this canon, Jarvlepp (1983–84, 220) observes the simultaneous existence of three different kinds of tempo layers: steady tempo, accelerating, and decelerating.

[12] The pitch content and relative rhythm of the lowest voice of Canon 1 are shown in Fig. 5. The first six notes—with the first three in the equivalent of quarter notes and the next three in eighth notes—along with the first rest, serve to set up a pattern that divides the line primarily into trichords. This trichordal division is maintained throughout much of the canon through rests and changing note values. The most prevalent pitch-class set represented is 3-4 [015], while sets 3-2 ([013], Carlsen’s “partitioned minor third”), 3-3 [014], and 3-9 [027] are presented three times each; melodic movement by half steps and P4/P5 prevails. The canon’s concluding pentachord, 5-Z18, is a subset of both the opening hexachord 6-Z10 and the hexachord that precedes it, 6-Z11.

[13] Fig. 6 shows a section from Canon 1 where the canon is in two voices. This passage is a rare instance in Study No. 27 where the voices are close enough together in stating the canon that the echo distance is audible. There are two leader-follower switches on this page, and these are made more clearly audible because of the rests that accompany the switches. The passage is especially interesting because of the symmetry of the intervals between the voices at these points.

[14] Canon 2’s pitch content is, like the clock line, limited to four contiguous chromatic pitches, as shown in Fig. 7. The canon concludes with a CP on the attack of the final pitch, which is the same as the first pitch of the canon’s pitch cluster (a minor 7th chord on E); these pitches, as shown in the figure, create a symmetrical pitch structure about the D# in the clock line.

[15] Thomas finds the CP at the end of this canon to be remarkable in its swift movement from extreme temporal dissonance to consonance:
What is so remarkable about this canon is its single-minded pursuit of convergence, and its rapid attainment of it. Within the span of approximately eighteen seconds the four related but temporally clashing canonic voices resolve their dissonance through their convergence upon a simultaneity, which represents consonance. Because the voices not only start at different speeds but also proceed at different rates of change, the convergence is dramatically faster than in a tempo proportion canon: the pace of the process itself is magnified through the use of gradually changing speeds. (Thomas 1996, 135–36)

[16] As Thomas notes, the four voices in the canon begin at “different speeds” (i.e., with different initial durations); however, the length of the final duration (the CP) is the same in all voices. Table 3 reports the actual and calculated measurements in millimeters of the twenty-three decelerating pitches in Canon 2; the final note in the series represents the CP, which occurs at the attack of the final note in the canon. Within a certain amount of tolerance, there is generally good agreement between the actual and calculated measurements. The canon’s final pitch, the CP, is 108 millimeters long in all four voices, and this value happens to relate to only one of the voices: it is roughly double the value in the series at that point in the 5% voice.

[17] It is interesting to speculate how the CP at the end of Canon 2 might have represented a greater achievement in convergence had the opening values in each voice been such that 108 was double the value in the series at the CP. The bottom part of Table 3 shows the calculated values for opening durations that would have resulted in this “durational convergence” on the value 108 at the CP. The resulting values in the 6%, 8%, and 11% voices are closest to the values that occur in the third note of each series (shaded area in table). It would have been a simple matter for Nancarrow to begin the 6%, 8%, and 11% voices with these values in order to facilitate this durational convergence, if he had chosen to do so.

table 3
 

Canons 3 and 4


[18] These two canons conclude the first section, in which the clock line is still iterating single pitches (Fig. 1). In Canon 3, the voices above the clock line (8% and 5%) first decelerate and then accelerate, while the voices below the clock line (6% and 11%) do the opposite. In Canon 4, each voice first accelerates and then decelerates, with the fastest-changing voice (11%) at the bottom and the slowest (5%) at the top. Fig. 8 shows Canon 3’s 23-note canon as it appears in the first voice (A6%/R6%). As Gann notes (p. 160), the canon has a narrow pitch focus of a major sixth. The canon’s opening contour is similar to that of Canon 1, and there is a fairly strong implication of first E major and then B minor in the last half of the canon.

[19] The halfway point of the canon, the twelfth note, is where the A6%/R6% voice is notated to switch from acceleration to deceleration, but in the score the R5%/A5% and A11%/R11% voices are marked to switch on the note prior to this, and the R8%/A8% voice on the note following the halfway point. Besides the fact that this seems a violation of the spirit of the canonic process, there are other reasons to believe that all the voices should be marked in the score to switch at this halfway point. Based on measurements in the score, it appears that the tempo change process occurs as follows: ten duration changes take place in one direction (either accelerating or decelerating), with the duration of the eleventh note used again for the twelfth note, followed by ten duration changes in the opposite direction, with the final (23rd) note a longer duration which is apparently unrelated to the prevailing duration series. If the switch takes place on the middle note and the same number of duration changes is taking place on both sides of this point, the opening value and the value preceding the long concluding note should be roughly the same. Indeed, this does seem to be the case with all four voices, leading to the reasonable conclusion that the switch in each voice should be marked at the twelfth note of the canon.

[20] Canon 4 unites an organic principle of organization in both pitch and rhythmic duration, organizing both around five-note groupings separated by barlines. The melody’s five-note pattern is a sort of major/minor upper trill, including a total of 24 groupings and 16 iterations of the trill figure as shown in Fig. 9; groupings of one to four of these “trills” are followed by a single short note. Fourths/fifths and triadic structures appear to dominate the melodic movement between the first notes of each “measure”; the trill in the fourth and fifth measures spells out the clock line’s tetrachord, and several other groupings include the chromatic tetrachord. The deceleration portion of the canon is two groupings longer than the acceleration portion.

[21] Like the pitch changes, the rhythmic durations of this canon change with each five-note grouping. Fig. 10 shows a representative portion of the score and duration measurements from this section.

 

Canon 5


[22] Canon 5 introduces two related texture changes: the canon includes many half-step trills, and the ostinato line is reinforced by octaves and registrally divided with the bottom half-step pair (D#/E) in the bass register and the top pair (F/Gβ) in the treble. Like Canon 4, this canon’s melody is arranged with variable groupings of trills followed by short notes.

[23] Fig. 11 shows a passage from Canon 5 where there is a near temporal convergence among the 11% and 8% voices and the ostinato. The switch to “rit.” is almost simultaneous in these voices (although in the 11% voice the switch is marked one measure earlier than in any of the other voices). At the ritardando, the 11% voice’s tempo approximates 880 and the 8% voice’s 440, while the ostinato is maintaining its steady 220 tempo, for an overall tempo ratio of nearly 4:2:1.

[24] Canon 5 concludes with the Study’s second CP, with the convergence taking place at the release of the final note. The convergence at the note release is made the more striking by the sudden change in dynamic from f to p and the change back to a single note texture in the clock line immediately following the CP. The final values in each voice at the CP are not equivalent and thus not a “durational convergence,” but in each voice the durations of the notes at the CP appear to be close to the next value in the deceleration series taking place at that point (although in some places, particularly in the 11% voice, the note lengths in the score do not match the series very well).

 

Canon 6


[25] At Canon 6, the ostinato line returns to its original single-note texture, and the dynamic level drops from Φ to Π. As shown in Fig. 3, each voice first decelerates in the bass register below the clock line, and then, in tempo, leaps five octaves plus a minor sixth to the treble register above the clock line for its accelerative portion. This idea of registral separation is an extension of the texture used for the clock line in Canon 5.

[26] Canon 6’s melodic line is shown in Fig. 12. Thanks to the registral separation within the canon, it covers the widest range in the Study—in fact, the lowest note of the lowest voice (8%) occurs at the last note of the ritardando portion and is the lowest note on the player piano keyboard (B0), and the highest note of the highest voice (11%) is at the end of the acceleration and is the piano’s highest note, A7. Nancarrow has thus included a rough parallel between frequency and speed—the slowing pitches sink to the lowest register and the accelerating pitches climb to the top of the keyboard.(3) Combined with the very close intervals of imitation (M2 – m2 – M2: another expression of the intervals in the clock line), this texture is an extreme contrast to any heard in the Study thus far. The tempo change scheme in Canon 6 is very straightforward, and each voice begins and ends with the same durational value.

[27] The canon opens with a chromatic descending line in which the intervals progressively widen (five m2’s, two M2’s, three m3’s, and a M3) in a manner reminiscent of the overtone series in retrograde. The first and last thirteen notes of the canon are the same melodic series in retrograde inversion; between these extremes, melodic movement again favors intervals of fourths and fifths. This canon’s use of retrograde inversion at the beginning and end of the melody and the use of the entire range of the keyboard highlight the significance of Canon 6 as the midpoint of the Study.

 

Canon 7


[28] Canon 7’s melody consists of sustained and staccato notes, all of which remain in tempo. The texture thickens dramatically in this canon. The clock line expands to major and minor thirds, and the canonic lines are doubled five octaves apart—a simultaneous (vertical) expression of the registral separation idea that was sequential (horizontal) in Canon 6. And, like Canon 6, which covered the entire range of the player piano keyboard, Canon 7 covers almost the entire range (minus the very top note). The extreme pitches, the B0 in the lowest octave of the 11% voice and the G#7/Aβ7 in the highest octave of the 8% voice, are reiterated over and over again, particularly in a passage of compound melody near the middle of the canon. The compound melody consists of the extreme registral pitch plus a descending chromatic line; these two extreme lines of compound melody are shown in Fig. 13. Note that the extreme repeated pitch is a long note and the other notes are staccato during the chromatic descent, while the repeated pitch is a staccato note (and its spelling changes) and the others sustained during the chromatic ascent section.

[29] As shown in Fig. 3, the tempo change pattern in the 8% and 11% voices is accel.–rit.–accel.–rit., while in the 5% and 6% voices it is just accel.–rit.; this is the only canon in this piece in which a different number of changes takes place in different voices. The 98 pitches of the canon are divided differently in each voice between acceleration and deceleration, with the pitches divided as follows:
8% voice:    A = 36    R = 34    A = 17    R = 11    /    = 98 total
5% voice:    A = 46    R = 52            /    = 98 total
6% voice:    A = 49    R = 49            /    = 98 total
11% voice:    A = 24    R = 27    A = 25    R = 22    /     = 98 total
 

Canon 8


[30] From here to the end of the piece, the percentages stay in voice order: first from slowest change (5%) in the bottom voice to fastest change (11%) in the top voice in Canons 8 and 9, reversing in Canon 10, and reversing back in Canon 11. This canon’s main textural idea is the minor/major triad. The clock line here expands to major and minor triads: the D#4 and E4 are the fifths of the G#/Aβ minor/major and A minor/major triads, while the F4 and F#/Gβ4 are the roots of F minor/major and F#/Gβ minor/major triads, forming a complex of eight triads in the clock line. The minor/major triad is also spelled out by the levels of imitation (B, D, D#, and F#—see Fig. 4). Additionally, occasional half-step trills take place in the canon melody on the third of the triad to create alternate minor and major triads (except for the final trill, which alternates between the major third and the fourth scale degree).

[31] Fig. 14 shows the canon melody from the bottom line (R5%/A5%). Canon 8 contains two rhythmic values in the ratio 2:1, a feature that was previously used only in Canon 1. The lowest note on the keyboard, B0, that was prominent in both Canons 6 and 7 is just as much so here: the lowest voice enters first, on a B major triad built on this note that is repeated twice more. The melody in the lowest voice is tonally organized around B, with the first section (to the point where the tempo direction changes) concluding on a supertonic triad, which near the middle of the second half of the melody becomes a minor/major triad and alternates in function between the supertonic and the V/V, finally becoming (enharmonically) V/V moving to V at the very end. In the highest voice (11%), the keyboard’s highest note (A7) is stated four times.

 

Canon 9


[32] Canon 9 is rhythmically the most complex canon, encompassing note values related by the ratio 4:3:2:1. Fig. 15 shows the canon melody as it would appear in rhythmic notation in the lowest voice; the treble and bass registers in each voice are rhythmically coordinated but there is never a simultaneous note attack between them. The trill in some triads in the treble register involving the chord third and the fourth degree is retained from the end of Canon 8.

[33] The clock line in Canon 9 states various kinds of seventh chords, again with D#4 and E4 at the top of the chord structures containing those notes, and F4 and F#/Gβ4 at the bottom of chords containing them. The levels of imitation in Canon 9 are the same as the pattern in the clock line: m2 – m2 – m2 – m2, descending in the bass from the highest voice’s D1 to the keyboard’s lowest note, B0, in the lowest voice, thus filling out the chromatic tetrachord below the clock line. The most extreme high pitch in the treble of the highest voice is Aβ7, just below the top of the keyboard.

[34] Like an arch canon, Canon 9’s voices are placed almost symmetrically about the durational center of the canon, but, unlike the arch canon in the middle of Study No. 22, the symmetrical placement of the voices here does not allow for a convergence in the middle because of the constantly changing tempos.


Canons 10 and 11


[35] The final two canons are each very brief and both use only acceleration. Canon 10’s 21-note melody is an ascending chromatic scale extending an octave plus a minor sixth, against a clock line that has returned to its original form of four chromatic pitches. The intervals of imitation progressively increase by a minor second: the interval between the top two voices is a perfect fourth, between the second and third voices an augmented fourth, and between the third and fourth voices a perfect fifth (see Fig. 4). The beginning pitches of the four voices—C#, G#, D, and G—add two notes above and below the chromatic space occupied by the clock line, and the A that concludes the first voice and the Bβ that concludes the third voice add two more pitches to the chromatic space.

[36] At the conclusion of Canon 10, both the clock line and the lowest voice of the canon conclude on the note Eβ, which is then expanded to four octaves and repeated four times in the clock line; the clock line states only the major second Eβ–F in Canon 11 (with the exception of the final two notes—see Figs. 3 and 4).

[37] Canon 11 is the most tonally-oriented canon in the Study; all the intervals of imitation are octaves, and the pitch content is entirely G major. The major second that is being expressed in the clock line is imitated in the canonic lines, where every note is expressing a major second trill (thus, no trills involving the intervals B–C and F#–G appear in the canon). The final CP (see Fig. 16), unlike the first two, incorporates the clock line and occurs on the final staccato attack; thus there is no opportunity for a “durational convergence” as there was in the first CPs.

Summary and Conclusions


[38] In Study No. 27, Nancarrow was able to express the same ideas in several different contexts, sometimes sequentially and sometimes simultaneously. The clock line’s chromatic tetrachord, for instance, is manifested in the pitch content of individual voices in Canon 2, in the opening pitches of the canonic voices in Canon 6, and in the levels of imitation in Canon 9. As another example, the concept of registral expansion moves through three consecutive canons: first in the clock line in Canon 5, then in the canonic voices of Canon 6, and then a simultaneous (vertical) expression of this idea in Canon 7.

[39] Thomas found Study No. 27 to be “decidedly non-tonal” (p. 17), but several tonal areas do achieve some significance. The final tonal goal is clearly G, the note immediately above the chromatic tetrachord of the clock line; G major is stated unequivocally in all the voices of Canon 11 and this canon is a jarring contrast to the tonal aimlessness and ambiguity of the rest of the Study. G is established almost immediately in Canon 1, being the first pitch in the lowest voice, and the lowest voice in Canon 2 begins on this same pitch. By the conclusion of Canon 4, however, with its F#–B final interval in the lowest voice, B plays a more prominent role—particularly with the emphasis on B0 in the bass, which reaches a peak in the compound melody section of Canon 7. The filling out of the chromatic tetrachord below the clock line (D, C#, C, and B) in the bass register of the four canonic voices of Canon 9 provides a final emphasis on B before a G tonal center is reasserted in Canons 10 and 11.

[40] Study No. 27’s convergence points do little to support the establishment of these tonal centers. Each of the CPs is structured differently. The first two CPs involve only the canonic voices, take place at the end of a decelerating section, and end with sustained notes; the first CP occurs at the beginning of this note, and the second CP is at the end of the sustained note. The first CP has a durational convergence on a value related to only one of the voices while the second CP’s final sustained note is of different durations and represents the next series value in each voice. The Study’s final CP, which concludes the piece, takes place in all the voices (including the clock line) at the end of an accelerating passage on the attack of the final staccato note; this CP also includes a pitch convergence on G.

[41] It is somewhat surprising to note in Study No. 27 how little control Nancarrow seemed to exert in areas where he had typically done so in many of his other tempo canons: for instance, in the establishment of tonal centers, areas of harmonic convergence, and various kinds of symmetry. There is also very little in this Study that could be considered a “collective effect”(4) other than the similar gestures in the canonic voices; these gestures are generally quite disjointed and rarely achieve a level of interdependence except at the CPs.

[42] When considering both Studies No. 22 and 27 together, it is apparent that Nancarrow had far less success in these two geometric acceleration canons in creating collective effects, symmetry, and harmonic convergence than he did in a piece such as Study No. 8, which is based on arithmetical acceleration. The arithmetical acceleration technique allows for a common background unit, while geometric acceleration does not. Without the common background unit, it is nearly impossible to overcome the independence of the voices to create any sense of interdependence. Nancarrow must have felt this, and one is left to wonder how satisfied he was with the results of the acceleration canon technique when considering Gann’s observation (p. 163) that Nancarrow wrote no more acceleration canons after Study No. 27.



REFERENCES


Gann, Kyle. The Music of Conlon Nancarrow. Cambridge: Cambridge University Press, 1995.
Jarvlepp, Jan. “Conlon Nancarrow’s Study #27 for Player Piano Viewed Analytically.” Perspectives of New Music 22/1–2 (Fall–Winter 1983/Spring–Summer 1984): 218–22.
Pickover, Clifford A. Keys to Infinity. New York: John Wiley & Sons, 1995.
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.




(1) Although Gann based his structural diagram on the first four canons comprising the first section because of the non-changing texture of the clock line, one could also make a case for the first section ending at the convergence point at the end of the second canon, and the piece as a whole being in three sections as demarcated by the CPs.
(2) See http://morsecode.scphillips.com/morse.html.
(3)  A hallmark of Nancarrow’s tempo canons is voices arranged so that higher voices are faster and lower voices are slower.
(4)  These are what Gann called areas in Nancarrow’s tempo canons where figures such as glissandi are first widely spaced and separated and, as a CP approaches, they become closer and closer together until they fuse into a single gesture.