Implied Polyphony in the Unaccompanied String Works of J.S. Bach:
A Rule System for Discerning Melodic Strata
Northwestern University School of Music
Research into the existing writings on J.S. Bach's unaccompanied string works reveals generally unenlightening attitudes and comments. For the most part, these writings raise this music up to an almost mythical status, containing phrases such as "unrivaled wealth", "infinite inventiveness", "incomparable challenge", and "unsurpassed masterpieces". As one author states,
A monumental challenge to violinists for over two hundred and fifty years, Bach's 'Six Solos for Violin without Bass Accompaniment' . . . happily show no signs of age and offer no hint whatever that their quasi-hypnotic sway over players and public will ever diminish. The reason is not hard to seek. Such music, and by such a master, would command and rivet attention no matter how it might be scored; but scored as it is for an instrument with but four strings (and the smallest member of its family at that), this set of sonatas and partitas calls for a mastery of technique and a gift of sheer virtuosity the like of which had never been known before Bach's time, and has rarely been exceeded since (Stevens 221).
This type of description certainly reveals the reverence that various writers have for this music, with much talk of their grand reputation, technical demands, and unique place in the string repertoire. Rarely, however, do they go on to describe in any depth the specific structural characteristics that might have brought about this great esteem or the influence that these features might have on both an expressive performance and a perceptual experience. This paper will respond to this lack of adequate research by addressing one distinct structural feature of this music through the use of both analytical and experimental research methods, all as a means of gaining greater insights into the unique characteristics of these works and acquiring a more comprehensive understanding of the ways in which this structure influences both performance and perception.
One of the most striking and oft discussed characteristics of these pieces is their polyphonic structure. Although there are certainly movements within these sonatas that reflect the predilection of 18th century German composers for multiple stops in solo string music, the majority of these pieces are almost completely monophonic. Still, countless performers, pedagogues, and theorists maintain that there is a sense of polyphony in these movements, and that Bach created counterpoint by outlining multiple voices within a single instrumental line. In a 1968 dissertation entitled Heinrich Biber and the Seventeenth Century Violin, Elias Dann made the following reference to Bach's unaccompanied violin pieces.
Any superficial examination of these solos, the most polyphonic pieces ever written for the violin, will reveal so many single notes rather than double-stops or chords that the musician unacquainted with these works (a hypothetical one, if necessary), may well wonder where the polyphony is to be found . . . If these movements, in which only one tone at a time is sounded, are to be considered polyphonic, it becomes obvious immediately that no usual definition of polyphony, predicated upon the combination of several sustained parts, will suffice. Any attempt at a coherent analysis from the standpoint of melody alone soon raises more questions than it can possibly answer. Careful study would seem to suggest that either these movements are polyphonic or they cannot be explained at all (196-97).
From this and many other similar comments, it is apparent that a general consensus about the existence of this implied polyphony has been reached. There is, however, very little explanation of how this monophonic music is actually parsed into these different voices. The few explanations that do exist typically only include a short excerpt of music (most often taken from some Bach instrumental piece), a brief definition of what is typically called "compound melody," and a diagram showing the melodic line separated into multiple voices. But there is rarely any description of how this separation was determined or any consistent statement about which specific musical features contributed to that particular parsing of the melodic line (cf. Piston, Kennan).
General principles of auditory stream segregation may help to explain some cases of this type of linear polyphony. In many ways, these principles coincide with the fundamental claim of Gestalt psychology. As Lerdahl and Jackendoff describe, this claim is that "perception, like other mental activity, is a dynamic process of organization, in which all elements of the perceptual field may be implicated in the organization of any particular part" (Lerdahl and Jackendoff 303). Two Gestalt principles that seem especially applicable to auditory perception are proximity and similarity. Proximity is basically the idea that listeners tend to perceptually group elements together that are closer to one another, while similarity refers to the tendency to group elements of similar shape or other likeness together. According to Albert Bregman, auditory stream segregation seems to follow most directly from the Gestalt law of grouping by proximity (20).
In audition, the two most important influences on the segregation of tones by proximity are the rate of the sequence and the frequency separation of different elements within the sequence (Bregman 643). Separations of this kind represent a bottom-up type of processing, with emphasis on the more detailed, note-to-note level of the music. A grouping determined according to similarity, however, can work on a variety of different levels. Examples in music might include the grouping of instrumental sounds by timbre (low level processing) and by motivic parallelism (larger level processing).
In creating melodies, composers have long realized the influence that these grouping principles have on perceptual coherence, especially the repetition rate and the frequency separation of tones. For instance, various studies have shown that much Western music is dominated by small melodic intervals, thereby reflecting the idea that notes closer together in frequency tend to produce stronger perceptual groupings (Ortmann 7). Even though many composers seek to achieve this melodic coherence by avoiding any extended use of those features which are apt to create segregation, others choose to purposely maximize the tendency for tone sequences to break apart (given a sufficient degree of frequency separation, for instance). In an interesting reference to the very style of music that Bach's unaccompanied string pieces represent, Bregman states,
Rapid alternations of high and low tones are sometimes found in music, but composers are aware that such alternations segregate the low notes from the high. Transitions between high and low registers were used by the composers of the Baroque period to create compound melodic lines - the impression that a single instruments, such as a violin or flute, was playing more than one line of melody at the same time. These alternations were not fast enough to cause compulsory segregation of the pitch ranges, so the experience was ambiguous between one and two streams. Perhaps this was why it was interesting (675-76).
The previously mentioned dissertation by Elias Dann contains one of the few published attempts to separate one of Bach's monophonic movements into multiple voices. Dann bases his analysis on the assumption that the melodic function of each individual tone in this music is dependent on the tones that surround it, its rhythmic placement within a measure or phrase, and whether its range ever crosses into the frequency space of another voice (199). He also points out that each individual tone has a dual role, first as part of a single melodic line simply because the tones actually are heard one after the other and then as a member of one of the many polyphonic lines that can be followed throughout the course of the piece (212-13). Dann then provides his interpretation of how this music could be separated into multiple voices, using the opening four measures of the Sarabande Double from Bach's B minor Partita as his material.
As can be seen in Example 1, Dann separates this brief excerpt of music into five different voices, with some instances of doubling reflecting times when two different lines have coincided or when a single tone functions as part of more than one polyphonic strand. At least initially, it seems that Dann's analysis is mainly determined by his interpretation of the opening three tones, a simple root position arpeggiation of the tonic triad. Although he does acknowledge that these three tones could be heard as a single entity due to the influence of harmony, he ultimately chooses to interpret them as "carving out an area of musical space in which they will start operation as three individual voices" (Dann 215). He then goes on to explain in detail the different lines that he sees emerging out of just the first tone. While it is seen as a sustained note in one voice that remains constant until the leading tone enters on the downbeat of the third measure, it is also considered the beginning point of both an ascending and a descending voice that he marks in the diagram with a : and an 6 , respectively.
While further explaining his particular analysis, Dann states the following:
In a polyphonic complex such as the one under consideration, no system of analysis can be expected to present more than a partial picture of the various voices and their interrelationship. The following analysis does not presume to be the only possible one, nor even to be entirely correct; it merely attempts to illustrate one way in which the inner ear may gather together the threads implicit in this piece of one-line polyphony . . . the five staves have been chosen for convenience, to bring out certain polyphonic relationships and not to argue that there are exactly five voices to be heard (215).
This brings up the important point of the perceptual relevance of this type of analysis and also returns us to issues of auditory stream segregation. Although Dann's interpretation does provide an interesting perspective on the implied polyphony based mostly on the harmonic and rhythmic functions of each individual tone, he admittedly has not taken into account how this passage might actually be heard by both performers and listeners.
One of the first issues that arises when principles of auditory stream segregation are applied to Dann's analysis is the way he parses the opening b minor arpeggiation into three different voices. In a 1975 study, Leo van Noorden presented subjects with an alternation of two tones in varying rates of repetition, with one tone remaining fixed and the other tone moving to various frequency differences. The subjects' task was to indicate the points at which the frequency separation became too large to hear one coherent stream and too small for separate streams to be perceived. Van Noorden essentially concluded that "the degree of association varies inversely as the pitch difference, or pitch distance", with streams played at high rates of repetition being heard as a single, coherent unity when the frequency separation was less than five semitones (13). It, therefore, seems unlikely that the opening arpeggiation of the Bach Sarabande Double would actually be perceptually segregated into three different strands based on frequency separations of only three or four semitones.
Another issue concerns the fact that Dann separated this brief passage into five total voices. In a 1989 study whose aim was to determine the number of simultaneously sounding polyphonic voices that a listener could identify and count, David Huron found that a threshold seems to exist at three voices. One subject even commented after the study that he found himself using two different techniques for determining the number of concurrent voices. The subject felt very confident in his ability to provide an accurate count when there was a small number of voices, but instead found himself comparing the density of surrounding textures and simply estimating the number of voices when the total number was greater than three. As Huron then concluded,
It appears that in the perceptual denumeration of sounds of homogeneous timbre, listeners do not follow the arithmetic sequence: one, two, three, four, etc. to infinity, but proceed in a manner similar to the counting language of the San bushmen: auditorily we may count: one, two, three, many - where one might admit only gradation of "manyness" rather than definite discrete values (378).
After taking into consideration both the lack of extensive research into this issue and the apparent disregard for fundamental perceptual tendencies, it is clear that a more detailed set of guidelines is necessary in order to shed greater light on this idea of implied polyphony. For this reason, a simple rule system was created to provide a concrete and consistent method for parsing these solo instrumental lines into multiple voices. This rule system focuses on bottom-up conditions, or note-to-note interactions, and intentionally does not take into account every possible musical parameter. For this reason, some voice changes will be blatant or obvious, while others will be harder to distinguish or perhaps not present at all. Some degree of such ambiguity certainly seems appropriate since even the most superficial glance at the movements consisting of mostly chords and multiple stops shows that Bach did not consistently maintain the same number of voices throughout a single piece. There are numerous instances were one voice seems to be suspended while other voices move around it, with that original voice only later reappearing for resolution and further melodic motion. It, thus, seems reasonable to assume that similar compositional techniques were applied in the monophonic movements. Again, this is something that Bregman recognized as stemming from fundamental principles of auditory stream segregation. As he stated,
The alternation of registers in a single instrument tends to produce a more ambiguous percept in which either the two separate lines or the entire sequence of tones can be followed . . . It is not certain that the composers who used this technique would have called for such an unambiguous separation of musical lines even if the players could have achieved it. It is likely that the technique was not used simply as a way of getting two instruments for the price of one, but as a way of creating an interesting experience for the listener by providing two alternative organizations (464-65).
In this rule system, weights are assigned to the transitions between different pitches based on the degree to which three basic features are present. These weights essentially only signify the extent to which these features might act in conjunction with one another at any single point of transition to suggest a clearer or more obvious change of voice. Transitions whose weight crosses a threshold of four points are seen to signal a change of voice, thereby generally ensuring that more than one of the following rules would be enforced. In order to facilitate the analysis of this entire repertoire, a computer program representing this rule system was created using the Humdrum Toolkit (Huron 1999). This program works from a file of each original score with all pitches translated into a succession of ascending and descending intervals.
Rule 1: Interval Size
Given a sequence of four notes (n1, n2, n3, n4), let Int2 = n3-n2.
If Int2 > 3, then score = Int2-3.
This first, and most important, rule centers around the parameter of pitch distance or interval size. As has already been discussed, it is assumed that larger intervals are more likely to signal a change of voice than smaller intervals. The perfect fourth was chosen as the dividing point between small and large intervals, with the previously cited van Noorden study providing collateral that there is some tendency for streams to divide perceptually at this interval. Points are then assigned according to the actual size of the interval, with points increasing by one with each expansion in diatonic interval size. This accounts for the fact that the voice changes become more obvious as the interval gets larger. In terms of this rule system, the presence of a large interval is absolutely required in order to even suggest a change of voice and to move on to the subsequent two rules.
Rule 2: Change of Contour Direction
Let Dir1 = the contour of Int1 (same applies for the contour of Int2 and Int3)
If Dir1 = "up" and Dir2 = "dn" (or vice versa), then score = 1.
If Dir2 = "up" and Dir3 = "dn" (or vice versa), then score = 1.
The second rule addresses the issue of contour, and more specifically whether or not one or both of the two notes that make up the large interval mark the point where the contour moves from ascending to descending (or vice versa). In other words, this rule accounts for the places where the sign of the slope of the contour changes between two successive intervals. One point is assigned per instance of a change in contour surrounding a given interval, thus making two points possible for this rule.
Rule 3: Influence of Conjunct Motion
Let conjunct = the number of consecutive instances of conjunct motion
If Int2 = 2, then conjunct+ = 1. If Int2 > 3, then score = conjunct and conjunct = 0.
The third and final rule considers the degree to which any given large interval is surrounded by conjunct motion on both sides. This takes into account the continuous or stepwise nature of the music, which contrasts with the discontinuous nature provided by the large leaps. It also accounts for the fact that two stepwise melodic lines separated by a large interval would more easily be interpreted as two different voices. One point is added for each instance of conjunct motion that precedes or follows a large interval, with directional changes still qualifying for points as long as the conjunct motion is not interrupted.
Another important issue in this repertoire is that of melodic continuity -- the question of whether or not a single voice can be followed throughout an entire piece. Although a melodic continuity rule has not yet been fully integrated into this simple model, this rule will eventually attempt to follow the path of each individual voice and connect every pitch with either a currently existing voice or a previously heard voice that was momentarily latent.
The computer-assisted implementation of this rule system is only a first step in the analysis of this music. Ultimately, this model serves as an aid in interpreting the results of various analytical and perceptual tasks that examine the degree to which listeners and performers are able to recognize this polyphonic structure. It basically provides a prediction or hypothesis against which the results of these activities can be compared, thereby also testing the validity of this particular set of rules.
One such activity required a class of fifteen sophomore music majors to analyze the implied polyphony in the Allemande from Bach's d minor Partita for Unaccompanied Violin. This is an almost entirely monophonic movement, with actual chords only occurring at the beginning and ending of each half of the piece. The students' task was to separate this single strand of notes into multiple voices and to then provide a detailed description of the musical features that most significantly influenced their decisions. In order to make their choices absolutely explicit, the students were required to reorchestrate the piece for multiple instruments (corresponding to the number of voices that they heard), which resulted in a single staff of music representing each implied voice.
Since the melodic continuity rule has not yet been fully implemented, the output of the rule system only provides information regarding the degree to which any interval might suggest a change of voice of some kind. Therefore, the first step in analyzing the results of this activity was to simply identify the specific places in the music where students most often notated a change of voice, regardless of which voice they placed that passage into within the overall texture. A quantitative analysis of the students orchestrations was then performed by marking each note in the score with the number of students that signified that note as the beginning of a new voice. These numbers can be found below each line of the score provided in Example 2. The upper row of numbers then represents the output of the model described above, with each score reflecting the degree to which interval size, contour, and conjunct motion combine to suggest a change of voice. The highest possible score for the students responses is fifteen, while there is no upper limit for the output of the rule system.
Since these two sets of scores are measuring two fundamentally different things, their significance does not ultimately lie in their actual numerical values. The numbers, instead, simply reflect the relative strength of any potential change of voice. In general, then, a comparison of these strengths shows a fairly high degree of correlation between the model's output and the students' responses. This correlation is most easily seen in the notes which have been circled in the score, which show places where both the model and at least 10 out of 15 students indicated an obvious change of voice. Due to the large leaps, the dramatic changes in register, and the instances of stepwise motion, the clearest example of this correlation is found in measures six and seven of the Allemande, where almost all of the fifteen students consistently indicated the same changes of voice that the rule system strongly suggests.
There are, however, places in the score which exhibit a much greater degree of ambiguity, where there is more discrepancy between the output of the model and the responses of the students. This is mostly due to the fact that the rule system described here essentially only analyzes note-to-note interactions in order to determine the extent to which bottom-up processing can adequately explain the implied polyphony in this type of music. Many other larger level structures and processes certainly influence the tendency for this music to separate into multiple voices, though. Based on the students' responses, rhythmic placement, motivic repetition, timbre, and articulation are some of the most powerful of these influences.
In measures four and five, for example, a majority of the students only indicated changes of voice in three places (the circled C, D, and E in Example 2). While these results certainly reflect the fact that larger intervals are more likely to suggest a change of voice than smaller intervals, they also reflect an awareness of articulation changes and motivic parallelism. It is evident that the students were recognizing the shift between slurred and separately bowed notes, as well as the repetition of a two beat motivic pattern (which is misaligned by one sixteenth note with the meter). Even though the model indicates additional voice changes inside this repeated pattern, fewer students chose to include them as part of their analysis. Perhaps this is because these changes are only influenced by the note-to-note details that the model addresses, not by either one of the larger level groupings that the students were taking into consideration.
One instance of the influence of meter then occurs in the sequence which begins in measure eleven. A large majority of the students consistently noticed that the first note of each leg of this sequence creates a descending melodic line and chose to signal a change of voice prior to each of those notes. The rule system also marks the same points as changes of voice, mostly because of the descending sevenths and the change of contour that occurs between each leg. It is interesting to note, however, that there is a corresponding descending melodic line occurring in the last note of each leg of the sequence. Perhaps this line was not recognized by as many students as a change of voice because it is not strengthened by a metric accent, thereby supporting Leonard Meyer's idea that an accent is "a stimulus (in a series of stimuli) which is marked for consciousness in some way" (8).
Although these larger level issues have certainly been recognized and addressed to some degree, their formalization as part of the model set forth here is still forthcoming. Perhaps the fundamental value of this model as it currently stands thus lies in its contrast to the ways that this music is typically discussed, especially by performers and pedagogues. As a final example, noted violinist and pedagogue Yehudi Menuhin stated,
Even though there can be no allowance in the music of Bach for arbitrary effects, personal indulgence, or changes of direction, as there are indeed in the romantic literature, there is every justification for a flexibility, a fluidity of line, a play of accent, colour and stress within a given series of notes, but only of course when these are justified by a sensitive and disciplined musical intuition and by an intellectual awareness . . . For instance, although many of Bach's movements for solo violin and particularly for 'cello are written in one voice, that is without counterpoint and harmony, the counterpoint and the harmony are in fact implied and every effort must be made to bring the different voices out clearly, even though there is never more than one voice sounding at a time (119).
Despite the fact that this statement does make reference to both the existence of this implied polyphony and the need for flexibility and sensitivity in its performance, Menuhin does not actually describe how to identify this polyphony or to "bring the different voices out clearly." Perhaps this phenomenon is related in some way to basic ideas about automaticity, which suggest that someone who has mastered any technique or skill often has a difficult time adequately describing what he is doing and instead prefers to teach by demonstration or imitation. It is possible that many musicians have developed their skill to the extent that they no longer consciously think about what might be considered the fundamental technical aspects of the music, instead turning their thoughts to larger level groupings or phrasings. In turn, this results is an underestimation of the power of these note-to-note details that are the very focus of the rule system presented here.
For this reason, it is important to have some method which can assist in clarifying the issue, thereby helping to identify those features of the music that might otherwise be overlooked, taken for granted, or left unexplained. Although it is unlikely that any single system could fully account for all aspects of this intensely complex music, this study has shown that a model based on a small number of simple guidelines is actually a fairly powerful indicator of how musicians might interpret the implied polyphony in this piece. As performers and pedagogues become more aware of these specific polyphonic potentials, they also become more conscious of the expressive potential in this music. Ultimately, an informed performer has utmost freedom to either let the implied polyphony emerge on its own or to provide added emphasis through the use of a variety of expressive techniques, thereby making this structure more perceptually relevant to audiences of all kinds.
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Kennan, K. 1987 Counterpoint: Based on Eighteenth-Century Practice, 3rd ed. Englewood Cliffs, N. J.: Prentice-Hall.
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Stevens, D. (1976) "Bach's Sonatas and Partitas for Violin." In Violin and Viola. New York: Schirmer Books.
Van Noorden, L.P.A.S. (1975) Temporal Coherence in the Perception of Tone Sequences. Unpublished doctoral dissertation, Eindhoven University of Technology.
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