Notation in Elliott Carterís Double Concerto




Robert Strizich



The Double Concerto for Harpsichord and Piano with Two Chamber Orchestras of 1961 is perhaps the most instructive of Elliott Carter's recent works to study from a notational standpoint, since the composer apparently had ample opportunity to hear the piece performed many times before the score was put in its final form and printed. In fact, Carter has remarked that as a result of these multiple hearings prior to publication, he was able to try to make all the indications in the scores "as foolproof as possible")1


The effectiveness of any notation depends on its ability to convey clearly first to the players, and then in turn to the listeners, the ideas that the composer had in mind and to which he wanted to give musical expression. In a work as difficult and complex as the Double Concerto, the need for notational efficacy is particularly great if the thought underlying the structure of the work is to be accurately projected and perceived. Thus, it is interesting to examine the piece from a notational standpoint, in order to test, and hopefully verify, Carter's assertions of notational "foolproof"-ness.


The major aspects of Carter's notation are perhaps best discussed in the context of the basic structural procedure underlying the entire work - the overlay of separate musical streams. Therefore, the main part of this paper (Section HO will focus on the notation of these in the Double Concerto, and will comprise an investigation of: a) how streams are defined in musical terms, b) how streams are made clear notationally, and c) how tempo fluctuation - an important element in Carter's manipulation of musical layers - is treated from a notational standpoint. In examining the musical definition of streams, only the Introduction, Adagio and Coda from the Double Concerto will be considered, since all of the basic procedures pertaining to stream-overlay are to be found in these three sections.


After this explication, a critical evaluation of the notational procedures in the Double Concerto will be presented (Section IV), dealing not only with the work's notational strengths and weaknesses, but also with a particularly controversial aspect of his notation - the notation of non-duple subdivisions. This evaluation will also discuss a number of smaller notational details not mentioned in the previous section, drawn from the Allegro scherzando, the Presto and the Cadenzas, as well as from the three sections already scrutinized in Section III.


Possible notational improvements and alternatives will be proposed in Section V, and some observations on Carter's compositional realization of the basic structural concepts behind the work will be made in Section VI. Finally, some general conclusions will be made in Section VII.


However, in beginning this study of notation in the Double Concerto, it will be helpful to set the stage by examining briefly the background of the work. The following section will look at Carter's general compositional preoccupations prior to the Double Concerto and his specific concerns in writing the Concerto itself, and will also give a brief outline of the work's form.


Background of the Double Concerto


Elliott Carter is best known for works of great intricacy and rhythmic complexity, and his important Double Concerto is one of the best known and most frequently-heard works of this type. However, the composer's concern with the dimensions of time and rhythm and the simultaneous occurrence of various layers or streams of music is something that he has only systematically explored in roughly the past 30 years. Indeed prior to the 1950s, Carter's works were structured along basically traditional lines, and it was only during that decade that he became preoccupied with the more radical and experimental approaches to temporal and rhythmic organization that have typified his. works in recent years.


In a number of articles, Carter has discussed the development that h underwent in arriving at his present musical perspective. He has described h' preoccupation during the fifties as being "a desire to find a new flow of music thought and expression - a tendency to which the previous efforts seemed be leading."2 Disregarding the serialistic trends of post-WW II Europe, which he considered a "return to old-fashioned avant-gardism" that was not "really 'experimental' or advanced" with regard to musical time, he instead began to tread his own path towards "a more significant temporal thought."3


This interest in the dimension of musical time expressed itself in two different, but complementary, ways. The first was Carter's interest in "the speeding up and slowing down of themes,"4 that is, the development of musical structures in the temporal sphere by means of tempo manipulation; such procedures were already at work in the String Quartet No. 1 (1951) and the Variations for Orchestra (1954). The second was what he termed "a special dimension of time, that of 'multiple perspective,' in which various contrasting characters are presented simultaneously."5 Such "double and sometimes manifold character simultaneities" led, in Carter's opinion, to moments of music that were richer in musical expression than had theretofore been achieved, except, perhaps, in opera.6


Coupled with these interests in the temporal aspect of music were two additional attitudes towards the creation of musical materials. One of these has been described by Carter as a "preoccupation with reduction of musical ideas to their simplest terms";7 it is this tendency to embody an idea with the most basic material possible that leads, in the Double Concerto, to a musical thought being consistently represented by merely a single interval, or a length of time. A second attitude towards musical material which Carter has emphasized again and again was his concern that the musical "expression and thought arise from the unique sound and performance techniques" of the instruments for which he was writing;8 his concern was never that several instruments share the same musical material, but instead that each instrument play material idiomatically suited to its particular technique and sonic characteristics.


The gradual development in the 1950s of Carter's ideas regarding musical time and the shaping of musical materials gave rise to the creation of the significant body of works for which he is now noted. The composer's obsession with "multiple perspective" and idiomatic instrumental writing led quite logically to a repertoire in which each instrument played different music and in which several musical streams occurred simultaneously; indeed, the phenomenon of several simultaneous layers of music relating to one another at times dependently, at times independently - has been a hallmark of his music since the mid-1950s.


However, Carter has never been interested in an aleatoric relationship between musical streams; in fact, he has specifically repudiated the relating of these streams "in any one of the possible uncoordinated ways that have been used either by Ives or by others in recent years,- feeling that such procedures produce "a form of entropy, a degrading of the possibilities of communication."9 Thus, he always has insisted on notating as exactly as possible the temporal relationships between layers.


The Double Concerto itself was a response to a request from harpsichordist Ralph Kirkpatrick for a piece for harpsichord and piano.10 Carter saw his task as joining compositionally two keyboard instruments of quite differing characteristics, while at the same time maintaining their individuality:


The harpsichord and piano in the Concerto . . are each given music idiomatic to their instruments, meant to appeal to the imaginations of their performers and cast them into clearly identifiable, independent roles.11


This concern to create a compositional unity in which the two different instruments could also be given individual treatment suggested quite naturally the idea of two musical streams, one centered around each of the two soloists. An ensemble was then attached to each soloist, which took on the musical characteristics of the solo instrument. The ensemble associated with the harpsichord (called "Orchestra I") consisted of flute (doubling on piccolo), horn, trumpet, percussion (two players), viola and contrabass, while the ensemble associated with the piano ("Orchestra II") comprised oboe, clarinet, bassoon, horn, and percussion (also two players), violin and 'cello.


Formally the Concerto took on a symmetrical seven-part form:


Cadenza for Harpsichord Allegro scherzando



Cadenza for Piano



The Introduction employs all the instruments and introduces the musical material of the work. The Harpsichord Cadenza introduces material particular to that instrument, and the following Allegro scherzando features mainly the piano ant Orchestra II, with occasional interjections from the harpsichord and Orchestra I. The central section, the Adagio, develops accelerating and decelerating material in the percussion, strings and solo instruments against a relatively static, slow-moving music in the woodwinds and brass. The Presto features predominantly the harpsichord and its ensemble (with frequent commentary from Orchestra ID, while in the two piano cadenzas (one actually in the middle of the Presto, and the other at the end), that solo instrument develops further its own particular material. Finally, while the Introduction gradually presented the musical material of the work, the Coda achieves its gradual dissolution; regarding this cycle of musical formation-dissolution with which the Double Concerto deals, Carter has mentioned two classical poetic works which he feels represent literary analogs to his musical thought - Lucretius's De rerum natura, and Alexander Pope's Dunciad.12




Example 1: There should be as wid a space between the two orchestras as the stage allows.


The seating plan of the Concerto reflects its symmetrical form. The two "orchestras" are to be placed as far apart as the stage allows, with the strings and winds encircled by the percussionists towards the rear and the soloists in the front. The percussionists are arranged in a convex arc at the back of the stage and should be evenly spaced; the strings, placed just in front of the winds and the first and fourth percussionists, should also be evenly spaced. Thus, the winds are surrounded by an elliptical ring made up of percussion, strings,harpsichord and piano (Example 1). The separation between the two orchestras is designed to bring into audible relief the frequent antiphonal structures in the score, while the elliptical - almost circular - ring of instruments around the winds not only symbolically reflects the symmetrical formal plan of the work, but also gives a circular, swirling shape to the accelerating and decelerating patterns that surround the slow wind music in the Adagio.


Double Concerto: How Streams Are Defined Musically In The Introduction, Adagio and Coda


The Introduction deals with ten musical strands, five of which are assigned to the harpsichord and its ensemble, and five to the piano and its group. The strands are introduced one by one at the beginning of the movement, some dropping out for a time, while others continue. Those that have dropped out reenter later, and a climactic point is reached about half-way through the movement at which all ten strands are heard together, converging rhythmically in mm. 45 and 46. A dissolution follows, although after some antiphonal exchange between the two orchestras, the section again builds to another high point in mm. 93-99, just before the harpsichord cadenza takes over in m. 100. Thus, the Introduction concerns itself basically with the density fluctuation produced by the superimposition of from one to ten musical strands.


Each strand or stream represents the articulation of a specific length of time, which is repeated continuously like a rhythmic "wave." Thus, each strand can be associated with a metronome speed, the space of time between two clicks of the imaginary metronome being the time unit pertaining to that particular strand; the speeds range from 35 (the value of a dotted half-note at = 105, the tempo of the entire Introduction) to 17.5 (the value of a dotted whole note). Each time unit is articulated either by continuous sound (Example 2), absence of sound (Example 3), or by a detailed musical structure (Example 4); the detailed structures frequently comprise material of a specific density, often expressed by the repetition of a specific note-value (Example 5).


                                        Example 2: (perc, 2, mm. 7-8)                                             Example 3: (perc. 3, mm. 8-9)



                        Example 4: (m. 7)†††††††††††††††††††††††††††††††††††††††††††                         Example 5: (piano, m. 16, speed 21)


Furthermore, each speed is associated with a particular interval;  for instance, in mm. 11-12, the intervals of a minor and major second are heard together, being identified with the two simultaneous strands with speeds of 24.5 and 25 respectively (Example 6). A chart listing all ten speeds and showing their lengths and corresponding intervals is to be found in Example 7.

Example 6


Example 7


Two additional factors are pertinent regarding the musical definition of streams in the Introduction. Often, a particular timbre is identified with a stream; for instance, in mm. 16-17, speed 21 is identified chiefly with the piano, although there is some additional reinforcement from percussion 4 (Example 8). Also, changes of texture and/or density often mark the beginning of a rhythmic wave, as in mm. 37-38, where each "wave" of speed 35 is articulated by a different rate of activity (Example 9)


                                                                                                Example 8



Example 9


There are a number of different ways in which the wave-like structure of the musical streams in the Coda are articulated musically. Often the solo instrument or the entire "orchestra" will initiate a passage at the beginning of an accented measure (mm. 620, 624); sometimes the passage will be preceded by a short upbeat figure (mm. 626). In other instances, the soloist or the group will build up to an accented beat for several measures and then cease playing just after the beginning of the accented measure has been reached (mm. 628, 632). In a few cases, the soloist or the ensemble again plays a passage building up to the accented measure, but then ceases suddenly, not actually playing on the accented beat itself (mm. 640, 652).


Carter seems to treat his basic structure with more and more freedom as the Coda progresses: as the end approaches, beats which according to the system should be accented are frequently left unaccented (mm. 638, 643) or even empty (m. 648) It is possible that this increasing freedom is in some way connected with the feeling of gradual dissolution that the composer wishes to project as the work draws to a close.


How Streams Are Defined Notationally


Various notational devices are used by Carter in the Double Concerto to make the rhythmic structures in the work clear to both the players and the conductor, to ensure that the rhythmic structure will be heard by the listener, and to clarify the dynamic and metric relationships between the streams of music.


The rhythmic structure of a musical stream is often made clearer for the player - especially for the soloists - by the direction of the stems; for example, the upward-turned stem in the harpsichord flourish in m. 18 corresponds exactly with the beginning of the rhythmic wave of speed 29 1/6 (Example 10).



Example 10


In the central movement of the Double Concerto - the Adagio - the musical structure is somewhat simpler than in the Introduction, as there are basically two simultaneous musical streams; these two streams are differentiated by virtue of the relative stasis or mobility of the sound material. The woodwinds and brass play a relatively slow, static music which comes to a climax about two-thirds of the way through the movement (at m. 406) and then subsides. Over this background of relative stasis, the percussion, strings and soloists play accelerating and decelerating patterns which move in elliptical configurations around the winds in both clockwise and counter-clockwise motion; there are altogether six such patterns. Towards the very end of the Adagio, the mobile material is brought to its own climax (mm. 434-465) which features simultaneous high-speed passage-work by both solo instruments. In this climactic passage, the mobile music itself breaks into two separate streams, represented by the contrasting pitch materiai and configurations in the piano and harpsichord.


        In fact, the independence of the soloists is further underscored by two sections of this passage (mm. 434-5 and 453-65) in which the piano actually accelerates independently of the harpsichord and the remainder of the ensemble.


Still working with the concept of separate musical streams, the Coda employs yet another method of juxtaposing two contrasting layers of music. Here, each of the two orchestras articulates a stream of music, each stream consisting of a regularly recurring pattern of accentuation in "wave" fashion: the harpsichord group has main accents every fifth measure with subsidiary accents every thirty-fifth dotted eighth, while the piano group has main attacks every seventh measure and subsidiary attacks every thirty-fifth quarter note. Thus, the Coda has largely to do with the antiphonal interrelationships between the two main ongoing "waves," which are at first articulated by a high degree of activity and intensity at the beginning of the Coda and then gradually subside in activity and dynamic level as the end of the work approaches.


The independence of the two opposing musical streams is further underscored by the metrical structure of the Coda; while the music of the harpsichord group is in 6/8 time, the material of the piano group is cast in 3/4 time (because of this metrical independence between the two groups, Carter recommends that two conductors may be used for the Coda). Accompanying these interrelating waves are long rolls played by all four percussionists, which seem to proceed somewhat independently of the other instruments.


        In order to clarify the rhythmic structure for the conductor, Carter occasionally places composite rhythms in the score. In the Adagio, for example, both the pattern of the accelerating structure in mm. 421-33 and the specific instruments that are articulating this structure are shown at the bottom of the score on a separate line (Example 11). In the Allegro scherzando, the interrelationship of the opposing musical streams of Orchestras I and II is made apparent by the composite rhythm placed in the center of the score (Example 12).

Example 11



Example 12


A number of indications are placed directly in the music in order to ensure the proper dynamic articulation of a stream's rhythmic structure. The most common indications are dynamic markings and accent marks; in mm. 248-50, for example, notes expressive of the two rhythmic streams (indicated by the composite rhythms in the center of the score) are marked with accent marks and are indicated to be played at a higher dynamic level than the surrounding non-structural notes (Example 12).


        Another indication frequently used by Carter for the same purpose is the bracket ( ). Brackets often mark the note or notes that begin a rhythmic phase, as in the woodwind and strings parts of mm. 248-50 (Example 12); they also sometimes enclose not only the note that begins a wave but also the figure forming an upbeat to it, as is the case in percussion 3, mm. 7-8 (Example 13). Brackets occasionally designate longer structures delineating a stream; for example, brackets enclose an entire wave in percussion 2, m. 7 (Example 14), while in mm. 421-8 of the Adagio brackets enclose every element expressive of the accelerating structure indicated by the composite rhythm at the bottom of the score (Example 11).



Example 13

Example 14


Finally, Carter frequently uses verbal descriptions in the score to clarify characteristics of streams and interrelationships between them. For example, in order to regulate the dynamic balance of one of the orchestras, or the balance between the two solo instruments, Carter indicates verbally which solo instrument should be in the fore; thus, the piano is indicated as being predominant when it introduces in the Introduction the speeds 21 in m. 16 and 31.5 in mm. 23-4 (Examples 7 and 15), while the balance between the harpsichord and piano is controlled by several verbal directions in the brilliant cadenza-like passage in the Adagio, mm. 434-50. In the latter case, the verbal directions indicate whether the piano should be slightly louder than, or evenly balanced with, the harpsichord.

                                        Example 15


In addition to regulating dynamic balance, verbal indications also clarify metric interrelationships between the musical streams. In the Adagio, the contrast between the mobile material characterized by tempo fluctuation and the more static slow music in the winds is underscored at one point by Carter's prescription at the foot of page 100 of the score that "the accelerating pattern should be clearly in the foreground." Further on in the same movement, verbal directions on pages 102 and 108 make it clear that at these points the piano is to accelerate independently of the rest of the ensemble.


How Tempo Fluctuations are Handled Notationally


Carter makes use of two methods of notating tempo fluctuation in the Double Concerto: metric modulation and very precisely notated accelerandos and ritardandos. While the former method is found in several sections (the solo cadenzas, the Allegro scherzando and the Presto), the latter is found chiefly in the central Adagio.


Metric modulation is a method of handling tempo fluctuation that Carter has used in many of his works since the 1950s. It is, in essence, a simple procedure that has three basic steps. First, a specific tempo or pulse is established. Then a new rhythmic figure is introduced that moves at a speed either faster or slower than the speed already established. Finally, this new figure then becomes the basis of a new pulse or tempo. A clear example of metric modulation can be seen in Example 16, taken from mm. 109-114 of the harpsichord cadenza; here, in the space of a few measures, the tempo fluctuates three times, from q = 98 to q= 56 to q = 70. The first tempo, q = 98, is clearly established by means of the continual eighth-note motion in the left hand. Then in m. 110 a new pulse is introduced in the right hand, expressed by "beats" equal to seven sixteenth-notes at the established tempo. Then in m. 111, these new beats become equal to one quarter-note of the new tempo, q = 58. In m. 112, quintuplet sixteenth-notes are introduced, which then are interpreted in m. 113 as regular sixteenth-notes, thus establishing the third tempo, q= 70.



                                                    Example 16


In both the harpsichord and piano cadenzas, the music undergoes many such metric modulations. In the Allegro scherzando, there are altogether six tempo fluctuations handled by this procedure, while in the Presto, metric modulation is responsible for three tempo fluctuations before the first piano cadenza and for two tempo fluctuations after the second piano cadenza.


The second method for handling tempo fluctuation - found only in the Adagio - involves the precise notation of accelerating and decelerating patterns. There are altogether five such patterns in this section. Although the accelerandos and ritardandos are actually notated for the entire ensemble, the tempo fluctuations are heard only in the most active instruments - the soloists, percussion and strings - while the slower-moving instruments (the winds) sound as if they are playing a static background.


Carter perhaps could have chosen to notate these gradual fluctuations in the usual way, indicating that the music should increase in speed gradually by means of an "accelerando" marking. Instead, he has chosen to indicate an exact tempo-marking for every measure of the pattern, each measure having a slightly faster tempo indication than the previous one. This method has the disadvantage of seeming overly picayune, but has at the same time the advantage of indicating the exact rate of speed at which the accelerando is to proceed.


For example, between mm. 421 and 434, the music accelerates from quarter-note motion at = 70 to eighth-note triplet motion at = 140. Carter does indicate this gradual increase in speed by writing "Accel. poco a poco" at m. 421 and extending a broken line up to m. 434. However, underneath this general indication is a much more precise one which proceeds as follows. The speed of the quarter-note motion (beginning at q = 70) is brought up to just under the speed of eighth-note triplets at .1 70 in mm. 421-428 by giving each of the eight measures a slightly faster tempo marking (q = 70, 80, 92, 105, 121, 139, 159, 183). Then at m. 429, the eighth-note triplet motion at q = 70 takes over, still accelerating up to m. 434, each of the measures in this part of the pattern being also given gradually increasing tempo markings  q = 70, 80, 92, 105, 122, 140). By means of this graphically complicated procedure, the effect of a gradual increase in speed in the instruments playing the "mobile" music is achieved. The addition of the composite rhythm at the bottom of the score (already illustrated in Example 11) makes the intent of this accelerating structure quite clear.


A Critical Evaluation of Carterís Notation in the Double Concerto




In much post-World War II music, a great deal (perhaps too much at times?) frequently has been left up to the executant, with the result that performers have often become confused regarding the intent of some new music, exasperated by the ambiguities it presents, and, as a result, hostile to many recent compositional developments. Given this context, it can be seen as a

strong point of his work that Carter has taken such pains to convey his ideas as exactly as possible to the performer. Indeed, even the brief foregoing discussion of the notation in the Double Concerto should make it apparent that his notation is highly detailed, meticulous and fastidious. A number of additional notational factors - not previously mentioned because they had no direct bearing on the notation of separate musical streams - are present in the Double Concerto which reinforce the above impression even further.


        For instance, at the very beginning of the work, Carter gives a clear explanation of his method of notating non-duple subdivisions the first time they occur; in mm. 3-4 he indicates in percussion 2 and 3 that a sixteenth-note septuplet equals a half-note, and that a sixteenth-note quintuplet equals a quarter note (Examples 17aand b),   Also at the beginning ay a curved dotted line with an arrow ( ) indicates that  a written-out ritardando is occurring in mm. 2-3 of percussion 2 and an accelerando in mm. 2-4 of percussion 3 (Examples 18 a & b).


                        Example 17a: (perc. 2, m. 3)††††††††††††††††††††† Example 17b: (perc. 3, m. 4)


                                Example 18a: (perc, 1 & 2. mm. 2-3)


                                   Example 18b: (perc. 3, mm. 2-4)


Carter also beams fast notes according to metric placement (e.g., beat-by-beat) - rather than according to musical organization or phrasing - in order to make reading easier. At several points in the work, overall dynamic patterns (that might otherwise not be apparent) are indicated by verbal description or wedge signs in the middle of the score. At one point in the second piano cadenza, short vertical arrows are inserted over the staves to indicate the presence of several equal pulses that are to be heard over a metric modulation (Example 19).



 Example 19


Finally, two more examples of Carter's meticulous notation are to be found in the last two sections. Towards the end of the Presto in mm. 608-11, composite accents applying to both orchestras are indicated in the appropriate instrumental parts by means of short vertical arrows. In the Coda, the sign * is placed over the score at a number of points, indicating that at those instants the entire ensemble must perform their decrescendos just enough to allow the harpsichord to be heard.




In spite, however, of the commendability of Carter's effort to make his notation as explicit as possible, the notation in the Double Concerto presents a great number of rather serious inadequacies.


One fault may be the fastidiousness itself; the proliferation of meticulous detail requires so much of the performer from the standpoint of technique and expression, that the demands may be overwhelming. Very often, the requirements of the notation are so explicit as to seem overly fussy and even unreasonable; for example, in the Introduction (whose tempo is q = 105), Carter several times requires the players to subdivide a quarter note into exactly ten parts, rather than prescribing an unmeasured roll or tremolo - which would, of course, be the aural effect of his overly-explicit notation anyway (Examples 20 a & b). It is for such over-exactitude that he has been justifiably criticized by Gardner Read:


The Double Concerto, the Piano Concerto, the Concerto for Orchestra, and all three string quartets bring to mind Schoenberg's remark - applied to his own music - that such music must be realized with "inexorable severity." The players of Carter's works cannot "interpret" the music; they can only follow as best they can his minutely conceived and rapidly fluctuating indicia. The scores, as they stand, vividly illustrate the pitfall into which the over-zealous notator can tumble: their hyper-exactitude becomes more inhibiting than the most calculated obscurity.13



                                            Example 20a: (perc. 3, m. 16)                              Example 20b: (m. 12)


Furthermore, there are many details of Carter's notation which are confusing, unclear, or contradictory. Particularly disturbing is Carter's frequent use of brackets ( ) in confusing and redundant ways, in which the applicability of their defined meaning ("principle voice") is dubious. For instance, there are a number of places in the Coda where one quarter-note or eighth-note out of a long percussion roll is bracketed, in spite of the fact that the piano (and often other instruments as well) is playing at a higher dynamic level than the percussion in question (Example 21); in fact, in one such passage, Carter has expliditly indicated "Piano and Harpsichord to the fore" (Example 22). In other instances in the Coda, occasional notes in the harpsichord part are bracketed, without any apparent reason for these few notes to be more important than others in the surrounding texture (example 23). Such usage of the brackets not only contradicts the definition of their given meaning, but also taxes the patience of the reader of the score.

Example 21

                                    Example 22


                                       Example 23


In several places, Carter's verbal instructions also unclear. A case in point is the 12-measure passage in the Adagio (mm. 452-65) in which the piano gradually accelerates while the remainder of the ensemble gradually decelerates; showing a surprising lack of his usual fastidiousness, Carter does not indicate at what speed the piano accelerando should terminate, at what rate the accelerando should proceed, or whether the indicated alignment between the piano and the rest of the ensemble in the score should be adhered to. Another unclear verbal description is the composer's comment that a passage in the second piano cadenza should be played "very precisely and as one line (not two);" a glance at this passage (Example 24) will show that due to The rhythmic and aarticulative independence of both hands, it is very difficult to understand how the pianist could render these two lines as one.

                        Example 24


Also puzzling is Carter's contradictory use of short vertical arrows; whereas these are used to indicate equal beats over a metric modulation in the second piano cadenza (mm. 558-60), the same symbol is used at the very end of the Presto (mm. 608-11) to indicate simultaneous accents in both orchestras. While there is perhaps little danger of confusing the separate meanings of this symbol - due to the fact that it occurs the first time in a solo cadenza and the second time in a tutti passage - it nevertheless might have been better from the notational standpoint to have used two separate symbols.


Another possible weakness of Carter's notation is his insistence on metric beaming; while this type of notation has the advantage of giving the player a beat-by-beat picture of the music, it has the disadvantage of not effectively indicating the real rhythmic groupings of notes. Kurt Stone has commented on this problem, and has proposed two stages of modification for Carter's notation, each of which would make the musical intent clearer. he has taken a measure from the Introduction (Example 25a), in which typical examples of Carter's metric beaming are to be seen, especially in the flute and percussion 2. In his first re-written version, Stone proposes that the beams covering entire beats could be removed, resulting in a visually clearer picture of the rhythmic groupings (Example 25b). In his second proposed version, Stone has gone further and beamed the groups of notes according to their rhythmic grouping, in spite of the fact that this involved beaming across beats (Example 25c). This latter method certainly has the advantage of making the rhythmic groupings as clear as possible, but might make reading of such passages more difficult; indeed it seems as if Stone's first proposal best balanced the factors of legibility and graphic depiction of musical intent.14

            Example 25a††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† Example 25b


                                                       Example 25c


Perhaps the most serious fault of Carter's notation in the Double Concerto, however, is that the notation does not depict the various musical streams in a simple, visually clear manner, nor does it convey to the players to which stream they belong and how their contribution fits into the overall context. This problem is certainly present in the score, for it takes a great deal of reading of the composer's own comments, combined with detailed study of the music, before the conductor, student, or listener can perceive the underlying structure of the work. Matters are not any clearer in the individual parts either; aside from the usual cues, the parts are basically a reproduction of the individual lines from the score, and do not give the player any idea of how his line fits into the overall structure.15


Kurt Stone has also commented on this problem, taking as an example a passage from Carter's String Quartet No. 2, whose visual complexity, though considerable, does not rival that of the Double Concerto (Example 26a). Stone then goes on to show how each of the individual lines in the passage could be notated in a far simpler, more visually direct manner, were it not for the problem that all four lines must be coordinated (Examples 26 b, c & W.16 Unfortunately, such vexing notational dilemmas are perhaps unavoidable to some extent, given the style of notation and the medium of expression with which Carter has chosen to work. Indeed, this problem is, at least in part, the inevitable result of using traditional notation and uniform tempo control for an entire instrumental ensemble.

                                                                         Example 26a


                        Example 26b: (yin. II)                                        Example 26c: (via.)


                                      Example 26d: (cello)


A Controversial Aspect of the Double Concerto: The Notation of Non-Rational Subdivisions


One aspect of Carter's notation in the Double Concerto that has excited considerable controversy is his notation of quintuplet and septuplet subdivisions of beats - often termed non-duple, "artificial" or "irrational" subdivisions. Carter himself has alluded to the controversial nature of his notation of such subdivisions:


In it [i.e., the Double Concerto] there are many problems of notation - the awkward dotted notations, used for quintuplets and septuplets in compound time, that have bothered performers in Warsaw and elsewhere. In defense of them, I do not think many passages which use dotted and undotted notes simultaneously could have been notated as clearly.17


The issue concerns two aspects of the notation of non-duple subdivisions - their notation in simple meters and compound meters. In simple meters, Carter has subdivided quarter notes into five sixteenth-notes when writing quintuplets, a practice which has met with no disagreement: however, in writing septuplets against a half-note, he has chosen to use sixteenth-notes instead of the more usual eighths. In compound meters, Carter subdivides a dotted quarter-note into five dotted sixteenth-notes when writing quintuplet figures, rather than using five undotted eighth-notes, which is the more usual practice.


He has explained that he based his practice on the system propounded by Renée Longy-Miquelle in a text dating from the early part of the century, the Principles of Musical Theory (Boston, 1925).18 He defends his dotting of irrational subdivisions in compound meters by pointing out that when dotted, they are immediately distinguishable from irrational subdivisions in simple meters:


The question of the notation of artificial divisions of dotted units has even less uniform agreement. It seems to me that Mme. Miquelle's method, cumbersome as it looks, has the advantage (in music that is in alternations of 6/8 and 3/4) of making the different groups of artificial divisions look different at once.19


A strong position against Carter's method has been taken by Gardner Read. Arguing in favor of Hindemith's method of writing non-duple subdivisions in simple meters, he states:


it is not possible, however, to go along with Carter in advocating the method of notation for artificial divisions expounded in an outmoded text by Renée Longy-Miquelle. This system uses identical note-values when writing quintuplets and sextuplets against duplets or quadruplets.  But the next smaller note-value is employed for septuplets.The author's reasoning - to me, completely fallacious - is that seven is closer to eight than to four (the first division of a quadruplet); therefore, one should use the note-value that would be employed for eighth-notes as the unit. Far more accurate in the mathematical sense, and more generally applied by composers today, is the principle advocated by Paul Hindemith. He would retain the same note-value for both the regular and the extraordinary groups until the notes in the irregular group exceed twice the value of the normal group. This is only another way of saying that the note-value in the extraordinary group does not change until the number of notes contained therein enters the second division of the basic unit.20


Thus, Read would advocate the use of eighth-notes (rather than sixteenth-notes) for writing a septuplet against a half-note, since seven does not yet exceed twice the normal number of eighths for the unit.


Read is not in disagreement with the practice of subdividing a dotted quarter-note in compound time into five or seven sixteenth-notes when writing quintuplets or septuplets. He is, however, strongly opposed to Carter's practice of dotting quintuplet and septuplet sixteenth-notes in compound meters, since the traditional meaning of the augmentation dot is thereby negated:


The patent illogicality of dotting each note (or rest symbol) within a quintuplet or septuplet, in either simple or compound time, is not a matter of personal taste and discretion but of simple mathematics. Whereas duplets and quadruplets can, and even should, consist of dotted notes in compound time, all other irrational figures cannot. If the augmentation dot still means what it always has since its inventionin the seventeenth century - the increase by one-half of the note value it follows - it cannot be applied to rhythmic figures whose total value would then exceed, or fall short of, the normal number of units within the prevailing pulse or measure.21


Especially when consulting a visual comparison between Carter's method of the notation of irrational subdivisions and Read's counterproposals (Example 27), one cannot help but side with Read, particularly where the matter of non-duple subdivisions in compound meters is concerned. The use of the augmentation dots is not only mathematically incorrect, but also gives a needlessly over-complete appearance to the eye. Indeed, one must further agree with Read that such practices are an unnecessary hazard for the conductor and performer, already burdened by rhythmic complexities and defy easy solution. In his zeal for meticulous notation the composer has perhaps become overly complicated, rather than clear and unequivocal.22 

                                Example 27


Some Possible Notational Improvements and Alternatives




In view of the notational difficulties mentioned above, it seems appropriate to suggest some possible improvements to Carter's notation in the Double Concerto.


The most obvious suggestion would call for a rectification of confusing, unclear or contradictory elements. In particular, unmeasured rolls or tremolos could be substituted for the over-fastidious fast subdivisions of quarter-notes, and the extremely confusing over-use of brackets could be eschewed. Also, some degree of rhythmic beaming - similar to that proposed by Stone in Example 25b - might make the musical intent clearer in many cases.


Furthermore, there are several improvements that might provide more assistance to the player and conductor in recognizing how musical streams are defined and related. For instance, more verbal explanations and/or composite rhythms might make the rhythmic structure more apparent. It is also thinkable that the placement of some simple symbol (perhaps an "x") over the score at the appropriate points might be a practical way of marking the location of pulses pertaining to each stream. Finally, some manner of showing the individual player to which stream he belongs would be especially helpful in making it clear how his line fits in to the overall context; such indications could take the form of cues, verbal statements, dotted lines in the score, numbering for each individual stream, etc.




In addition to the above suggestions for improvement, it is possible to envision several proposals for notational alternatives, although some of these either might involve styles of notation to which Carter would not ascribe, or might not be as effective in musical situations as complicated as those found in the Double Concerto.


In any case, one could imagine transcribing at least some portions of the work into proportional notation, using a general conductor's beat to coordinate the entire ensemble. Such a solution would certainly contribute greatly to visual clarity of the parts and ease of playing; it would, however, have the disadvantage that at least some rhythmic ideas could not be conveyed with as much exactitude, and not nearly as much ensemble precision would be possible.


Another possibility would be actually to notate the various musical streams in different tempi, to notate the score traditionally and bar each line according to the natural pattern of recurring accents found in each stream, to lay out the lines in the proper proportional relationship to one another, and then to have the various streams conducted by several conductors. In this way, rhythmic ideas could be conveyed as exactly as they are in the present score, with the added advantage of increased visual clarity, since the actual metrical structure of each line would be readily apparent. Still, it is uncertain whether the precision of coordination desired by Carter would be possible, unless the conductors had an extremely precise sense of tempo.


Emmanuel Ghent has actually made a proposal for a renotation of the Double Concerto much along the lines of the second of the above suggestions, also making use of his method of electronic cueing. He has renotated a short passage from the Introduction (mm. 12-16) with proportional spacing, using only the four percussion lines for purposes of clarity (Example 28a); a second version (Example 28b) renotates the lines in four different tempi, each line barred according to its individual metrical structure, with an "x" over the staff marking the location of each pulse pertaining to any given musical stream. The players would wear earphones, and would be cued electronically just before each necessary tempo change.23




                                            Examples 28 a and b


This method certainly has the theoretical capabi!ity of dealing with 11 work as complex as the Double Concerto. However, the music would have to be written with electronic cueing in mind, since the players need some time to assimilate the cue for a new tempo before any tempo change. The feasibility of this method as a viable and workable alternative to the present notation of such a work would have to be carefully tested in practical trials.

Observations on the Compositional Realization of the Basic Structural Concepts in the Double Concerto


Aside from the numerous notational difficulties in the Double Concerto, there seem to be additional problems involved with Carter's realization of his basic compositional ideas which tend to obscure the musical structure for the listener.


For instance, Carter's presentation of the overlapping musical strands in the Introduction often seem purposely ambiguous, as can be seen by comparing the basic rhythmic plan of the opening of the movement (Example 29) with the score itself. For instance, the entry of the first strand in m. 6 does not seem sufficiently set apart from the previous introductory material to enable the listener to detect the beginning of a new structure (Example 30). Then, instead of leaving (in percussion 3) a clear space between the second pulse of the first strand in m. 7 and the upbeat figure to the third pulse in m. 8, Carter immediately confuses the matter by the insertion of an additional role in percussion 4 (Example 31).

Example 29


Example 30

Example 31


Notice also that the numerous accented notes in the quintuplet and septuplet upbeat figure in mm. 6-8 are just as strong as the note that mark the beginning of each "wave," thus making it almost impossible for the listener to hear which of these many accents is structural; furthermore, given the speed of the upbeat figures (especially those in quintuplets), it is difficult for the listener to perceive a difference between anticipatory upbeats and rolls.


Also, it will be seen that the two bracketed notes in percussion 4 at the end of m. 12 and the beginning of m. 13 confuse the audibility of the two musical streams at that point; while the second note at the beginning of m. 13 does fall at the beginning of a "wave," it is overshadowed by the previous note - marked with an accent and a louder dynamic level - which does not coincide with a structural accent at all (Example 32).



Example 33

Finally, the third musical stream (speed 21 7/8) is not fully expressed when it enters in mm. 13-14; although its characteristic interval (a major seventh) is present here and returns several times in subsequent measures, this one inaccurate statement is all the listener is given until much later in the section (Example 33).

Example 33


These departures from, and obfuscations of, the structural system of the Introduction are only a few of the myriad liberties taken in the compositional realization of the structure in this and other sections. Indeed, one wonders not only why the realization of the underlying musical and rhythmic structure is not more clear and emphatic, but also whether Carter actually intends the structure of the work to be understood by the performers or perceived by the listeners. Is the basic structural framework - which he emphasizes again and again in his writings - not meant to be heard as such, and instead intended to serve as a means to some other musical end that he has not mentioned?


In any case, if one takes the point of view that the structure must be audible for the work to be understood, it is tempting to imagine that the comprehensibility of the Double Concerto could be greatly improved if certain changes were made in the compositional realization. For example, dynamic suppression of non-structural notes would greatly facilitate the listener's ability to perceive the rhythmic organization of the work; possibly a visual distinction between ornamental/non-structural notes (different colors, size, etc.) and structural ones could be of considerable help to the player in this regard. Even more helpful to both player and listener might be the actual removal of many non-structural notes and figures; indeed, greater clarity of texture could lead to much greater structural audibility.




There is little doubt that Elliott Carter's inventiveness, integrity and discipline are impressive, and that his musical concepts are strong, provocative, and rich in possibilities. His compositional realization of these concepts is highly elaborate, and his notation is exacting, meticulous and even painstaking.


Yet it is precisely the great intricacy of the realization that paradoxically tends to obscure the clear projection of much of the work's musical thought. The implosive force of the notational and compositional detail not only weighs the piece down, but also strains the abilities of both player and listener alike.


Thus, while the Double Concerto opens new vistas and explores exciting musical territory, it at the same time raises serious questions regarding the realization, notation, projection and comprehension of highly complex musical ideas.



Carter, Elliott. Program note for the "Epic" recording (LC 3830) of the Double Concerto (no date); Ralph Kirkpatrick, harpsichord, Charles Rosen, piano, Gustav Meier, conductor.

__ . Letter in "Letters to the Editor," Journal of Music Theory, Vol. 9, No. I (1965), pp. 270-273.

__ . "The Time Dimension in Music" in Kurt Stone (editor), The Writings of Elliott Carter (1977), pp. 246-247.

___ "The Orchestral Composer's Point of View" in Stone, op. cit., pp. 290-298.

__ . "Double Concerto for Harpsichord and Piano with Two Chamber Orchestras (1961), Duo for Violin and Piano (1974)" in Stone, op. cit., pp. 326-330.

________ . "Music and the Time Screen" in Stone, op. cit., pp. 350-357.
Ghent, Emmanuel. "Programmed Signals to Performers: A New Composi
tional Resource" in B. Boretz and E.T. Cone (editors), Perspectives on Notation and Performance (1976), pp. 134-144.

Read, Gardner. "Some Problems of Rhythmic Notation" in Journal of Music Theory, Vol. 9 (1965), pp. 153-162.

______ . Modern Rhythmic Notation (1978), pp. 46-49.

Stone, Kurt. "Problems and Methods of Notation" in B. Boretz and E.T. Cone (editors), Perspectives on Notation and Performance (1976), pp. 14-15, 18-26.


 1  Kurt Stone (ed.), The Writings of Elliott Carter (1977) p. 296

 2 Stone, op. cit., p. 351.

 3 ibid., p. 351.

 4 ibid., p. 246

 5 ibid., p. 355.

 6 ibid. p. 356.

 7 ibid., p. 350.

 8 Stone, op. cit., p. 326.

  9 ibid., pp. 353 and 355

10 ibid., p. 326.

11 ibid.. p. 328.

12 Stone, op. cit., pp. 296 and 329.

 13 Gardner Read, Modern Rhythmic Notation (1978), pp. 48-49.

 14 Cf. Kurt Stone, :Problems and Methods of Notation," pp. 23-5 in B. Boretz and E.T. Cone) eds.) Perspectives on Notation and Performaonce (1978)

 15 The parts fcr vo n and percussion 1 were examined for this study.

 16 Kurt Stone. op. cit. (in footnote 14), pp. 1 8-1 9.

 17 Stone, The Writings of Elliott Carter (1977), p. 296.

         18 Cf. Elliott Carter, "Letters to the Editor,- in Journal of Music Theory, Vol. 9, No. 1 (1965), pp. 270-272.

         19 ibid., p. 272.

         20 Gardner Read, 'Some Problems of Rhythmic Notation,- in Journal of Music Theory, Vol. 9 (1965), p.

         21 Gardner Read, Modern Rhythmic Notation (1978), pp. 46-47.

         22 Gardner Read, "Some Problems of Rhythmic Notation" (cf. footnote 20), p. 159.

         23 Emmanuel Ghent, "Programmed Signals to Performers: A New Compositional Resource," in B. Boretz and E.T. Cone (editors), Perspectives on Notation and Performance (1976): op I 37-8.