Introduction

In an age when the ephemeral is the rule, the stylistic coherence of Carter is remarkable. The influence of great personalities of the artistic world is reflected in a traditionalism transformed by a very personal vision of their art. Carter has created works which are at the same time conservative and innovative - a dialectic achieved by his conception of rhythm. Carter brings the notion of time, of pulsation, of a simple common denominator (to which an ensemble conforms in proportionally unified durations), to a fluctuating set of durations and by the use of metric modulation which he initiated at the end of the '40s, and developed subsequently. Carter thus possessed the means of opening an immense gamut of continually transformed temporal relationships. Maintaining an extraordinary suppleness, they never lose their precise coordination. Instead of one unique "time," there would be multiple simultaneous or successive times according to the chosen combinations. The opposition of different times in the same work is reflected in the influence of other disciplines, notably literature, dance and the cinema.

The rhythmic innovations of Carter support a tendency to create a music which makes freer use of durations than has been traditional in the western metric system. Post-war composers arrived at this freedom in completely abandoning metric support, either in recourse to durations defined strictly without any apparent relation between themselves, (thus destroying any sense of common measure) or, in the case of indeterminacy, in explicitly avoiding the rigidity of measures. Carter, without renouncing the traditional metric system, has rethought and redeveloped it in terms of relativity and adapting it to contemporary notions of psychological time and of multiple narrative succession. He has progressively abandoned what he called the "square articulation" of the neo-classical style and oriented himself towards a more complex and personal pursuit marked by his evolution of musical form.

Musical discourse results from the relation of a given element to a process. In this process each element is merely the present moment to which we have access at each instant of our observation. We always have a past, a present and a future. The entirety of these diverse instants is the musical work. When we arrive at the 20th century, we find a great number of musical languages resulting from the constant compositional research. In this context, we see the birth of of several currents which, nevertheless in themselves, do not have the capacity to survive in this century's atmosphere of constant development and global socio-political and economic events.

After a long period in which compositional research was interested in harmony and pitch, composers had arrived at a point where they then needed to study rhythm which had maintained certain esthetic limitations. The history of music shows us that we had already reached a very high degree of elaboration in the domain of the harmony of pitches. Thus, composers, and notably Carter, took care to introduce into musical composition relationships with other forms of organization of rhythm and time.

At the beginning of his career, Carter wrote in a style strongly influenced by his time in Paris and by his work with the composer and teacher Nadia Boulanger. In the course of the '40s, he developed a new rhythmic language. He moved towards non-western and pre-tonal rhythms. Carter also manifested a genuine interest in the tempi of Balinese gamelans, in the music of the Durub arabs, the Talas of Indian music and in African music, in particular that of the Watusi. He uses these elements in order to present new developments of rhythm and of time. One can see results of them in the metric and temporal modulation which he introduces in his works.

Carter's career developed in a more marked fashion beginning in 1945. He wrote at this time the Sonata for Piano (1945). By the end of the '40s, Carter turned toward the heterogeneous superposition of different layers of material. The use of this process of composition is seen especially beginning in the String Quartet n^o1 which he composed in 1950-51. Carter also superimposed different pulsations. He took extreme care to control the different speeds which existed in themselves. In fact, they could never coincide because that would lead to a regular metric which was not at all his objective. In Carter, the rhythmic relations interlock in order not to give the listener the sensation of a distinct character. He regularly uses the superposition of different rhythmic figures but in a way which could not be considered anarchistic since his compositional techniques are always very elaborate.

Time is also a very important parameter in the elaboration of a musical work. The latter can perhaps be considered solely in function of time. It exists in time, with - and in function of - time. Time, thus, exerts a considerable power in its perception. It is one of the elements present in the conception of a work, perhaps the most important one. It is comparable to the space of a canvas upon which the composer projects his imagination.

Carter's compositional technique is strongly influenced by the films of Eisenstein and the ballets of Balanchine.² Concerning the films of Eisenstein, Carter is influenced especially by the processes of collage and continuity. In relation to the ballets of Balanchine, he has shown a genuine interest in the continuity of motion. His thought on musical time and his conception of musical composition also constitute significant relationships with 20th century music. He has profoundly modified it by his simultaneous use of diverse elements as well as by the notion of the time screen³ offering us an analogy between space and time. These dimensions, so remote, yet at the same time so near, permit an interconnection between the notion of space in the plastic arts and the notion of time in music.

In relation to European music, Carter was influenced in the course of his career by different composers: for example, Scriabine, Stravinsky, Schöenberg, Berg and Bartók. We see this influence, not only in his compositional techniques, but also in the different relations which he establishes between the rhythms, the tempi and even between the different parts of his works. Thus, we see the influence of Stravinsky in the Sacre du Printemps (1913) l'Histoire du Soldat (1918) and Les Noces (1923). These works bring new manners of changing time and rhythm and consequently of superimposing them. We also see in them the superposition of small instrumental ensembles, that is to say the integration of small sections within the musical work. These combinations give us a mixture of oftentimes distinct situations, which ultimately appear as superimposed contrasting tableaux. We can also note in reference to this, the example of the superposition of different tableaux in the opera Les Soldats (1965) of Alois Zimmerman where a very complex method combines different materials which come from very different historical periods. In the collage of these different materials, the composer arrives at a multiple conception of the temporal phenomenon. Sinfonia (1968) of Berio realizes the same approach to the temporal phenomenon. In the beginning of the '60s we see also the influence on Carter of the music of Ligeti, Xenakis, and Penderecki. He also undergoes an influence of the music of the Middle Ages and of the Renaissance in composers like Perotin, Dunstable and Gabrielli. The late works of Beethoven similarly influenced Carter in the domains of rhythm and time.

The Double Concerto

This work dedicated to Paul Fromm, was commissioned by the Fromm Music Foundation and presented in concert for the first time on Setember 6th of 1961 in the Grace Rainey Rogers Auditorium in New York. It was played by Ralph Kirkpatrick at the harpsichord, Charles Rosen at the piano, and Gustav Meier conducting during the Eighth Congress of the International Musicological Society. Written in 1959-61, this concerto is based on groups of materials whose texture, rhythmic figures and linear patterns are continuously transformed. The Double Concerto written for two orchestras, is divided into seven contrasting sections: Introduction, Cadenza for harpsichord, Allegro Scherzando, Adagio, Presto, Cadenzas for piano and Coda.

The Introduction which develops up to measure 103, presents material in a progressive manner. Carter creates different strata as he superimposes contrasting melodico-rhythmic lines and introduces new instruments each time. In analyzing the table below, we see that the two orchestras are not both complete until measure 19, where finally the oboe and the clarinet of Orchestra II are presented.

In this movement, Carter changes very little in the rhythmic figures. He uses only one type of figure per line. The diversity results from the use of other types of figures in other lines, in which the superposition of different divisions of time may be regular or irregular. Measures 19-22 of the Introduction reveals this process of composition. The composer uses here eighth-note triplets, the quintuplet and the septuplet in sixteenths - and also tremolos and other figures which divide time in a regular fashion - thus arriving at a very complex structure yet whose constituent parts are very simple.

This compositional procedure is altered in measure 31. The transition is effected in the harpsichord in measure 30. In using the quintuplet sixteenths, Carter demonstrates great care in the transition from one texture to another; he uses here, in addition to the held notes, figures which are multiples of five - either quintuplets or ten thirty-seconds. This section unfolds up to measure 46 where the composer comes to a moment of great tension in the third beat of the measure. This tension results from a crescendo poco a poco which began in measure 41. Carter uses very simple rhythms here, but he combines them in such a way as to arrive at very complex rhythms: percussion 1 and 2 in a relation of 5:7, percussions 3 and 4 in a relation of 5:3. The other instruments follow the same type of rhythm of the percussion, that is to say the quintuplets or septuplet sixteenths, and also, longer rhythmic figures. From measure 46 to measure 58, the harpsichord and the piano play together. The harpsichord takes more complex rhythms which result from the elaboration of the septuplet, either in dotted sixteenths or in dotted thirty-seconds: the piano gives harmonic support. Then events multiply up to the intervention of the harpsichord in measures 64 - 68. Three rhythms arise based on quintuplets in dotted sixteenths, then in dotted sixty-fourths. Measures 69 to the end then recall the essence of the movement.

Concerning the use of dynamics. Carter gives preference to pp, p and mp but he does not exclude the mf, the f and even the sff. In using contrasting dynamics, the composer creates a circulation of sound between the four percussionists and the two instrumental ensembles which results in the association of percussions 1 and 2 of Orchestra I with percussions 3 and 4 of Orchestra II. In oscillating between pp and ff marcato sf, the composer employs them in layers. While certain instruments use dynamics based in piano, others play dynamics based in forte. Nevertheless, in certain passages the two ensembles play the same types of dynamics. This procedure unifies the character of the two ensembles and in certain cases, one group uses the dynamic which the other has just left behind. Thus the listener is drawn to sounds which come from different parts of the space. In fact, there exists in the piece, a "field of play" which Carter uses to attract our attention to events which are so close, yet at the same time so far apart. In measure 19, Orchestra II arrives at a f which is taken by Orchesta I in shifting right away to piano-pianissimo. There is a very strong interpenetration of dynamics despite the rhythmic, timbral, and intervallic contrasts between the two orchestras. After an extensive use of dynamics in the p and pp range, we see beginning in measure 41, a crescendo poco a poco up to the third beat of measure 46, the moment of greatest tension of this section. At measure 93, a contrasting part begins and the texture of the two ensembles changes totally. The composer uses ff and f ; this change announces the entry of the harpsichord in measure 98, with the dynamics of pp and mp. Up to the beginning of the harpsichord cadence in m. 103, we find only indications of p, pp, and ppp.

Cadenza for Harpsichord

The cadenza for harpsichord which unfolds from measure 103 to 156 entails two distinct parts: a first up to measure 143 and a second which follows through to the Allegro Scherzando.

The first part, where the harpsichord plays completely alone, is based essentially on rhythmic figures of very short values and consequently, has a very rapid quality. The beginning is prepared by the use of rhythmic figures based on the triplet eighth as well as the use of a texture which emphasizes the harmony. The beginning of the cadenza includes melodic figures which progress very quickly into the low register of the harpsichord where a number of pitches are highlighted in leading to the confrontation of contrasting figures within the same measure - melodic figures in very short values and harmonic figures in longer values. Carter also creates a harmony within the melody. He explores the technical possibilities of the instrument using different modes of attack, contrasting phrases and the combination of diverse registers and timbres. Otherwise, the succession of figures of sixty-fourth note septuplets, sixteenth-note quintuplets, and four sixteenths allows him to progressively decelerate in relation to the figures based on the triplet eighths at the end of the introduction.

In using quite varied figures, Carter presents very heterogeneous rhythms, and, apart from the succession of changing rhythmic figures, momentarily stabilizes one of them. The change of meter is linked to the appearance of rhythmic figures of longer duration which create a simpler texture in the middle of this part (m. 119 - 133). Carter favors the harmonic aspect here using chords. There are no rhythmic figures of short values - the small number of events contrasting with the previous processes.⁴ Then Carter changes once more the process; he progressively introduces figures based on shorter values. We see that, apart from the shorter values, he does not abandon the procedures based on the chords; He introduces the wind instruments (he has already introduced the percussion in measure 138) which announce the end of the cadenza and introduce a new part of this section. In using rhythms of longer values (except for the rhythms of the piano in measure 146 and the harpsichord in measure 151) Carter arrives at a texture in which the harmonic element emerges because of the change of texture and the longer durations - in opposition to the first part of this passage which favored the rhythm. Carter uses complex rhythms which result not only from the irregular division of the beat, but also from the superposition of different lines within the same instrument. In fact, he uses in the first part, the harpsichord solo to execute rhythms in very short values, and in the second part, he mixes the two instrumental ensembles succeeding via the use of longer values, in emphasizing the harmony.

In the score, we see a very strong break in the discourse in measure 197: the composer uses only Percussion 2 to which he adds Percussion 3 and Orchestra II beginning in measure 199. (This is the only moment in this movement where Carter uses percussion alone.) Rhythmically we see the use of groups of seven figures essentially in the harpsichord line while the groups of five figures appear in the piano line; there is therefore a superposition of different periodicities. In measure 248, the texture changes. The composer gives more importance to the percussion and emphasizes the rhythmic patterns based on the quintuplet eighths. Then he reinforces the triplet and we can see the frequent use of rhythmic figures which divide the beat in an irregular fashion: the triplet, the quintuplet, the sextuplet and the septuplet. In measure 258 a new section of the movement begins. The harpsichord continues the rhythmic figures based on the sixty-fourth note septuplet to which the piano opposes its triplet eighths and quintuplet and sextuplet sixteenths. In measure 265 the composer changes again and places the two orchestras in opposition. Against a more harmonic texture in Orchestra I based on figures of longer values, he sets a texture in which the piano takes the principal role and maintains a very lively dialogue. Rhythmically, this passage is very rich, the piano realizing changing figures based on regular and irregular groups of sixteenths.

After measure 271, the two instrumental groups play in alternate fashion. Carter abandons the essentially harmonic texture of Orchestra I and works with the dotted sixteenth-note quintuplet presented in measure 272 in Percussion 2. The counterpoint between the viola and double bass lines passes to the harpsichord. It is only in measure 307 that the quintuplet and septuplet figures are superimposed in the same instrumental group. In measure 314, the duration of figures is augmented in such a way as to give, in measure 315, a new texture in which the harpsichord plays a principal role. Thus Carter uses the dotted sixty-fourth note septuplet in a 9/8 measure (m. 313) which he replaces in the following measure (in 14/16) by figures of seven sixteenths and seven dotted eighths. Carter conserves the figure of seven notes either in septuplets or in groups of sixteenths or dotted eighths. Beginning in measure 316 everything is developed around the harpsichord; while the music becomes sparser, the tempo increases in the accelerando poco a poco of measure 329 to 341.

Adagio

In measures 403 - 407 the superposition of these different elements alters very slowly their rhythmico-melodic characteristics. The rhythms dissolve in long durations which stabilize the discourse after a period of strong acceleration. This moment is very important since, beginning in measure 421, Carter begins a new tempo acceleration which ends in measure 434, the last tempo in the elaboration of different rhythms. He presents a texture in which long values are used not only in the strings but also in the winds and brass. Carter introduces the percussion beginning in measure 416 in narrowing the time intervals between the shorter notes which he circulates in Percussion 2 and 3. In using these elements, he introduces a new texture which begins in measure 421.

This part constitutes a very successful example of a measured acceleration, rigorously indicated by the metronomic indications at the top of the page in the score. We have first an acceleration which goes from 70MM to 210MM for the quarter. Next, the composer takes the quarter at 70MM and accelerates to 140MM which justifies the indication of q = 70 MM of tempo giusto from measure 434. Via the change of measure and the accelerando of measures 434 and 435, the piano arrives at tempo guisto (m. 436) in a completely different meter from the other instruments: a 6/8 ternary meter, while the other instruments use a 2/4 binary measure. But Carter specifies that for the piano, the tempo of the 6/8 measure is equal to that of the preceding 6/4. In accelerating to q = 210 MM, Carter arrives at h. = 70 MM (beat = beat) hence the indication q.= q. = 70 MM in measure 436. The use of triplet eighths in measures 434 - 435 enables the transition to sixteenth-note triplets of measures 436 and following. The composer changes once more the metronomic indication in measure 436 where the half note of the 4/4 measure equals the quarter of the 2/4 measure. We emphasize that all of the accelerandi use the same progression of metronomic indications: 70-80-92-105-121-139-159-183-210. In the same way all of the ritenuti use a single progression of metronomic indications: 140 -118 - 99 - 83 - 70.

The musical notation changes also; the composer uses groups of triplet eighths in the harpsichord and triplet sixteenths in the piano. Carter joins into one tempo what he was doing in two tempos; hence the use of smaller values. The association which he develops by the superposition of the triplet sixteenths in the piano line (the 6/8 measure) and the figures of sixteenth and thirty-second note triplets in the harpsichord line (the 2/4 measure) gives, for the piano, groups of three eighths which are correct for the tempo in ternary division and triplet sixteenths which subdivide each eighth in three. Carter therefore introduces irregular divisions in the midst of each division of the beat which are opposed to the regular divisions of each part of the 2/4 measure. Beginning in measure 439, the composer uses the triplet sixteenth up to measure 443 where he changes meter once more. He uses an 18/16 measure and a more elaborate technique in which he abandons the sixteenth-note triplet and he adds a new melodic line which derives from the emphasis of several pitches in the piano line. He superimposes different rhythms which create other lines - rhythmic layers within the single melodico-rhythmic line. (See Figures 2 and 3).

Figure 2: Double Concerto rhythmic schema in the piano mm. 436 - 446.

Figure 3: Double Concerto rhythmic schema in the harpsichord mm. 436 - 446.

In the following examples, we make three observations:

- the subdivision into three of each note - itself resulting from a division into three #9; of each beat

- the change of measure (m. 443 to 18/16); in the 18/16 measure each group of three sixteenths divides the time in a regular way. We thus have six groups of three sixteenths and not six triplet sixteenths (the irregular division of each part of the #9; beat).

Similarly between measures 434 - 436 we find a new texture in the piano line. The composer prepares this new texture beginning in measure 434 where he uses:

- the eighth-note triplet in 3/4 and 6/4 measures (in the piano);

- the quintuplet sixteenths then the groups of four sixteenths (in the piano)

- the same number of figures in the 2/4 measure implying the use of figures of smaller values

Thus, by the use of shorter values, the composer links measures 434 - 435 to 436 and following. He accomplishes this by dividing the value of the eighth in two. We will illustrate next the procedure in the transition from measure 435 to 436. The perception of figures remains equal independently of the use of figures of shorter values (Figure 4).

In fact, the acceleration which goes from q = 140 MM to q = 210 MM make each quarter the value of an eighth in the following measure (q = q ) and consequently, the tempo of the 6/4 measure (q.) is equal to the tempo of the 6/8 measure (q.). The indication q.= 70 MM corresponds exactly to q. = 210 MM.

We also cite the transition of the 4/4 measure into 2/4 for the harpsichord. There is no change in texture. In the 4/4 measure the composer indicates that the half note equals 70 MM and afterwards, he uses the indication q = q = 70 MM. We continue to have each group of figures with the metronomic indication of 140 MM and eight figures in each beat. In the 4/4 measure we have eight sixteenths; in the 2/4 measure we have eight thirty-seconds (Figure 5).

This procedure is evident not only in the piano line. The composer uses it also in the harpsichord line where several melodic lines take shape. We have examples in measures 438, 441, 443 - 446, either in the use of thirty-seconds or dotted sixteenths or the use of sixteenths or eighths. The continuous opposition of the piano against the entire orchestra in the use of a different time signature (measures in 6/8 against measures in 18/16) reveals its importance in this dialogue.

From measure 453 to measure 466, the composer superimposes a continuous ritenuto for all of the instruments of Orchestras I and II, to a continuous accelerando for the piano. He opposes two different speeds. We have two layers which displace each other, interpenetrate and play together in different ways. Beginning in measure 454, Carter superimposes a 2/8 on all of the instruments except for the piano which plays in a 3/8 measure. Carter realizes the accelerando over the quintuplet thirty-seconds. In opposition, he bases the ritenuto on figures which are successively longer each time. The opposition of the two meters results in a very complex procedure since the piano line is out of phase ahead of the orchestras. This section which ends in measure 466 is very difficult to play (and conduct) as the instruments do not go together. In measure 465 Carter changes once more. He uses a 4/4 measure for all of the instruments of the two ensembles except once more for the piano part in which he uses an 8/8 measure. In measure 466, the composer establishes a 4/4 measure for all of the instruments.

 Presto

The Presto succeeds the Adagio in an uninterrupted manner via the attacca indication in the harpsichord which is playing continually and in the violin sounding in harmonics (Orchestra II). This movement has a lighter texture than both the preceding movement and the following piano cadenza. The harpsichord takes a more important role; the piano enters only in measure 513 (maestoso) in dialogue with the harpsichord and contributes a more harmonic texture in opposition to the harpsichord which is always more melodic. Despite the differences which exist in the materials, the dialogue and the interpenetration of the two orchestras is more evident, the latter merely providing some minor harmonic support for the solo instruments.

This movement introduces a particular compositional process - metric modulation. Often utilized by Carter, this process enables the change of meter without change of rhythmic figure. In analyzing measures 511 - 512, we see that Carter maintains the figure of four dotted eighths in the upper line and the figure of seven eighths in the lower line of the harpsichord. Via the change of measure, Carter is obliged to change the figure, that is to say, to use the septuplet of dotted sixteenths and the figure of four dotted eighths. Nevertheless, the relation remains the same: four figures for the upper line and seven figures for the lower line of the harpsichord.

Cadenzas for Piano

The cadenzas for piano (there are two cadenzas with indications at measures 526 and 557) have a texture full of events from which there emerges great rhythmic richness. The conception diverges a bit from the traditional schema since the piano does not play completely alone; excluding measures 527 to 540 and 562 to 576, it is inserted within a tutti. Carter announces in the change of process in measure 523, a new section of the work. The use of rhythms based on sixteenths in Percussion 1 and the septuplet thirty-seconds in the harpsichord and Percussion 2, announce what will happen next. In fact, in the course of the cadenzas, the piano realizes only rhythms based on the irregular division of the beat, either the triplet, the quintuplet or septuplet. Nevertheless, rhythms are not excluded which divide the measure in a regular fashion.

The harpsichord is opposed to the piano beginning in measure 540 playing very fast septuplet sixteenth figures which attract attention. The piano, on the contrary, only realizes short figures in triplet eighths. It stops playing completely beginning in measure 544 in favor of the harpsichord. The composer brings out the orchestral ensemble in opposition to the instrumental soloist - the piano. The latter only plays up to measure 556 where it announces the entrance of its new cadenza which reveals an admirable richness of rhythmic design. In it, the composer utilizes the same type of rhythms but superimposes them in continually changing measures and in a different sequence from the preceding. The superposition of septuplet thirty-seconds in the harpsichord line on the sixteenth-note quintuplet in the piano as well as the creation of diverse rhythmic layers within these lines brings forth a strong degree of entropy. The use in the piano line of the septuplet thirty-seconds (mm. 528, 529, 564, and 575) as well as in the harpsichord line (mm. 540 - 545) does not convey the same result: it is less rich. Beginning in measure 576, the composer introduces once more the two orchestras in superimposing three different sonorous layers:

- the strings executing sustained sounds

- Percussion 1 and 2 support the rhythms of four sixteenths of the septuplet of thirty-seconds

- Percussion 3 and 4 supporting the rhythms of sixteenth-note quintuplets and triplet eighths

Next, the held notes in the strings are taken up by the winds (Flute and Horn 1 and Bassoon and Horn 2 in mm. 581 - 583). The two solo instruments continue to oppose each other in their rhythmic figures, and the intervals and registers which they use. Beginning in measure 584, Carter changes again the texture but continues to superimpose three different types of material:

- the strings and winds providing harmonic support

- the soloists playing different parts

- the percussion ensemble developing the most complex rhythm: beginning in measure 603 the percussion plays alone.

The texture which emerges from the first cadenza for piano is less dense than that which emerges from the second. In the first cadenza, Carter uses first the superposition of rhythms of the same type; that is to say, the quintuplets of sixteenths or septuplets of thirty-seconds. In fact, it is not until measure 532 that he superimposes two different rhythms: the triplet in eighths and the quintuplet in sixteenths. He introduces the harpsichord in measure 540 playing septuplet figures - either septuplet sixteenths or septuplet thirty-seconds (m. 560). Carter thus creates a very complex texture; he continues to bring about rhythmic oppositions beginning in measure 562, but only in the piano line. He prepares the 5:3 opposition of the measure 562 by already using in m. 561 the sixteenth-note quintuplet and triplet eighths to modulate metrically from one measure to the other. We find two relationships: the quintuplet sixteenths part to the quintuplet of dotted sixteenths, and the triplet eighth in both measures (Figure 6).

Carter uses this procedure in different meters, thus creating different types of relationships.

We emphasize also the creation of several sonorous layers by the sustenuto in different pitches in the piano, either in chords or melodically. The composer thus creates a new rhythm within the basic rhythm. We find examples of this in measures 559 - 560 (Figure 7).

From measure 576 to measure 579, Carter introduces figures each time in a successively more elaborate form. He finally superimposes in the piano, the triplet eighth and quintuplet sixteenth rhythms in measure 579. This superposition accomplishes the transition to the next texture. Carter takes care to create, using the processes to which we have just referred, more complex rhythms. He uses, for example, the quintuplet thirty-seconds against a triplet eighth. We emphasize that he only uses the quintuplet within the triplet, specifically on the second eighth (m. 576 in the piano, Figure 8.)

Carter also uses this compositional procedure in measures 579 - 580 but this time on the septuplet sixty-fourths figure. Within the four sixteenths figure, the composer introduces irregular division of the measure. He uses the septuplet sixth-fourths on a sixteenth (Figure 8a). In the course of this passage, Carter employs certain compositional procedures used previously, either the superposition of different rhythms or the creation of different sonorous layers within the same line.

 From measure 579 to measure 583 the thirty-second note septuplet figure in the harpsichord and the quintuplet sixteenths in the piano are constantly in opposition in a 7:5 relationship. This opposition is seen not only in the soloists, it exists also in the other instruments. We see it, for example, through the four-sixteenth rhythm and the septuplet thirty-seconds in Percussion 1 and the quintuplet sixteenth and the triplet eighths in Percussions 3 and 4. Carter thus creates a very complex texture where different subdivisions of the tempo form an intense polyphony (Figure 9).

From measure 603 to measure 611 we find another very interesting passage. It uses a 6/8 measure (note that the piano plays in 6/8 only from measure 604), but certain instruments play rhythms which contradict the ternary division of the beat; this is the case of Percussions 1 and 2. The use of the sixteenth-note triplet in Percussion 4 also contributes to this asymmetry placing Percussion 4 in a 2:3 opposition to Percussion 3 (Figure 10).

In the above example four lines are superimposed (Percussion 1, 2, 3, and 4) creating a very complex texture by the use of irregular subdivisions of the measure and beat.

From measure 611 to measure 618 the texture changes once more, announcing the shift in tempo created in measure 619 by the Coda. In using a more sustained harmonic idiom from measure 614, the composer prepares the big opening of the Coda, a large orchestral tutti and arrays of great percussive resonance. In order to arrive at this point, the composer follows a well defined course. He superimposes different lines and places different orchestral groups in alternation. From the confrontation of the wind instruments with the strings, Carter arrives at a process which is dense, but at the same time, more fragmented followed by a grand pause (mm. 617 - 618) which introduces the Coda.

It begins with a stroke of the gong taken up in succession by the orchestra. The sound in this way gives birth to the form of the coda. Carter divides this coda in two parts of 35 measures each. The first unfolds to measure 563 and the second from measure 564 to the end of the work in measure 689. Carter underlines this division by annotations at the bottom of the page: the first indicates that the percussion should dominate, nevertheless without smothering the sound of the piano and harpsichord; the second specifies that the percussion should dominate more and more nevertheless without smothering the sound of the soloists where they play solo.

This movement superimposes four different rhythmic patterns based on the numbers 3, 4, 5, and 7. This superposition has a periodicity of five measures for Orchestra I (harpsichord) and seven measures for Orchestra II (piano). Because we feel these periodicities, Carter takes care to introduce different dynamics, registers, rhythms and combinations of instruments. For the piano he often uses a new dynamic, another register or a new phrasing. For the harpsichord, he often uses new rhythmic figures and new instrumental combinations. Secondary accents are added to the principal accents: there are in all 35 sixteenth-notes for Orchestra I and 35 quarters for the piano.

In the first section of this movement, we have a clear confrontation of the two instrumental ensembles. Orchestra I plays in 6/8 and Orchestra II in 3/4. In measures 661- 662 the piano momentarily changes meter to 18/16. This opposition already reveals the intention of the composer to oppose different interventions as well as the two orchestras. Thus, the two solo instruments are opposed as much as soloists as members of the ensemble. We continue to have diverse ensembles within each orchestra. The wind instruments, the strings, the percussion and the piano or the harpsichord realize different lines from each other, each contributing in this way to a very heterogeneous texture. The harpsichord opposes figures of five and four in a 6/8 measure: the piano opposes figures of seven and three in a 3/4 measure.

Carter often uses in the elaboration of his works the superposition of different rhythms in a periodic fashion: resulting in more or less polyrhythmic textures according to the rhythms utilized. If we analyze what happens, for example, in measures 633-636, we see that he achieves this very type of superposition. Thus, the rhythms applied to the triplet eighths, the four dotted sixteenths, the quintuplet sixteenths and the septuplet thirty-seconds are used in a relationship of 4:5 for the harpsichord and 7:3 for the piano (Figure 11 below).

The manner of arriving at this relationship denotes great care in the elaboration of musical material. The composer, already using the quintuplet sixteenths in the Percussion 3 line replaces them with triplet eighths at the moment where they are passed to the harpsichord. A different procedure is used in the notation of the septuplet thirty-seconds. This figure was already presented in the piano line of measure 633. Carter introduces the septuplet sixty-fourths in place of the triplet eighths. It suffices therefore to double its value to obtain the septuplet in thirty-seconds. The composer thus passes from triplet eighths to the superposition of two irregular divisions of different tempi: a relation of 7:3. We emphasize the use of the septuplet sixty-fourths in the two lines of the piano in measure 633. In using a figure which involvew a rest in its second part, Carter prepares the listener's attention to this polyrhythm. Next, he simultaneously opposes the figures in a 7:3 relationship, either in thirty-seconds or eighths (Figure 12).

In the harpsichord line of measure 634, Carter utilizes another compositional procedure whose result is the same; he underlines the relationship of 4:5 in using the quintuplet and quadruple sixteenth figures in ascending motion. In using this figure, he arrives first at the figure in four dotted sixteenths then at the quintuplet in dotted sixteenths. (Example 12a).

In measure 642-644, a very complex process results from:

- the superposition of two different measures (6/8 for Orchestra I; 3/4 for Orchestra #9; II) whose metric divisions are different.

- the use of figures which divide the beat in a regular or irregular fashion and their superposition

- the use of all of the instruments in both orchestras.

Thus in measure 642, we find rhythms which go from the quarter-note figure to the triplet eighth. Carter only uses the irregular division of the beat in Orchestra I, in the 6/8 measure. He contradicts the ternary division of the beat using rhythms of four dotted sixteenths, two dotted eighths or the quintuplet of dotted sixteenths. In Orchestra II, Carter uses regular and irregular divisions of the beat thus arriving at a very complex structure. Measure 642 bears an enormous richness of rhythms and techniques:

- the superposition of two different measures which, independently of the same number of eighth-note figures, have different accents; in the 6/8, on each dotted quarter, in the 3/4 on each quarter (an opposition of 2:3 in accented values)

- the simultaneous use of regular and irregular divisions of figures within the same line

- the superposition of different divisions of the beat. The last beat of this measure contains figures which go from eighths to thirty-seconds. We have thus the superposition of 2, 3, 4, 5, 6,7 and 8 figures (see Figure 13).

Figure 13: Double Concerto Rhythmic Schema, Orchestras I and II m. 642.

Measure 643 is seen to be less dense, but has very complex rhythms for the harpsichord because of the subdivision of dotted sixteenths in the quintuplet. The existence of a rhythmic fade-out in Orchestra II helps to lighten this texture. The harpsichord introduces the quintuplet sixteenths before the final figure of this "fade-out" (see Figure 14).

Figure 14: Double Concerto Rhythmic Schema, harpsichord and Orchestra I m. 643.

From measure 668 to measure 671, we find a texture in which new rhythmic figures are created within existing figures. The basic texture is already familiar. It is based on the relationship of 5:4 for the harpsichord and 7:3 for the piano. Carter introduces here new lines which result from the emphasis of certain pitches and the basic figures. We thus arrive at the superposition of different rhythms and the creation of different velocities (Figure 15, p. 57).

The Double Concerto presents an increasing stratification of sound. Rhythmic figures are introduced progressively in the percussion ensembles in the following order: 2, 1, 3, 4. In using distinctive events, Carter achieves a spatialization of sound; our attention is directed to a specific point in the performance space. The composer repeats at certain intervals, the small rhythmic figures in arriving in measure 11 at a continuous tremolo in all of the instruments. The tremolo constitutes the strongest degree of repetition of a sound. In the first three measures, Carter writes very simply with little superposition. Next he superimposes different rhythmic figures in each line of percussion ensembles 1, 3, and 4. Conversely Percussion 2 has an uninterrupted line in which the composer attains a higher degree of elaboration. Carter introduces here supplementary lines within the existing line. He thus achieves, through the prolongation of certain sounds, different speeds of execution in the same line. In measure 11, the introduction of wind instruments and strings implies a different type of technique: the introduction of the piano in measure 16 and harpsichord in measure 17 bring a further degree of differentiation. Each sonorous layer is identified by its sonority resulting from its combination of dynamics, timbres, rhythms and intervals.

Harmonically, the works which Carter composed from the Sonata for Violoncello and Piano (1948) up to and including the Double Concerto (1961) use certain chords as pivots from which all of the harmonic and melodic material of the work is derived. They are used as factors of unification in compositional technique. In the Double Concerto, there exist two -

The use of these types of chords "isolates" the two instrumental soloists. Nevertheless it allows another chord to be heard while the two instruments play together. This chord of the two ensembles is present throughout the length of the work. It brings coherence and global unity to the discourse.

This work has very particular colors which derive from the two instrumental groups. The choice of instruments, the chords and the sonorous material which derives from the intervals themselves in each section of this work, dictate their characteristics. These compositional procedures reveal the antiphonal character of the work. Concerning the elaboration of this work, Carter confirms that -

The characteristics of the two solo instruments, the division in two of the chamber orchestras and their disposition in the performance space, play an essential role in the perception of the work. As Carter states in his interview with Allen Edwards:

Time is also a very important element in the conception of the Double Concerto. In simultaneously presenting diverse melodico-rhythmic lines, Carter superimposes different tempi which lend a great multiplicity of expressions and characters. The character which emerges from each ensemble results from the association of different materials which are as much rhythmic as melodic or dynamic. These materials are responsible for the expressivity of the piece. The association of certain intervals, pitches or rhythms leads to different states of mind. In addition, the tempo, and in particular, the agogic indications utilized in the elaboration of a work, are responsible for the resulting pathos. This music gives the listener the same sensations as an opera; the superpositions of different tableaux, the co-existence of different times as in the diversity of persons and characters leads to quite particular results which illustrate what we have just described.

Carter was, in addition, conscious, that "one of the great problems of contemporary music was that irregular and other kinds of rhythmic devices used in it tended to have a very small-scale cyclical organization."¹¹ Thus Carter is enormously interested in periodicities. He is preoccupied more and more with giving "the feeling of both smaller and larger-scale rhythmic periods"¹² but he often insists on the fact that he is not especially concerned with knowing if the listener is or is not capable of hearing the polyrhythms. He affirms "The effect I am interested in producing is, as I have said, one of perceived large-scale rhythmic tension, sometimes involving the anticipation of an impending final coincidence of all the disparate rhythmic layers at some key moment."¹³ The polyrhythmic passages, according to Carter can produce quasi-hysterical effects of disorganization. According to Carter, the conflict of chaos and order is particularly significant because it seems to be at the root of so many of the things important to us."¹⁴ He made experiments to arrive at -

Carter also uses repetition of certain rhythmic structures to arrive at varying intensities of polyrhythms. These polyrhythms produce a tension which implies at times a culmination in a (final) point of coincidence so that all of the strata would then be modified in such a way as to progress to another type of structure. These passages produce momentary effects of disorganization, but in the end, participate in an intelligible order. The conflict between order and chaos is very important. It is present in a great number of daily events. The elaboration of polyrhythmic passages obeys the superposition of materials, hence the stratification of elements. This procedure allows the identification of different figures which are the basis of these passages (i.e. the strata).

The Double Concerto maintains two orchestras which are characterized in different ways. Having different instruments and consequently different timbres, these ensembles sustain the solo instruments lending to each a distinct color. The disposition in the performance space dictates the importance of each ensemble. Because of the metric independence of the two orchestras in measures 93 -103 and 619 - 689, the composer suggests the presence of two conductors of which the principal conductor should be placed in such a location as to be seen by the assistant. The composer also wishes that the ensembles be separated as far as possible (especially for the wind instruments). The two ensembles have the same organization, but their completely different characteristics are placed in relief. Carter attributes to each ensemble different rhythmic figures and lines which have different tempi thus characterizing them differently. The tempo which emanates from each ensemble either from the indicated tempo or from the agogic indications also contributes to their differentiation. The different chords and secondary voices at the heart of each ensemble (we emphasize that each almost always has its own speed) contribute to this differentiation as well. Unity results from the interdependence of the two ensembles. Despite the great differences of process and the moments of great metric independence, Carter succeeds in this to the extent that each orchestra needs the other for its rhythmic richness to be heard.

Concerto for Orchestra

The Concerto treats the instruments of the orchestra as veritable soloists. Vents suggests colors and gestures from which Carter takes the dynamism of the Concerto. Similarities with the Concerto (1942-43) of Bartók or La Mer (1903-05) of Debussy can be easily detected. Carter demonstrates his perfect mastery of the technique of collage of different textures and different rhythms in a continuous flux. The orchestra is divided into four groups according to tessitura:

The four groups are given different characters in terms of musical time. Groups 1 and 2 possess a metronomic character; 3 and 4 play rubato; 1 and 3 in ritardando ; 2 and 4 in accelerando. From the rhythmic character of each group there results a different type of transformation. As in the Double Concerto the percussion plays a very important role. It plays almost throughout the entire work.

The form of the Concerto is perhaps the most complex structure which Carter has ever conceived. It entails a sequence of Allegro, Presto volando, Maestoso and Allegro agitato with which the above groups of instruments are associated in the order: 3, 1, 4, 2. We note the technique of collage. The fragments of each group are repeated using variation. We find also vast polyrhythms. The important points of change in these polyrhythms appear in measures 141, 284, 488 and 550.

The introduction presents a very strong variety of rhythmic and temporal elements discerningly distributed. Certain rhythms predominate in certain instrumental lines which characterize and differentiate the instruments. In the distribution of rhythms we see relations of 7:6:5:3 in the wind instruments and the strings. The piano and the harp create diverse rhythmic relationships. The piano is characterized by the septuplet and the harp by the septuplet and sextuplet. In measures 4 and 5, Carter superimposes rhythms which result from the use of an irregular number of figures in each beat (or each part of the beat). We find the triplet and quintuplet eighths in the strings, the septuplet in dotted sixteenths in the marimba and the harp, and the sextupet in sixteenths in the clarinets. From this superposition is born a polyrhythm giving rise to the divisions of 2+2+2 or 3+3 in a 6/4 meter. The phrasing underlines this type of division. The tremolos in the strings emphasize the lines of the woodwinds and the harp. The double bass sustains this texture playing long notes and in measure 8, the dotted sixteenth-note septuplet in Percussion II and the harp passes to the 'cellos. Percussion I and II use again the septuplet figure but in subdividing each dotted sixteenth into two dotted thirty-seconds. The harp continues to use the septuplet in dotted sixteenths and the strings effect relationships of 5:4:3. In the course of this passage, Carter arrives at the opposition of 7:6:5:4:3 in measure 7 where these events proliferate.

Carter accomplishes, beyond the very complex polyrhythms, subdivisions of the 6/4 measure. He groups pulsations into 3+3 or 2+2+2. The latter group is found in particular in the wind instruments. We see the triplet eighths and the sextuplet sixteenths in the clarinets (m.9) third beat; we see also the change of color made possible in the options of instrumentation. Nevertheless in measure 9 we see again a certain ambiguity: in the trumpet line the measure is divided into 3+3.

In measures 11 and 12, Carter again opposes the subdivisions 2+2+2 and 3+3. The phrasings becomes very important in the definition of different layers. We see for example that the Violins II, first desk play, after a group of five eighths, a phrase of four eighths which leads (m.12) to groups of 2+2+2. On the other hand, in the second and third desks, the phrase is based on the staccato accented on each quarter beginning with the last

eighth of measure 11. The violas, 'cellos and double basses play the same type of figures grouping them 2+4, that is figures which derive from the group of four sixteenths in the first two beats, followed by quintuplet sixteenths in the four beats which follow. A phrasing applied to the 3+3 sixteenths or the subdivision of quintuplet sixteenths in 3+3 is evident. Similarly in the Violins I, Carter changes pitches on every third dotted sixteenth (except for desks 4,5a). The piano uses (m. 11-12) the septuplet of dotted sixteenths; beginning with the second, each septuplet presents an accent on every second sixteenth either via the introduction of a new line in the xylophone or by the difference in density in the piano. Similarly, Carter gives an accent to each group of three sixteenths (Percussion V) or, more indefinitely, to each group of figures (Percussion IV). We see also at some times articulations via changes of register, and at others by the simultaneous appearance of numerous events (m.12, see Example 1 above.)

In the wind instruments, Carter continues to use this compositional procedure, but this time in groups of two or three figures. In the trumpet lines and the clarinets, Carter emphasizes the quarter note in the triplet eighths, and in the flutes and oboe, the group of three sixteenths in the quintuplet of sixteenths (he does the same with the dotted eighth - see the piccolo, flutes and clarinet in E flat in the last beat of the measure.) There are similarly regular pulsations in the horns which enforce the quarter note. We are exposed to several reprises via the accentuation in different lines in several different points of the performance space. The passage abounds in events!

In measures 12 and 13, Carter places several layers in relief:

- three layers in the strings (double basses and altos: quintuplets; violin II: triplets Violin I: binary.)

- two layers in the percussion (triplets and quintuplets), piano and xylophone (septuplets)

- two in the brass (quintuplets and binary) and woodwinds (quintuplets).

This profusion of elements creates a tension which is resolved in measures 14, the beginning of Movement I.

Movement I (mm. 14 - 151) Allegro non troppo q = 84

The changing densities lead to a strong contrast between the different parts. Movement I constantly uses irregular rhythms. Diverse types of combinations are used; we emphasize the quintuplet in all the instruments, the triplet for the harp and the double basses, the triplet for the wind instruments and the quintuplet for the strings. We note the successive use of new rhythms, in the piano, the septuplet with which Carter chose to distinguish it. The lighter textures result from the use of longer values and silences which attract attention to the uncommon events.

In measures 16 to 26 several events are emphasized either by the percussion or by the meter. Carter distinguishes the piano with the septuplet thirty-seconds, sixteenths and eighths (m.26). The rhythm is underlined by the percussion, notably the marimba which

often plays the quintuplet (an opposition of 7:5). Carter also uses the sextuplet in sixteenths in the harp. He applies to the figures the dynamic and the phrasings which alter the perception of the 5+5 and he makes us hear other rhythms within the quintuplet. Carter uses other figures beginning in measure 22, not only in the marimba, but also in the piano which he places in a different measure from that of the orchestra beginning in measure 24. The marimba stops playing. Carter superimposes (mm. 24 -26) different measures in the orchestra and the piano. While the orchestra passes from 3/8 to 7/8 and to 3/4, the piano passes from 9/16 to 21/16 and to 9/8. Different pulsations are superimposed and emphasized in the piano line. The composer creates within the same measure several rhythms by the emphasis of certain pitches. He also opposes different rhythms in the piano line. The upper voice creates an asymmetry (sextuplets in sixteenths) in relation to the lower voice which divides the beat in a regular fashion (use of the figure of seven sixteenths - an opposition of 6:7). Carter uses groups of dotted eighths for the 9/16 measure, of seven sixteenths for the 21/16 and of three eights for the 9/8 measure (see Example 2 below).

Phrasing also has an important role in this passage. Carter uses it differently in creating another level of interpretation; as other groups of figures emerge in relation to the initial line. We emphasize that Carter uses almost all the instruments in the orchestra up to measure 23 before the appassionato entry of the piano. The moment is very powerful because of the dynamics (f < ff for the strings, mf <f for the brass and ff < for the woodwinds) - the phrasing (accented on all the notes) and the 4:3 opposition resulting from the use of rhythms of four sixteenths (strings) and of triplet eighths (winds). The harp, Percussions II and IV and the piano are absent on the third beat of this measure (Example 2 above mm. 22-24).

 In measures 40-43, Carter superimposes different rhythms. He does not only use rhythms which subdivide the beat in a contrasting manner, but he uses unusual metrics in the 4/4 measure (m. 41): he alters via a triplet the number of figures which belong, not only to the beat but to the measure. He opposed very simple rhythms (4:6; 7:4: 6:2; 3:2) but arrives at very complex results since he introduces little by little freer subdivisions to arrive at an extreme complexity in measure 41. When the piano abandons these opposition, the orchestra continues immediately.

Between measures 57 - 59, Carter superimposes different measures. The piano takes one one meter, and the other instruments, another. Similarly, the composer changes the meter in each measure. Thus in measures 57 - 58, Carter uses 3/4 and 5/4 measures for the orchestra and 9/8 and 15/8 measures for the piano. Not content to superimpose two different pulsations, Carter uses a group of nine sixteenths instead of septuplet eighths and he superimposes figures which divide the measure in a regular manner either quarter/eighth or three eighths. These elements are again altered by the phrasing used for the group of nine sixteenths (5+5+8). We note that Carter uses in the harp (3/4 and 5/4 measures). In measure 59, he moreover employs, beyond the septuplet eighths which alter the number of eighths in the 4/4 measure, two different measures for each line of the piano part. The 12/8 measure uses a figure which divides the beat in a regular fashion, the group of three eighths. Carter uses a phrase which alters the basic rhythms. The phrasing creates two gestures, the second ending only in measure 61. We note the ascending character of the first in contrast to the second which rises before descending to the lower register. The gestures given alternatively to the strings and to the wind instruments, give the impression of different currents (i.e. "winds") which move across the orchestra. It is not only the orchestral color which changes but also the density, the rhythmic figures, the melodic designs and the dynamics of the string instruments and winds (Example 4 below).

In measure 97 Carter employs in the 'cellos a figure of six sixteenths grouped 3+3 in a phrasing of 4+4. The displacement of the phrase to lines 3b and 4 of the 'cellos leads to other perceptions of these rhythmic lines. Carter creates diversity in unity. He uses the same procedure (mm. 121-122) in violin ensemble I divided in seven stands. He alters the quintuplets by the phrasing grouping them into 4+8 (1,3a ; 4,3b), 3+8 (2 ; 5), 3+7 (5) and 4+6 (6) (Figure 17).

In measure 126 Carter superimposes again measures with different beats. He even juxtaposes two different meters in the piano; 6/8 and 2/4 which together complete five beats of the 5/4 measure. In this measure the composer places in relief a third line in the lower voice of the piano in which he suggests a reading in 9/8. The phrasings prompt a second level of interpretation. Thus Carter opposes four dotted thirty-seconds against three sixteenths (in the 6/8 measure), and two triplet eighths against a rest of eighth/septuplet in dotted thirty-seconds for the 2/4 measure. In spite of the change of metronomic value, the sixteenth note keeps the same value, 378 MM (Figure 18).

Movement II (mm. 142 - 286) Allegro non troppo q = 84

Carter alters the perception of rhythms by the use of contrasting phrases. In measures 146 and 147, diverse types of phrasing group the septuplets thirty-seconds in the violin 1 producing a very elaborate passage. It is not only the uniform treatment of the septuplet which matters, we have different readings because of different phrasings (Figure 19).

In measures 186 to 189, Carter creates relationships of 6:4:3: or 5:4:3 where contrasting phrases create a different ensemble effect. Carter moves easily from beyond the limit "imposed" by the measure and creates a very soft legato (legato) in figures suggestive of a much more energetic character (played mf and pizzicato ; Example 5 above).

In measures 209 to 212, Carter uses the triplet in such a way as to create a new rhythm, that is to say that he alters the perception of 3+3 by linking the last eighth of one triplet to the first of another. He arrives at a displacement of the quarter within the triplet . Elsewhere as well, Carter's phrasing emphasizes this rhythm emerging from the use of the triplet (Figure 20 below).

In measures 222 to 224, Carter opposes the winds against the strings of the ensemble. He uses the quintuplet sixteenths in the woodwinds and the triplet eighths in the strings. The quite contrasting phrases alternate in pulsations of 5+5+5 or 3+3+3 according to the figures used. We see also that the strings play between pp > ppp or pp <p > pp using pizzicato or legato, and the winds legato between pp or mf (Example 6 above).

In measures 244 to 247 Carter continues to use the septuplet in the piano. Nevertheless, he introduces triplets within the septuplet which increase the complexity of the figure.

Once more (mm. 329-330) the septuplet is used in a very particular manner, either in dotted thirty-seconds or in sixteenths or dotted sixteenths. The use of this type of figure in a 6/8 measure implies the irregular division of the beat or of the measure. Carter creates a third degree of interpretation. In fact, he divides two 6/8 measures into 2+2+2 by the use of the septuplet in sixteenths. The septuplet in dotted thirty-seconds implies an irregular subdivision of the beat and the septuplet in dotted eighths, an irregular subdivision of the measure. The phrasing is also very important in this passage since the composer creates other groups of figures within the septuplet. He even succeeds in linking two measures by the phrasing (Figure 21).

A very important change of texture arises in measure 339 where Carter remains on the tremolo chords. This texture has already been prepared by the mixture of trills and tremolos in measure 338. It emphasizes harmonic passages applied to very long values which are opposed to the preceding passages where rhythms applied to shorter values abound (see Example 7 below).

Movement IV (mm. 420 - 517) Allegro Agitato q = 144

In the course of measures 471- 478, Carter proceeds to an acceleration of time in a relationship of measure = beat. He accelerates from q = 60 to q = 180, then he takes the triplet eighth which he considers equal to the preceding beat (180MM) and proceeds once again through the same type of acceleration. He resumes q = 60 (180 divided by 3 = 60) and he arrives at q = 180. Then he accelerates once more but only in the course of one measure to q = 81, tempo giusto. The preceding accelerations are accomplished in three measures.

Commentary

Carter often conceives his music from images of films which show several planes of one scene before passing to the next. In his music, this is translated by the combination of different sorts of tempi. In the Concerto, beginning from a whole, the work directs attention to the small points or small parts without losing sight of the essential. In this concerto, there emerges the sensation of a perpetual movement where different elements do not cease to exist and change. The works of Carter oppose different characters and their form arises from the variety of instrumental combinations:

Carter superimposes not only different textures which result from the use of different rhythmic elements, but also different metric structures. This process allows him to superimpose different accents and so to create another degree of differentiation of elements in the musical discourse. He superimposes four orchestral groups; he juxtaposes them also. He thus obtains intersecting collages. The fragments of each group are never literally repeated. They are subjected to continuous variation. The groups are characterized by different rhythms, harmonies and expressions. Thus Carter obtains intersections of contrasting materials, notably the tempi between groups 1 and 3 in creating a reversal (mm. 350-353), and so neutralizing the tempo difference. The polyrhythmic passages provide a means of evolving disorganization in the discourse from an intelligible order. In fact, the conflict which results between order and chaos is very significant and effectively illustrates much of the social and political outlook at the end of the '60s. In the Coda Carter reverses the model of the Introduction and the work ends via transformations of the opening.

The tempi of the first and third movements are constant while those of the second and fourth change constantly. In the Coda, the movements are presented in a kaleidoscope between measures 518-600. Carter notes that the metronomic indications are not rigid. Nevertheless, he shows that the relationships indicated in each change of speed should be rigorously respected. The rubato serves to emphasize the character of certain instruments but should not be exaggerated. Certain sections, in particular the solo passages of the 'cello (mm.29 - 125) and the timpani, tuba and double basses (mm. 289 - 418) are played according to these directives.

translation: John MacKay

Selected Bibliography:

articles:

Barret, Richard. "The Notation of Time: a Reply," Contact, a Journal of Contemporary Music, Spring 1987, n^o. 30 pp. 33 - 35.

Bernard, Jonathan W. "Problems of Pitch Structure in Elliott Carter's First and Second String Quartets," Journal of Music Theory , vol. 37.2 1993, pp. 231 - 266.

Boretz, Benjamin. "Conversation with Elliott Carter," Perspectives of New Music, Princeton: Princeton University Press, Spring-Summer 1970, pp. 1-22.

Boykan, Martin. "Elliott Carter and the Postwar Composers," Perspectives on American Composers New York, W. W. Norton & Company, Inc. ed. Benjamin Boretz and Edward T. Cone, 1971, pp. 213 - 216.

Carter, Elliott. "American Intellectual Composers & the "Ideal Public," All American Music: Composition in the Late Twentieth Century, London: John Rockwell, Kahn & Averil, pp. 37 - 46.

___________ The American Composer Speaks, a Historical Anthology 1770 - 1965 Baton Rouge Lousiana: Louisiana State University Press Gilbert Chase, ed.

___________ "Expressionism and American Music," Perspectives on American Composers New York: W.W. Norton & Co. Inc. Benjamin Boretz and Edward T. Cone ed. 1971 pp. 217 - 229.

__________ "Gentility and Apocalypse," Composer to Composer, Converstaions About Contemporary Music , Great Britain: Quartet Books, 1993 pp. 1- 9.

__________ "Il ponto di vista des compositore di musica para orchestra," Elliott Carter - a cura di Enza Restango, Torino: E.D.T., 1989, pp. 196 - 213.

__________ "La Base Rhythmique de la Musique Américaine," Contrechamps Lausanne: L'Age d'Homme, April 1986 n^o 6 pp. 105 - 111.

__________ "Music and the Time Screen," Current Thought in Musicology: Symposia in the Arts and the Humanities Sponsored by the College of Humanities and the College of Fine Arts, University of Texas Press, Austin and London ed. John W. Grubbs, 1976 n^o 4.

__________ "Parla un compositore americano," Elliott Carter - a cura di Enza Restango, Torino: E.D.T., 1989, pp. 185 - 195.

_________ "Ricordo de Charles Ives," Elliott Carter - a cura di Enza Restango, Torino: E.D.T., 1989, pp. 214 - 223.

__________ '"Shop Talk by an American Composer," Problems of Modern Music The Princeton Seminar in Advanced Musical Studies New York: W. W. Norton & Company, Inc., Paul Henry Lang ed. pp. 51 - 64.

__________ " The Milieu of the American Composer," Perspectives of New Music Princeton: Princeton University Press, 1962, pp. 149 - 151.

___________ "The Orchestral Point of View," Norman: University of Oklahoma Press, 1970, pp. 36 - 61.

Derrien, Jean-Pierre. "Elliott Carter Aujourd'hui," Entretemps Paris J. Clattès, June 1987, n^o 4, pp. 51 - 54.

Durieux, Fréderic. "A Mirror on Which to Dwell, Domaines d'une Écriture," Entretemps Paris, J. Clattès, June 1987, n^o 4 pp. 55 - 67.

Escot, Pozzi. "Elliott Carter - A Forward," SONUS: a Journal of Investigations into Global Musical Possibilities vol. 14 Spring 1994, n^o 2 pp. 1 - 4.

Knussen, Sue. "Elliott Carter in Interview with Sue Knussen," Tempo London: Boosey and Hawkes, 1996, n^o 197, pp. 2 - 5.

Link, John. "The Composition of Carter's Night Fantasies, " Sonus A Journal of Investigations into Global Musical Possibiliiies, vol. 14, Spring 1994, n^o 2, pp. 67 - 80.

Mead, Andrew W. "Le Troisième Quatuor à Cordes. Structure des Hauteurs," Entretemps Paris: J. Clattès, June 1987, n^o 4, pp. 77 - 96.

Perlis, Vivian. "Elliott Carter (talks about Ives)" in Charles Ives Remembered. New Haven: Yale University Press, 1974, pp. 131 - 145.

Pozzi, Raffaele. "Elliott Carter Talking," Tempo London: Boosy & Hawkes, Dec. 1988, n^o 167, pp. 14 - 17.

Rosen, Charles . "Entretien avec Elliott Carter," Contrechamps Lausanne: ed. l'Âge d'Homme, April 1986 n^o 6, pp. 112 - 122.

Schiff, David. "Elliott Carter's Harvest Home," Tempo London: Boosey & Hawkes Dec. 1988, n^o 167, pp. 2 - 13.

Shreffler, Anne. "Elliott Carter and his America," SONUS: A Journal of Investigation into Global Musical Possbilities vol. 14, Spring 1994, n^o 2. pp. 38 - 66.

books:

da Silva Santana, Maria do Rosário Elliott Carter: Le Temps et le Rythme dans les Deux Premiers Quattuors à Cordes. paper given at the DEA Musique et Musicologie du 20ième Siècle, Paris EHESS/IRCAM - Centre Georges Pompidou, 1995, 121 p.

Edwards, Allen, with Rosen, Charles, and Holliger, Heinz. Entretiens avec Elliott Carter. Geneva: ed. Contrechamps, 1992, 109 p.

Edwards, Allen. Flawed Words and Stubborn Sounds: A Conversation with Elliott Carter. New York: W. W. Norton, 1971, 128 p.

Harvey, David. The Later Music of Elliott Carter: A Study in Music Theory and Analysis. New York: Garland Publishing Inc., 1989, 343 p.

Rosen, Charles. The Musical Languages of Eliott Carter. Washington: Library of Congress, 1984, 87 p.

Schiff, David. The Music of Elliott Carter. London: Eulenburg Books, New York: Da Capo Press, 1985, 371 p.

Stone, Else and Kurt. The Writings of Elliott Carter. Bloomington: Indiana University Press, 1977, 390 p.

Notes:

2 See Edwards (1971) pp. 56 and 99.

3 See Carter (1976) p. 63.

4 The harpsichord has a weaker sonority hence the writing in rapid figures so that the decay of the sound is "hidden" by the immense proliferation of notes.

5 The instruments are not the same in each orchestra.

6 The lines are created by the emphasis of various pitches within the sixteenth-note triplets. The composer uses this compositional procedure most evidently beginning in measure 443 where he creates a second melodic line, then a third beginning in measure 449. This procedure disappears in measure 453 where we note very important changes in technique.

7 Charles Rosen, The Musical Languages of Elliott Carter, Washington Library of Congress. 1984, p. 35

8 See Allen Edward Flawed Words and Stubborn Sounds A Conversation with Elliott Carter New York: W.W. Norton, 1971, p. 107.

9 Allen Edwards op. cit. p. 104.

10 Allen Edwards, ibid.

11 Allen Edwards, op. cit. p. 111.

12 ibid.

13 Allen Edward, op. cit. p. 113.

14 Allen Edwards, op. cit. p. 115.

15 Alken Edward op. cit. p. 111. For Carter's detailed account of this procedure, see the foot-note on this page.

16 See Schiff (1985) pg. 246.

17 ibid.

18 See Schiff (1985) p. 246.

19 See Schiff (1985) p. 247.

20 Allen Edwards, op., cit p. 102.

21 See Schiff op. cit. p. 46.

22 Allen Edwards op. cit. p. 101.

23 Allen Edwards op. cit p.112.

ex tempore
as published in the Summer 2000 issue