Offrande III by Serge Garant, De subitement lointain by François Morel and Quatuor a Cordes No. 2
"Ad Pacem" by André Prévost
As pianist of the ensemble of the Societé de Musique Contemporaine du Québec (SMCQ) I played in the first performance of Offrande III on March 28th, 1971. I was absolutely fascinated by this work and, as I wished to analyze a work of Serge Garant, our major composer of serial music, in my 20th‑century music class at McGill University, I asked Serge to allow me to photocopy his sketches. Thus, I was able to make a detailed analysis of the work and for my final class on his work Serge was present to answer our questions.
In the program notes for the vinyl recording of Radio Canada International which I quote below, Serge wrote " Wanted matter to come alive and sing". One can say that he succeeded in making numbers sing."
Written in March 1971, first performed at the 36th concert of the SMCQ on March 28th, 1971, Offrande III is the third of five works Offrande I, II, III, Circuit I, II ‑ that exploit a vast network of proportions taken from the them of Bach's Musical Offering".
In Offrande I Bach was abundantly quoted, a little of Offrande II and not at all in this work.. On the other hand, one note in particular around which gravitate all the events in the work, is given a truly thematic function: that is the Bb, the one missing note in the original (Bach’s) theme which uses, as is well known, eleven. It is the very absence of this note which has increasingly fascinated me since Offrande I and this quotation "in absentia" is yet another way of showing my admiration for this summit of achievement of the human mind that is the "Musical Offering".
In Offrande III, macro‑ and micro-structures rigorously obey the series of proportions, even thought the latter may be very freely applied ‑ according to the sequences - both to events and harmonic fields and to different parameters: pitch, registers, durations, intensities, tempi.
The symmetry of the instrumentation 3 'cellos, 2 harps and a piano, flanked by 2 percussion ‑ and its precise structure make it a "classical" work. However Offrande III would seem to me to be above all an expressive work" it was written at any rate, in that frame of mind. I wanted matter to come alive and sing. And that is how I hope it will be heard.
This deficient row of 13 notes (see Figure 1) contains neither F nor B but 3 pitches appear twice, D, Ab and Bb. The first of 7 notes consist of a tritone followed by 3 descending chromatic notes (E, Eb, D), a perfect fourth and 3 notes forming a chromatic group (G, Ab, F#). Starting with F# the same pattern is repeated in inversion. Since 2 pitches are missing, there are only 109 different versions of the original row (O1, O2, O3, ... O10) and tem forms of the inversion ("renversement" in French ‑ R1, R2, ... R10). Garant divided the row into 5 cells of 1, 4, 3, 3, and 2 notes (I, II, III, IV, V) which are used for the system of multiplication of the basic chords (see below) and to produce the durations of 8 of the 9 sections of the work with the formula o1 equals 20 seconds. Section B takes its duration from the interval between cells I and II, a tritone or semitones. Section C is calculated from the sum of intervals in cell II (1+1+5=7). Section E takes its duration from the interval between cells II and III i.e. 1(a semitone). Section E is calculated from the sum of the intervals in cell III (2+6+8). Section F takes its duration from the interval between cells III and IV, I.e. 1 (a semitone). Section G is calculated from the sum of intervals in cell IV (1+5+6).
Figure 1: The Row of Offrande III
For Section H the interval between cells IV, V is 1 (one semitone). The interval of cell V gives 2 semitones for Section I. Thus the proportions of the section B to I are 6,7,1,8,1,6,1,3 which gives the durations of 120 seconds (B), 140 seconds, 20 sections (D), 160 seconds (E), 20 (F), 12 seconds (G), 20 seconds (H), 40 seconds (I).
The Four Basic Chords OI, OII, RI, RII and System of Multiplication of the Basic Chords
OI (original I) is composed of the 7 notes of the theme of the Musical Offering of Bach which produced the intervals giving the numbers of the first line of the magic square (see Figure 2) C, D flat, G, A flat, B, A, F. OII is composed of the 5 complementary notes and RI and RII are the inversions ("renversements") of OI and OII.
This system of multiplication (see Figure 2). by the 5 cells of the row produces the harmonic fabric of the work. OI on OI ‑2‑3‑4‑5 produces chord of 7, 12, 9, 11 and 8 notes. The multiplications on R (the inversion of the row) are an octave higher. This enriches the harmonic vocabulary in terms of tessitura. Figure 2 shows the system of multiplication with two examples, OI on O2 and OI on O3. Chord OI is transposed on the 4 notes of cell 02, i.e. E, E flat, D and G. After the first chord on E only the new notes are retained. Those that double preceding notes at the octave are suppressed. Thus the transposition on E flat gives 3 new notes, C, A and F, the transposition on D, on new note, C natural, and the transposition on G no new note.
Figure 2: System of Multiplication by the Five Cells of the Row
q = 64 68 72 76 80 92 96
The Magic Square
The succession of 7 different intervals of the theme of the Musical Offering of Bach gives the 7 numbers of the first horizontal line of the magic square 3,4,1,9,8,3,5. The same numbers appear in the first vertical line. The second horizontal line is identical to the second vertical line and so forth. The square can also be read diagonally, the two "diagonals" being 3,9,4,8,5,1 and 5,1,9,8,9,1,5. These "readings" can be made to the left or to the right of the diagonal giving 4 possibilities in all; in this work Garant uses three of these possibilities:
Figure 3: Square-Reading Diagonals
Garant uses also the square by addition and the square by subtraction of the addition. He also uses 2 of the 4 possibilities of the diagonal of the addition (see Figure 2). The diagonal squares have 7 lines of 7,6,5,4,3,2,1 numbers. Here is the first line of these two squares:
Figure 4: Squares by Addition and Subtraction of the Addition from First Line of Magic Square
Figure 5: The Magic Square of Offrande III and Them from the Musical OFfering
Figure 6: Sectional Structure of Offrande III
Analysis by Section
Section A, of variable length, is dominated by the note Bb, the only note that is absent in the theme of the Musical Offering. At the beginning only 3 'cellos play it, normally or in artificial harmonics, on the fingerboard, with the wood of the bow, or normally with 1/4 tone variations. At rehearsal number 1 the piano starts an improvisation on B flat and E (2nd note in the row OI( with varied dynamics and types of attack. At the same time the two percussion start playing quietly on the skin and metal instruments with phrases of the number of notes in the first line of diagonal 'a' (Perc I, 3‑9‑4‑8‑2‑5‑1) and in the first linen of diagonal by addition 'a" (perc II, 7‑11‑9‑9‑5‑14‑6).
At 3 the piano and the 3 cellos play as at number 1 and the 2 harps play the chord OI or O# on the 2nd sign of the conductor. On the first sign perc I plays phrases with the 4th line of the diagonal (9‑5‑8‑2) and percI with the 5th line (8‑1‑3). On the 2nd sign of the conductor the two percussion play fast phrases on the 5th line (2‑8) and the 7th line (5).
At letter B the 3 'cellos play a 3‑part counterpoint with occasional doublings in the 2 harps and piano. The counterpoint uses the pitches of seven successive chords whose registers display an alternation of high and low spreading out like a fan as follows:
Figure 7: The Seven Chords of Section B
The 3 'cello parts resemble rhythmically a 3‑part species counterpoint with a different note value for each part. Thus 'cello I uses in sixteenths the first line of the square (3‑4‑1‑9‑8‑2‑6) alternating with sixteenth-note rests using the retrograde (5‑2‑8‑9‑1‑4‑2, see Example 1). 'Cello II uses in eighths the second line (4‑9‑2‑3‑5‑1‑8, see Example 2) 'cello III uses in thirty‑seconds the third line (1‑2‑4‑5‑9‑8‑3) alternating with dotted sixteenth‑note rests using the retrograde (1‑2‑4‑5‑9‑8‑3 , see Example 3). At 2 measures before 4 'cello i uses the 5th line in eighths, 'cello II the 5th line in thirty-seconds alternating with dotted sixteenth‑note rests and in retrograde. 'Cello III uses the 4th line in sixteenth alternating with rests in sixteenths and in retrograde.
At 5 measures after 5 'cello I uses the second line in retrograde in units of 1/2, 2/5 or 3/5 of a beat. 'Cello II uses the second line in dotted sixteenths, alternating with rests in thirty-seconds and in retrograde. 'Cello III uses the second line in units of 1/2, 13 or 2/3 of a beat.
At 6 'cello I uses the third line in dotted sixteenths alternating with rests in thirty-seconds and in retrograde. 'Cello II uses the third line in units of 1/2, 1/3 and 2/3 of a beat. 'Cello III uses the third line in retrograde in units of 1/2, 1/5 and 3/5 of a beat.
At 4 measures after 6 'cello I uses the 4th line in retrograde and in units of 1/2, 1/3 and 2/3 of a beat. 'Cello II uses the fourth lie in units of /25, 1/2 and 3/5 of a beat. 'Cello III uses the 4th line in dotted sixteenths alternating with rests in retrograde in dotted thirty-seconds.
Concerning the use of durations, first in the harp and piano parts, at 5 Garant introduces the idea of expressing the middle of a duration, the beginning and the end passing in silence. Thus in the harp I part the first line of the square in retrograde is counted in halves or thirds of a beat. The first duration being of 5 eight notes, the harp plays on the third one (Example 6). In the Harp II part the first line is counted in 1/2 or 2/5 of a beat (Example 7).
In the piano part the first line is expressed in sixteenths, read like a closing fan from the extremes to the center (3‑5‑4‑2‑1‑8‑9). All these durations are expressed in silence except for the last one, 9, which provokes a fast run in grace notes of 17 notes. In the square by addition 17 is the fourth number of the first line, similar to 9 in the original square. Subsequently the first line is read like an opening fan from the center to the extreme (9‑8‑1‑2‑4‑5‑3, see Example 8).
At five measures after 5 harp I plays at the middle of the duration of the second line, counted in 1/2 or 1/3 or a beat. Harp II plays at the middle of the durations of the second line (in retrograde), counted in 1/2 or 2/5 of a beat.
Example 1: 'Cello I in Section B
Example 2: 'Cello II in Section B
Example 3: 'Cello III in Section B
Example 4: 'Cello I at 5
Example 5: 'Cello II at 5
In the piano part the second line is expressed in sixteenths, read from the extremes to the center and then from the center to the extremes. At the beginning the number 8 provokes a run of 12 grace notes. Later the number 9 provokes a run of 11 notes. As at 5 the run contains the number of notes of the corresponding number in the square by addition.
At 6 harp I plays in the middle of the durations of the 3rd line (in retrograde) as above. Harp II also pays in the middle of thoe durations of the 3rd line (in normal direction) as above. In the piano part the 3rd line is expressed in sixteenths, read as a closing fan from the extremes to the center (1‑3‑2‑8‑4‑9‑5). The numbers of 8,9 and 5 provoke grace‑note runs of 11, 17 and 14 notes respectively as above. Then the numbers from centre to the extremes pass in silence.
At five measures after 6 harp I plays in the middle of the durations of the 4th line as above. Harp II plays in the middle of the durations of the 4th line (in retrograde) as above. In the piano part (starting at the 3rd beat) the 4th line is expressed in sixteenths, taken from the extreme to the center (9‑2‑3‑4‑5‑8‑1). The number 4 provokes a grace‑note run of 5 notes as above. Then in the 4th line from center to the extremes (1‑8‑‑5‑4‑3‑2‑9) the number 8 provokes a run of 9 notes.
Figure 7 summarizes the rows used in Section B:
beginning of Section B:
starting at 5:
Figure 8: Rows Used in Section B
Example 9 (above) gives the rows of the ‘cello I part. The notes in brackets are omitted because they are note present in the chord used at the time. Notes that are out of the range of the 'cello are played by the percussion. The short section at 7 uses pitches of a single chord, on , presenting 3‑note chords in the 'cellos as well as more complex chords in the harps, piano and percussion.
The 3 'cellos, using rows R8 and OI express the first line of the diagonal square 'a' in 1/3, 1/2 or 2/3 of a beat. The last number, 1, is replaced by 9 (see Example 10). The two harps, the piano and the percussion play 3 low chords in the middle of the duration 9,8, and 9 of the first line of the diagonal square in retrograde, counted in 1/2, 2/5 or 3/5 of a beat. The first number, 1, is replaced by 9 (see Example 11).
12 4 9 72 16 10 15
Figure 8: Multiplication of First Line of Magic Square at Sign 1
The dynamics relate to the numbers of the original square according to two formulas, the numbers being either the loudest of the softest dynamics:
ppp pp p mf f ff fff
1 2 3 4 5 8 9
fff ff f mf p pp pp
The numbers for 'cello II and 'cello III are derived from the second and third lines respectively.
At sign 2 the numbers for 'cello I, II and III are derived from lines 4, 5 and 6 respectively. In the dynamics there is a diminuendo from mf to p. At sign 3 the numbers for the 3 'celli are derived from line 6. There is a diminuendo from p to ppp.
In percussion I at sign 2 the gong strokes are played pp and with the durations of the two circles which come from the diagonal of the square by addition, line 1 (7‑1‑9‑9‑5‑14‑6) line 2 (13‑6‑13‑3‑8‑10). The unit is 240. In other words one counts the sixteenth with the quarter at 60. At sign 2 the repeated gong strokes are played fff (dry) or ppp (with resonance) according to the gestures of the conductor and at sign 3 the gong strokes are played pp and with durations of the 5th, 6th and 7th lines of the diagonal square 'a". The unit equals 160.
In percussion II at sign 1notes chosen from within a high cluster (G to F# except for Bb) are played on the glockenspiel ppp or fff. The numbers in the circles come from the 3rd, 4th, 5th, 6th and 7th lines of the diagonal square by addition. The unit equals 320; in other words one counts sixteenths with the quarter at 80.
line 3 3‑8‑10‑7‑13
Figure 9: Numbers in Percussion II from Lines 3, 4, 5, 6 and 7 of Magic Square at Sign 1
Example 6: Harp I at 5
Example 7: Harp II at 5
Example 8: Piano at 5
Example 9: Rows in 'Cello I at Five Measures after 6
Example 10: Durational Values in the Three 'Cellos at 7
Example 11: Durational Values in the Harps and Piano at 7
At sign 2 the glockenspiel continues with line 3 (1‑3‑9‑4‑4) and line 4 (9‑5‑8‑2) of the diagonal square counting in sixteenths with the quarter at 160. At sign 3 the gong and tam‑tam strokes are played in the same way as percI.
At sign 1 after a pause of 6 seconds the conductor gives repeated cues following the numbers of the circles formed from the 1st and 2nd lines of the diagonal square 'a'. The unit is 160 and these repeated cues apply only to the harps and the piano. At sign 2 after a pause of 6" the conductor gives repeated cues following the numbers of the circles formed from the 3rd and 4th lines of the diagonal square. These cues apply only to percussion I and at sign 3 after a pause of 5 seconds the conductor gives cues following the numbers of the 5th, 6th and 7th lines of the diagonal square 'a'. The numbers of the 7 lines of the diagonal square gives a total of 128. With the tempo of 160 per unit, this gives 52'. Adding fours pauses (7". 6", 5", 12") and the two chords of 9 seconds (end of page 18) one arrives at a total of 100".
The passage at 8 without the 'cellos lasts 64 quarters at 96 MM, which gives a duration of 40 seconds. The pitches are defined by the content of six chords: OI on O4, OI on R4 (m.5), RI on O4 (m.8), RI on R4 (m.11), OII on R4 (m.15). OII on O4 (m. 16). The 1st, 2nd and 4th chords are modified in order to create a gradual compression of the range. Example 12 shows the modified version of these three chords followed by their normal form. Here are the ranges of the 6 chords:
Example 12: The Chords at 6 with their Respective Ranges
In the piano and harps the upper staff uses the diagonal of the square by addition 'b' horizontally in sixteenths alternating forward and retrograde directions.
® 7 ‑ 11 ‑ 9 ‑ 9 ‑ 5 ‑14 ‑ 6
¬ 5 ‑ 5 ‑ 14 ‑ 5 ‑ 7 ‑11
® 10 ‑ 8 ‑17 ‑ 6 ‑12
¬ 10 ‑ 9 ‑ 4
® 7 ‑ 12
Figure 10a: Durational Values in Upper Staves of Piano and Harps at 8
The lower staff uses the diagonal square by addition 'b' vertically in sixteenths alternating forward and retrograde directions:
® 8 ‑ 7 ‑ 10 ‑ 7 ‑10 ‑ 5 ‑7
¬ 12 ‑ 9 ‑ 6 ‑ 8 ‑ 5 ‑11
® 4 ‑ 11 ‑17‑14 ‑ 9
¬ 11‑ 6 ‑ 5 ‑ 9
® 12 ‑ 7‑ 5
¬ 11 ‑14
Figure 10b: Durational Values in Lower Staves of Piano and Harps at 8
¬ ® ¬ ® ¬ ®
The rows at 8 in the piano and harps are R3, O5, R5 in the upper staff, and R4, O8 RI in the lower staff.
In the Percussion I at this point (in quintuplet sixteenths, see Example 13) the odd-numbered durations come from the diagonal 'b' of the square by addition, the first two lines and the first numbers of the 4rd line (see Figure 11). In Example 13 these numbers are below the staff:
Example 13: Percussion I at 8
® 7‑11 ‑ 9 ‑ 9 ‑ 5 ‑14 ‑ 6
¬ 5 – 5 ‑14 ‑ 5 ‑ 7 ‑11- 10
Figure 11: Odd-numbered Durations in the Percussion 1 Part at 8
The even numbered durations come from the diagonal 'b' of the square by addition starting with the 7th line. In example 13 these numbers are above the staff:
® 7 ‑ 12
¬ 10 ‑ 9 ‑ 4
® 17 ‑ 6 ‑ 11‑11
¬ 10 ‑ 8 ‑ 17‑ 6 ‑12
Figure 12: Even-numbered Durations in the Percussion 1Part at 8
In percussion II the durations are those of percussion I in retrograde.
Figure 13: Orchestration of Chords at Fourth Measure of Section D
This longest section of the work is perceived on hearing, as being in four part: (i) a texturally complex passage with all instruments (p. 25), (ii) a piano solo at 12, (iii) a cluster in the higher register a 13 and (iv) chords in harps and piano (see Figure 12) However, structurally it consists of 8 parts which have the proportions of the whole work in retrograde using the formula of 5 seconds per unit which gives the following plan : 10 seconds, 5 seconds, 30 seconds, 5 seconds, 40 seconds, 5 seconds, 35 seconds, 30 seconds. (total 160 seconds).
In the texturally complex first part (up until 12) the durations come from the square by addition. One must note that the beginning of a duration may be in silence. The number of notes in each duration (or its density) is expressed by two numbers, one coming from the first four lines of the original square, read alternatively forward and retrograde (i.e. line 1 forward, line 2 retrograde, etc.) and the other one starting with the retrograde (i.e. line 1 retrograde, line 2 forward etc.). The tempi are organized according to the first line of diagonal 'a' as follows:
q = 72 96 76 92 68 80 64
The tempi change with the long durations (12, 13, 14, 17 - see Figure 14) In order to see how this system work, let us examine some precise cases in Example 15:
The harmonic structure contains 5 chords (see Example 14).
Example 14: Harmonic Structure of Section E
Figure 14: Relationship of Duration, Density and Tempo at the Beginning of Section E
Example 15: Harmonic Materials in Beginning of Section E
(a) 10 (1/1) one pitch, C, is played arco and then pizzicato on the 4th and 5th sixteenth of the
measure. The beginning of the duration is silent.
(b) p. 25 17(9/8) The 3 sustained notes of the 'cellos, the 2 groups of 2 notes on the marimba
and the two notes of the harp at the end make a total of 9 notes. In the low register piano
and the harp II play 2 times 2 notes, giving a total of 8.
(c) p. 27 8 (5/9) The sustained A of the 'cello plus 2 groups of 2 notes in the harps make 5.
The figure in grace notes in the marimba has 7 pitches plus 2 in silence, making 9.
d) p. 28 8 (3/5) The sustained B flat of 'cellos I and III plus the two pizzicato notes of 'cello II
make 3. The marimba plays a figure of 5 notes of which one is in silence.
(e) p. 29 3(1/8) The A of the 'cello is accompanied by a figure of 8 notes in the cow bells.
(f) 14(5/3) Starting at the second beat there are 2+2+1 (5) notes in the harps, marimba and
piano. The A flat of Harp I and the E flat of the 3 'cellos make only 2 notes, so the possibility of
a 3rd note is not used.
(g) p. 32 9(8/9) The two half notes in the 'cellos (f and E) plus the two thirty-seconds of the
marimba plus the 4 notes of Harp II make a total of 8. The figure in grace notes in the
cowbells has 9 notes.
This passage at 12 for piano and percussion lasts 33 beats at q = 52, which give us 38 seconds plus a pause of 2 seconds, then 40 seconds in total.
In the piano the durations in sixteenths of the upper staff come from the diagonal 'b' square (5‑1‑9‑8‑9‑1‑5 etc.). The duration in sixteenths of the lower staff come from the diagonal 'c' square read from the bottom starting with the 7th line (3‑4‑4‑1‑9‑1‑9‑2‑2‑9 etc.).
® ¬ ® ¬ ® ¬
The rows for the upper staff are R5, O3, R3 (starting with the 8th note) and the rows for the lower staff are O8, R4, O6. Although the registers are not controlled by the system of chords as in other passages, at the beginning there is a wide range and at the end only the extreme low register.
In percussion I the durations come from the first line of the square, counted in quarters (almost) but the performer plays only the centre of the durations, calculated in quintuplet sixteenths. In percussion II the durations come from the first line of the square in retrograde, counted in quarters (almost) but the performer plays only the centre of the durations, calculated in triplet eighths.
For a duration of 5 seconds at 13 we have the chord OII on O1 in the harps and piano. The harmonics in the 'cellos, which constitute a link by their register with the high cluster which follows, come perhaps from the 3 highest notes of OII on O3. The "cluster" of all chromatic notes except for G# in the harps, piano, glockenspiel and crotales lasts 32" followed by a pause of 3 seconds, i.e. 35 seconds in total. The durations with 92MM as a unit, come from reading clockwise or counter‑clockwise, of the square by subtraction of the addition. Here is the first line.
square by addition 7 5 10 17 10 7 8 (7)
square by subtraction 1 5 7 7 3 1 1
Figure 15: First line of Durational Values in Harps and Piano at 13
At 14 the four basic chords played by all except the 'cellos last 10 seconds (OI), 5 seconds (OII), 7 seconds (RI) and 8 seconds (RII) which gives a duration of 30 second in total.
This short section is dominated by the five chords of the 'cellos, extracted from OI on RI,2,3,4,5. The durations of the percussion (Chinese cymbal, tam0tamand claves) are the retrograde of the proportions of the sections of the complete work (2‑1‑6‑1‑8‑1‑7‑6) played at a tempo of 96 MM given by the left hand of the conductor, which gives 32 quarters or 20 seconds.
With his right hand the conductor cues the 5 choirs following a circle of 5 numbers read clockwise or counterclockwise starting with any number. The 5 numbers are obtained from the proportions of the complete work thus:
2 1 6 1 8 1 7 6
└─┘ └─┘ └─┘
3 7 9
Figure 16: Cue Numbers for Conductor in Section F
This long section is perceived in 3 parts, the first, very fast with the 3 'cellos playing the same rhythm until 18, a slower second part with the 'cellos in unison and, at 19 a solo for the two harps. Here the durations come from the square by subtraction of the addition, read forwards for the 'cellos, alternating with a retrograde version for the percussion, always in sixteenths. The different tempi for the 8 lines of the square are based on the first line of the original square in retrograde thus:
q = 112 100 124 128 96 108 104
Figure 16: Tempi of the Eight Lines of Section 8
To calculate the total duration of the seven lies (60 seconds), let us take the first line for example. The numbers total 26 or 52 sixteenths with the retrograde or 13 quarters at 112 MM or 7 seconds (approximately). Here are the 7 lines:
Figure 17: Derivation of Durations from the Seven Lines of the Square at Beginning of Section G
In order to add interest and rhythmic variety Garant modifies the sixteenth notes, treating them as irrational rhythms, for example, the transformations of the first line of the square for the 'cello part (p.38).
Example 16: Transformations of the First Line of Magic Square in ‘Cello Part in Section G
And here are the transformation of the first line of the percussion I part (temple blocks):
Example 17: Transformations the First Line of Magic Square in Percussion I in Section G
Percussion II starts only with the 4th line (p.44) using the 4th line of the original square (9‑3‑5‑8‑1‑4‑2) treated in a free tempo with groups of grace notes. Starting with page 46 percussion II uses the 5th and 6th lines treated in quintuplet eighths. The 7th line is in sixteenth notes (see Example 18). At the 3rd measure of p.51, the addition of 8 triplets of quarter notes foreshadows the very end of the work.
From page 44 on in the piano the ff attacks on B flat use the 4th line of the original square expressed in values of 1 2/3 quarters (see Example 19).
The 3 'cellos have generally the same melodic design (Example 20). The pitches are chosen freely from following chords in Figure 18.
Figure 18: Chords Providing Pitch Material for 3 ‘Cellos in Section G
The passage at 18 lasts 34 beats at 72 MM, or 28 seconds, almost the planned 30 seconds. For their durations, the 3 'cellos in unison and the harps use the diagonal 'a' of the original square in sixteenths. The piano uses the diagonal of the square by addition of 'a' in eighths (first line only, 7‑11‑9‑9‑5‑14‑6). The following are the 4 chords used: RI on O4 at 18 , OI on O4 p.53, measure 2, 3rd beat, RI on R4 p. 54 OI on R4 pl 54 measure 2, second beat.
Example 18: Percussion II Rhythmic Structure at 16
Example 19: Rhythmic Structure in Piano from p. 44 onwards
Example 20: Three 'Celli at Beginning of Section G with Example of Derivations from Oi on O2 and Oi on O3
Example 21: 'Cellos and Harps at 18
The 'cellos respect the registers of the chords and the harps respect the register when they are expressing a duration, but not when they double rhythmically a duration expressed by the 'cellos. The piano often extends the register, especially with low notes.
® ¬ ® ®
>Celli and harps employ rows O8, RI, OI from the 6th note( see Example 21) and the piano R10.
The passage at 19 which is dominated by the two harps lasts 34 beats at 73MM, thus 28 seconds. Harp I uses the retrograde of the diagonal 'a' (5‑8‑2‑3‑1‑8 etc.) with the rows R2, O7,O1. Harp II uses the retrograde of the diagonal 'b' (1‑9‑9‑4‑5‑4 etc.) with the rows O9, R0, O10. The two figures of the piano have 9 and 8 notes (the largest numbers of the squares) and use rows O9 and R10. The numbers of the circle used for the "cluster" come from the first line of the square by addition (7‑5‑10‑17‑10‑7‑8).
These rapid section at H with 9 7/8 measures of 4 beats ( q =120) lasts about 20 seconds. At the end a pause of 6 to 8 seconds is added. The staccato attacks use the last line of the square by subtraction (2‑6‑1‑7‑3‑4‑7) in sixteenths and the rests in eights use the first line of the square by addition (7‑5‑10‑17‑10‑7‑8). The harmonic vocabulary of the entire work, 30 chords in all, are presented in rapid succession in the order OI on O1,2,3,4,5, RI on O,1,2,3,4,5, OI on R1,2,3,4,5, RI on R1,2,3,4,5, OII on O1,2,3,4,5 and RII on oI1,2,3,4,5. The 'cellos provide rhythmic continuity and the harps and the piano alternatively complete the chords.
The section at I is dominated by 7 chords (played by the 'cellos and the harps) lasting 30 seconds, followed by a pause of 2 seconds and a final chord lasting 8 seconds, for a total of 40 seconds. The seven chords are two of the basic chords, OI (only four of the 7 notes and at the octave) OII (2 octaves higher) followed by OI on R1, OI on R2 (9 of the twelve notes), OI on R3 (7 of the 11 notes). OI on R4 (5 of the 12 notes). OI on R5 (only 3 of the 10 notes). The final chord of the work is the basic chord RI. The percussion gives 72 regular strokes at 108 MM on a brake drum with appogiaturas creating a grouping of 8‑7‑5‑10‑17‑10‑7‑8. The last 7 numbers are those of the first line of the square by addition. At the same time the piano plays repeated B=s using the first line of the square by addition (7‑5‑19‑17‑10‑7‑8) counted in 2/3 of 2/5 of a beat at the tempo 1, q = 48.
The Listener's Perception of the Large‑Scale Form
Having examined in detail the complexities of such a highly structured and calculated work in which number reign, it is interesting to look at the large scale form as perceived by the listener. Although Offrande III is conceived in 9 sections, the listener perceives 16 sections. Let us first examine the elements which recur in the course of the work.
i) The B=: At the beginning the B= is alone, several chords appearing only after one
minute. In section C it is accompanied by low chords and by a cluster in the high register.
In section D it is alone but alternating with E, a tritone lower. In section G and I it is present
but in the background.
ii) Slow chords in the harps and piano accompany the B= in sections A and C. At the end of
section E (p.36) they appear alone as in section F where they are played by 3 'cellos in harmonics.
This constitutes a change in timbre, register and density. In section I they are
played in the high register ('cellos and harps) in a filtered version with few notes.
iii) The high cluster appears twice, first in the background in sections c and then all alone
toward the end of section E. Other striking features are the two fast section G and H and
three passages without 'cellos at 8, 12 and 19.
De Subitement Lointain by François Morel
This work of François Morel demonstrates great mastery on all levels. In this study I have put the emphasis on the mastery of harmony since it is so rare in the present day. We shall examine the variable density of the harmonies, their ranges, their spacings and their intervallic structures.
The great inventive wealth is all the more impressive as everything is derived from a singe non‑transposable mode (Example 22). This mode is characterized chiefly by an alternation of tones and semitones. It is a very simple system used in such a way as to give a rich and varied result. This is preferable to what one sees most often, that is, a very complex system that produces a simplistic and monotonous result.
In a statement at the beginning of the score Morel divides his work into 4 sections, A, B, C and D. These sections are of very different lengths, 11, 35, 11 and 12 measures respectively. The first section is a mosaic of 4 "musics""
(1) chorale in the brass at number 3 (2 phrases), 8 (3 phrases) 12 (4 phrases) and 14 (5 phrases)
(2) "solos@ by the English horn (at the beginning and at 15) and by the alto flute (at 5 and at 10)
(3) "chords andn arpeggios" at 2 (3 chords), 4 (2 chords) 7 (2 chords) and 11 (4 chords)
(4) "lines and arpeggios" at 9 (57) and at 13
We shall study these four "musics" separately.
Chorale Melodic Line
It is interesting to study the admirable construction of the melodic line of the chorale, so coherent and yet so varied (Example 23). Two constant elements are the use of anacrusis and ending ("desinence") figures in sixteenths as well as durations of 1,2,3,4,5,6 or 7 sixteenths.
At number 3 the first phrase, rising with large intervals (P44 and x4) with durations 5,4,3,1,2,3 sixteenths, is preceded by an anacrusis of two sixteenths followed by an ending of 3 sixteenths. The second phrase, very static, is centered around the note B.
At number 8 the first two phrase rise, the third descends. The first phrase has a 3‑note anacrusis and a 4‑note ending. The second phrase has a 2‑note anacrusis but the third phrase has none at all. The first has increasing duration 3,4,5,6, the second separates even and odd duration 6,1,3,5,7,2,4 and the third has increasing durations 22.214.171.124.7. The third phrase is static, moving around B like the second phrase at 3.
At number 12 the first two phrase are in the low register and the next two in the high register. The first phrase descends, the second rises. As for the note‑values, the first phrase (after its two sixteenths anacrusis) has two even and two odd numbers (4,6,1,7) while the second phrase presents the retrograde of the 7 number (7,6,5,4,3,3,2,1). In the first phrase the succession of intervals is symmetrical (x4,M2 P4 M2 x4). The third phrase uses the same durations as the first (4,6,1,7) and the fourth phrase the same as the second phrase of 8 (1,3,5,7,2,4). The 5 notes of the ending of this fourth phrase 'echo' the notes of the beginning of the phrase in retrograde, with 2 common notes (B and A) and the tritone E‑A# replaced by D‑G#.
Example 22: Mode used in De Subitement Lointain
Example 23: The "Chorale" Melody in De Subitement Lointain
In the five phrases at 14 the anacrusis and endings in sixteenths are more developed, the first phrase having two sixteenths at the end, the second w two sixteenths at the beginning and four sixteenths at the end, the third three sixteenths at the beginning and five at the end, the fourth with five sixteenths at the beginning and six sixteenths at the end and the fifth with six at the beginning and seven at the end.
Concerning the register, the first phrase has a large range (an octave plus a fourth), the second and fourth are in the middle register, the third and the fifth in the high register (except for the anacrusis and the ending of the fifth). In the fifth phrase the notes F and B are very important. As for the note‑values the first phrase presents the seven numbers in increasing order (1,2,3,4,5,6,7), the second uses 3,5,6,7 (like the third phrase at 8), the third uses 5,6,7 and the fourth is identical to the second phrase at 8 (6,1,3,5,7,2,4)
Harmonization of the Chorale
The second phrase of number 14 (measure 100, see Example 3) is especially rich. We shall examine it in detail. The density of the 11 chords is relatively constant 9,6,6,8,7,66,7,5,6 notes) but their ranges are variable, from an octave and a half for chords 1,2,3,8,9,10,11 to more than two octaves for chords 4,5,6,7 (see Example 24).
Example 24: Chords in Second Phrase of "Chorale" Melody, m. 100
Notable in the first chord is the opposition of two whole‑tone groups.
Example 25: Whole-tone groups in first chord at m.100
The second chord, in thirds, can be seen as a thirteenth chord on F:
Example 26: Second chord at m.100 as Thirteenth Chord on F
The third chord opposes two augmented triads, one on F, the other on G=. In their construction chords 4,5,6,7 feature major and minor seconds. Their spacing can be shown graphically thus:
Figure 19: Spacing of the Chords 4, 5, 6, 7, at m. 100
The fourth chords features an opposition between a chord in whole tones and a chord in fourths around a dissonance D‑E=.
Example 27: Components of Fourth Chord at m. 100
The fifth chord features two minor second of which one (D‑E=) is in common with the fourth chord. It can be seen as an augmented ninth chord with two appogiatura notes.
Example 28: Components of Fifth Chord at m. 100
In the sixth chord the five low notes form a whole‑tone chord (with two major seconds) which is contradicted by the dissonances of the high F. The seventh chord contains both a major and a minor second. It can be seen as a dominant ninth chord (on G) with two appogiatura notes. Note the three tritone, F‑B, D‑A=, A‑E‑=.
Example 29: Components of Seventh Chord at m. 100
The eleventh chord is symmetrical with the structure x4, M2, m3 on E and on E=. The spacing of chord 8,9,10,11 are very disparate with "clusters" of 2,3 and 4 notes.
Example 30: Components of the Eleventh Chord at m. 100
Figure 20: Spacing of Chords 8, 9, 10, 11
Example 31 a, b, c and d: Chords of 1st, 2nd, 4rd and 4th Phrases at m. 79
The six chords of the second phrase at 12 (measure 79, Example 31a), starting in the low register have durations of 6,6,6,7,5,8 notes respectively and diminishing range from 2 octaves to a major ninth. The great variety of spacing can be represented thus:
Figure 21: Spacing of Chords in Second Phrase at m.12 The numbers indicate the number of notes in a cluster of tones or semitones.
The six chords of the third phrase at 12 (measure 82, Example 31b) have densities of 5,,6,7,6,8,7 notes respectively and increasing range from a minor seventh to a minor seventh plus an octave. Here are the spacings:
Note the symmetry of chords 5 and 6:
Figure 22: Spacing of Six Chords of Third Phrase at M.12 and Symmetrical Structures of Chords 5 and 6
The first six chords of the third phrase of 14 (m. 103, Example 31c) are very static, the bass notes being A, C, B‑=,, B=. In these six chords C appears 5 times, E= five times and F six times. It is not a criticism to say that the chords are static. On the contrary it brings necessary contrast.
In the fourth phrase of 13 (measure 104, Example 31d), after an anacrusis of 5 chord of 5 notes, there is a great variety of densities (8,7,4,6,6,5,6 notes) and of range (3 octaves, 2 1/2 octaves, 2 octaves, P11, P12, m10, P12).
The English Horn Solo (beginning of the work)
The 5 phrases of this accompanied solo are ornamented with groups of grace notes and are all in the same register. The fifth phrase is is renderd by the brass.
Between the two extreme note, C and B, there is a descending line of 4 notes, G#-F#- F natural, E. Starting with C these notes are preceded with 1,2,1,3,1 grace notes. This phrase is accompanied by a sustained symmetrical chord built with a M2 and m3 with the addition fo an E= (see Example 32).
Example 32: Notes and Durations of Phrase 1of English Horn Solo with Accompanying Sonority
Phrase 2 (measure 5)
Except for D this phrase uses the same notes as phrase 1. However they are arranged int the form of an opening fan. Theses notes are preceded by grave‑note groups of 2,2,1,4,2,2 notes. As for the accompaniment it precedes the solo with three chords in the brass constructed with perfect and augmented fourths (Example 33b) and follows it with two symmetrical chords (Example 33c).
Example 33: Notes and Durations of Phrase 2 of English Horn Solo with Accompanying Sonorities
Phrase 3 (measure 10)
The five notes of this phrase are arranged with expanding intervals (M2, x4, M6m m6) with one new note (compared with the previous phrase), a low F#. The last three notes are preceded by groups of 3,5, and 3 grace notes. The accompaniment consists of two chords similar to those of the first phrase preceded by an anacrusis of eight thirty-seconds in four parts (2 clarinets and 2 bassoons, see Example 34) Each of the eight chords is different. A constant feature is the interval between the two clarinets and the intervals between the two bassoons, usually a major or minor third. This brief anacrusis is a jewel cut by a master.
Example 34: Notes and Durations of Phrase 3 of English Horn Solo with 32nd-note Anacrusis
Phrase 4 (measure 12)
The five notes of this phrase using short values (1,4,3, 2,1) and alternating directions, are preceded by an anacrusis of 8 thirtisecond notes alternating of m2 and x4. The accompaniment (Example 35) consists of two symmetrical chords played by flutes and clarinets.
Example 35: Notes and Durations of Phrase 4 with Accompanying Chords
Phrase 5 (measure 14)
This final phrase is given to the brass (Example 36) with four chords of durations 1,2,3,4 and 4 sixteenths and of densities 6,7,6 and 7 notes. The spacing of the four chords can be shown as follows:
Example 36: Accompanying Chords of Phrase 4 with Spacings
"Chords and Arpeggios"
This music brings a more relaxed mood to the work. At 2 the first and second chords are presented in two parts, a final attack in the low register with harp (resonating) and piano (staccato) followed by a second attack by vibraphone and tubular bells. In quarter notes the durations of 5 (2+3) and 6 1/2 (4+2 1/2) respectively. The third chord is presented in three parts with the piano‑harp attack, the vibraphone‑bell attack and a final piano attack for a total duration of 10 1/2 quarters (2+4 1/2/+4, see Example 37).
Example 37: Chords at 2
The first attack of the first chord is symmetrical with two groups of three notes (m3,m6). The second attack uses three tritones of which two intersect to create 4‑note chords (P4,m2, P4).
The second chord has the opposite intervallic structure, beginning with two symmetrical groups (P4, m2) followed by tow more in the high register (m3,m6). The third chord has nix notes in common with the second chord and, like the first two thirds, uses mainly tritone and minor seconds.
The slow lines with the indication "irrégulier, flexible" are in general arpeggiated version of the chords. The number of notes in these lines are taken from the numbers 1 to 7. Thus the first chord has "arpeggios" of 3 (clarinet), 5 (flute) and 7 (bass clarinet) notes and the third chord "arpeggios" of 3 (bassoon), 5 (harp), 6 (piano) and 7 (English horn) notes.
At 4 (measure 28) the two chords (see Example 38) have duration of 10 1/2 quarter (7+3 1/2) and 7 1/2 quarters (5+2 1/2). What is different here (from number 2) is the presence of arpeggios in sextuplets (piano, clarinet, flute) in addition to slow lines of 3 and 5 notes (first chord) and of 5 and 7 notes (second chord). The first chord starts with an attack in the low register (harp, piano) with two juxtaposed tritones (F‑B, C‑F#). With the second attack one can see the entire chord as a succession of perfect and augmented fourths ( F,B, F, B=, E=, A, D, G, C F#). The second chord is similar to the first with three common notes in the high register (G, C, F#) and eight notes transposed a semitone lower.
Example 38: Chords at 4
The two chords at 7 (measure 40, see Example 39), in the middle register, have durations of 10 1/2 quarters (8+ 2 1/2) and 12 1/2 quarters (6+6 1/2). As for the first chord, the first attack with harp and piano (staccato) is followed by a second attack with the piano (with resonance) and vibraphone (staccato). There are slow lines "irrégulier, flexible" of 4 (piano) and 7 (bells) notes but also sextuplets (piano and vibraphone) and a line in eighth notes (harp and piano).
Example 39: Chords at 7
As for the second chord, the first attack (piano, harp) is followed by a second attack with oboes and bassoons (held notes) and flutes and clarinets (staccato). There are slow lines "irrégulier, flexible" of 7 notes (piano) and 5 notes (harp) and sextuplets in the winds and the bells. A new element here are the sustained notes in the horn.
The symmetrical first chord contains a group of 3 notes (P4, m6) and its inversion around a central semitone (E‑F) and two whole‑tone groups of 4 notes (M2, M3, M2). The second chord very similar to the first with eight common notes, contains two groups of three notes (M3, x4) around a central semitone (A‑B=) and two whole‑tone groups of 3 notes (M2, x4).
The 4 chords at 11 (measure 71, see Example 40) have shorter durations as follows:
Example 40: Chords at 11 with Rhythmic Relationships
The first attacks (harp, piano) have a constant range and descend progressively. The second attacks (winds) have widely varying ranges ‑ 3 1/2 octaves, 1 1/2 octaves, 2 1/4 octaves, a major seventh respectively. Their spacings can be shown thus:
Figure 23: Spacing of the Chords at 11
The slow lines are fewer here, 6 notes in the English horn, 4 notes in the flute and 3 notes in the bassoon in the 3 first chords respectively.
In the first chord the first attack is similar to the second chord at 7, with 6 notes in common. It forms an augmented ninth chord. The second attack (with an anacrusis in sextuplets) is characterized by the tritones at the extremes.
In the second chord the first attack has 5 notes in common with the first attack of the first chord. It also forms an augmented ninth chord. The second attack has 4 notes in common with that of the first chord.
the opposite of the spacing of the third chord.
"Lines and Arpeggios"
At 9 (m.56) the lines use durations 1 to 7 thirty-seconds, notably at measure 57 in the piano (7,6,5,4,3,2,1) at measure 59 in the English horn, clarinet, flute and vibraphone (2,3,4,5,6,7) at measure 62 in the oboe (4,5,6,7,1,2,3) and at measure 68 in the woodwinds (5,6,7). The harmonic structure (Example 41) descends toward the low register with various colors, such as a chord in fourths (measure 65) and a chord with three augmented triads (measure 67).
Example 41: Harmonic Structure at 9
At 13 (m. 87) the principal melodic line assumes more importance. It uses more a restricted mode than that used for the work as a whole. This mode creates a symmetrical mirror around the central notes C# and D. In the first phrase one sees the durations 3,5,7, in the second 2,4,6,1,3,5,7, in the third 1,.3,5, in the 4th 7,4,5,5,4, in the 5th 6,5,4,3,2,1 (Example 42).
Example 42: Mode and Melodic Line at m. 87
One notes also several melodic formulae which return with different note values, a (B, E=, F), b (E,C#,,F) or (C#,E,F) c (B,C#,C natural, F#) and d (C,B= E). At measure 87 the Japanese bowl on A= uses the durations 1,3,5,7,2,4,6,1,2,2,1,4,5,6. The harmonic structure is similar to that of number 9:
Example 43: Harmonic Structure at 13
"Section B" (measures 110‑146)
An English horn solo serves as a bridge to this section which alternates a scherzo' (with tempo indication 'vif') in 6 phrases with slow chords (with the indication "Libre et très calme".) The overall form can be represented as follows:
(i) 5,7,8,1 (5) (ii) 8,1 (6) (iii) 5,4 (5) (iv) 2,4 (3) (v) 8,1,3 (2) (vi) 5,1,4,1,10
Figure 24: Formal Scheme of B Section
The numbers indicate the groups of eighths separated by rests. The numbers in brackets indicate the number of slow chords Now we can examine these two elements.
In the rhythmic plan for the 6 phrases of the scherzo (below, the upper stems indicate the attacks of the woodwinds. The lower stems indicate the percussion. The accents (>) indicate the brass.
Usually the percussion plays all attacks but at measure 129 it omits the first of the 4 attacks and, at measure 132, our of 8 attacks it plays only 2,3,4,6,7,8. Usually the woodwinds start with or after the percussion except at measures 129, 132, and 143 where the percussion starts after the woodwinds. At measure 135 there is no percussion. All three elements contribute to the vibrant and exciting result:
Example 44: Rhythmic Structure at mm. 112 - 140
Example 45: Harmonic Structures at MM. 119, 123, 127, 131 and 136
The five harmonic progressions are absolutely fascinating. At measure 119 (see Example 46) the five chords are notable for the chromatic "sliding" between them and for the ambiguity of the "cluster". The first chord can be seen as a superposition of two consonances, a seventh chord and a diminished triad with the following spacing:
Example 46: Spacing and Components of First Chord at m. 119
A first slide of a minor sixth (F#‑F natural, B= ‑ A) between the first two chords is followed by another one of a tritone (E= ‑ D, A ‑ A=) between the 3rd and 4th chords. One can see the 5th chord as a superposition of an F minor triad and a diminished triad on A# (see Example 47).
Example 47: Components of Fifth Chord at m. 119
The progression of six chords at measure 123 is extremely dissonant. In the second chord one finds the six chromatic notes from A# to E=. The 3rd chord adds a minor 9th (G‑A=) and the 4th chord another minor ninth (f‑F#). In the 6th chord there is a "resolution" into 3 tritones (F#‑C, F‑B, A‑E=). In fact the 5 upper notes form a "relaxed" whole tone chord.
The progression of 5 chords at measure 127 starts with a relatively consonant chord: (2 minor sixths, one major sixth, one minor third) with equal spacing:
Figure 25: Spacing of First Chord at m. 127
By the third chord the spacing is uneven (2+2+2+1 notes):
Figure 26: Spacing of Third Chord at m. 127
The fourth chord is absolutely symmetrical. As from the extreme notes, this 8-note chord is very consonant, the 6 inner notes forming a whole-tone chord.
Figure 27: Spacing of Fourth Chord at m. 127
The fifth chord is almost symmetrical:
Figure 28: Spacing of Fifth Chord at m. 127
At measure 131 the first chord has equal spacing for its 8 notes and opposes a diminished triad and a diminished seventh chord at a distance of a semitone.
Example 48: Components of Second Chord at m. 131
The second chord (with 5 notes) is diatonic (D= major) with this spacing .
The third chord (with 8 notes) is notable for its 2 minor third and its major second. At measure136 the first chord, very dissonant with its two minor ninths, opposes one consonance of 4 notes (a whole‑tone chord) to another consonance on C# (an incomplete dominant 7th).
Example 49: Components of First Chord at m. 136
The second chord features 3 tritones (F#‑C, F‑B and C# G).
This short section of 12 measures at 16 (m. 147) features highly ornamented solo in the flute and the piano as well as held notes in the brass and in the clarinets with bassoons. The harmonic structure (Example 50) resembles that of the "lines and arpeggios".
Example 50: Harmonic Structure at 16 (Section C)
This passage at 17 (measure 159) acts as a Coda. First we hear a fff chord played 5 times with durations of 11, 11, 17 and 9 eighths, separated by lines much as at 2. Then there is a recall of the 6 first notes of the English horn solo from the beginning accompanied by 4 sustained chord (Example 51) which reminds us of those of "lines and arpeggios" and of which the last one is identical to the first chord of the piece with the addition of 3 notes in the low register (A, E=, F).
Example 51: Harmonic Structure at 17 (Beginning of Section D)
String Quartet #2 "Ad Pacem@ by André Prévost
The music of André Prévost in general and this work in particular is distinguished by a very pronounced structuralism and by and great expressivity in marked contrasts and nuances of tessitura, of speed and by the clarity of the large form. Before getting into the details of the technique it seems appropriate tot describe the large form in its eight sections in the first part and the six sections of the second part. It is a question of two parts and not of two contrasting movements, and as the composer specifies that the pause between the two parts should be very brief, we perceive aurally a work in fourteen sections.
In any case, the reference numbers in the score, which curiously d not correspond to the structure of the music but more generally to 9 measure intervals, continue in the second part; that is 1‑ 46 for the first part, and 47 to 63 for the second part.
Figure 29: The Large Form of Quattuor à cordes no. 2 (Timings given are those from the Alcan Quartet recording)
Example 52: Series Used in Quattuor à cordes no. 2
Example 53: Harmonic Texture of Fourth Measure of 3, Octave in Darkened Notes
The quartet is in twelve‑tone technique but in a very special twelve-tone technique. First, contrary to typical twelve‑tone technique, there are no changes of octave. In other words, the interval between the two first notes of the series used at the beginning is always a major seventh, never a minor second.
Example 52 shows the series used in this section. There are only two, the one on C (for the second violin and 'cello) and its transposition on E= (for violin and viola). It is necessary to note that pitches 7‑12 are the inversion of notes 1‑6 and notes `‑1, 5‑6, 7‑8 and 11=12 form major sevenths.
If one examines the series of the 'cello part one sees that this series is present 4 times but in permutations of its four groups of three pitches: I (1,2,3), II (4,5,6), III (7,8,9), IV (10,11,12). If ones applies this to the 'cello part, the result is thus: I II III IV | II I IV III | III IV I II | IV III II I. So from the third presentation of the series there is a retrograde (in Roman numerals).
One notices several canons (in pitches, not in rhythms). At the beginning there is a canon at the minor third between 'cello and viola and between violn 2 and violin 1. In the allegretto there is a canon at the octave between the 'cello and the second violin, and between the viola and the first violin. In the @vif A there are two canons at the third as at the beginning. In the Atrès vif@ there are two canons at the octave as in the the allegretto.
We should now look at the organization of registers. The >cello rises over four octaves and the first violin also descends four octaves. This a clear relationship of inversion. The same principle applies for the viola which rises an octaves and the second violin which descends an octave.
Next, there is the progressive appearance of double stops with successive notes of the series (1-2-3-4-5- etc.) which arise at the beginning in the >cello, in the allegretto in the viola, in vif in the second violin and in the très vif in the first violin.
In typical 12‑tone technique one of the reasons for octave changes in the unfolding of the series is to avoid octave relations. But Prévost loves octaves. He has been able to integrate them perfectly into his chromatic language. In Example 53 from the fourth measure of 3 I have indicated in black the notes in octave relations.
We see in Example 48 that the first violin and viola finish on B=, the 9th note of the series on E= and that the 'cello and second violin also finish on B=, the third note of the series on C. Because of the organization of registers (see above) we have B= in 3 octaves, the 'cello above, the second violin and viola at the unison in the middle and the first violin below.
Example 54 shows the 5 chords linked by the glissandi. The 5th chord is repeated (1 before 15) first an octave lower and then 2 octaves lower (except for the F#). Note the disposition of the instruments (from high to low) for the three last chords: vc, vla, vln2, vln1 / vln2, vln1, vc vla / vln1, vln2, vla, vc.
Example 54: Chords Linked by Glissandi in Section 2
Example 55 shows the 12 chords which utilize the beginning series of the Second Part (see Example 54). In other words there are only 3 chords of 4 tones but their dispositions change.
Example 55: Chord Utilizing the Beginning of the Series in Second Part
This transition section has a dynamic profile of ff > pp < ff and in parallel, a tempo profile of * = 100, rall. * =66 accel. * = 360. After a rhythm on C#‑F# in the first violin above a 4 note chord we have (from 1 before 17) imitations with notes 7 ‑ 12 of the initial series beginning on F (viola, C# ('cello), E (viola), G (second violin) and B= (first violin). The two last notes (11, 12) of the four last entries form a rhythm in major 7ths on D, E=, G=, A.
This passage is uni‑rhythmic. The complex rhythm is presented 4 times: (i) 2 before 19, (ii) at 22 (iii) two after 25 (iv) five after 28. The complete rhythm is presented in Example 56.
Example 56: Complex Rhythm of Section 5
Example 57: Rhythmic Motives and Variants from Section 5
Examination of this rhythm reveals 4 motives which return with variations. Motive 'b' possesses two dotted quarters. In motive 'c' the first beat is divided and in motive 'd' the second beat is divided. The 4 motives and their variants are presented in Example 57.
Example 58 gives the series of the second violin part from 3 measures before 19. The series unfold in normal order (1 to 12) with notes 7‑12 first. Because of the construction of the series one could also explain this by the use of the original series or its inversion. I have indicated the first note of the series with an arrow. The texture of this section is a series of canons in pitch (not in rhythm) at the octave. The first canon begins 4 measure before 10 with the 'cello, followed by the viola, second violin and first violin over 4 octaves and uses the series on c, E=, G and E. he second canon begins 1 before 22 with the viola followed by the second violin, violin 1 and 'cello and uses the series on E=, G=, B= and G. The third canon begins in the second measure of 25 with the second violin followed by the first violin, 'cello and viola with the series on C, A, F, and A=. The fourth canon begins in the 5th measure of 28 in the first violin followed by the 'cello, viola and second violin over 3 octaves only with the series on E=, C, A= and B.
Example 58: Series in Second Violin Before 3 Before 19 Section 6
This section beginning at 32 is the exact retrograde of section 4 except that the 'cello takes the first violin part (two octaves lower), the viola takes the second violin part (same tessitura), the second violin takes the viola part (2 octaves lower) and the first violin takes the 'cello part (4 octaves higher).
This section beginning at the sixth measure of 34 serves as a transition. Example 59 shows the last measure of section 6 and the three chords of section 7 with a narrowing of the range. The notes F# and D# pass through the 'cello, viola, second violin and first violin. The notes D and B pass through the second violin, first violin, 'cello and the viola. The note C passes from the viola to the 'cello, to the first violin and second violin (with the addition of F). The note B= passes from the first violin to the second violin, to the viola, to the 'cello (with the addition of A).
Example 59: Last Chord of Section Six and Three Chords of Section 7
In section 8 at 35 the 4 canons are identical to those of section 5 but the order of entry is different. We could compare them thus:
first canon second canon third canon fourth canon
Section 5 vc, vla, vln2, vln1 vla, vln2, vln1 vc vln2, vln1, vc, vla vln1, vc, vla, vln2
Section 8 vln1, vln2, vla, vc vln2, vla, vc, vln1 vla, vc, vln1, vln2 vc, vla, vln2, vln1
Figure 30: Comparison of Canons in Section 5 and 8
From the fourth canon (second measure of 44) the preceding rhythm is abandoned for a succession of continuous eighths. The repeated notes from 46 use the same pitches (four major sevenths on C, E=, G= and A) as the passage at 18 before section 5.
Section 1 Adagio q = 50 dynamic level f
This first section of 8 measures greatly resembles section 3 of the First Part. The series (see Example 60) is present in the first violin. There are only 3 chord of 5 tones with the following changes of disposition:
Example 60: Series in First Violin and Superpositions at Beginning of Second Part
Section 2 Vif
A series of quarters is established with 4 phrases of 12 measures making a progressive crescendo: p (pizz), mf (arco), f, ff
(1 before 48) The 'cello line is the inversion of that of the first violin. The viola's is the inversion of the that of the second violin. The line of the first violin, for this first phrase is limited to 12 fixed pitches (see Example 61). Above and below four degrees of whole-tone scales frame a chromatic group.
Example 61: Fixed Pitches of First Violin Phrase at Beginning of Section 2, Second Part
Example 62 shows the parts of the first and second violin for the first four measures. We note in each part two major sevenths, two perfect fourths and three major seconds. In writing the intervals of the two parts we note the imitations started from a single note or simultaneity.
Example 62: Pitches of First and Second Violins in First Four Measures of Section 2, Part 2
vln1 x4 M7 m9 M7 M2 P4 M2 m7 M2 P4 M2
ú ú ú ú ú ú ú | |
vln2 m9 x4 M7 m9 M7 M2 P4 M2 M3 P4 M2
Figure 31: Intervallic Imitation Between First and Second Violin
Example 63 shows the first violin part for the following 4 measures (from the fourth measure of 48). We note twice a minor third followed by a minor sixth and twice the opposite, a minor sixth followed by a minor third. The second violin keeps the same notes in relation to the first violin as in the first four measures. In other words, an A in the first violin will be accompanied by a G in the second violin and an E= in the first violin will the accompanied by a G.
Example 63: Pitches of Violin Parts from Fourth Measure of 48
Example 64 shows the first violin for the following four measures (from the eighth measure of 48). This is a case of a retrograde inversion of the first violin part from before 48 (see Example 62).
Example 64: Pitches of Violin Part from Eighth Measure of 48
Phrase (ii) at 49
The viola line is the inversion of that of the first violin. The 'cello's is also the inversion of that of the second violin. These relations remain up to the end of the section. Example 61 shows the fixed pitches for the first violin parts and second violin parts in this second phrase. Here we find three time the formula of tone‑semitone‑tone on G, E= and B.
Example 65: Fixed for First and Second Violin at 49
Example 66 shows the violin part for the first four measures. This is a case of an inversion of the line at the beginning of the first phrase (1 measure before 48). In writing the intervals of the first and second violin parts we note the same imitations as at the beginning of the first phrase.
Example 66: Actual Pitches of Violin Part at 49
Example 67a shows the first violin part for the following four measures. This is a case of the (transposed) retrograde of Example 63. Example 67b shows the first violin part from one measure before 50. This is a (transposed) retrograde of the line before 48 (see Example 62).
Example 67: Pitches of First Violin Part of Subsequent Four Measures and at One Measure before 50
Phrase (iii) (fourth measure of 50)
In this phrase the fixed pitches are those of Example 57. Example 64 shows the first and second violin part for the first four measures. In relation to the two violin parts at one before 48 (see Example 58), these parts give the following figures:
Example 68: First and Second Violin Pitches at Fourth Measure of 50
Example 69 shows the first violin part for the four following measures (from the fourth measure of 50). In relation to the first violin part at the fourth measure of 48 (see Example 59) this part gives the following figures: 3,4,5,6 11,12,12,2 7,8,9,10
Example 69: Violin Part from Second Measure before 51
Example 70 shows the first violin part from the third measure of 51. In relation to the first violin part at the eighth measure of 48 (Example 63a) this part give the following figures: 5,6,7 1,2,3 8,9,10,11,12,4.
Example 70: First Violin Part from Third Measure of 51
Violin 2 keeps for all of this phrase the same notes against the first violin as in Example 64.
Phrase (iv) (3 measures before 52)
In this phrase the fixed pitches are those of Example 65. Example 71 shows the violin 1 part for the first four measures. This is a retrograde of the first violin part at one before 50 (Example 67b).
Example 71: First Violin Part at Three Measures before 52
Example 72 shows the first violin part for the four following measures. This is an instance of the retrograde of the first violin part at the fifth measure of 49 (Example 67a).
Example 72: First Violin Part from Second Measure 51
Example 73 shows the first violin part from the sixth measure of 52. This is a case of the retrograde of the first violin part at 49 (Example 66). In all this phrase the second violin plays the same notes against the first violin as at 49.
Example 73: First Violin Part from Sixth Measure 52
Section 3 (53)
Example 74: First Violin Part at 53
This passage in its glissandi is in the form of a double canon (of pitches, not of rhythms) at the minor third and inversion. Here are the four entries:
'cello (series form the First Part on C m 7‑12, 1‑6)
viola (second measure before 54, series on E= 7‑12, 1‑6)
second violin (55, series on C 1 to 12)
first violin (fifth measure of 55, series on E= 1 to 12)
What lends great interest to this passage is the variety of speeds in the glissandi. For example, here are the duration of glissandi in Major 7ths in half: 6 (vc), 9 (viola), 1 (vln2), 4 (vln1), 1 (vc), 7 (viola), 3 (vln2), 6 (vln1), 6 (vln2), 9 (vln1), 3 (vc), 6(viola), 4(vln2), 7(vln1).
Section 5 (third measure of 57)
The first pattern, beginning in the 'cello in the third measure of 57 alternates with A (the first note of the series of 1 before 48 in the first violin, see Example 58) and C# (the sixth note of the series of the First Part on C).
The four entries make a crescendo from ppp to ff finishing all on B= as in section 1 of the First Part. Example 75 give the notes of the entries of the 'cello, first notes 1 ‑ 6 of the series of 1 before 48 (Example 62) followed at measure 4 of 59 by notes 1 to 6 of the series of the fourth measure of 50 (Example 68).
Example 75: Pitches from >Cello Part at Third Measures of 57
Example 76 give the notes of the entry of the viola, first at three before 55, notes 1 to 6 of the series at 49 (first violin, Example 66) followed at two before 60 by notes 1 to 6 of the series of 3 before 52 (vln1, Example 71).
Example 76: Entry Pitches of the Viola at Three Before 55 and Two Before 60
Example 77 gives the notes of the entry of the second violin, first at the fourth measure of 59, notes 1 to 6 of the series of 1 before 48 (fist violin Example 62) followed at 3 before 61 by notes 1 to 6 of the series of the fourth measure of 50 (second violin, Example 68).
Example 77: Entry notes of Second Violin at Four Measures of 59 and Three Before 61
Example 78 gives the notes of the entry of the first violin, first at the second measure of 60, notes 1 to 6 of the series at 49 (first violin, Example 66) followed at the third measure of 61 by notes 1 to 6 of the series of 3 before 52 (violin 1, Example 71).
Example 78: First Violin Part from Sixth Measure 52
Section 6 w = 50
The final glissando of B= over four octaves passes at four speeds:
for the first violin in eighths for semitone from B= to A=
for the second violin in quarters for the semitone from B= to F
for the viola in halfs for the semitone from B= to E=
for the 'cello in whole notes for the semitone from B= to C
This quartet is dedicated to John Roberts and his wife Christine. I should underline the importance of this remarkable man who, as director of music of the English network of the CBC, greatly maintained and encouraged, not only André Prévost, but many other Canadian composers and performers.
February 1, 2004