Offrande III by Serge Garant, De
subitement lointain by François Morel and
Quatuor a Cordes No. 2
"Ad
Pacem" by
André Prévost
Bruce Mather
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.
The Row
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
The Tempi
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).
Section B
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).
Section C
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
4 12‑14‑12‑6
5 13‑9‑5
6 3‑11
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
¬ 8
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
6
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:
8
® 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.
Section D
Figure 13:
Orchestration
of Chords at Fourth Measure of Section D
Section E
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
of addition
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.
Section F
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
Section G
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 1.3.5.6.7. 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.
Phrase 1
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.
Scherzo
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
Slow Chords
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:
. m3
. M7
. M6
. m7
. m6
. m6
Figure 25:
Spacing
of First Chord at m. 127
By the third chord
the spacing is uneven (2+2+2+1 notes):
: P4
: m3
: m3
.
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.
P4 ┐
m6 │
x4 ┘
M2
x4 ┐
m6 │
P4 ┘
Figure 27:
Spacing
of Fourth
Chord at m. 127
The fifth chord is
almost symmetrical:
M6
M2 ┐
m6 ┘
m2
m6 ┐
M2 ┘
m6
m6
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).
Section C
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)
Section D
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.
First 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
Section 1
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.
Section 2
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
Section 3
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
Section 4
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.
Section 5
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).
Section 7
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
Section 8
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.
Second Part
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
Phrase (i)
(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.
Bruce Mather
February 1, 2004