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Culture and Creativity: Impact of Combinatorics on the
Webern Concerto for Nine Instruments (Opus 24)
S. Kojo ENNINFUL1 & W. OBENG-DENTEH2
1Centre for Cultural and African Studies, Kwame Nkrumah University of Science and Technology,
Kumasi, Ghana
skenninful63@yahoo.com 2Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi,
Ghana
ABSTRACT This research work is an analysis of Bars 41 – 55 of the third movement of Webern’s Concerto for nine
instruments (Opus 24), which is a composition under Serialism. It brings to the fore, the creativity in
Webern’s compositions. It analyses the Combinatoriality of four groups of three notes of constant
Melodic Intervals of <1, 4>; and the symmetrical inversions of the groups of notes using Imagery. The
research adds new knowledge in the interplay between Mathematics and Musical Composition. It then
came out with a name known as Musician Combinatorialist.
Keywords: pitch, nine instruments, angular mathematical music, partition, bell numbers, serialism,
musician combinatorialist
INTRODUCTION
Anton Webern was an Austrian Composer and Conductor, born on December 3, 1883 at Vienna, Austria;
and died at Mittersill, Austria, on September 15, 1945. Anton Friedrich Wilhelm von Webern was the
only surviving son of Carl von Webern, a civil servant, and Amelie, née Geer, who was a competent
pianist and accomplished singer—the only obvious source of the talent of a future composer (Hayes,
1995). He lived in Graz and Klagenfurt for much of his youth. But his distinct and lasting sense of Heimat
was shaped by his reading Peter Rosegger (Johnson, 1999); Webern's father inherited an estate, upon the
death of Webern's grandfather in 1889 (Johnson, 1999).
Webern's music, along with that of Berg, Křenek, Schoenberg, and others, was denounced as "cultural
Bolshevism" and "degenerate art" by the Nazi Party in Germany, and both publication and performances
of it were banned soon after the Anschluss in 1938, although neither did it fare well under the preceding
years of Austrofascism (Taruskin, 2008; Moldenhauer and Rosaleen, 1978; Bailey, 1998). As early as
1933, an Austrian gauleiter on Bayerischer Rundfunk mistakenly and very likely maliciously
characterized both Berg and Webern as Jewish composers (Bailey, 1998). As a result of official
disapproval throughout the '30s, both found it harder to earn a living; Webern lost a promising conducting
career which might have otherwise been more noted and recorded and had to take on work as an editor
and proofreader for his publishers, Universal Edition (Bailey, 1998). His family's financial situation
deteriorated until, by August 1940, his personal records reflected no monthly income (Bailey, 1998). It
was thanks to the Swiss philanthropist Werner Reinhart that Webern was able to attend the festive
premiere of his Variations for Orchestra, op. 30 in Winterthur, Switzerland in 1943. Reinhart invested all
the financial and diplomatic means at his disposal to enable Webern to travel to Switzerland. In return for
this support, Webern dedicated the work to him (Shreffler, 1999).
There are different descriptions of Webern's attitude towards Nazism; this is perhaps attributable either to
its complexity, his internal ambivalence, his prosperity in the preceding years (1918–1934) of post-war
International Journal of Innovative Social Sciences & Humanities Research 4(1):28-39, Jan.-Mar. 2016
© SEAHI PUBLICATIONS, 2016 www.seahipaj.org ISSN: 2354-2926
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Red Vienna in the First Republic of Austria, the subsequently divided political factions of his homeland
as represented in his friends and family (from Zionist Schoenberg to his Nazi son Peter) (Bailey, 1998), or
the different contexts in which or audiences to whom his views were expressed. Further insight into
Webern's attitudes comes with the realization that Nazism itself was deeply multifaceted, marked "not
[by] a coherent doctrine or body of systemically interrelated ideas, but rather [by] a vaguer worldview
made up of a number of prejudices with varied appeals to different audiences which could scarcely be
dignified with the term 'ideology(Fulbrook, 2011).'"
He was a member of the Second Viennese School. As a significant follower of Arnold Schoenberg, he
became one of the best-known exponents of the twelve-tone technique.In the increasingly hostile political
climate of the early C20th, the idealistic symphonies seemed more than a little out of place. Musicians
like Schoenberg, Berg and Webern reacted against the establishment and stripped music backed to its bare
bones. By putting each of the 12 notes in the scale in a particular order, they produced angular
mathematical music known as Serialism.
COMBINATORICS IN MUSIC
It is an important fact that Combinatorics is a stem of mathematics relating to the learning of finite or
countable discrete structures. Facets of combinatorics comprise counting the structures of a specified
variety and size namely enumerative combinatorics, deciphering when convinced criteria can be met, and
constructing and analyzing objects meeting the criteria as in combinatorial designs and matroid theory,
finding "largest", "smallest", or "optimal" objects as in extremal combinatorics and combinatorial
optimization, and studying combinatorial structures arising in an algebraic framework, or applying
algebraic techniques to combinatorial problems as in algebraic combinatorics.
Combinatorial problems crop up in several areas of pure mathematics, notably in algebra, probability
theory, topology, and geometry (Björner and Stanley, 2010) and combinatorics also has many
applications in mathematical optimization, computer science, ergodic theory and statistical physics. Many
combinatorial questions have, in times gone by, been well thought-out in isolation, giving an ad hoc
clarification to a setback arising in some mathematical perspective. In the later twentieth century,
however, influential and wide-ranging speculative methods were developed, making combinatorics into a
self-determining twig of mathematics in its own right. One of the oldest and most accessible parts of
combinatorics is graph theory, which also has numerous natural associations to other areas.
Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of
algorithms.
A mathematician who studies combinatorics is called a combinatorialist or a combinatorist.
Fundamental combinatorial concepts and enumerative consequences appeared all the way through the
ancient world. In 6th century BCE, ancient Indian physician Sushruta asserts in Sushruta Samhita that 63
combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc., thus computing
all 26 − 1 possibilities. Greek historian Plutarch discusses an squabble between Chrysippus during 3rd
century BCE and Hipparchus in 2nd century BCE, of a rather fragile enumerative problem, which was
later made known to be linked to Schröder numbers(Stanley, 1997; Habsieger, Kazarian and Lando). In
the Ostomachion, Archimedes of 3rd century BCE considers a tiling puzzle.
The Indian mathematician Mahāvīra (c. 850) provided formulae for the number of permutations and
combinations(O'Connor and Edmund; Puttaswamy,2000), and these formulas may have been
recognizable to Indian mathematicians as early as the 6th century CE(Biggs, 1979). The philosopher and
astronomer Rabbi Abraham ibn Ezra (c. 1140) established the symmetry of binomial coefficients, while a
closed formula was obtained later by the talmudist and mathematician Levi ben Gerson,enhanced known
as Gersonides, in 1321(Maistrov,1974). In Medieval England, campanology provided examples of what is
now known as Hamiltonian cycles in certain Cayley graphs on permutations (White, 198).
Combinatorial analogs of concepts and methods in topology are used to study graph coloring, fair
division, partitions, partially ordered sets, decision trees, necklace problems and discrete Morse theory. It
ought to not be mystified with combinatorial topology which is an older name for algebraic topology.
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WEBERN’S CONCERTO FOR 9 INSTRUMENTS (OPUS 24)
The following is an extract from the third movement of the Concerto (Measures 41 – 58)
Figure 1
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The measures 41 to 55 of the 3rd movement of Webern’s Concerto for 9 instruments (Flute, Oboe,
Clarinet, Horn, Trumpet, Trombone, Violin, Viola and Piano), Op. 24 is a make-up of Twelve-tone rows
with combinatoriality of four groups of three notes of constant melodic intervals of <1, 4>. Each row is
made up of interval Prime form (P)<+1. -4>, a Retrograde (R)<-4, +1>,a Retrograde Inversion (RI) <+4, -
1>, and an Inversion(I) <-1, +4>.
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Figure 2: 12 X 12 row matrix
PRIME FORM (P)
0 1 9 8 4 5 3 7 6 11 10 2
11 0 8 7 3 4 2 6 5 10 9 1
3 4 0 11 7 8 6 10 9 2 1 5
4 5 1 0 8 9 7 11 10 3 2 6
8 9 5 4 0 1 11 3 2 7 6 10
7 8 4 3 11 0 10 2 1 6 5 9
9 10 6 5 1 2 0 4 3 8 7 11
5 6 2 1 9 10 8 0 11 4 3 7
6 7 3 2 10 11 9 1 0 5 4 8
1 2 10 9 5 6 4 8 7 0 11 3
2 3 11 10 6 7 5 9 8 1 0 4
10 11 7 6 2 3 1 5 4 9 8 0
RETROGRADE (R)
INVERSION (I) RETROGRADE INVERSION (RI)
Fig. 3 Rows in the section
ROW 1: (I5) – mm. 41.4 – 44.2
ROW 2: (P4) – mm.44.3 – 46.2
ROW 3: (I9) – mm.46.4 – 48.3
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ROW 4: (R3) – mm.48.4 – 50.2
ROW 5: (P7) – mm.50.3 – 52.1
ROW 6: (I) – mm.52.2 – 54.2
ROW 7: (RI9) – mm.54.2 – 55.4
Webern’s choice of rows in this segment (mm. 41 – 55) of the piece is very relative. Considering the first
two rows, he interchanges two notes in the first two motives, retrograde the motives and does the same in
the last two motives.
CHOICE OF ROWS IN MM. 41 - 55
FIRST ROW: < (5, 4, 8), (9, 1, 0), (2, 10, 11), (6, 7, 3) >
SECOND ROW: < (4, 5, 1), (0, 8, 9), (7, 11, 10), (3, 2, 6) >
From the above, it is seen that the middle pitch of the second motive of the first row interchanges with the
last pitch of the first motive. The middle pitch of the last (fourth) motive interchanges with the first pitch
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of the third motive; and the pitches which were not tempered with, just retrogrades in each motive: and
that arrangement is used as the second row.
The third row has an idea of retrograding. The first half of the second row is retrograded to begin the third
row; and the second half of the second row is also retrograded to continue the third row.
SECOND ROW: < (4, 5, 1), (0, 8, 9), (7, 11, 10), (3, 2, 6) >
THIRD ROW: < (9, 8, 0), (1, 5, 4), (6, 2, 3), (10, 11, 7) >
The fourth row is a retrograde inversion of the third row.
THIRD ROW: < (9, 8, 0), (1, 5, 4), (6, 2, 3 ),(10, 11, 7) >
+ + + + + + + + + + + +
Retrograde of the 4th Row < (3, 4, 0),(11, 7, 8), (6, 10, 9), (2, 1, 5) >
0 0 0 0 0 0 0 0 0 0 0 0
FOURTH ROW: <(5, 1, 2), ( 9, 10, 6), (8, 7, 11), (0, 4, 3)>
The fifth row is an inversion of the first row.
FIRST ROW: < (5, 4, 8), (9, 1, 0), (2, 10, 11), (6, 7, 3) >
+ + + + + + + + + + + +
FIFTH ROW: < (7, 8. 4), (3, 11, 0),(10, 2, 1 ), (6, 5, 9) >
0 0 0 0 0 0 0 0 0 0 0 0
In the sixth row, Webern interposition the second motive of the fourth row with the first motive ; and the
fourth motive with the third of the same row. He then finds the addend of each pitch to get the sum of 10
in base 12, or get the pitch class 10.
THE FOURTH ROW: < (5, 1, 2), ( 9, 10, 6), (8, 7, 11), (0, 4, 3) >
< ( 9, 10, 6), (5, 1, 2), (0, 4, 3), (8, 7, 11) >
+ + + + + + + + + + + +
THE SIXTH ROW: < (1, 0, 4), (5, 9, 8), (10, 6, 7), (2, 3, 11) >
1012 1012
The seventh row is a retrograde of the third row.
THIRD ROW: < (9, 8, 0), (1, 5, 4), (6, 2, 3 ), (10, 11, 7) >
SEVENTH ROW: < (7, 11, 10), (3, 2, 6), (4, 5, 1). (0, 8, 9) >
There are also intervallic similarities in motives as played by other instruments apart from the piano. In
mm. 41.4 – 42, the motive by the woodwinds has ordered interval of <-1 +4>; and this is inverted and
transposed (T+1) in mm. 43 – 44 by the same instruments to <+1 – 4).
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Horn Horn
< -1 +4 > Trombone < +1 – 4 > Trombone
Inversion/Transposition
In mm. 45.2 – 46.2, the trumpet’s motive is ordered pitch interval < -8 +11>; and this is transposed a
tritone down in mm. 47 – 48.1 by the Horn.
Trumpet Horn
< - 8 + 11 >< - 8 + 11> T – 6
In mm. 48 – 49, the Viola has ordered pitch interval of <+8 +13>; and this is retrograded and transposed
(T+4) by the Violin <+13 +8>.
Viola< +8 + 13 >
Violin < +13 + 8 > T+4
Also in mm. 50.3 – 51.3, the Flute has ordered interval of <+1 – 4>; and it is transposed (T-19) and
retrograded by the Clarinet.
Flute < +1 – 4 >
Clarinet< - 4 + 1 >
The Horn/Trombone have <+11 – 8> in mm. 52 – 53 and retrograde <-8 +11> in mm.53. In m.54, the
Bass has <+4 +11>; and the Violin retrogrades and inverts to <+11 -8>.
There are also recurring pitch class groupings between two instruments or sometimes resounded by the
same instrument. Samples are provided in the table below:
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Table 1: Recurring Pitch class groupings
Pitch Class
Motives
Instrument
Measure
Instrument
repeating
Motive
Measure
Form
<4, 5, 1> Piano 44.3 Horn
Piano
47 – 48.1
55.2
RI
P
<0, 8, 9> Piano 45.1 Piano
Piano
46.4
55.4
I
R
<7, 11, 10> Trumpet 45.2 – 46.2 Piano
Viola
48.3
54.2 – 54.4
P
RI
<3, 2, 6> Piano 46.2 Piano 48.1 R
<5, 1, 2> Viola 48.4 – 49.2 Violin 54.4 – 55.2 I
Motivic linear melodic events exist within the winds and the strings; whilst the piano has motivic
trichords. In the entire section (mm. 41 – 55), it is evident that whenever there are two other instruments
playing alongside the piano in a row, the piano takes two motives and allows the instruments to share the
other two motives or the pitches in the motives of the row. Also, when there is only one other instrument,
the piano takes three of the motives in the row and allows the instrument to play only one motive.
In mm. 41.4 – 44.2 and mm. 52.2 – 53.4, there are similarities instrumentally, rhythmically also in the
intervallic structure. In the former, the Horn plays two quarter notes (Crotchets) and the Trombone
responds with a half note (Minim) in each motive; and the two motives are separated from each other by
the two Trichords of the Piano. But in the latter, the Trombone plays a half note and it is responded by the
Horn with two quarter notes. Here, the ending of the first motive is overlapped by the beginning of the
next; and these appear in between two separate Trichords of the Piano. Probably the overlapping caused
the distortion or the delay of the constant whole note (Semibreve) rest after each two Trichords of the
Piano to six beats rest. When the intervallic structure of the latter is changed to ordered pitch class
interval, they stand as the same as that of the first motive in mm. 44.4 – 42.3 and its retrograde
(ie. < - 1, +4 >< +4 – 1 >).
The rhythmic texture of the section really relates to the beginning of the piece. At mm. 1 – 6, there is the
use of half notes in motives; and this rhythmic timingis decreased by half constants from mm. 7 – 13
except that of the piano in mm. 9 – 11. From mm. 41.4 – 48.1, the melodic motives have a quarter note
rest after each quarter note, which in effect when added to the note, makes it a half note. From mm. 14.1 –
28.1, after every three Trichords there is a melodic motive. In mm. 41.4 – 55.4, after one or two melodic
motives, there are two Trichords.
The Piano in a way serves as response to single or double question(s) posed by other instruments.
Basically, after each set of motive or two by other instruments, the Piano comes in with two Trichords
which stand as two separate motives with a quarter note rest between them. From mm. 42 to the first
quarter of mm. 52, the entry points of the Piano are consistent (ie, after a whole note rest). However, it is
also clear that each grouping of two Trichords by the Piano, the next Trichord begins a beat earlier than
the one just before. In mm. 42 – 43, it begins on the last quarter of mm.42 and ends on the second quarter
of mm.43. after a whole note rest, the second grouping begins on the third quarter of mm.44 and ends on
the first quarter of mm. 45. In mm. 46, it begins on the second quarter and ends on the last quarter of the
same measure. Mm.48 sees the grouping beginning on the first quarter and ending on the third quarter.
From mm.49 – 50, it continues the same trend. But in mm.58, where it is expected to begin on the second
quarter and end on the fourth, Webern inverts the entry points. He begins on the fourth quarter and ends
on the second quarter of mm.54. it is also expected that he would continue with the first quarter to the
Enninful & Obeng-Denteh.... Int. J. Innovative Soc. Sc. & Hum. Res. 4(1): 28-39, 2016
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37
third quarter, but he again reverses the entry in mm. 53 – 54 in a way; bringing the actual grouping
supposed to be in mm.53 – 54. Therefore, the grouping in mm.53 – 54 stands as a delay.
Again in each grouping, the second trichord serves as an inversion and transposition of the first. However,
in mm.46 the unexpected happens; instead of inverting and transposing, the trichord is just transposed
tritone above. In mm.45 – 48.1, the last two trichords stand as exact repetition of the first two. Also in
mm.49 – 52and mm. 53 – 55, the four trichords of the Piano respectively serve as a row.
In each trichord of the Piano from mm.42 – 55, there is a constant pitch interval of (11, 8, 3), creating
intervallic vector of (101100).
MATHEMATICAL PERSPECTIVE IN RELATION TO COMBINATORIAL ASPECTS OF
PARTITIONS
A partition of a set X is a set of nonempty subsets of X such that each element x in X is in precisely one of
these subsets (Halmos, 1960).
Consistently, a family of sets P is a partition of X if and only if all of the subsequent state of affairs hold
(Lucas, 1990):
1. P does not enclose the empty set.
2. The union of the sets in P is equal to X. The sets in P are said to cover X.
3. The intersection of any two distinct sets in P is empty.
In mathematical notation, these conditions can be represented as
1.
2.
3. If and and then .
where is the empty set. The sets in P are given the name blocks, parts or cells of the partition (Brualdi,
2004).
The above concept can be applied mathematically to showcase the convergence of mathematics and
music
FIRST ROW: < (5, 4, 8), (9, 1, 0), (2, 10, 11), (6, 7, 3) >
SECOND ROW: < (4, 5, 1), (0, 8, 9), (7, 11, 10), (3, 2, 6) >
The corresponding pitches are P1 = (2, 10, 11) and P2 = (0, 8, 9) do satisfy the three axioms above:
For P1 = (2, 10,11 )
The set { 2, 10, 11 } has these five partitions:
{ {2}, {10}, {11} }.
{ {2, 10}, {11} }.
{ {2, 10}, {10} }.
{ {2}, {10, 11} }.
{ {2, 10, 11} }.
Similar process can be carried out for P2 = (0, 8, 9).
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As a consequence the accepted wisdom of equivalence relation and partition are essentially equivalent
(Schechter, 1997).
It is assumed and established in this context that the empty set is not part because in music every part
counts.
Partitions can be counted by using the formula where the first several Bell
numbers are B0 = 1, B1 = 1, B2 = 2, B3 = 5, B4 = 15. Rhyme schemes were considered in Gardner (1978).
The number of noncrossing partitions of an n-element set is the Catalan number Cn, given by
.
The interplay of combinatorialism and music arising out of Music of Webern then gave rise to the name
Musician combinatorialist.
CONCLUSION
The research work carried out an analysis of Bars 41 – 55 of the third movement of Webern’s Concerto
for nine instruments (Opus 24), which was a composition under Serialism. It actually brought to the fore,
the creativity in Webern’s compositions by analysing the Combinatoriality of four groups of three notes
of constant Melodic Intervals of <1, 4>; and the symmetrical inversions of the groups of notes using
Imagery. The research adds new knowledge in the interplay between Mathematics and Musical
Composition based on partitioning and the usage of non-crossing partitions as well. It then came out with
a name known as Musician Combinatorialist.
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Press. ISBN 0-521-57336-X (cloth) ISBN 0-521-57566-4 (pbk).
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doi:10.1016/0315-0860(79)90074-0.
Björner, A.; and Stanley, R. P.; (2010); A Combinatorial Miscellany
Brualdi, R. A. (2004). Introductory Combinatorics (4th ed.). Pearson Prentice Hall. ISBN 0-13-100119-1.
Fulbrook, M.(2011). A History of Germany 1918-2008: The Divided Nation, third edition. Hoboken: John
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rhymes". Scientific American 238: 24–30. doi:10.1038/scientificamerican0578-24. Reprinted
with an addendum as "The Tinkly Temple Bells", Chapter 2 of Fractal Music, Hypercards, and
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Work. New York: Alfred A. Knopf. ISBN 0-394-47237-3 London: Gollancz. ISBN 0-575-02436-
4.
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Shreffler, A. C. 1999. "Anton Webern". In Schoenberg, Berg, and Webern: A Companion to the Second
Viennese School, edited by Bryan R. Simms, 251–314. Westport CT and London: Greenwood
Press, 1999. ISBN 9780313296048.
Stanley, R. P.; "Hipparchus, Plutarch, Schröder, and Hough", American Mathematical Monthly 104
(1997), no. 4, 344–350.
Taruskin, R.. (2008). "The Dark Side of the Moon". In his The Danger of Music and Other Anti-Utopian
Essays, 202–16.
White, A. T.; "Ringing the Cosets", American Mathematical Monthly, 94 (1987), no. 8, 721–746; White,
Arthur T.; "Fabian Stedman: The First Group Theorist?", American Mathematical Monthly, 103
(1996), no. 9, 771–778.
Enninful & Obeng-Denteh.... Int. J. Innovative Soc. Sc. & Hum. Res. 4(1): 28-39, 2016
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