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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

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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.

REFERENCES

Bailey, K. 1998. The Life of Webern. Musical Lives. Cambridge and New York: Cambridge University

Press. ISBN 0-521-57336-X (cloth) ISBN 0-521-57566-4 (pbk).

Biggs, N. L. (1979). "The Roots of Combinatorics". Historia Mathematica 6: 109–136.

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

Wiley & Sons. ISBN 9781444359725

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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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Mathematical Monthly 105, no. 5, 446.

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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.

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