Mathematical Analogies in Atonal Music Composition

Jon Warwick
London South Bank University

Introduction

There are many examples from the Arts where elements of mathematical structure combine with the visual arts to produce quite beautiful results. One thinks here of, perhaps, the beautiful pictures generated through the study of fractal geometry, an example of which is illustrated below in Figure 1. Many others are published in the literature (see for example Peitgen and Richter, 1986)

Figure 1:  A Mandelbrot Set (Pickover, 2000)

There are also many examples in which mathematical ideas are investigated to try and provide a window of understanding into the world of art, design or naturally occurring phenomena. The Golden Ratio, the irrational number:

is a good example here (Huntley, 1970).  It is claimed to have many occurrences in nature and is said to have strong links with composition in works of art (Boles, 1987) although some claims are disputed (Markowsky, 1992).  In this article we take a brief look at musical composition and the way that one particular branch of musical theory challenged conventional ideas of classical composition by adopting, in all but name, simple mathematical ideas related to modular arithmetic and group theory to produce some of the most important musical compositions of the 20th Century.

A Little Musical History

In conventional western musical theory, the basic building block of musical notation is the octave – two notes that sound the same to the human ear but at different pitches.  In terms of the Physics, the note A, for example, can sound at 110 Hz, 220 Hz, 440 Hz, 880 Hz and so on, each pitch sounding an octave higher.  The octave is divided into 12 equal intervals of pitch (semi or half-tones) and the half-tones played in ascending pitch sequence form a chromatic scale.  The classical major or minor scales spanning the octave each use eight of the half-tones.  Individual notes of the major or minor scales are separated by either one or two half-tones.  Playing music in a particular key uses only the pitches making up the major or minor scale of that key, so only a sub-set of the 12 half-tones making up an octave would be used.  Changing key, or modulation, can bring some of the other half-tones into play but then others will cease to be used.  As described in very simplistic terms, this approach of structuring music around individual keys which occasionally change (often according to well defined patterns) has been used to great effect many of the great classical composers.

A quick glance through typical classical concert programmes indicates that the most enduringly popular European classical composers would probably be those of the Viennese School (Hayden, Mozart, Beethoven) or perhaps those coming a little later such as Brahms, Wagner or Mahler.  These six were separated in time by a little under 130 years (Hayden was born in 1732 and Mahler in 1860) and yet in this relatively short period of time musical composition and what came to be understood as the classical style of the time changed quite considerably.  It is true that all of these masters composed using the diatonic scales and diatonic harmony (the system in which harmonies are made up using the notes of the prevailing major or minor key) and yet the music of Mahler sounds quite different to the music of Hayden.  Primarily this was due to changing compositional conventions and successive slackening of the rules of diatonism so that, leaving aside the length and structure of works and the orchestral forces used,  Mahler’s work makes far greater use of the chromatic scale i.e. all the half-tones.  Indeed, the chromatic scale is so heavily employed by Mahler that it is sometimes difficult to determine which key a passage of music has as its base.  At his most adventurous, the rules of composition were stretched further by Mahler to the extent that a piece of music may not end in the same key in which it started which would have been unheard of in the days of Hayden or Mozart!

In the early 20th century, Arnold Schoenberg (1874 – 1951) decided that a logical extension of theory would be to loosen the restraints of tonality completely and abandon the notion of key and diatonism.  In other words, rather than have melody and harmony using the half-tones specified by a particular key, he would give all 12 half-tones of the chromatic scale equal importance so that no particular key was implied at all and by 1908 he had developed the idea of this ‘atonality’ (Perle, 1991).

Unfortunately, if the rules and conventions of classical composition are removed completely then composition becomes extremely difficult.  For one thing, the natural inclination is to put certain tones together that would often give the sound of conventional keys, so by 1923 Schoenberg had fully developed his own set of rules designed to force the use of all 12 tones equally and so avoid the inclination of conventional keys and harmonies to appear.  This became the ’12-tone’ method which was of great significance in the development of 20th century classical music and which was taken up by the likes of Alban Berg,  Anton Webern and Igor Stravinsky.

The 12-Tone Method

Having abandoned the traditional rules of composition, Schoenberg devised a set of rules to replace them with the intention of ensuring the atonality of the composition and to engender some consistency in the use of the 12-tone method.  These rules may be summarised as follows (amended from Kelley, 2002):

• The composer begins by devising a row, or series, of notes that must contain all 12 chromatic pitches without repetition of any individual pitch.  These 12 notes need not be played as a sequence as some may be used as ‘harmony’ but no pitch may be repeated until all the chromatic pitches have been played.  This initial row is termed the prime form;
• The prime form may be developed as the composition proceeds and allowable developments include transposition, inversion, retrograde and retrograde inversion;
• To ensure consistent atonal treatment of the series no notes may be doubled as an octave, and tonal, melodic or harmonic intervals (thus giving the impression of a particular diatonic key) are best avoided.
• The composer should try and make exclusive use of one series per composition.

These rules helped to establish the 12-tone method (or serialism as it was also sometimes called) as an unique compositional process quite distinct from all classical methods employed before.  It helped composers to give structure to their atonal compositions but without having to make use of conventional ideas such as musical themes or musical forms.  The rules produced a ‘democracy of tones’ (Rudhyar, 1927) and allowed, through manipulation of the prime form row, new musical structures and forms to evolve.

Modular Arithmetic

In fact, the rules devised by Schoenberg for his 12-tone method have quite simple mathematical analogies which allow the computer to be a significant aid in producing 12-tone compositions.  We begin, though, by first reviewing the idea of modular arithmetic.

Modular arithmetic is simply arithmetic that takes place using a restricted set of (whole) numbers.  If we are working modulo 12, then this is also called clock arithmetic and can be best described using the clock face of Figure 2.

Figure 2:  Clock Arithmetic – Modulo 12

We can see from this that we are only using the digits 0 through 11 and other numbers not in this range can be re-written as clock numbers so that 13 = 1 (modulo 12) and 14 = 2 (modulo 12) etc.  It should be noted that in order to convert any number to modulo m then we simply need to write down the remainder when the number is divided by m.  For example, 14 divided by 3 leaves a remainder of 2 so 14 = 2 (modulo 3).

The standard arithmetic operations may be defined using modulo arithmetic with addition, subtraction and multiplication providing little problem.  For example, 6 x 3 (modulo 7) would be calculated as 6 x 3 = 18 which is 4 (modulo 7).  The only slightly odd notion is that of division.  As an example, dividing 4 by 3 (modulo 7) requires us to solve the equation (4/3) = y (modulo 7) which by re-arrangement gives the equation 4 = 3y (modulo 7).  From the multiplication example above we know that y should be 6 since 3 x 6 = 18 which is 4 (modulo 7).  Some division problems have no solution (2  3 modulo 9) and others have multiple solutions in modular arithmetic (3  3 modulo 9).

In this description of the linkage between the 12-tone method and modular arithmetic we can restrict ourselves only to addition.  In fact we have, in effect, defined a mathematical structure called an abstract group.  To define a group we need a set, which we shall call G, and a binary operation on G (denoted by * ) which exhibits the properties of closure, associativity, identity and inverse.  In mathematical terms we have the following:

a, b  G  a * b  G  (the property of closure)

a, b, c  G  (a * b) * c = a * (b * c) (the property of associativity)

e  G  a  G  e * a = a = a * e  (the identity property)

a  G b  G  a * b = e = b * a  (the inverse property)

Now if we define our set to be G  =  Z12  =  {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 } then we can see that doing addition modulo 12 is essentially a mapping from Z12 x Z12  Z12 and that Z12 forms an abstract group under addition in which the identity element is 0 and the inverse of an element a would be –a .

Connecting the 12 –Tone Method and Modular Arithmetic

First we note that there are 12 half-tones in an octave and we will number these from 0 to 11.  The original tone row or series on which the whole atonal composition will be based is thus a sequence of twelve numbers between 0 and 11 with no number repeated.  Note that there are a possible 12! tone rows which is 479,001,600 so atonal composers have plenty of choice!.  Now, as we have seen above, Schoenberg’s method allows this tone row to be altered in a number of ways:

Transposition:  this involves moving every member of the tone row up by one, two, three etc half-tones so that there are 11 possible transpositions of the original row.  Note that if the number 11 is transposed up by two half-tones then it becomes 2 (not 13 which does not correspond to a tone in the row) so in making transpositions we are simply carrying out the calculations modulo 12 i.e. in clock arithmetic.  If we denote a member of the original tone row as x, then transposing it by n half-tones would be achieved using the expression Tx(n) = x + n (modulo 12).

Inversion:  this involves ‘turning the tone row upside down’.  Mathematically, this is just a reflection of the tone row in the vertical line joining 0 to 6 on the clock face as in Figure 3 below.

Figure 3: Examples of Inversion

In mathematical terms, we can invert any member of the row, x,  by carrying out the operation described by Ix(12) = -x + 12 (modulo 12).

Retrograde:  this is simply the reverse of the row which is to say that the row is simply read from right to left rather than from left to right.

Retrograde Inversion:  this is a combination of the last two types of manipulation and consists of simply reading the inversion row in reverse order.

Matrix Representation

Using these simple mathematically based ideas it is now possible to generate all the transpositions and inversions for a tone row and represent these as a matrix for ease of use.  In the matrix representation, the rows represent possible transformations and the columns represent possible inversions and these can be produced easily by computer spreadsheet.

We shall begin by defining a tone row: C#, A, B, G, G#, F#, A#, D, E, D#, C, F.  This is a tone row used by Schoenberg in his opus. 23, No. 5 Five Piano Pieces (London, 2000).  If we convert this to the numbering system then we get: 0, 8, 10, 6, 7, 5, 9, 1, 3, 2, 11, 4 using the C# as the base tone labelled 0 and numbering the others in terms of their half-tone difference from the base tone (so that D = 1, D# = 2, E = 3 and so on).  This set of numbers forms the first row of the matrix.  We know work out the inversion of this row and write it as the first column.  Thus we get:

 0 8 10 6 7 5 9 1 3 2 11 4 4 2 6 5 7 3 11 9 10 1 8

Table 1: First Inversion of the Tone Row

Now we complete each row of the matrix by transposing the first row by the number given at the start of the row.  So the second row of the matrix is the first row transformed upwards by 4 (i.e. 4 half-tones) so that we are performing Tx(4).  This exercise is easily  completed by using a computer spreadsheet.

 0 8 10 6 7 5 9 1 3 2 11 4 4 0 2 10 11 9 1 5 7 6 3 8 2 10 0 8 9 7 11 3 5 4 1 6 6 2 4 0 1 11 3 7 9 8 5 10 5 1 3 11 0 10 2 6 8 7 4 9 7 3 5 1 2 0 4 8 10 9 6 11 3 11 1 9 10 8 0 4 6 5 2 7 11 7 9 5 6 4 8 0 2 1 10 3 9 5 7 3 4 2 6 10 0 11 8 1 10 6 8 4 5 3 7 11 1 0 9 2 1 9 11 7 8 6 10 2 4 3 0 5 8 4 6 2 3 1 5 9 11 10 7 0

Table 2:  Complete Matrix of Transpositions and Inversions

Now every transposition and inversion can be found by inspecting the rows and columns, with retrograde versions found by reading the rows and columns from right to left or bottom to top respectively.  The composer can now simply read these transformations of the original row as desired for the composition.

Some Later Developments

Many musicians (including Schoenberg himself) have developed the 12-tone method further.  Examples would be the use of two tone rows simultaneously or perhaps the use of partial tone rows where not all 12 tones are used (Stravinsky employed fewer than 12 tones in some of his atonal work in the 1950’s).

We have looked at tone row operations that involve addition (transposition and inversion) but composers have also sought to use multiplication as a means of transforming their tone rows.  Multiplication does, however, produce some difficulties since if x is a member of the tone row and we carry out the multiplicative operation Mx(y) = x.y (mod 12) in which y is the multiplier then we only get tone rows produced which contain each distinct tone once if y is relatively prime to 12, i.e. y takes the value 1, 5, 7 or 11.   Other values of y will result in tone rows in which tones are repeated.  However, multiplication has been used by atonal composers and is often combined with the transposition operations.

Another example of multiplication is that of pitch multiplication, which has been applied to reduced sets of tones (sets not including all twelve tones) and is a form of Cartesian product of two sets.  The composer Pierre Boulez (Boulez, 1971) has used pitch multiplication in some of his works and in mathematical terms it is defined as:

X  Y = {(x + y) mod 12 : x X, y Y}

For example, if X = {0, 5, 6} and Y = {1, 4} then pitch multiplication yields the set {1, 4, 6, 7, 9, 10}.  Any duplicate tones produced by the operation would be ignored.  In this way, shorter tone rows are combined and developed.

Conclusion

There has been a great deal of material published linking musical analysis with mathematics.  Indeed there are university courses that cover just this subject.  This article has just skimmed the surface of the topic to illustrate that the mathematics involved is not necessarily complex and hopefully interested readers will follow some of the references listed in order to discover more.  Indeed, we have only looked at one form of atonal music composition involving the 12-tone method, and within this only considered how tone rows may be transformed.  We have said nothing about other aspects of musical composition related to the duration of notes, rhythm and structure.

Particularly interesting is the influential work of Allen Forte (Forte, 1973) who further developed the set theory of Milton Babbitt, establishing a complete theory of pitch class set analysis or musical set theory.  Although there are many differences between musical set theory and mathematical set theory there are also interesting comparisons and the reader may wish to become acquainted with the work of Forte and discover more about the links between serial composition and mathematics.

References

Boles, M., (1987), The Golden Relationship: Art, Math, Nature, 2nd ed., Pythagorean Press.

Boulez, P., (1971), Boulez on Music Today. Translated by Susan Bradshaw and Richard Rodney Bennett. Cambridge, Mass.: Harvard University Press.

Huntley, H. E., (1970), The Divine Proportion: A Study in Mathematical Beauty, Dover.

Kelley, R., (2002), “Introduction to Post-Functional Music Analysis: Set Theory, The Matrix, and the Twelve-Tone Method”, viewed 8th October 2006 at http://www.robertkelleyphd.com/12-tone.htm and accessed on 12/10/06.

London, J., (2000), “Row Forms in the Serial Works of Schoenberg, Berg, and Webern”, viewed 16th October 2006 at http://www.people.carleton.edu/~jlondon/2ndviennese.htm

Markowsky, G., (1992), “Misconceptions about the Golden Ratio”, The College Math. J. 23, 2-19.

Peitgen, H.-O., and Richter, P. H., (1986), The Beauty of Fractals: Images of Complex Dynamical Systems, Springer-Verlag, Berlin and New York.

Perle, G., (1991), Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern, University of California Press.

Pickover, C. A., (2000), Wonders of Numbers: Adventures in Math, Mind, and Meaning, Oxford University Press, viewed 16th October 2006,

Rudhyer, D., (1927), “Carl Ruggles, Pioneer - As Seen by a Fellow Modernist”, viewed 15th October 2006 at  http://www.khaldea.com/rudhyar/rugglespioneer.html

Forte, A., (1973), The Structure of Atonal Music, Yale University Press.