'overall movement proportions of most of his three-movement piano works' but also the internal form of each of the three songs in his song cycle *Trois Mélodies* of 1916. She found that the usage here was 'both too consistent and too precise to attribute to chance or instinct.'164 The three songs of *Trois Mélodies* are 'La Statue de bronze', 'Daphénéo' and 'Le Chapelier'. 'Daphénéo' is used here as an example (see below). The song opens with both the vocal line and the piano accompaniment formed largely of quavers.

'Daphénéo' is 39 bars long so we would expect the GS division to occur shortly after bar 24 and for there to be a new turn in the music. As Adams points out, 'the long GS (after bar 24) marks the end of two bars which include the lowest note in the piece; moreover it introduces a sudden semiquaver motion that leads to a new register at the reprise.' So a case could be made here for the use of GS.165

164 Adams. 'Erik Satie and Golden Section', 250-251.

165 Adams, 'Erik Satie and Golden Section', 248.

Erik Satie *'Daphénéo'* bars 17 - 28

## **Béla Bartók, Claude Debussy and Golden Section**

Other claims for the use of GS in music, notably by Bartók and Debussy, have aroused more controversy. In his book *Béla Bartók: An Analysis of His Music* (1971), the Hungarian music analyst Ernő Lendvai argued that the composer used formal principles based on GS and the Fibonacci series, in, for example, his *Sonata for Two Pianos and Percussion* and *Music for Strings, Percussion and Celesta* which were both composed in 1937.166 Lendvai studied the latter piece in detail and cited, amongst other examples, the use of Fibonacci numbers as structural devices in the first movement. Unfortunately he made a few errors and fudged some of his calculations. The music analyst Roy Howat discredited the work and he countered these claims in his own detailed analysis of the work which exposed some of the inaccuracies. He found for example that 'numbers of bars begun are sometimes confused with bars completed'. Howat argued that the use of proportional systems was most likely to have been subconscious given that there was 'no definite

166 Ernő Lendvai. *Béla Bartók: An Analysis of His Music*. (London: Kahn and Averill, 1971).

proof of conscious application'.167 He did however make a case for the use of Fibonacci numbers in the slow third movement of Bartók's *Music for Strings, Percussion and Celesta* (music to familiar to those who have seen the film *The Shining* where it is used to create an eerie atmosphere). The Fibonacci series is quite clear to see in the xylophone solo of the opening bars where the rhythmic pattern uses 1–2–3–5–8–5–3–2–1.

Howat found further examples of the GS and Fibonacci series in this movement by dividing the movement up into crotchet beats, rather than bars as Lendvai had done, revealing clear structural examples.168

The question remains as to whether Bartók was making conscious use of these mathematical proportions. The musicologist Paul Griffiths is very dismissive writing that the 'appearance of the Fibonacci series in a rhythmic pattern at the start of the slow Movement of Bartók's *Music for Strings, Percussion and Celesta* (1–2–3–5–8–5–3–2–1) is suggestive, but no more'.169 Lendvai acknowledged that Bartók said very little about his compositional techniques stating 'Let my music speak for itself; I lay no claim to any explanation of my works'.170 As Laszlo Somfai wrote, in his research into Bartók's extant sketches and autograph, he did not find 'a single calculation of the proportion of a composition - with Fibonacci or other numbers'.171 Howat wrote that Bartók was secretive and reluctant to divulge the 'use of highly

167 Roy Howat. 'Bartók, Lendvai and the Principles of Proportional Analysis' *Music Analysis.* Vol. 2, No. 1 (March 1983): 69-95.

168 Howat, 'Bartok and Proportional Analysis': 80-82.

169 Paul Griffiths, 'Numbers in music' in *Grove Music.*

170 Mario Livio. *The Golden Ratio: The Story of Phi, the World's Most Astonishing Number.* (New York: Broadway Books, 2002): 190.

171 Laszlo Somfai. *Béla Bartok: Composition, Concepts and Autograph Scores*. (Berkeley: University of California Press, 1996).

abstract constructions in what many critics already regarded as over-cerebral music' but he goes on to point out that 'More positive manuscript evidence of proportional calculations exist in Bartók's transcriptions of folk music' citing as evidence Manuscript 80FSS1 in the New York Bartók Archive, a sketchbook devoted largely to Bartók's first drafts of Turkish folk songs. These include a metrical sequence of numbers which is easily identifiable as the Lucas summation series (1, 3, 4, 7, 11, 18 …) where, in the same way as the Fibonacci sequence, each term is the sum of the two previous terms, and the ratios of successive terms approach the Golden ratio.172 Howat argued that this demonstrates that the song's 'metrical organization is analysed wholly in terms sequence numbers'.173

In 1983, Howat went on to make a detailed case for GS as a structural device in Debussy's music in his book *Debussy in Proportion: A Musical Analysis* where he set out to demonstrate Debussy's use of 'intricate proportional systems', in particular the Golden Section .174 His study included detailed analyses of the piano pieces *Reflets dans l'eau* (1905), *L'isle joyeuse* (1904), and the symphonic poem *La mer* (1908), and demonstrated how the pieces are built around GS and the Fibonacci series. In *Reflets dans l'eau*, the first of three pieces in Debussy's first volume of *Images*, as part of the detailed analysis Howat found that the principal climax at bars 56-61 tallied with the overall point of GS i.e. after 58 bars out of a total of 94. He also discovered that Fibonacci numbers are often found at 'strategic places' such as in the 55 bars of introduction to the final section, 'Dialogue du vent et de la mer', of *La Mer* (where the bar groups follow the Fibonacci numbers 3, 5, 8, 13, 34 and 55) and the 34 bars of 'build-up to the climactic coda' of *L'isle joyeuse* (bars 186-219)*.* Howat considered the question of whether Debussy's proportional patterns were 'designed consciously' or 'intuited unconsciously' but came to no definite conclusion he did however acknowledge that Debussy very probably knew about GS and Fibonacci through a series of articles in which the mathematician Charles Henry discussed such structural principles in

172 Lucas numbers are named after the French mathematician François Édouard Lucas (1842-

91) who launched the *American Journal of Mathematics* in 1878 with a long paper dealing with the Fibonacci sequence along with the sequence starting 1, 3 which he named after himself.

173 Howat, 'Bartok and Proportional Analysis', 85.

174 Roy Howat. *Debussy in Proportion: A Musical Analysis* (Cambridge: Cambridge University Press, 1983).

the Symbolist journals which Debussy 'read avidly'.175 Howat later added a caveat 'How consciously this came about is unproven, and the subject can still be contentious; its main interest lies in linking different aspects of the structure into a naturally balanced dramatic flow'.176

In the 1940s, the Russian/American teacher and composer Joseph Schillinger (1895-1943) devised his own system of composition based on mathematical principles.177 His thinking behind this was that a mathematical approach would be more natural because nature was intrinsically mathematical. In his words, 'Music imitates nature, and particularly the forms of motion in the universe, i.e. the growth and evolution of natural forms. The forms of organic growth appeal to us as a form of beauty when expressed through an art medium'.178 In his books *The Schillinger System of Musical Composition* (1941) and *The Mathematical Basis of the Arts* (1948) he suggested creating pitch structures based on the Fibonacci sequence.179 His idea is based on counting semitones where each semitone represents one unit. So, for example, starting on C and moving up one semitone would mean that the second note would be Db. Schillinger's system alternated between ascending and descending intervals. In this way the Fibonacci series 1, 1, 2, 3, 5… starting on C would create the following sequence of notes.

**C** up one semitone **Db**, down two semitones **B**, up three semitones **D**, down five semitones **A,** as below.

175 Roy Howat. *The Art of French Piano Music: Debussy, Ravel, Fauré, Chabrier*. (London: Yale University Press, 2009).

176 François Lesure and Roy Howat, 'Debussy' in *Grove Music*.

177 After Joseph Schillinger had emigrated to the USA he taught his mathematical methods to a large number of pupils, many of these were popular musicians and included George Gershwin, Glen Miller and Quincy Jones amongst their number. Another pupil, Lawrence Berk, became the founder of the Berklee College of Music where Schillinger's methods were taught in its early days. Schillinger's ideas included various mathematical ways in which to produce melodies, rhythms and chords and, although they were littered with miscalculations and misunderstandings of mathematical concepts, for many years they were in high demand.

178 Joseph Schillinger: *The Schillinger System of Musical Composition* (1941).

179 Joseph Schillinger: *The Schillinger System of Musical Composition* (1941) and T*he Mathematical Basis of the Arts* (1948).

**Darmstadt International Summer Courses for New Music in the 1950s and 1960s**

In 1854, Eduard Hanslick had argued in *The Beautiful in Music* that 'No mathematical calculation ever enters into a composition, be it the best or the worst. Creations of inventive genius are not arithmetical sums'180 One hundred years later, however, there were several composers associated with the centre of new music in Darmstadt who made conscious and open use of GS and the Fibonacci series. Central to the Darmstadt International Summer Courses for New Music were premieres of new works and the teaching of composition.181 It soon became what Alex Ross describes as the 'principal showcase for the avant garde'.182 In his 1957 lecture on the development of serial technique the Italian composer Luigi Nono (1924-1990) coined the term 'Darmstadt School' for a group of composers that included Bruno Maderna, Karlheinz Stockhausen, Pierre Boulez and Nono himself.

Stockhausen (1928-2007) went on to make extensive use of the Fibonacci series in his music of the 1960s and, according to the American music theorist Jonathan Kramer, used it 'prominently and systematically'. Kramer argues that in Stockhausen's music the Fibonacci series contributes 'more significantly to the total form than it does in the [music of] Bartók'; the Fibonacci series is used proportionally in many ways, some simple and some complicated'. In *Klavierstuck IX* (1961), for example, majority of bars have Fibonacci-related time signatures and in the coda, each bar contains either 1, 2, 3, 5, 8, 13, or 21 attack points. A larger scale use of Fibonacci proportions can also be found in *Adieu* (1966) for wind quintet where the durations of all the bars 'except the five which contain tonal references and the four with unmeasured silences' are given by Fibonacci numbers from 1 to 144. Kramer wrote that the 'properties of the series are readily heard because each

180 Hanslick, *The Beautiful in Music, 92.*

181 Darmstadt International Summer Courses for New Music (*Ferienkurse fiir neue Musik*) were initiated in 1946 by Wolfgang Steinecke.

182 Alex Ross. *The Rest is Noise.* (London: Fourth Estate, 2008): 391.

duration contains a largely static … unfolding of a sonority' which changes when the music moves to the next bar.

In Nono's cantata *Il Canto Sospeso* (1956) he utilised the Fibonacci series in a different way. Rather than generating formal proportions Nono determined individual note lengths by means of the series. In the second movement, for example, the durations of notes are generated by the Fibonacci numbers 1 2 3 5 8 13.183 The Danish composer Per Nørgård also experimented with the properties of the Fibonacci series in some of his music. In the programme note for his orchestral piece *Iris* (1968) he compares his use of the Fibonacci series to the way it can be found in nature:

The basic idea is … a network of lines where each one represents a rather simple melody or rhythm. The whole thing is not unlike the flowers of the daisy, where you see 21 spiralling lines moving in one direction and 34 spiralling lines in the other, in other words the proportion of The Golden Mean (as expressed in the Fibonacci series.184

The Austrian composer and writer Ernst Krenek (1900-1991), was a regular attendee at Darmstadt. In his 1964 piece *Fibonacci Mobile* Op. 187 for string quartet and piano duet, he uses the Fibonacci series in the construction of the proportions.

Although the ideas of the radical Greek composer Iannis Xenakis (1925 – 2001) were often at odds with the serialism of the Darmstadt School, his orchestral piece *Metastaseis* (1954) was the subject of a lecture given there in 1962 by György Ligeti (1923-2006). Following his arrival in Paris in 1947, Xenakis had worked as an architect and engineer for 12 years with Le Corbusier where proportion was central to his work (see also pages 187-8). This was to have a strong influence on the composition of *Metastaseis.* In the words of Xenakis, "In my composition *Metastasis* … the role of architecture is direct and fundamental by virtue of the Modular. The Modular was applied in the very essence of the musical development"185 Here he is referring to a system devised by Le Corbusier, an architectural approach

183 Jonathan Kramer. 'The Fibonacci Series in Twentieth-Century Music'. *Journal of Music Theory*, Spring, 1973, Vol. 17, No. 1 : 110- 148.

184 Per Nørgård, programme note for his orchestral piece *Iris*, 1988.

185 Sharon Kanach*. Iannis Xenakis. Music and Architecture.* (New York: Pendragon Press, 2008).

to form and proportion and used by Xenakis to shape pitch envelopes and time structures in *Metastaseis. Metastaseis* is built in four sections. The first section begins on a sustained single note G played very quietly by all 46 strings gradually sliding outwards by means of slow glissandi as the dynamic intensity increases, arriving at a massive note cluster for full orchestra with each instrument sustaining its own note. This large cluster is treated as one sonic unit where note lengths are derived from the Fibonacci/Modulor series by means of dynamic and timbral changes.186

## *P***roportional parallelism in the music of J S Bach**

As we have seen, GS is closely linked to proportion therefore readers may also be interested in the theory of proportional parallelism in the music of J S Bach that has been put forward by the musicologist Ruth Tatlow. This asserts that 'all Bach's revised fair copies and published collections have three characteristics in common: the total number of bars are a multiple of ten, one hundred, and sometimes one thousand; there are perfect proportions such as, 1:1 or 1:2 on at least three different levels; and there is an embedded signature.' Tatlow discovered that the 'total number of bars in the six Brandenburg Concertos is exactly 2220, Concertos 4, and 5 having an exact total of 1110 bars, and Concertos number 1, 2, 3, and 6 also with 1110 bars, creating a parallel 1: 1 (1110: 1110 bars) in 1: 2 (4: 2 concertos)'. Bach created these parallel proportions when he revised his works and collections in preparation for publication by adding a 'few bars to movements and movements to collections'.187According to Tatlow, proportional parallelism offers an explanation for these revisions, hitherto unexplained and with no apparent musical objective; she argues that Bach was motivated to make these changes because of his Lutheran beliefs in the theory and practice of music that leant heavily on universal harmony.188.

186 James Harley, *Xenakis. His Life in Music*. (London: Routledge, 2004): 10.

187 Daniel R Melamed.'Parallel proportions in J S Bach's music'. Published online by Cambridge University Press: 05 February 2021

188 Ruth Tatlow. *Bach's Numbers: Compositional Proportion and Significance*. (Cambridge: Cambridge University Press, 2015).

Tatlow's theory challenges a long-held and influential theory of the German Bach scholar and theologian Friedrich Smend (1893-1980) which argues that J. S. Bach's works contain hidden symbolic references based on the German 'natural-order' number alphabet (A= 1 ... Z=24).189 Smend's work spawned many further studies of Bach's presumed extensive use of number symbolism which was seen as in keeping with his 'mystical' musical outlook. The response to Tatlow's work has been mixed, with some scholars being critical, notably Daniel Melamed.190 Alan Shepherd explored Tatlow's theory further in *Let's Calculate Bach* (2021) which sets out to apply theories of probability and statistical analysis to numbers in music.191

Golden Section and the associated Fibonacci series give rise to satisfying natural proportions. Many examples of their use can be found in music, sometimes these adhere strictly to the exact ratios and at other times they are approximations. The question remains as to whether they are used by composers consciously or subconsciously. A study of first movement sonata form across Mozart's piano sonatas reveals that, although more than a third of them fall within what might be considered to be a reasonable GS margin, it is more likely that Mozart lent towards the pleasing proportions intuitively through his innate sense of form. Cases have been made for various pieces using GS by Satie, Bartók and Debussy, but these have been inconclusive in the absence of documentary evidence to prove that they were based on mathematical calculations. This is not the case with several composers associated with the centre of new music in Darmstadt during the 1950s and 1960s; Stockhausen, Nono, Krenek and Nørgård, for example, all both consciously and openly integrated the use of the Fibonacci series into their compositional techniques.

189 Friedrich Smend. *Luther und Bach*. (Germany: Verlag Haus und Schule, 1947).

190 Daniel R. Melamed, "'Parallel Proportions' in J. S. Bach's Music," Eighteenth-Century Music 18, no. 1 (2021): 99–121. 4 See Melamed, "'Parallel Proportions,'" 100–103.

191 Alan Shepherd. *Let's Calculate Bach: Applying Information Theory and Statistics to Numbers in Music* (Cham, Switzerland: Springer, 2021).

#### 176 MATHS & MUSIC

# 11 **Randomness and chance**

## **Random numbers in mathematics**

generally speaking, randomness means having no specific pattern, purpose or principle of organization. Random sequences of events have no order and single random events are unpredictable. Chance is the unknown and unpredictable element that causes an event to have one outcome rather than another. Nowadays random numbers are computer generated. They are widely used in both sample selection and in allocating treatments to units in designed experiments. Randomized control trials are used on two groups – a study group and a control group. In medical trials, for example, scientists test the study group (those who receive a new treatment) and a control group (those who are given an existing treatment or a placebo). Trials could be biased if it is known which group the patient belongs to so random selection is used to avoid this bias.

Just as random selection is used to avoid bias in scientific experiments by taking control away from the project leader, so too in music the use of chance operations means that total control is taken away from the composer. The terms indeterminacy, chance music and aleatoric music are often used interchangeably to describe this random element. Music using chance processes falls into three main groups: the use of random procedures to produce a fixed score; indeterminate notation, including graphic notation and texts to be interpreted by the performer(s); and scores where the indeterminate element arises from choices made by the players about elements of the performance.

This chapter explores the different ways in which composers have used randomised elements in their music from the use of rolls of dice by eighteenthcentury composers to the computerised chance operation of Iannis Xenakis who coined the term 'stochastic music'. The leading composer of aleatoric music was undoubtedly John Cage who used chance procedures in almost all the music he created after 1951.

## **The use of dice in eighteenth-century novelty music**

Dice (singular: die) were one of the earliest means of generating random numbers and examples can be traced back to 24,000 BC. During the eighteenth century a fashion developed where dice were used to introduce an element of chance into musical compositions. This genre was known as *Musikalisches Würfelspiel* (musical dice game). It was a system for randomly generating compositions from pre-composed bars of music. The games consisted of a number of pre-composed bars along with a method for selecting their precise sequence usually the throwing of a number of dice. Random numbers were used to order the melodic fragments thus generating a new piece. One of the first such games was published by the German composer and music theorist Johann Kirnberger (1721-1783) in 1757. *Der allezeit fertige Polonoisen und Menuettencomponist* (The at all times prepared composer of polonaises and minuets.) comprises a 'score' and a sequence of possible bars for both of the musical forms. The sequence of bars is determined by the throwing of either one or two dice.

Some pieces were attributed to well-known eighteenth-century composers such as C P E Bach, Mozart and Haydn. One such piece attributed to Mozart, but not authenticated, is *Anleitung zum Componieren von Walzern so viele man will vermittelst zweier Würfel, ohne etwas von der Musik oder Composition zu verstehen* (Instructions for the composition of as many waltzes as one desires with two dice, without understanding anything about music or composition) (1792). Rolls of the dice are used to select bars of music to be used in sequence and then played as a complete waltz. Each number rolled corresponded with a bar number given in a table. There are 176 numbers given in two tables (one table for the first eight bars of the waltz, and the second table for bars 9 to 16 of the composition). Each of the numbers from 1 to 176 corresponds with a bar of music. To select the first bar of the waltz, two dice are thrown and then the corresponding bar is chosen from the first column, for the second bar two more dice are thrown and the appropriate bar is selected from the second column and so on. Given that there are 11 different options for each of the 14 bars along with two options for one of the bars (bar 8) and only one option for the final bar (bar 16).192 This means that there are 2 x 1114 possible variations on the waltz – 759, 499, 667, 166, 482 trillion. Here is the first table and the first 47 bars of the music.

## **Mozart dice game**

192 Giovanni Albini. 'Combinatorics, probability and choice in music composition'. Bridges 2018 Conference:

The appeal of the games was that they appeared to make it possible for anyone to be able to compose a piece of music. In fact composers used their knowledge of compositional procedures, harmony and formal design, coupled with a basic understanding of mathematics to design the games in such a way that whatever options were arrived at 'by chance', the resultant composition would sound convincing. Many bars are identical (bars 8 and 13, for example), many have the same notes in the left hand (3, 6, 32, 40, 41 and 42, for example) and the number of chords implied in the harmony is limited. So, for example, bars 3, 6, 7 and 8 are all based on the tonic chord of C major (C

10.5920/mathsandmusic.fulltext 10.5920/mathsandmusic.fulltext

E G), whereas bars 2, 4 and 5 are all based on the dominant chord - bar 2 is based on G (G B D) and bars 4 and 5 have an added 7th (G B D F#). This means that the table of bar numbers can be manipulated so that the devised piece makes harmonic sense. An example of this is Column H; this gives the selection for the eighth bar of the piece. According to the number of the dice thrown the bar selected will be either 30, 81, 24, 100 or 107, and so on. In fact each of these bars is based on a dominant chord with either an F natural or an F#. This implies an imperfect cadence which, in this Classical style of music would be expected half way through the 16 bar melody at the end of an eight bar phrase. Similarly, all the possible options for bar 16 (14, 73, 89, 170 and so on) are identical, meaning that whatever dice are thrown, the piece ends on a tonic chord as would be expected and sounds finished.

These compositions involving chance were novelties. Until the twentieth century, Western classical music had traditionally been a field in which chance had little or no role to play. As John Paynter writes, 'Musical action involves feeling, responding, thinking and making' and until the early years of the twentieth century, mainstream composers took this to mean that they should 'have overall control of their musical ideas in order to express the consciousness of their being'.193 When chance music, often known as aleatoric music, emerged in the mid-twentieth century it challenged the traditional idea of a composition being a closed entity fixed by its composer; elements of chance which were undetermined by the composer were introduced into the process of composition and/or in performance. As such it explored what Paynter describes as 'ways of creating music which, if possible, would be divorced from the will and the taste of the composer – the very opposite of the traditional expressing and communicating objective'. Composing rather than 'being a matter of putting together all the details of a carefully calculated and evaluated structure' instead 'set up opportunities for "sound events" to happen'.194

All sounds and any sounds could come in and find a place in the music; and the accidental structures they created could be as delightful in their own way as the random sounds of nature – which we can enjoy without their having been 'Composed'. 195

193 John Paynter. *Sound and Structure.* (Cambridge: Cambridge University Press, 1992): 166.

194 Ibid

195 Ibid

## **Chance in twentieth-century music – Charles Ives, Henry Cowell, Morton Feldman and John Cage**

The American composer Charles Ives (1874–1954) allowed a greater degree of freedom for performers in some of his pieces through the experimental use of alternatives, free passages and obscure notations inviting the performer's interpretation. A friend of Ives, the American composer Henry Cowell (1897-1965) continued this line of experimentation in aleatoric music. His *Mosaic Quartet* (*String Quartet No. 3*, 1934) is an experiment in musical form; a collection of five short movements in five different tempos where the order of the movements, and the number of times each is played, is left up to the performers. The piece also calls for some improvisation, another chance element. However it was Cowell's sometime pupil, John Cage (1912- 1992), who, starting in the early 1950s, was to make the fullest exploration of chance operations. Cage's exploration of chance processes was partly inspired by *Extensions 1* (1951) by Morton Feldman (1926-1987) where sounds are written as numbers on graph paper, the figures indicating only the number of notes to be played and the register in which they are to be performed. Cage admired Feldman's use of 'whatever sound comes along'196 and in his 'Lecture on Something' (written in early 1951), he described Feldman's graph pieces as having 'changed the responsibility of the composer from making to accepting.'197

In 1951 Cage completed his *Music of Changes,* a piano piece whose title makes reference to the *I Ching,* the Chinese oracle text sometimes known as *The Book of Changes*. The *I Ching* is concerned with divination – predicting the future through a systematic process, in this case using random processes to produce and interpret a series of symbols. The *I Ching* represents the binary poles of reality as Yin and Yang (negative/positive, night/day, winter/ summer…) these are often discussed as female and male in the text of the *I Ching*. In the *I Ching i*mages are selected at random from a set of 64 by means of tossing yarrow sticks or coins. The *I Ching* is structured around an idea which is essentially combinatorial, that is it uses principles which stem from

196 Barry Russell and Julia Winterson. *Everything we do is Music. Cross curricular Experiments Based on the Music of John Cage.* (London: Edition Peters, 2016).

197 John Cage. 'Lecture on Something' in *Silence. Lectures and Writings by John Cage*. (Middletown, Conn.:Wesleyan University Press, 1961): 129.

combinatorics, the mathematical study of the enumeration, combination and permutation of sets of elements and the mathematical characteristics of their properties. A broken line represents Yin, and an unbroken line represents Yang. These lines are combined in sets of three to produce the eight basic trigrams, which are in turn combined to give 64 hexagrams. Each hexagram describes a different state of mind and is accompanied by an explanatory text. The text discusses each of the six lines as separate entities as well as explaining the meaning of the hexagram as a whole. Once the hexagram is cast, the *I Ching* is consulted for insight into its meaning. Cage made extensive use of the *I Ching* in the composition of *Music of Changes* using chance operations to create charts for tempi, dynamics, sounds and silences, and durations.

He went on to use chance operations in many other compositions and designed several different chance-controlled systems. In the *Music for Piano* series (1952–6) he translated the imperfections on paper into musical notes by the application of staff lines and clefs. *Atlas Eclipticalis* (1961-2) used templates drawn from maps of the constellations. *Concert for Piano and Orchestra* has no score, but rather consists of 63 highly detailed parts. It was composed using chance operations as well as the use of the imperfections found in the paper upon which the music was written. The notation of each part uses a system wherein space is relative to time. The amount of time is determined by the musician and then altered during performance, by the conductor, whose arms simulate the movement of the hands of a clock. The pianist may play the material in whole or in part, choosing any notations, elements, or parts, and playing them in any order.

In *HPSCHD* (1969) Cage used a computer to generate random musical material for a multi-media event that involved harpsichords and computergenerated sounds. The first performance lasted five hours and included slide projections. Three large posters were created featuring images chosen by chance operations, copies were sold at different prices arrived at by using the *I Ching*. The music comprises 20-minute solos for 1-7 amplified harpsichords and tapes for 1-51 monaural machines distributed unpredictably to make an indeterminate concert of any length having 2 - 58 separate channels with loudspeakers around the audience. The harpsichord solos used music by composers including Mozart (primarily the music of the aforementioned Mozart dice game), Beethoven, Chopin and Schoenberg.

In 1952 John Cage was one of a group of artists, musicians, poets, film makers and dancers to take part in what is said to be the first 'happening', a title which covers a range of multi-media performance activities where many things happen simultaneously with no narrative structure. Cage's first happening was at Black Mountain College. For that event, later called *Theatre Piece No. 1*, Cage gave all the participants time-brackets during which they were to dance, read poetry, show film and/or play music. The time brackets were used to specify the intervals during which sounds must start and stop. He then went on to use the concepts of both time brackets and happenings in many other works.

Cage was a prolific composer and most of his mature pieces included some element of chance. He stated that his goal was to be 'free of individual taste and memory'.198 In some ways this is similar to the use of random selection to free bias in scientific experiments by taking control away from the project leader. The introduction of chance into a piece of music undermines the idea that creation requires a definite choice at every level on the part of the composer. With Cage this abandonment of control reached its extreme point *4' 33''* (1952),or 'My silent piece,' as Cage called it, an empty structure in which the only sounds – those of the environment – are the most full of possibilities. As Cage once stated in his lecture 'Indeterminacy'

…the purpose of this purposelessness music would be achieved if people learned to listen; that when they listened they might discover that they preferred the sounds of everyday life to the ones they would presently here in the musical program; that was alright as far as I was concerned.199

## **Fluxus, Terry Riley and Cornelius Cardew**

Cage's use of chance operations, particularly those used in multi-media performances, influenced many other composers. As Michael Nyman writes 'By the late sixties indeterminacy … had made itself available to a larger

198 Cage. *Silence,* 59.

199 Cage. *Silence,* 267.

number of people with a wider range of abilities and experience'.200 It was central to the work of the New York group Fluxus which formed in 1963 and went on to stage performances which were 'characterised by simplicity, experimentalism, chance, playfulness and humour'.201 The indeterminate scores of Cage and his contemporary Christian Wolff often demanded technical and musical expertise, but with Fluxus this was no longer the emphasis, instead it encouraged unskilled players thereby, it could be argued, increasing the element of randomness. An advertisement for a concert in 1965 read as follows

FLUXORCHESTRA PERFORMS 20 WORLD PREMIERS! Of avant-gagist music, ying yang music, Donald Duck music, antineobaroque music, pataphysical music, no music, La Monte Young conducting an orchestra of twenty unskilled instrumentalists.202

Fluxus counted amongst its members Dick Higgins, La Monte Young, Joseph Beuys, Yoko Ono and Terry Riley and eventually became an international movement with offshoots in Germany and elsewhere. In 1964 the American composer and performer Terry Riley (b 1935) went on to compose one of the most frequently played pieces where chance is central to its performance – *In C*.

The score of *In C* comprises 53 'repeating patterns' which should be played in consecutive order starting with 1 and ending with 53. All performers (any number playing any instrument) play from the same part apart from the 'pulse'. A steady quaver pulse is played on the top octave of a piano, marimba or vibraphone. 'Each performer decides for himself when to enter' and 'is free to move from figure to figure at his own rate'.203 This element of choice makes the detail of each performance of *In C* unique. It could also be regarded as a precursor of the Minimalism movement.

In London Cornelius Cardew (1936-1981) formed the Scratch Orchestra which had a similar philosophy to that of Fluxus. All members were encouraged to participate on an equal footing regardless of skill or previous

200 Michael Nyman. *Experimental Music. Cage and Beyond*. (Cambridge: Cambridge University Press, 1999): 110

201 *Every Day is a Good Day. The Visual Art of John Cage*: 62.

202 *Village Voice*, 23 September 1965.

203 Score of Terry Riley *In C*.

experience. Formed in 1969, it brought together trained and untrained musicians from various backgrounds and was dedicated to free musical exploration. The band's repertoire included simple classics as well as Cardew's own compositions. Probably the most famous of his compositions is *Treatise* (1967) which has a totally graphic score (apart from the very occasional musical symbol). Cardew also worked as a graphic designer and the score – a melange of geometrical forms – raised graphic notation to the level of visual art as does the score for *Octet '61 for Jasper Johns* (1961). At the same time the score of *Treatise* provided the performer with little or no information as to how it was to be interpreted, consequently the ways that the sounds can be realized are endlessly diverse. As Cardew writes in the score 'Any rigidity of interpretation is automatically thwarted by the confluence of different personalities'. There are many recorded interpretations of *Treatise* and these are not confined to performances by experimental music groups. In 1999 the American rock band Sonic Youth included page 183 on their album *SYR4: Goodbye 20th Century.*

A key piece in the repertoire of the Scratch Orchestra was Cardew's *The Great Learning* (1971) based on the first chapter of the Confucian text with the same name. It was developed collectively by an experimental music group which Cardew ran at Morley College in South London. The piece consists of seven extensive movements - 'Paragraphs'- and is a blend of improvisation, formal composition and Maoist propaganda. In order to fulfil his intention that anyone could enjoy collective music-making, Cardew uses a variety of notations predominantly graphic notation and simple verbal descriptions of performance processes. 'Paragraph 7' is for voices alone, any number of trained or untrained singers. The text instructs the performers to start on a random note and with each breath to move to a new note sung by one of their neighbours.204 This means that in performance, having started with a completely unpredictable chordal texture, the pitch content of the piece gradually narrows as notes are shared among the ensemble, until there's just a single note remaining. As a result every performance of Paragraph 7 is different in terms of its notes, its length and the overall sound. The outcome is random. As Cardew once pointed out, one of his standards was 'not to make a sound that's like something, but to make a sound that is just that …

204 Harris, Tony. *The Legacy of Cornelius Cardew*. (Farnham, Surrey: Ashgate, 2013): 66-67.

I want the feeling that everything you do is for the first time'.205

## **Iannis Xenakis and stochastic music**

Although the above chance methods rely on random selection of one sort or another, none of them are as strictly mathematical as what became known as 'stochastic music'. Stochastic music was pioneered by the Greek composer Iannis Xenakis (1925 – 2001) who coined the term. In mathematics a stochastic process is one in which the steps are governed by rules of probability: a sequence of random objects often ordered in time and space. Xenakis used stochastic processes in his composition to determine musical parameters such as pitches, tempi, durations and timbres. Such numerical calculations were usually undertaken on computers. He wrote several articles and essays on stochastic processes, game-theory and computer programming in music. 1963 saw the publication of his seminal work on mathematics and music: *Formalized Music: Thought and Mathematics in Composition* (1971) in which he explains the thinking behind his techniques for composing music with stochastic mathematical functions.206

No discussion of the work of Xenakis, composer, architect and engineer, would be complete without mention of his extraordinary life. Of Greek parentage, he studied engineering in Athens and later, in Paris, composition with Olivier Messiaen. Messiaen advised him to take advantage of his architectural background 'You have the good fortune of being Greek, of being an architect and having studied special mathematics. Take advantage of these things. Do them in your music'. 207Towards the end of World War II he fought for the Communist Greek resistance and in 1944 was severely injured in street fighting with British tanks. He was hit in the face by shrapnel receiving severe facial injuries and losing the sight in his left eye. When the new non-Communist government came to power in Greece, he was exiled to Paris under a death sentence which was not lifted until the end of the Greek junta in 1974. Following his arrival in Paris in 1947, he

205 Cited in Nyman, *Experimental Music*, 127.

206 Iannis Xenakis. *Formalized Music: Thought and Mathematics in Composition*. (New York: Pendragon Press, 1992).

207 Nouritza Matossian. *Xenakis.* (London: Kahn & Averill, 1986): 48.

worked for 12 years with the architect Le Corbusier. In 1958 Le Corbusier was commissioned to design the Philips Pavilion for the Brussels World Fair. He gave complete creative control of the project to Xenakis. As part of the World Fair, the Philips Pavilion staged a multimedia event in the Xenakis's steel and concrete structure which was based on parabolic curves. As visitors entered and exited the building they heard the music of Xenakis.208 Once inside the auditorium *Poème* électronique by Edgar Varèse was projected from 350 speakers via 20 amplifier alongside ever-changing visuals creating a ground-breaking random simultaneity.

Xenakis's theories about stochastic synthesis were published in *Formalized Music*, *Thought and Mathematics in Composition*. 209 Here he advocated the use of stochastic processes to efficiently produce sonorities with 'numerous and complicated' transients arguing that the 'transient part of the sound is far more important than the permanent part in timbre recognition and in music in general'.210 Stochastic processes are random and non-deterministic, that is, the next state of the environment is not fully determined by the previous one. Probability theory is used to determine what should happen next. In *Formalized Music*, Xenakis makes references to macroscopic (large scale) structures and the microscopic finer details of stochastic music. His music is often extremely elaborate in its detail, but that detail is usually at the service of the whole. This is well-illustrated in his orchestral piece *Pithoprakta* (1955- 56) where Xenakis focuses on the overall block of sound, using probability theories to build different clouds of sound.211 The finer detail, the individual notes and timbres are 'unpredictable' but are used to make up the combined mass of sound.212

*Pithoprakta* draws on probability theories including Gaussian distribution.213 It is scored for string orchestra (with 46 individual parts), two trombones, xylophone, and wood block. The title translates as 'actions by means of probability'. The piece is based on the kinetic theory of gases

208 Xenakis' sound work *Interlude Sonore* which was subsequently titled *Concret PH.*

209 'New proposals in microsound structure' in Xenakis, *Formalized Music*, 242-254.

210 Xenakis, *Formalized Music*, 243-244.

211 Xenakis, *Formalized Music*, 12.

212 Gareth Roberts. *From Music to Mathematics {Exploring the connections}.* (Baltimore: John Hopkins University Press, 2016): 280.

213 Gaussian distribution (often known as normal distribution) is a very common continuous probability distribution and is the basis of a large proportion of statistical analysis.

where the temperature of a gas derives from the independent movement of its molecules. An analogy could be drawn between the movement of the molecules through space and that of the string instruments through their pitch ranges. At a macroscopic level, the work is made up of four sections which differ from each other in their use of textural and timbral characteristics, such as glissandi and pizzicato. As composer, Xenakis designed these large-scale features, but the individual components of sound such as which pitches and dynamics to use are generated by probability theories. Each sound-particle of the score is precisely defined through stochastic distribution functions at the same time as contributing to the overall sound impression. Xenakis created a graph representing a set of speeds of temperature and time; the pitches were drawn as the ordinates on the vertical scale and each of the instruments was represented by a jagged line representing a speed taken from the table of probabilities according to Gaussian distribution. The graph was then transcribed into traditional notation and the score was fixed. Continuous sounds are heard in the glissandi in the strings and trombones, for example, and discontinuous sounds include pizzicato plucking in the strings, tapping the strings with the opposite side of the bow, and the use of the wood block.214

In 1962, Xenakis composed his first string quartet *ST/4‐1,080262*. The coded title translates as ST (stochastic composition), 4-1 (four instruments, first of its kind) created on *080262 (*8 February 1962). The work's material was calculated by an IBM 7090 Computer. The computer was programmed with compositional parameters that determine the point in time that a sound sequence occurs, next its timbre (arco, pizzicato, glissando and so on), choice of instrument, pitch, direction of glissando, duration of notes, and dynamics. The string quartet also made use of pioneering extended techniques such as playing with the wood of the bow, bowing the body of the instrument and other new tapping and plucking techniques. In 1963-4 the composition of *Eonta* for piano, two trumpets and three trombones was also based on the theory of probabilities where some of the instrumental parts were calculated on the IBM 7090 computer at the Place Vendome in Paris.

Xenakis used the idea of random walks for the creation of *Mikka* (1972), a short work for solo violin. A random walk is a process in which a sequence

214 Gareth Roberts lecture. 'Composing with Music. Iannis Xenakis and his Stochastic Music '. Gareth E. Roberts. Math/Music: Aesthetic Links, Montserrat Seminar, Spring 2012.

of discrete steps of fixed length is described in terms of the movement of a particle so in a one-dimensional random walk the state of the process is described as a position on a straight line. After starting at, say, 0 the first step takes the particle to either +1 or -1 with equal probability, after two steps it is +2, 0 or -2 and so on.215 In *Mikka*, computer printouts of synthesized sounds are directly transformed into a continuous sequence of glissando curves for violin. The pitch is in continuous movement, with the left hand continuously sliding on the strings brushing past the microtones marked in the score. Plotted graphs of stochastic synthesis are also used in *N'Shima* (1975) and *Mikka S* (1975) where Xenakis read the horizontal axis of the graphs as time and mapped the vertical axis onto a grid of quarter-tone pitch values.

The music of *N'Shima* is based on the principle of Brownian motion, a continuous-time version of the random walk, displayed by minute particles of solid matter when suspended in a fluid or a gas resulting from their collision with the fast-moving molecules in the fluid. This translates in the music as short glissando figures sliding form one microtone to another. *N'Shima* is a Hebrew word meaning 'breath' or 'exhalation'. Scored for to mezzo sopranos and five instruments, the text of *N'Shima* text is taken from 'The Emperor's daughter and the King's son' a parable about two rival families whose children fall in love. Xenakis does not use whole words however; the vowels are used for tone colour and the consonants for articulation.

In the early 1980s, Xenakis's research, together with that of other colleagues specialising in electronics, software and signal processing, found a home at the *Centre d'Etudes de Mathématique et Automatique Musicales* in Paris along with the newly developed Unité Polyagogique Informatique du CEMAMu (UPIC) system completed in 1977. The UPIC system enabled composers to create sounds directly by 'moving an electromagnetic "pencil" across a sloping sensitised "drawing board"', and to hear the sounds they had created almost immediately. In the words of Richard Steinitz, the founder of the Huddersfield Contemporary Music Festival where the works of Xenakis were frequently performed, 'It thus united the visual and aural domains of Xenakis's career both as architect and composer'.216

215 A two-dimensional random walk is represented by a set of steps along a grid or lattice. Brownian motion is a continuous-time version of the random walk.

216 Richard Steinitz. *Explosions in November: The First 33 years of Huddersfield Contemporary Music Festival*. (Huddersfield: University of Huddersfield Press, 2011): 91.

Each of the composers above combined mathematical devices with musical considerations in their chance compositions. The eighteenth-century dice composers manipulated and controlled the harmonic and melodic possibilities to ensure that whatever pieces arose 'by chance' would be musically convincing. In contrast, the aleatoric compositions of the twentieth century deliberately avoided musical control. Ives and Cowell experimented with episodes of chance and one of Cage's aims was to be 'free of individual taste and memory'. By the late sixties Cage's use of indeterminacy influenced many other composers amongst these Terry Riley whose piece *In C* could be regarded as a precursor of the Minimalism movement. However, the strictest mathematical approach was taken by Xenakis who coined the term stochastic music in which the compositional steps are governed by rules of probability.

#### 192 MATHS & MUSIC

# 12 **Fractals and chaos theory**

fractals belong to a class of curves or complex geometric shapes in which each part has the same statistical character as the whole, that is, it is made up of smaller scale copies of itself. Their properties are found in many irregularly shaped objects and non-uniform phenomena in nature such as coastlines and mountain ranges. The word fractal is derived from the Latin word *fractus* meaning broken.

Can fractals be found in music? This chapter examines the evidence. It opens by looking at fractals in mathematical terms describing the characteristics and properties of the von Koch curve and the Cantor set and outlining the early work of Benoit Mandelbrot in this field. Over the years various attempts have been made to identify fractal patterns in music, and several ideas have been proposed including the use of nested sequences and self-similarity. These include the prolation canons dating from the fifteenth century by composers such as Ockeghem and des Prez. More recently the Hungarian composer György Ligeti showed a great interest in both fractals and chaos theory, inspired by the computer-generated illustrations of Heinz-Otto Peitgen and Peter Richter. Several compositions of Ligeti are examined and analysed in terms of their fractals characteristics. The American composer Charles Wuorinen was also fascinated by fractal geometry and he used some of its mathematical properties in the construction of his music. In recent years several composers have taken advantage of the capabilities of computers to generate fractal inspired pieces, amongst these the Norwegian composer Rolf Wallin. Robert Sherlaw Johnson was one of the first to explore this field and he provides an illuminating critique of his method. Nowadays there are increasingly sophisticated software packages specifically designed to created fractal music. The chapter concludes with a brief discussion of the original question 'Can fractals be found in music?'

## **What is a fractal?**

Fractals commonly have 'fractional dimension' a concept which was first introduced by the mathematician Felix Hausdorff in 1918 and a measure of their complexity. Although fractional dimension, and other key concepts associated with fractals, had been known about for some time, it was not until the French-American polymath Benoit Mandelbrot (1924-2010) coined the term 'fractal' in his 1975 book on the subject that more interest was shown in the field.217 Later Mandelbrot's seminal 1982 book, *The Fractal Geometry of Nature* catalogued the ubiquity of geometric patterns found in nature and is widely credited for bringing fractals to popular attention. He highlighted the need for fractal mathematics and pointed out that they could be useful in applied mathematics for modelling a variety of phenomena from physical objects to the behaviour of the stock markets. Since the 1980s virtually every scientific field has been explored from a fractal viewpoint and fractal geometry has become a major area of mathematics. Fractals are useful in modelling structures in which similar patterns recur at progressively smaller scales, as well as describing partly random or chaotic phenomena such as crystal growth and galaxy formation.

## **The von Koch curve**

The properties of fractals can be illustrated in the von Koch curve (see Figure 1). An intricate and complicated object, the main von Koch curve contains many tiny von Koch curves which are made up of smaller scale copies of itself. This property is known as self-similarity. A self-similar object is one whose component parts resemble the whole. Details or patterns are reiterated at progressively smaller scales so that in effect, it remains invariant under

217 Benoit Mandelbrot. *Les Objets Fractals: Forme, Hasard et Dimension*(1975) later translated in 1977 as *Fractals: Form, Chance and Dimensio.* London: W H Freeman and Co,1975).

changes of scale, that is, it has scaling symmetry. In classical mathematics curves are usually described in terms of tangents but the von Koch curve does not have a well-defined slope or direction at any one point, hence it is too irregular to be described in traditional mathematical language. Mathematicians disagree on a precise definition, but a fractal is typically described as exhibiting self-similarity,

*The von Koch curve*

The curve is constructed by repeatedly replacing the middle third of each line segment with the other two sides of an equilateral triangle. Because the same step is repeated over and over again this is known as a recursive construction. When three von Koch curves are fitted together they form a snowflake curve (see below).

## *The von Koch snowflake*

The main tool of fractal geometry is fractional dimension. The von Koch curve has dimension

log 4/log 3 = 1.262.218

## **The Cantor set**

In 1883 the German mathematician George Cantor (1845-1918) introduced what he referred to as the middle third Cantor set. This was obtained by repeatedly removing the middle thirds of intervals which results in a basic self-similar fractal made up of two scaled ½ copies of itself (see Figure 3).

*The middle third Cantor set*

218 Kenneth Falconer. *Fractal Geometry. Mathematical foundations and Applications*. (Chichester: Wiley, 2003): xix.

Each stage of the construction is obtained by removing the middle third of the previous stage. Take the closed interval [0, 1]. First delete the open interval that forms the middle third [1/3, 2/3] then delete [1/9, 2/9], then delete [7/9, 8/9] and so on.

In 1918 Felix Hausdorff showed that the middle third Cantor set had dimension of log2/log3=0.631.

In summary the main features of fractals (*F*) as given by Kenneth Falconer in his classic text *Fractal Geometry. Mathematical Foundations and Applications* are:

## **Early attempts to find fractals in music**

Although there is no widely accepted consensus as to what constitutes a fractal pattern in music, a variety of statistical tools have been proposed over the years. The earliest contributions came from the American physicists Richard F. Voss and John Clarke in the 1970s who analysed the spectral density of audio power (effectively loudness) in a range of recordings from news radio, to Scott Joplin rags and Bach's Brandenburg Concerto No. 1.

219 Falconer, *Fractal Geometry*, xxv.

Through this they identified what they referred to as scaling behaviours (the word fractal had not yet been coined) in the fluctuations of volume as well as some pop music melodies**.** 220 They observed that both pitch and volume fluctuations showed 1/*f* distribution, a type of distribution often associated with the hierarchical structure of fractals.221

Twenty years later, Henderson-Sellers and Cooper proposed a definition of musical self-similarity based on note lengths analogous to the Cantor set. They specified how the faster passages are to be related to the slower ones arriving at a definition of a musical fractal based on the development of a melodic line; different time scales appear sequentially, in a single line of music, so that the entire passage is built from copies of the original idea written in shorter notes. Each faster copy is transposed in such a way that it begins with a note from the previous order so that every new order retains the memory of all previous orders, creating a self-similar structure. The melody is thus constructed through layers of nested sequences in a single line of music, which becomes faster and faster at each order, but retains the outline of the slower version in the leading notes of each scaled copy.222 This algorithm is clearly quite mechanical and prescriptive; it cannot be sustained musically over any length of time and examples found in music depend on creative adaptations. Such an example can be found in the opening bars of 'Uranus, the Magician' from Holst's *The Planets* (1917). Here the opening statement is made by trumpets and trombones in dotted semibreves, this is followed by a second statement by tubas, but this time the note values are halved. The note values are halved again for the last statement by the timpani. As can be seen in this final statement the pitch order of the notes is no longer followed strictly.

220 Voss RF, Clarke J. 1/f noise in music and speech. Nature 1975;258:317–8 and Voss R F, Clarke J.. J Acoust Soc Am 1978; 63(1). 258-163.

221 In terms of noise 1⁄f noise (or fractal noise) is a signal with a frequency spectrum such that the power per frequency interval is inversely proportional to the frequency of the signal.

222 Henderson-Sellers B, Cooper D. 'Has classical music a fractal nature?: a reanalysis. Comput Hum 1994, 27(4): 277–84.

The American musician and mathematician Harlan Brothers has been exploring fractals in music for some years and has written extensively on the topic. He has pointed out that identifying fractals in music is more problematic than seeing them in an image. 'Unlike a picture, which is all laid out so that you can instantly see the structure, music is fundamentally a serial phenomenon… With music, the whole piece takes shape in your mind. This makes it more challenging to identify the self-similarity'.223 It may be difficult to perceive fractals in music without looking at the score or listening to the piece repeatedly.

In 2007 Brothers published a study of the fractal structure of Bach's *Cello Suite No. 3* where he found that, at various points, patterns of long and short notes within bars reappeared as patterns of long and short phrases at larger scales. He referred to this as 'structural scaling' noting that this self-similarity bore a striking resemblance to the Cantor set. According to Brothers, Mandelbrot had suspected that music contained fractal patterns but that he didn't have time to investigate further. Subsequently Brothers contributed a chapter to *Benoit Mandelbrot: A Life in Many Dimensions* in which he lists the seven types of statistical self-similarity that he had found in music: duration scaling, the distribution of note lengths; melodic interval scaling, the distribution of changes in pitch; melodic moment scaling, the distribution of the changes in melodic intervals; harmonic interval scaling; structural scaling; and motivic scaling. Motivic scaling involves the simultaneous rendition of the same motif or theme at various tempos resulting in an intricate interplay of the theme with its faster or slower versions. An example of this musical self-similarity is the prolation canon (see pages 150-1) where each voice in the canon sings or plays the same music, but at different speeds with voices entering either successively or simultaneously.

223 Harland Brothers. 'Structural scaling in Bach's Cello Suite No. 3'. *Fractals. 15, 2007: 90.*

## **The prolation canon**

The fifteenth-century Franco-Flemish composer Johannes Ockeghem was the master of the prolation canon and possibly its inventor.224 His Missa prolationum (c. 1450) presents a series of prolation canons each offering a different variant of the form and whose interval of imitation moves from the unison progressively through to the octave. Many of his compositions are grounded in some rational, often ingenious, conception, some also serving a didactic as well as musical purpose. 'Kyrie Eleison I' is a double prolation canon, where in the opening bars, two separate pairs of voices, soprano and bass, alto and tenor use two different motifs. The four parts begin together, each of the four voices follow the entire six-bar theme exactly once. For each pair, the ratio of tempos is 3:2. This is clearest to see in the alto and tenor parts where the alto uses four minims and a crotchet (all divisible by 2), whereas the tenor part uses four dotted minims and a dotted crotchet (all divisible by 3). After this passage, the music continues as a regular canon (see page 150).

Prolation canons continued to be composed in the twentieth century in, for example, Arvo Pärt's vocal work *Cantus in Memoriam Benjamin Britten (*1977) and the first movement of Shostakovitch*'s Symphony No. 15 (1971). Cantus in Memoriam Benjamin Britten*, scored for string orchestra and bell, is a prolation canon in five voices, with a ratio of speeds of 16: 8: 4: 2: 1 (violin I, violin II, viola, cello, and bass). The theme is based on the descending octave of an A minor scale, developed sequentially with one note added at each repetition. Thus, the theme proceeds as A-A-G-A-G-F-A-G-F-E-A thus creating what could be described as an ever-elongating fractal sequence. In the same way as Ockeghem and des Prez, the composers were using this compositional device consciously, albeit being unaware of the concept of fractals. Once the concept of fractals became well-known, a number of composers used fractal mathematics as part of the compositional process. Amongst these are the Hungarian György Ligeti, the Norwegian composer Rolf Wallin and the American Charles Wuorinen.

224 John McDonough and Andrzej Herczynski. *Chaos, Solitons and Fractals* (Science Direct 170, 2023).

## **The use of fractals by György Ligeti**

György Ligeti had a keen interest in fractals: he was well-informed and knowledgeable, and he took some of the ideas he had learnt and applied these to his compositions. In a 1990 interview, Ligeti commented on the influence of mathematics and fractals in his compositional process.225 He said, '… in my music, there are mathematical considerations of fractal geometry. Its kind of growth and its processes of pattern transformation have occupied me for a great deal'.

In 1985 Ligeti first met the German mathematician Heinz-Otto Peitgen (b 1945). Peitgen was a pioneer of fractal geometry who had helped to introduce fractals to the broader public, partly through the publication of his lavishly illustrated book of computer-generated images *The Beauty of Fractals*226*.* Their friendship lasted until Ligeti's death in 2006. In Peitgen's words 'He introduced me to new music and to him I opened the door to the study of fractals, fractal geometry and other fields of mathematics … However dissimilar Ligeti's origin and life as a musician was from my scientific background, we both saw that what we did was, in some way, remarkably connected and interwoven'.227

It is important to note that Ligeti did not apply fractal geometry to his music with systematic mathematical rigour; rather he took some of the principles and formulated compositional procedures based loosely upon them. He once said that only two of his compositions were 'deliberately based on ideas from contemporary mathematics, the first piano etude 'Désordre' (1985) which is self-similar - an iterated structure based consciously on the Koch snowflake' - and the fourth movement of the *Piano Concerto* - 'a fractal piece'.228

225 Stephen Satory. 'Colloquy: An interview with Gyorgy Ligeti'. *Canadian University Music Review*. 1990.

226 Heinz-Otto Peitgen & Peter Richter. *The beauty of fractals* (Berlin & New York, 1986).

227 Heinz Otto Peitgen. 'Continuum, Chaos and Metronomes – A Fractal Friendship' in Louise

Duchesneau and Wolfgang Marx. *György Ligeti. Of Foreign Lands and Strange Sounds*. (Martlesham, Suffolk: Boydell Press (2011).

228 Ligeti in conversation with Heinz-Otto Peitgen and Richard Steinitz, Huddersfield Contemporary Music Festival (November 1993) cited in Richard Steinitz. 'The Dynamics of Disorder'. *The Musical Times,* Vol. 137, No. 1839 (May, 1996): 8.

## **Ligeti – 'L'escalier du diable (The Devil's Staircase)'**

These two pieces will be looked at later in more detail, but first some mention should be made of another piano etude - 'L'escalier du diable' (The Devil's Staircase)' (1995-2001), the thirteenth etude which makes direct reference to a construction based on the Cantor set. The devil's staircase, sometimes referred to as the infinite staircase or the Cantor function, has unequal ascending steps constructed by using, for example the recursive 1/3 to 2/3 proportions of the most common middle-thirds Cantor set (see below).

## *The Devil's staircase*

'L'escalier du diable' is a flamboyant virtuosic piece, *presto* with polyrhythms and rising chromatic scales getting progressively higher. Richard Steinitz describes it thus:

The rotating repetitions of the principal motif climb up an endless staircase, flight upon flight, octave by octave. Meanwhile, the pattern restarts in low register, as if emerging from dark subterranean dungeons. Like the synthetic spiral - the computer-generated illusion - it seems to rise ad infinitum, yet remains enigmatically the same. Its patterns can be represented numerologically, but are intended first and foremost to generate polyrhythmic energy. Here we have Ligeti at his most melodramatic, in music overwhelmingly powerful and thrilling.229

Although it has recursive qualities, it does not follow the strict mathematical formula prescribed by the devil's staircase. As Steinitz writes, 'Ligeti constructs his musical staircase using his own numerical system', and although 'Its patterns can be represented numerologically' they 'are intended first and foremost to generate polyrhythmic energy'.230

## **Ligeti – 'Désordre'**

'Désordre' is the first of Ligeti's piano etudes and, as was mentioned above, Ligeti described it as having an 'iterated structure based consciously on the Koch snowflake'. The piece is marked *molto vivace*, very fast, and could be described as a study in rapid polyrhythms moving up and down the keyboard. The right hand plays only white keys and the left hand is restricted to the black keys in a pentatonic mode. Although both the right hand and left hand have eight quavers to the bar, the barlines and accents are placed differently. Hence the material of the two hands unfolds simultaneously but independently. As Steinitz puts it, 'At first glance their musical material looks alike. However, it is their dissimilarity which is crucial.' Later in the piece the same phrase structure is resized, the melodic material is similar but the phrase lengths are shorter and the accents appear more frequently.

In the words of Steinitz, 'Repeating an operation over and over again, on ever smaller scales culminates almost inescapably in a self-similar structure, a classic example being the 'Koch curve''. How do these musical procedures measure up against Falconer's fractal characteristics that were listed earlier? The music has elements of a 'fine structure' with small scale detail and it exhibits some form of 'self-similarity'. However, although it could be argued

229 Richard Steinitz. 'Maths and Chaos'. *The Musical Times, Vol. 137, No. 1837 (March, 1996): 19.* 230 Ibid.

that it has a 'recursive construction', this is not created in an orderly fashion in the way that a von Koch curve would be by repeating the same steps and more specifically replacing middle thirds.

## **Ligeti – The Piano Concerto**

Ligeti was particularly influenced and inspired by computer fractal images particularly those in *The Beauty of Fractals. Indeed* he was so fond of one of the images of a Julia set that Peitgen dedicated it to him naming it the Ligeti fractal, and Peitgen's '29 Arms at Seahorse Valley', a mathematical image of the Mandelbrot set, was an inspiration for the *Piano Concerto.*231 Although Ligeti understood the mathematical calculations behind fractals, these were not directly involved in his compositional methods. When describing the fourth movement of his *Piano Concerto*, he wrote

The whole structure of the piece is self-similar and the impression we get is that of a giant interconnected web … This vast self-similar maelstrom can be traced back - indirectly – to musical ideas which had been unleashed by beautiful fractal imaged of the Julia and Mandelbrot sets … Not that I use iterative calculations when I composed this piece … rather, it originated out of an intuitive synthetic correspondence on a poetic level.232

Steinitz describes the movement thus,

Indeed, we do find in this movement a particular musical gesture repeated endlessly at different magnifications... The movement is literally flooded with iterations of a three-note fragment, its extensions, contractions and derivatives - all progressively superimposed in varying tempos, with varying articulation, coloration and magnitude. The musical design builds up like a complex graphic image scanned,

231 Richard Steinitz. *György Ligeti*. *Music of the Imagination*. (London: Faber and Faber, 2003).

232 Gyorgi Ligeti 'Zu meinem Klavierknonzert (1988)' in Gesammelte Schriften, Vol. 2 :299,

translated by Louise Duchesneau cited in *György Ligeti. Of Foreign Lands and Strange Sounds*. Boydell Press (2011): 103.

pixel by pixel, on a computer screen. By the end of the movement we've arrived at an elaborate, multi-dimensional fantasy of self-replicated whirls, riotous colour and polymetric energy, an exuberant vortex of order and chaos, symmetry and freedom, like the magnificent fractal images of Mandelbrot and Peitgen.233

We are reminded of the problems in identifying fractals in music where, as Brothers pointed out, 'Unlike a picture, which is all laid out so that you can instantly see the structure … With music, the whole piece takes shape in your mind'.234 If, as Hans Lauwerier writes in the introduction to *Fractals. Images of Chaos*, 'A fractal is a geometric figure in which an identical motif repeats itself on an ever-diminishing scale' then this movement of the *Piano Concerto* movement cannot be described as fractal.235 It would not be true to say that it unfolds progressively over time moving systematically from large statements of the three-note fragment to ever smaller more intricate patterns although some aspects of this are evident, the ever-decreasing values are not systematic. The movement opens with loud statements of the three-note motif interspersed with bars of silences. Over the course of the movement the music does become more complex but reaching a climax about three quarters of the way through, rather than the end, before petering out to a pianissimo conclusion. Perhaps the fractal images are more evident in the dynamics and orchestration where the finer detail is represented by the high notes of the glockenspiel and piccolo, and towards the end of the movement the piano makes increasing use of the higher register, moving upwards, higher and higher to the very top of the keyboard register.

## **Tom Johnson and Naryana's Cows**

Much of the music of the American composer Tom Johnson (b. 1939) follows mathematical principles and logical procedures such as permutations,

233 Richard Steinitz. 'Weeping and Wailing'. *The Musical Times,* Vol. 137, No. 1842 (Aug, 1996): 17-22.

234 Brothers. 'Structural scaling', 90.

235 Hans Lauwerier. *Fractals. Images of Chaos.* (London: Penguin. 1991): xix.

206 MATHS & MUSIC

combinations, tiling, block design and fractals.236 He too was inspired by Benoit Mandelbrot's book *Fractals* (1977) with its 'pictures of the Koch curve, the Minkowsky sausage, the Sierpinski sponge, the Peano island and the Cantor triadic bar'. For Johnson they were 'essentially maps showing the route that I was about to follow'.237

When interviewed by New Music USA in 2003, Johnson was asked 'What role has theory played in your compositions and how important is it for people to know the theory behind the music in order to appreciate it?' He replied by saying 'Theory is when you know everything about it but it won't work. Practice is when it works but you don't know why' adding that good composers not only compose but also think a lot about how they compose. This was the thinking behind what he describes as his 'purely technical' book, *Self-Similar Melodie*s 238(1996), a didactic study with specially composed examples based on this characteristic of fractals.239

Many of his pieces use a variety of techniques in self-similarity, although as he pointed out in his IRCAM lecture of 2006, 'true self-similar structure is relatively rare.' In the lecture he defines these pieces as 'music that somehow contains itself within itself and does so on at least three different levels of time. The use of self-similarity is quite evident to the listener his piece, *Narayana's Cows.* It was 'derived from a 14th-century Indian problem in which a mother cow gives birth to a female calf at the beginning of each year and each calf becomes a mother and does the same thing beginning in her fourth year. By the 10th year, the herd numbers 60 mothers and daughters… repeating one process over and over always results in some kind of self-similarity' The following illustration shows the 60-note Narayan sequence that has evolved by the 10th year. The first line of numbers below represents the 60 cows of the herd in its 10th year. The second line singles out the 19 mothers, that is, the quarter-notes, of the 10th year, and this happens to be the notes of the 19

236 For more information about Johnson's use of tiling procedures see - Tom Johnson. 'Tiling in my music'. https://web.northeastern.edu/seigen/MusicDIR/Tiling\_in\_my\_music\_Johnson.pdf

237 Tom Johnson. 'Self-Similar Structures in my Music: an Inventory lecture' presented in the MaMuX seminar, IRCAM, Paris, Oct. 14, 2006 in context with a lecture on self-similarity by mathematician Emmanuel Amiot

238 Tom Johnson. *Self-Similar Melodie*s. (Two Eighteen Pr, 1996)

239 https://newmusicusa.org/nmbx/What-role-has-theory-played-in-your-compositions-andhow-important-is-it-for-people-to-know-the-theory-behind-the-music-in-order-to-appreciateit-Tom-Johnson/ April 2003

cows of the 7th year, and the mothers of the 7th year are equivalent to the six mothers and daughters of the 4th year, shown on the last line.

1 2 2 2 2 3 2 3 3 2 3 3 3 2 3 3 3 3 4 2 3 3 3 3 4 3 4 4 2 3 3 3 3 4 3 4 4 3 4 4 4 2 3 3 3 3 4 3 4 4 3 4 4 4 3 4 4 4 4 5 1 2 2 2 2 3 2 3 3 2 3 3 3 2 3 3 3 3 4 1 2 2 2 2 3

In Johnson's musical work *Narayana's Cows* there are two layers: one verbal and the other musical. In the verbal layer, the speaker transposes Narayan's numbers into words: 'The first year there is only the original cow and her first calf. The second year there is the original cow and two calves. The third year there is the original cow and three calves. The fourth year the oldest calf becomes a mother, and we begin a third generation of Narayana's cows.' And so on until the seventeenth year is reached. The second layer is made up of musical sounds where the pitch correlates with the cows' generation, and the rhythm correlates with the difference between a cow (long note) and a calf (short note). Clearly unfolding to the listener, together they evolve into an accumulative and complex melody.

## **Chaos theory**

In the words of Peitgen, 'Ligeti was just as fascinated by the discovery of deterministic chaos as he was by Benoit Mandelbrot's fractal geometry'. Chaos theory (the science of dynamical systems) focuses on the behaviour occurring in a system under iteration and is the study of apparently random or unpredictable behaviour in systems governed by deterministic laws. Fractals form part of the visual identity of chaos. In recent decades a diversity of systems has been studied that behave unpredictably, despite the fact that the forces involved are governed by well-understood physical laws. The common element is sensitive dependence to initial conditions and to the way in which they are set in motion. The connection between chaos and fractals is the strange attractor. In the field of dynamical systems an attractor is a set of states towards which a system tends to evolve from a wide variety of starting conditions. A strange attractor is a complicated set with a fractal structure. One effect of chaos is commonly known as the Butterfly Effect, where a small change in initial conditions can lead to a large change in the behaviour of a system. The term comes from the meteorologist Edward Lorenz who accidentally discovered the effect in 1961 while trying to model the weather. In one run-through he entered 0.506 into his data instead of 0.506127 and was surprised to see that the original results were vastly altered by such a small change in one variable; where there was originally sun, there was now rain, a windy day became a calm day and so on. The Butterfly Effect therefore describes how a small change in one state of a deterministic system can bring about large differences in a later state with the potential to render long-term predictions impossible.

According to Steinitz 'Ligeti realised that the new theories which sought to explain the precarious balance between order and disorder, pattern and chaos, and the apparent origin of both conditions in measurable deterministic situations, had intriguing parallels with the way he composed.'240 In a later article he writes 'That he [Ligeti] now views this interaction from the vantage-point of current mathematical thinking is indicated by his naming the first study after a crucial issue in the science of dynamical systems (more commonly known as chaos theory), the concept of 'disorder'.241 *'Désordre',*  writes Steinitz 'demonstrates how tiny discrepancies quickly breed confusion. Albeit in microcosm and in a finite context, Ligeti illustrates a fundamental idea of chaos - that small differences in initial conditions rapidly lead to dramatic outcomes.' In his 2012 lecture on 'Chaos and Fractals in Music' the mathematician Gareth Roberts explained how this is achieved:

Each hand opens with identical 8-beat rhythmic patterns (3 + 5). In the fourth measure, the right hand drops a beat, playing a 7-beat pattern rather than an 8-beat one, but continues the 8-beat pattern for the next three measures. This small change starts to cause a big shift, audible for the listener due to the shifting accents in each hand. In the eighth measure, the right hand drops another beat, playing 7 instead of 8. Now, the left hand is two beats ahead instead of one. Again, the right hand only drops a beat in this one measure. The "iterative"

240 Steinitz, 'Maths and Chaos', 15.

241 Steinitz. 'Dynamics of Disorder', 8.

process of dropping a single beat continues, as the right hand drops a beat approximately every four measures so that the two hands become completely out of synch, and the butterfly effect is realized.242

Closer analysis of *'Désordre'* shows that this system is not rigorous, however, so it is not obeying the deterministic laws of chaos theory. In the same lecture, Roberts argues that the American composer Steve Reich employs a similar Butterfly Effect device where 'small changes in rhythmic structure (e.g. slight phase shifts) lead to big changes in the music' in, for example, *Clapping Music*  (see pages 32-4). In *Clapping Music* (for two players) it could be argued that Reich applies deterministic laws in the way that the phase shifts he uses are applied systematically. In phase shifting constantly repeated patterns are subjected to gradual changes, one part repeats constantly and another gradually shifts out of phase with it, a compositional device that can be seen in terms of a mathematical translation. In *Clapping Music* the phase shifting is systematically applied quaver by quaver, with Player 2 shifting further out of phase with Player 1, effect is one of huge rhythmic diversity until eventually the parts come together again. It is interesting to note that *Clapping Music* was composed in 1972 so it pre-dates the period when chaos theory was widely known.

## **The use of fractals and chaos theory in the music of Rolf Wallin**

The Norwegian composer Rolf Wallin (b. 1957) combines computergenerated systems and fractals with his intuitive approach to composition. He is dismissive of what he refers to as Ligeti's 'supposedly 'fractal' music'. In a lecture given in 1989 he said

… when one investigates Ligeti's supposedly 'fractal' music, for instance his piano etudes from 1985, one finds that his music could very well be achieved without having heard of the existence of chaos theory or

242 Gareth Roberts. 'Chaos and Fractals in Music'. Maths and Music: Aesthetic Links, Montserrat, April 16, 2012.

having seen the Peitgen-Richter pictures. The techniques he uses are derived exclusively from traditional compositional practice, and the music is neither more nor less fractal than African drum rhythms or a Bartok string quartet.243

Wallin outlined a more mathematical approach to composing with fractals in a lecture at the Nordic Symposium for Computer Assisted Composition, Stockholm in 1989. Wallin describing several ways to apply fractal mathematics and chaos theory to music using different mathematical formulas. As he points out there are 'countless ways of applying fractals to music'. His rationale for the use of computers and chaos theory is that he can give the computer a 'task where the direction is clearly defined, but where the details are unpredictable, and therefore generating a balance between global order and local disorder'. As he says, 'this has kicked me out of some of the tracks I have been stuck in'.

*Stonewave (1990)*, one of his most popular and frequently performed percussion pieces, is made purely from fractal mathematics and in his programme note to *Stonewave* he writes that 'These formulas, used in the fast growing field of "Chaos theory", are relatively simple, but they generate fascinating and surprisingly "organic" patterns when shown graphically on a computer screen, or played as music.' He adds that 'One should think that such a mathematical approach would lead to sterile and 'theoretical' music. The sound world of *Stonewave*, however, is not one you would associate with math books. The steady, insistent pulse, and the use of sequences put squarely up against each other or divided by long rests suggest an invisible ritual.'

## **The use of fractals in the music of Charles Wuorinen**

Much of the catalogue of the American composer Charles Wuorinen (1938- 2020) was constructed using idea taken from fractals; the generation of infinite and chaotic structures through the iteration of finite elements. He was fascinated by fractal geometry, in particular the work of Benoit Mandelbrot

243 Rolf Wallin. Lecture given at the Nordic Symposium for computer Assisted Composition, Stockholm, 1989.

whose ideas seemed to confirm his own ideas about musical structure and form. Mandelbrot had an interest in Wuorinen's music and, just as Ligeti struck up a long friendship with the mathematician Peitgen, so too did Wuorinen when he got know Mandelbrot. In his essay, 'Music and Fractals', Wuorinen identified the ways in which fractals arose in his music – acoustically, rhythmically, structurally and in terms of pitch generation. Acoustically, the signal has a 1/*f*  distribution (as identified by Voss and Clarke). Rhythmically, his music has a self-similar, hierarchical division of time, and pitch generation is similar across different levels. In terms of structure, a single harmonic progression can be used to determine a single phrase, part of a section, or a whole section.244 In this way Wuorinen developed a logical structure made up of scaled units on which he built his compositions. As he writes, these fractal techniques act as a preparation for composing adding that 'Having made such preparation, then I have found it possible to compose with a kind of intuitive freedom which still assures macrostructural coherence'.

## **Computer generated fractal music**

There have been many instances of computer generated compositions based on fractals as well as computer programmes and software specifically designed to create these. Robert Sherlaw Johnson (1932-2000) was one of the first composers to explore computer generated fractal music. He speculated that 'if meaningful visual patterns can be created by fractal generation, then it should also be possible to create aural ones'. In 'Composing with fractals' he describes the process whereby he takes various iterative formulas to generate musical patterns leading to completed compositions. Echoing the words of Harlan Brothers, he points out that one of the difficulties in composing in this way is that although in visual patterns it is the 'accumulation of values that gives rise to a sensible pattern' in the case of music, 'the whole is not perceived simultaneously and only localised patterns can make sense'. He ends by posing the question 'Can the computer be said to have composed anything?' concluding that in his compositional process although the

244 Michael Frame and Amelia Urry. *Fractal Works. Grown, Built and Imagined. (*Yale University Press, 2016): 104.

computer, using various formulas, generates a stream of variables, 'it is only when the stream of variables is harnessed in a particular way that musical … sense is derived from it otherwise it remains a chaotic sequence'.245

So, can fractals be found in music? To recap, in simple terms, a fractal is a geometric figure featuring self-similarity where an identical motif repeats itself on an ever-diminishing scale. So it could be argued that a great deal of music has a fractal element to it in terms of repeated motifs and rhythms used at different pitches or durations, augmentation (scaling up) and diminution (scaling down) being long-established compositional techniques. As we have seen, Wallin criticised what he refers to as Ligeti's 'supposedly 'fractal' music' on the grounds that the techniques that Ligeti uses can be found in traditional compositional practice.246 Fractals are rarely used systematically, mathematically they are often incomplete or their appearance is only fleeting. Much has been written by musicians about chaos in music, but when aligning compositions with chaos theory, confusion often arises between the everyday definition of chaos as a state of utter disorder or a total lack of organization rather than the scientific definition whereby dynamical systems display unpredictable behaviour at the same time as obeying deterministic laws.

Nested sequences, for example, are necessarily fragmentary because they cannot be sustained across a whole piece. The algorithms used to produce nested sequences could be described as musically mechanical and prescriptive hence they need to be adapted with some creative licence. Of the examples given in this chapter, the prolation canon, the oldest form, seems to be the most consistent with fractal characteristics in its use of motivic scaling and self-similarity. There is a certain irony in this of course: prolation canons were at their most popular over 500 years ago, long before the concept of fractals had been discovered.

245 Robert Sherlaw Johnson 'Composing with fractals' in *Music and Mathematics. From Pythagoras to Fractals*. (Oxford: Oxford University Press, 2003).

246 Rolf Wallin. Lecture given at the Nordic Symposium for computer Assisted Composition, Stockholm, 1989.

# 13**Musical cryptography**

there was much talk when Pink Floyd's album *The Wall* concealed a secret message, a message which could only be heard when the record was played backwards. This was achieved through back masking, a technique where the reversed recorded sound rendered the unintelligible noise when played forward as clear speech. This was in 1979, but concealing secret messages in music is not a new idea. Various cryptographic techniques have been used in music, based on mathematical rather than technological premises, and going back about 500 years.247 Although in everyday language there is little distinction made between coding and cryptography, in mathematical terms coding theory refers to the reliable transmission of data and to the detection of and correction of errors in its transmission whereas cryptography is the branch of mathematics which is the practice of writing in code or cipher aiming to conceal information. The original message is called the plaintext and the enciphered message is called the cryptogram. The process of converting the plaintext to ciphertext is called encryption, with decryption denoting the reverse process. A code is a way of hiding meaning by changing whole words or phrases by giving them some different significance, often substituting one word for another, whereas a cipher uses an algorithm which usually replaces a letter or other single character with another to convert the plaintext, a message, for example, into another text known as the ciphertext.

This chapter examines various uses of cryptography in music from an early example by the Renaissance composer Josquin des Prez to Olivier Messiaen's

247 Track eight of *The Wall, 'Empty Spaces'*, conceals a hidden message which cannot be understood when the record is played normally. However if the record is played backwards a secret message can be heard which includes the words 'Congratulations. You have just discovered the secret message.'

use of a coded musical language in the twentieth century. It explores the use of monograms as cryptograms by composers including Schumann and Shostakovitch. At the same time it looks at some of the limitations of cryptograms in music, questioning whether musical encryption can produce a convincing piece of music and work as a successful encryption, emphasising the relative simplicity of musical encryptions in comparison with those used in mathematics. The chapter concludes with an examination of one of the most famous pieces of music embodying a secret message – Edward Elgar's *Enigma Variations*.

One of the first instances of a musical cryptogram can be found in *Missa Hercules Dux Ferrariae* a setting of the mass dedicated to the Duke of Ferrara by the Renaissance composer Josquin des Prez (c. 1450 – 1521). Josquin takes the Latin name of the dedicatee 'Hercules Dux Ferrariae' and matches the vowel sounds to the vowel sounds of the solmization system. Solmization was devised by the eleventh-century monk Guido of Arezzo. He created a sight-singing system where a set of syllables is matched to the degrees of the scale - ut, re, mi, fa, sol, la (a system very similar to the one used today - do, re, mi, fa, sol, la, te, do). Josquin coupled the musical pitch of the solmization syllable with the vowel of text he wanted to represent. Thus 'Hercules Dux Ferrariae' translates as:

The encryption becomes the pitches D-C-D-C-D-F-E-D in Josquin's *Missa Hercules Dux Ferrariae.* These pitches are used as the *cantus firmus*, the basis of Josquin's polyphonic mass. They can be heard in the opening bars of Kyrie I in the superius part, the top vocal line, and then moving to the tenor line in bar 9 (see below).

This cryptographic system which was later named 'Soggetto cavato' by the music theorist Gioseffo Zarlino in 1558. 'Soggetto cavato dalle vocali di queste parole' in full, translates as a subject 'carved out of the vowels from these words.' The *Missa Hercules dux Ferrariae* is the most famous example of a soggetto cavato and is also the first.248

Musical symbols, usually pitches, have been often used to create ciphers by matching up single notes with individual letters of the alphabet. Early examples can be found in the work of the sixteenth century scientist

248 Lewis Lockwood, '*Soggetto cavato*', in *Grove Music*.

and polymath Giovanni Battista della Porta (1535–1615). During the Renaissance, at a time when political intrigue and war coincided with new scientific developments, there was a lot of interest in secret communication. Della Porta's seminal work on cryptography, *De Furtivis Literam Notis* (On the Secret Symbols of Letters, 1563) is primarily about codes and ciphers. One of his best-known ciphers uses a system where 11 different pitches are allotted letters of the alphabet.249

This system formed the basis of many subsequent systems across the years that followed. The musicologist Eric Sams, who has written extensively on cryptography in music, writes that it 'recurs in readily recognizable adaptations throughout the 16th and 17th centuries'. He goes on to list some of the different ways in which it evolved: the combination of crotchets and minims suggested by the British author Philip Thicknesse in 1772; the use of a cipher wheel where the notes and corresponding letters are written on two wheels, one fixed and one movable in the late eighteenth and early nineteenth centuries; the random allocation of musical notes to cipher letters; and the representation of one letter by a two-note group. As Sams writes, all of these methods were in the pursuit of a 'policy of analogy with real music'.250

John Wilkins (1614–1672) was the first to write a major book on secret communication in English -*Mercury, or the Secret and Swift Messenger. Shewing how a Man may with Privacy and Speed May Communicate his Thoughts to a Friend at any distance.* Chapter XVIII 'Concerning a Language that may consist only of Tunes and Musical Notes, without any articulate Sound' includes a cipher where musical notes are matched up with letters of the alphabet, no clef is given (see below).

249 Eric Sams. 'Musical cryptography' in *Grove Music.*

250 Sams, Cryptography, *Grove Music.*

Wilkins writes 'Where the five Vowels are represented by the Minnums on each of the five lines, being most of them placed according to their right order and consequence, only the letters K and Q are left out because they may be otherwise expressed… By this you may easily discern how two Musicians may discourse with one another, by playing upon their Instruments of Musick, as well as by talking with their Instruments of Speech.'251

It is easy to see the limitations of this seventeenth-century cipher - there are only two note values and seven letter names- A B C D E F G. Philip Thicknesse (1719 – 1792) criticises the above example in his 1772 book *A Treatise on the Art of Deciphering, and of Writing in Cypher: With an Harmonic Alphabet.*252 He argues that Wilkins system will 'instantly appear to anyone, the least conversant with music; that being without harmony or time, it must have no meaning or that some hidden meaning is *therefore disguised*'. He then puts forward his own method, an 'alphabet of musical notes' so that 'even a master of music shall not suspect it to convey any meaning'. 'I am persuaded an alphabet of musical notes may be so contrived, that the notes shall not only convey the harmony, but the very words of the song'. Thicknesse's proposed system is more sophisticated and uses a treble clef, minims and crotchets as well as a key signature to add authenticity (see below).253

251 John Wilkins . *Mercury, or the Secret and Swift Messenger. Shewing how a Man may with Privacy and Speed May Communicate his Thoughts to a Friend at any distance*. (London: printed for Richard Baldwin, 1694): 75-76.

252 Philip Thicknesse: *A Treatise of the Art of Decyphering, and of Writing in Cypher (London: W. Brown, 1772).*

253 H. Neville Davies. 'The History of a Cipher, 1602-1772*.*' *Music & Letters.* October, 1967, Vol. 48, No. 4, Oxford University Press: 325-329

The musicologist David Loberg Code questions whether musical encryption can be both a convincing piece of music and a successful encryption, coming to the conclusion that 'Musical encryption can rarely be both because the attributes needed for convincing musicality and strong encryption are not mutually conducive.' He finds that the music of those composers whose 'primary goal' was to 'create music that embedded extra-musical content by means of musico-alphabetic correspondences' used systems which 'were usually so superficial they should not really be considered encryption; whereas the systems used by the latter were so mechanical that most would not consider the results to be music.'254 As Thicknesse observes, if music notation is to be used successfully to conceal information then it needs to look like a convincing piece of music.255

This lack of authenticity led to the downfall of a gambling group in 1940s New York. At the time, gambling was illegal and ingenious methods were devised to get around the law. One of the forms of gambling was a numbers game where bets were taken based on the U.S. Treasury Balance or baseball scores, for example. Those who recorded the numbers were known as policy collectors.

Back in 1940, a Brooklyn detective kept a policy collector under observation for the greater part of the day…he made the arrest but to his surprise, the pad on which the prisoner had been recording the bets did not contain the usual notations. It appeared to be the score of a musical composition…The detective was positive that the music represented some code…'For one thing, the whole piece was written on a range of ten notes. There seemed to be no variation in the value of the notes… So, I took it home and tried it out on a friend's piano. When I played it, I was sure that its atonalities were not music, either traditional or modern. So, I set to work to decipher it as a code…Not only were different musical notes used but their

254 David Loberg Code. 'Can musical encryption be both? A survey of music-based ciphers', *Cryptologia*, May 2022: 318-364

255 Davies, 'The History of a Cipher', 327.

positions on the scale changed their values...Each measure, indicated by the straight line, constitutes a separate wager'.256

## **Monograms as cryptograms**

Known examples of musical ciphers being used for espionage are rare. However there are many pieces of music where composers have used ciphers to create motifs or themes which are there to be interpreted rather than as codes to be decrypted. A common device has been to use a motif comprising notes whose letter-names spell words. These are often based on the composers own name or one of their friends. The problem in translating letter names into musical notes is the limiting nature of having only seven named pitches. The question arises as how to cipher the remainder of the alphabet. In German music however Bb is known as B and B natural is named H, hence the B-A-C-H motif which was used by J S Bach in several of his compositions, and then by many other later composers. Sams writes that 'Beethoven, Schumann, Liszt, Rimsky-Korsakov, Busoni and several others' used the BACH motif. This sequence of notes is not uncommon, so many of its appearances in Bach's work may be a coincidence.257

Further note names can be derived from their sound, for example E-flat, 'Es' in German, can be used to represent 'S' and other names can be arrived at by using the sol-fa system. A combination of the above offers the following possibilities of notes for the letters A, B (Ger. Bb), C, D, E, F, G, H (Ger. B), L (la), M (mi), R (re), S (Ger. Eb, 'Es') and T (te). Some composers took advantage of these extra letter names or used musical notes for more than one letter whilst others simply ignored any letters in their name which did not represent a musical note or made up their own systems.

256 Anon. 'Codes are Fragile', *The Journal of Criminal Law. Criminology, and Police Science.* Spring 3100, 23 (New York, 1952), 10–13.

257 Sams, 'Cryptography', *Grove Music.*

One well-known example of a thematic motif based on a person's name can be heard in Schumann's *Variations on the name "Abegg"* which are dedicated to the (possibly fictional) Pauline, Comtesse d'Abegg. In this instance the cryptogram is not used to conceal Abegg's name, rather it is used as a compositional technique.

Schumann (1810-1856) uses the notes A – Bb – E - G - G as the basis of his opening theme, quoting the notes directly in the opening bars over a dominant seventh chord and then using them to build a sequence. As the piece goes on, less attention is focused on the ABEGG motif and more attention is given to the developmental possibilities of its first two notes A – Bb.

The German composer Johannes Brahms (1833 – 1897), however, included a cryptogram in his 1868 *String Sextet No. 2 in G major* which quite possibly conveyed a secret thought. In the summer of 1858, he had fallen in love with Agathe von Siebold. They planned to get married, but Brahms broke off the engagement in order to focus on his musical career. Brahms never married and although Von Siebold later married someone else, Brahms continued to long for her. The notes A-G-A-(T) H-E can be

found as the music rises to a heart-wrenching climax in the first movement (bars 162 - 168). In a letter to his friend Josef Gänsbacher, Brahms wrote 'by this work, I have freed myself of my last love.'258

In 1909 six composers were commissioned to celebrate the life of the composer Joseph Haydn by the *Revue musicale mensuelle de la Société Internationale de Musique*. Haydn had died 100 years earlier and the composers – Maurice Ravel, Claude Debussy, Paul Dukas, Vincent d'Indy, Charles-Marie Widor and Reynaldo Hahn – were invited to compose a piece based on Haydn's name. Each of the composers used the same cipher system.

In Ravel's short piano piece of only 54 bars, *Menuet sur le nom d'Haydn*, H was given its German equivalent, B natural and was built on the following motif.

Similarly, in 1976, the cellist Mstislav Rostropovich commissioned 12 prominent composers to write a piece based on a musical cipher, this time for

258 Jacquelyn E C Sholes. *Allusion as Narrative Premise in Brahms's Instrumental Music*. (Indiana: Indiana University Press, 2018): 33.

the 70th birthday of the Swiss conductor Paul Sacher. Rostropovich invited each of the composers Conrad Beck, Luciano Berio, Pierre Boulez, Benjamin Britten, Henri Dutilleux, Wolfgang Fortner, Alberto Ginastera, Cristobal Halffter, Hans Werner Henze, Heinz Holliger, Klaus Huber, and Witold Lutoslawski to write a variation. In *12 Hommages à Paul Sacher* each of the solo cello pieces is based on this cryptogram of Sacher's name.

E-flat (Es), A, C, B-natural (H), E and D (Re) which spell out eSACHERe

The variations are technically demanding and were composed in a variety of styles, some use traditional notation, others use extended cello techniques. Some of the pieces 'cleverly disguise' the hexachord whilst in others the pitches are used in a 'clearly recognisable manner'.259

The Russian composer Dmitri Shostakovitch (1906 – 1975) used his monogram D-S-C-H throughout most of his career. The notes transliterate to D Sch(ostakovitch) a musical encryption of his name. The monogram can found in several of his works including Symphonies No. 1 and 10, String Quartets No. 5 and 8, Violin Concerto No. 1 and Cello Concerto No. 1.

The signature motif permeates his *String Quartet No. 8* in C minor, Op. 110 (1960). Rather than disguising a secret message it contributed to what Shostakovitch described, in a letter to his friend Isaak Glikman, as the string quartet's 'superlative unity of form'.260 It is an anguished work written after a

259 Ryane Dunnagan. An examination of compositional style and cello technique in *12 Hommages à Paul Sacher*. Unpublished PhD thesis. 2017.

260 Michael Mishra*. A Shostakovich Companion*. (London, Bloomsbury, 2008): 230.

visit to Dresden where he witnessed the devastation which had been caused by World War II bombing. In some ways it could be described as an autobiographical work, as well as the signature motif being woven seamlessly into nearly every page, the work is full of quotations from earlier works. The piece is dedicated 'To the memory of the composer of this quartet'. His use of the four-note motif is ingenious and imaginative. The piece opens with the personal motto heard in the cello part initially and then moving up through the strings in imitative counterpoint (see below). The parts are all in a low tessitura contributing to the sombre mood.

© Copyright by Boosey & Hawkes Music Publishers Ltd. Reproduced by permission of Boosey & Hawkes Music Publishers Ltd.

Later on in the first movement (bar 79) the DSCH motif is heard in all the parts but this time the texture is homophonic (chordal) rather than contrapuntal. The resultant chords are G major, Eb minor and F major moving on to the harshly dissonant tone cluster of G, Ab, B. This time the theme is in augmentation using notes of twice the value (see below)

Rather than enciphering single words or names, the twentieth century French composer Olivier Messiaen (1908 - 1992) enciphers whole phrases and uses both codes and ciphers in his music. He devised what he referred to as a '*langage communicable'*. In the preface to the score of his 1969 organ work *Méditations sur le Mystère de la Sainte Trinité,* Messiaen outlines some of the compositional techniques he has used in the encryption of the work. Each letter of the alphabet is assigned a musical note and this cipher is used to encode quotations from the thirteenth-century *Summa theologica* of St Thomas Aquinas (a compendium of the theological teachings of the Catholic church). He also uses symbolic leitmotifs which act as codes, referring to these as *formules musicales*. Messiaen's works are imbued with religious significance. The following two melodic leitmotifs are used to represent the verbs 'to be' and 'to have'. Messiaen explains in the preface to the score that 'To be' uses a descending movement, because all that is comes from God (the Being par excellence, He who Is)'.

'To have' is an ascending movement because we can always gain more by raising ourselves towards God'.

As Andrew Shenton points out, in sharing the cipher, Messiaen has removed any 'problem for the *performer* discerning the 'hidden' message', but this still leaves the more complex problem of 'comprehension for the listener'.261 This brings us back to Code's earlier question as to whether musical encryption can function successfully as both music and code. Part of his criticism is that the 'musico-alphabetic correspondences' frequently used by composers are often superficial to the extent that they should not really be considered as encryption.262 It is certainly true to say that in comparison even the simplest systems of mathematical coding are usually much more complex and sophisticated: they can involve, for example, probability theory, frequency analysis or set theory; some use random number generators; and others utilise concepts based on number theory.

## **The mystery of Elgar's** *Enigma Variations*

This chapter would not be complete without some reference to Edward Elgar's (1857 - 1934) famous orchestral piece *Enigma Variations* (1899) which has attracted, as the BBC radio presenter Tom Service once put it 'more musicological and pseudo-mathematical sleuthery than any other work of orchestral music before or since'.263 There are two enigmas. The piece is made up of 14 variations on an original theme, each being a musical sketch of a friend of Elgar's. In all but one cases the dedicatees are named, but the penultimate variation is simply marked with three asterisks. The first muchdiscussed enigma (which does not involve a cryptogram) is the identity of this anonymous dedicatee. The second enigma takes the form of a sophisticated cryptogram created by a skilled cryptologist. As Sams writes of Elgar,

261 Andrew Shenton. *Olivier Messiaen's System of Signs: Notes Towards Understanding his Music.*  (London: Routledge, 2008).

262 Ibid.

263 https://www.bbc.co.uk/music/articles/cb7ac9cf-207e-4244-8302-2436f2c2ba5a

He successfully solved a well-known challenge cipher, to which eminent experts later thought it worthwhile to publish their own solutions; he constructed a difficult if not impossible cryptogram; he made cipher entries in diaries and notebooks. One of his earliest works was an Allegretto for violin and piano on G–E–D–G–E, the name of a friend. It seems reasonable on the facts to conjecture that he used private ciphers in some of his compositions, and that suggestion has often been made in respect of the 'Enigma' Variations.264

Elgar suggested that the real theme of the Enigma Variations is not the theme that you can clearly hear, rather it is a familiar melody that can be played at the same time but never appears in full. In the programme note for the first performance Elgar is quoted as saying about the hidden theme that it is "so well-known that it is extraordinary that no one has spotted it".

The Enigma I will not explain – its "dark saying" must be left unguessed, and I warn you that the connexion between the Variations and the Theme is often of the slightest texture; further, through and over the whole set another and larger theme "goes", but is not played . . . . So the principal Theme never appears… the chief character is never on the stage.265

In 1900 Elgar gave another clue in an interview he gave to the *Musical Times*  where he said thatthe 'heading Enigma is justified by the fact that it is possible to add another phrase, which is quite familiar, above the original theme that he has written'.266 So the key to the decryption would involve finding a well-known theme that would fit both harmonically and rhythmically in counterpoint with the original theme - no mean feat. Years of speculation followed with suggestions including God Save the Queen, Pergolesi's *Stabat Mater*, Rule Britannia and Auld Lang Syne as potential hidden melodies working in counterpoint with the stated theme, but none of these quite works. Furthermore, Elgar accepted none of the solutions proposed during

264 Sams, 'Cryptography', *Grove Music.*

265 Turner, Patrick (2007). *Elgar's 'Enigma' Variations – A Centenary Celebration*. (London: Thames Publishing, 2007): 49.

266 F G Edwards. 'Edward Elgar'. *The Musical Times*, May 1900, Vol. 41.

his lifetime. The question remains as to whether this matters when the piece can be enjoyed without any consideration of its inner puzzle. Indeed part of its attraction may lie in what the conductor Norman del Mar describes as 'the impenetrability of the riddle itself.267

267 Norman Delmar.*Conducting Elgar*. (London: Clarendon Press, 1998).

#### 228 MATHS & MUSIC

# 14 **Magic squares and Latin squares**

a number of composers have used magic squares to generate material for their compositions. The first section of this chapter explores the way that magic squares have been used by three composers; the English composer Peter Maxwell Davies, and the French composers Pierre Boulez and Phillipe Manoury**.** The chapter then goes on to explore the use of Latin squares in music, in particular the ancient palindromic 5 x 5 Latin square known as the Sator Square which Anton Webern is thought to have used in his *Concerto Op. 24*. Inspired by the work of Webern, the English composer John Tavener also employed the Sator square in his music. The ways that he built it into his compositional technique are explored in two of his compositions; the song cycle *To a Child Dancing in the Wind* and his piece for double choir and string trio - *Ikon of Light.* The Italian composer Bruno Maderna used both Latin squares and magic squares as part of the creative process, notably in his *Serenata No. 2* which is looked at in some detail in the final section.

## **Magic squares**

A magic square is a square matrix where a different positive integer is written in each cell, each occurring only once. The 'magical' quality is that each horizontal, vertical and diagonal row adds up to the same number. Here is a 3 x 3 magic square using the consecutive numbers 1 to 9 (Table 1). Each row and column adds up to 15 and each of the two diagonals also adds up to 15.

#### **Table 1 – Magic square of order 3**

The constant number of the sum, in this case 15, is sometime referred to as the 'magical number' or the 'magic constant'.

The origins of magic squares are rooted in religious and esoteric contexts. One of the most famous magic squares is the Lo Shu 3 x 3 square (Table 1) which was known in China as early as 650 BCE. Legend has it that it was first seen on the back of a turtle emerging from the Lo river at a time of floods and the number pattern was seen as an omen. Every other 3 x 3 magic square can be derived from the Lo Shu by reflecting numbers of the square in the middle row or middle column and/or rotating numbers around the middle.

Magic squares have been a source of fascination for centuries and played a significant role in the development of numerology. Numerology attaches meaning and influence to numbers and number patterns. It is a belief in their divine, mystical power. Seven magic squares were dedicated to a planet or deity. The 3 x 3 square (Table 1) is known as the Square of Saturn. The 4 x 4 square is known as the Square of Jupiter where each row adds up to 34 (Table 2).

**Table 2 – The Square of Jupiter**

The others are:

Square of Mars 5 x 5 Square of Sol or the Sun 6 x 6 Square of Venus 7 x 7 Square of Mercury 8 x 8 Square of Luna or the Moon 9 x 9

## **Peter Maxwell Davies and magic squares**

The British composer Peter Maxwell Davies (1934 - 2016) cultivated various styles of composition ranging from the monodrama *Eight Songs of a Mad King* (1969) which shocked audiences at its premiere, to a series of symphonies and occasional lighter pieces many of them inspired by Orkney (his place of residence) such as the ever-popular 'Farewell to Stromness'. His contribution to music education, particularly his part in the introduction of composing in schools, was significant. In 2004 he was made Master of the Queen's Music.

Davies used magic squares to generate material in several of his pieces. He was attracted partly by the internal rhythms of the squares and the way that they generated symmetric patterns, transpositions and inverted figures through both note values and pitch. His first piece based entirely on a magic square is *Ave Maris Stella* (1974). The composition is for six players and is based on a plainsong of the same name. Towards the end of the work, each aspect of the musical material is based on the 9 x 9 magic square - the Square of the Moon. The piece itself is in nine sections. He described the plainsong as being projected through the magic Square of the Moon and felt instinctively that the piece assumed some of the healing qualities associated with the square.

In his chamber piece *A mirror of whitening light* (1976-77) he uses the 8 x 8 Square of Mercury (Table 3).268 The title makes reference to the alchemic process known as whitening when a base metal is transformed into gold. The Square of Mercury is associated with quicksilver and represents the perfect balance between the soul and the body. In Davies' words, the title

268 In various lectures and writings Maxwell Davies mistakenly referred to his use of the Square of the Sun.

refers to the mirror of whitening light outside my own window...there the bay is in fact like a crucible of ever changing miraculous light. I took as a starting-point the plainsong *Veni Sancte Spiritus* which, if one knows it, one can hear quite plainly recurring throughout the work. I also took as a principle of design the magic square … which I used as a kind of rhythmic and tonal grid-plan throughout the work…. the numbers 1 through 64 are arranged, on a square…they make very interesting patterns … the fact that these are numbers is of no consequence whatever. One is dealing in fact with rhythmic lengths and with pitches. For the listener, this is not of prime importance but I must mention it in passing to reassure the listener that the structure, underneath even the wildest passages, is very rigidly controlled.269

**Table 3 – The Square of Mercury**

The number 8 governs the whole structure of *A mirror of whitening light*. Here are some of the techniques Davies uses.270

The plainsong *Veni sancta spiritus* was used to generate much of the pitch material.

269 MaxOpus (http://www.maxopus.com/works/mirror.htm)

270 Jonathan Cross. 'Composing with numbers: sets, rows and magic squares' in *Music and Mathematics. From Pythagoras to Fractals*. (Oxford: Oxford University Press, 2003).

From the plainsong he derived this eight-note phrase G E F D F# A G# C.

Next he took the eight-note phrase and placed it down the furthermost left hand column (Column 1) and across the top row (Row 1).

**Table 4**

Using the notes in Column 1 as the starting points, he then transposed the phrase seven times and inserted the transpositions into the 8 x 8 grid (Table 5).

#### **Table 5**

The next step was to map the notes onto the Square of Mercury matrix in the order that the numbers appear (Table 6).

Notice that four of the numbers were altered by Davies at his discretion (and with no explanation); 45 and 48 have been interchanged, as have 25 and 28. These have been shown in brackets.271 Having created the new matrix Maxwell Davies then took different routes through it to generate melodic lines, the 'very interesting patterns' he refers to: left to right across; right to left across; top to bottom; bottom to top; diagonals and spirals.272 In this way the notes produced would be, for example,

left to right across C A A# F# D D C# G

top to bottom C G# A G B F E F

The first two instruments to play in *A mirror of whitening light* are the trumpet and the flute. Their pitches are taken from the top two lines of Table 6 (the first 13 notes). They use the following pitches

Trumpet C A A# F# D

At times throughout the piece Davies uses enharmonic equivalents so the trumpet part is notated as

271 Gareth Roberts. *From Music to Mathematics {Exploring the connections*}. (Baltimore: John Hopkins University Press, 2016): 263.

272 The manipulation of these rows and columns bears some similarity to the way that the note row is treated in Serialism (see Chapter 8).

C A Bb Gb D273

Flute D C# G G# F F# D# G This is notated as D C# G Ab F Gb Eb G

In his analysis of *A mirror of whitening light*, the musicologist Peter Owens gives further examples of the ways in which Davies traced pathways through the magic square in order to generate pitches. The word '*Hauptstimmen'* is used to denote the principal parts.274

For each of the work's main subsections, beginning at rehearsal letters F, J, Q, W, Z, El, H1 and two bars after J1, *Hauptstimmen* are created by tracing pathways through the matrix so that all cells are realised without repetition or omission. Bassoon and cor anglais, for example, share a presentation of the rows of the matrix - all read left to right - between F and J; violin 2 presents its columns - alternating ascending and descending readings - between J1 and Q1; other realisations trace zig-zags through the diagonals, or 'L' shapes, all, significantly, beginning and ending in different corner cells of the matrix, two of which contain the pitch class C.275

The Square of Mercury is also utilised at points to generate rhythms. The numbers in the Square of Mercury go from 1 to 64 (see Table 3). The range of note values is clearly impractical i.e. if number 1 was allocated a semiquaver

273 Notes that sound the same but are written (or 'spelt') differently are said to be enharmonic e.g. E# and F where E# is the enharmonic equivalent of F.

274 *Hauptstimme* is the name given by the serialist composers Schoenberg and Berg to a principal part in a complex texture, usually in serial or other rigorously non-tonal music.

275 Peter Owens .'Revelation and Fallacy: Observations on Compositional Technique in the Music of Peter Maxwell Davies'. *Music Analysis* , July - October, 1994, Vol. 13, No. 2/3: 161-202.

(sixteenth note), 2 a quaver (eighth note), 3 a dotted quaver (dotted eighth note) and so on, then the number 64 would be the equivalent of two breves. In order to overcome this potential problem, Davies used modular arithmetic to convert the numbers to those between 1 and 8. Modular arithmetic gives the remainder when dividing the magic square numbers by the eight. So, for example, 58/8 is 7 x 8 remainder 2 and 43/8 is 5 x 8 remainder 3. In mathematical notation this is shown as

58 ≡ 2(mod 8) and 43 ≡ 3(mod 8)

The value of a quaver is then allocated to each integer so that

1= a quaver (eighth note) 2 = a crotchet (quarter note) 3 = a dotted quaver (dotted eighth note) 4 = a minim (half note) 5 = a minim tied to a quaver (half note tied to an eighth note) 6 = a dotted minim (dotted half note) 7 = a dotted minim tied to a quaver (dotted half note tied to an eighth note).

There is a slight flaw in this system in that 8, 16, 24, 32, … 64 are all divisible by 8 so there is no remainder. So 8 takes the value of 8 quavers and becomes a semibreve. The resultant Square of Mercury can be seen in Table 7 which takes into account the alterations Davies made in his original transcription of the Square of Mercury (Table 6). It also assumes that a rest following a note counts towards it, a strategy that gave Davies more creative freedom.276

276 Roberts. *From Music to Mathematics*, 265.

### **Table 7**

Cross (2003) questions whether any of this can be heard when listening to *A mirror of whitening light* noting Davies' comment that the 'sequences of pitches and rhythmic lengths … [are] easily memorable once the "key" to the square has been found'. Although Cross observes that Davies might argue that the 'logic' given to the transformations may be 'subconsciously perceived' he goes on to state his own doubts as to this possibility.277

## **Pierre Boulez and magic squares –** *Structures 1a*

In the early 1950s, several composers including Pierre Boulez (1925-2016) extended the compositional procedures of serialism to the other aspects of the music beyond pitch, a method which has become known as total serialism or integral serialism (see pages 133-4). Boulez based his piece *Structures 1a*  (1951) entirely on a note row taken from Messiaen's piano piece *Mode de valeurs et d'intensités* (Mode of Durations and Intensities) (1950). From this he created two number matrices which represented all 48 versions of the row. These can be represented as a pair of magic squares (Table 8 and Table 9). In these squares, the number 1 always refers to Eb (pitch 1 of the original series) and to a demi-semiquaver (thirty-second note) - the first degree of the chromatic scale of durations. The number 2 always refers to D (pitch 2 of the

277 Cross, 'Composing with numbers', 140.

original series) and to a semiquaver (sixteenth note) - the second degree of the durational scale, and so on.

#### **Table 8**

#### **Table 9**

If the twelve columns in Table 8, reading from left to right across or down from top to bottom, represent the twelve possible transpositions of the original row, then the same columns read from right to left or bottom to top represent the twelve possible retrogrades. Similarly the columns of numbers in Table 9, (from left to right or top to bottom) represent the twelve inverted row forms, and (from right to left or bottom to top) the retrograde-inversions.

## **Phillipe Manoury - the Square of Jupiter in** *Melencolia (d'apres Dürer)*

In medieval alchemy special qualities were assigned to each of the magic squares defining the relationship between the soul and the body. Each square was associated with a metal as well as its corresponding planet. The Square of Jupiter is associated with tin and represents the soul emerging from the body. The German artist Albrecht Dürer featured a 4 x 4 square (Table 10) with the magic number of 34 in his engraving *Melencolia I* of 1514. As can be seen, this is a variant on the Square of Jupiter.

**Table 10**

Notice how the date 1514 is embedded on the bottom row and the way in which the four squares at the centre and each of the four 2 x 2 square grids at each corner also add up to 34. The French composer Phillipe Manoury (b. 1952) used Dürer's engraving as part of the basis for his Third String Quartet *Melencolia (d'apres Dürer)* (2013). He used the tabulated data for generating collections of pitch classes – often tetrads (a set of four notes) – and sometimes used it for other purposes, such as creating rhythmic patterns.278

## **Latin squares**

Another type of square grid (or matrix) is the Latin square: a square array of symbols arranged in rows and columns such that each row or column of the array contains each symbol precisely once. In mathematical terms this would be described as an *n x n* array of order *n* with entries from a set of *n* distinct symbols. So if *n* = 4 then a Latin square of order 4 using letters from the alphabet could be as seen below in Table 11. The word 'Latin' refers to the symbols which were taken from the Latin alphabet.

**Table 11 – Latin square of order 4**