In mathematical terms a frieze pattern can be described as a plane figure with a symmetric pattern bounded by two infinite parallel lines. The patterns are repetitive in one direction according to the symmetries of the pattern. Frieze patterns can be classified in terms of their groups of symmetries, hence the mathematical study of frieze patterns focuses on group theory. Group theory is a branch of abstract algebra, which looks at the main features of a group, from both the point of view of its elements and its group operations. Symmetrical patterns are classified according to the types of symmetry which are used. There are exactly seven different frieze symmetry types. All frieze patterns have translation symmetry i.e. you can slide a horizontal frieze pattern along a fixed distance (to the left or right, up or down) and it will appear unchanged. As well as translational symmetry, the other six types also have different combinations of rotational, horizontal, vertical and glide symmetry. This chapter looks at the way that the symmetries of the seven different frieze patterns can be found in music.

Here is a list of the frieze patterns along with the International Notation.

Guide to the international notation. The second symbol refers to reflection along the vertical line – where this exists the letter 'm' is used, otherwise there is a 1. The third symbol refers to reflection along the horizontal line – where this exists the letter 'm' is used, otherwise there is a 1. If there is a letter 'a', this means that there is glide symmetry (where the figure is reflected (inverted) and then translated by being shifted horizontally). Where the fourth symbol is 2 this means that there is rotational symmetry, otherwise there is a 1.

In the 1970s, the mathematician John Conway created names relating to footsteps for each of the frieze groups. The footsteps represent the following symmetries.

These can be seen in the table below along with a musical illustration of each frieze pattern. In each case a simple musical motif has been used to illustrate the symmetric pattern.

## **The seven frieze patterns with musical illustrations**

**STEP** translational and glide symmetry. **TYPE 2 p1a1**

**JUMP** translation and reflection along the horizontal line.

**HOP** translation. **TYPE 1 p111**

**TYPE 3 p1m1**

**SIDLE** translation and reflection along the vertical line.

**SPINNING HOP** translation and 1800

**SPINNING JUMP** translation, 1800 rotation, horizontal and vertical symmetry

**SPINNING SIDLE** translation, 1800 rotation, glide, and reflection along the vertical line

rotation. **TYPE 5 p112**

**TYPE 6 pmm2**

**TYPE 7 pma2**

## **TYPE 1 HOP Translation**

The Type 1 frieze pattern is the most common. It is based on translation in a horizontal direction - it is simple repetition. Repetition is the easiest symmetry for the listener to perceive and as listeners we are often attracted to repetition; repeated ideas become familiar, they set up expectations, allowing us to anticipate and thus to reap the reward. As Arnold Schoenberg (1874- 1951) wrote in *Fundamentals of Musical Composition* (1967): 'Intelligibility in music seems to be impossible without repetition.'81 The repetition of ideas helps bind music together. One idea is not enough, but if there are too many ideas following each other with no unifying factor then the music may sound shapeless and without a sense of direction.

Schoenberg went on to write 'While repetition without variation can easily produce monotony, juxtaposition of distantly related elements can easily degenerate into nonsense, especially if unifying elements are omitted'. The French composer Erik Satie (1866 – 1925) risked potential monotony with his 1893 piece *Vexations.* The music is only half a page long and the piece bears the inscription *In order to play the theme 840 times in succession, it would be advisable to prepare oneself beforehand, and in the deepest silence, by serious immobilities.* It was 70 years before anybody took up the challenge of a 'full' performance.82 In 1963, the American composer John Cage (1912-1992) organised a team of players for the first performance at the Pocket Theater

81 Arnold Schoenberg. *Fundamentals of Musical Composition* (London: Faber & Faber, 1982): 103.

82 In his study of performances of *Vexations*, Gavin Bryars writes that the first performance of the piece that he knows of was given in 1958 by Richard David Hames at his school when he was 13 years old. For a list of most of the performances of this work up to 1979, see Gavin Bryars, "Vexations and its Performers," Contact 26 (1983): 12-20.

in New York. In the event there were twelve pianists and the performance lasted eighteen hours and forty minutes.83 From then on there have been several marathon performances by teams of players. Here is the theme on which it is based. It uses every note of the chromatic scale other than G#/Ab and is marked 'Tres lent' (very slow).

The theme is then harmonised by a series of chromatic chords mostly unresolved, consequently any sense of the harmonic progression, normally found in Western classical music, is absent. The musicologist Robert Orledge believes that this was not Satie's intention.

Satie was not concerned with through-composition and the normal perception of music "getting somewhere" through functional forms and harmonies. [...] His concern lay in the way our perception of time could be expanded and telescoped, and how music could function as a spatial element in time.84

This reinforces Cage's response to his involvement in its performance. He recalled that after about 90 minutes 'we all realised that something had been set in motion that went far beyond what any of us had anticipated'.85

Constant repetition can be used for dramatic effect. In *Stay On It* (1973 the African-American composer Julius Eastman (1940-1990) repeats the opening seven-note theme throughout most of the 24-minute work. The memorable ostinato stays the same, but the dramatic effect comes from the way the music around it changes. For the first half of the piece the ostinato

83 The twelve pianists were: John Cage, David Tudor, Christian Wolff, Philip Corner, Viola Farber, Robert Wood, MacRae Cook, John Cale, David Del Tredici, James Tenney, Howard Klein and Joshua Rifkin. Gavin Bryars, "Vexations and its Performers," Contact 26 (1983): 12-20https:// gavinbryars.com/work\_writing/vexations-and-its-performers/

84 Robert Orledge, *Satie the Composer (Cambridge: Cambridge University Press, 1990), 142-43.*

85 Luckett. *Every Day is a Good Day. The Visual Art of John Cage*. (London: Hayward Publishing, 2010).

stays in the same form but eventually improvised licks are heard around it until the piece descends into exuberant chaos and the ostinato briefly collapses. For a few bars it is no longer there, but it is still firmly imprinted on the mind. When, after a few bars, it returns it is more a disfigured shadow of itself.

A frequently played piece that uses much repetition is the one-movement orchestral work *Bolero* by Maurice Ravel (1875-1937). The opening two-bar snare drum ostinato is repeated throughout the 17-minute work (see below). An ostinato is a short musical pattern repeated throughout a section or complete piece. In *Bolero* two alternating melodies are passed around the instruments above the relentless rhythm.

$$
\Gamma^{\mathsf{h}^{\mathsf{h}}}, \Gamma^{\mathsf{h}^{\mathsf{h}}}, \Gamma^{\mathsf{h}^{\mathsf{h}}}, \Gamma^{\mathsf{h}^{\mathsf{h}}}, \Gamma^{\mathsf{h}^{\mathsf{h}}}, \Gamma^{\mathsf{h}^{\mathsf{h}}}, \Gamma^{\mathsf{h}^{\mathsf{h}}}, \Gamma^{\mathsf{h}^{\mathsf{h}}}, \Gamma^{\mathsf{h}^{\mathsf{h}}}
$$

Before its first performance Ravel 'issued a warning' to the effect that' the piece was an exercise in orchestration with 'no contrasts, and practically no invention except the plan and the manner of execution'.86 The interest lies in the brilliant orchestration, the hypnotic power of the unflagging rhythm, and the steady crescendo which leads to an exciting climax. The piece was an instant success with audiences. Ravel, however, once described it wryly to his friend the composer Arthur Honegger as 'a masterpiece…without any music in it'.87

There are, loosely speaking, two types of repetition used in music. These are the repetition of short units (a few bars, a theme or a motif, for example) and repetition of large sections of a piece of music as a structural device (in binary, ternary and sonata form, for example). Repetition allows themes to be heard again and for patterns to be displayed.

The Italian composer Domenico Scarlatti (1685 – 1757) wrote over 500 keyboard sonatas, originally intended to be played on harpsichord or fortepiano. Virtually all the pieces are written in simple binary form. However, they are abundant with musical ideas, interesting rhythms and harmonies, remote keys (for the time) and unprepared modulations. One

86 *Daily Telegraph,* July 11, 1931.

87 A. Honegger, *Incantation aux fossils* (Lausanne: Editions d'Ouchy Lausann, 1948): 91–2.

of the musical characteristics found in many of the sonatas is the use of repetitions of the same musical phrase, one after the other – as in the Type 1 frieze pattern. Here is an example taken from his Keyboard Sonata K 27 (bars 11-16) where the same two bars are played three times consecutively.

A commonly found use of repetition popular in Baroque music is the ground. A ground is a recurring melody, usually in the bass, accompanied by continuous variation in the upper parts. The term 'ground' may refer to the bass line, or to the entire musical scheme including the harmonies and upper voices. A ground bass is a repeating bass line which forms the basis of a set of continuous melodic and/or harmonic variations. The bass line repeats many times whilst the variations usually get more and more complex as the piece goes on. It was particularly popular during the Baroque period in music and often formed the basis for the passacaglia and the chaconne.88 The bass line of the famous *Canon for three violins and continuo* by Johanne Pachelbel (1653-1706) has only eight notes forming a two-bar phrase (see below). This is repeated throughout the piece – 28 times in all.

88 The passacaglia is typically in triple time, in a minor key with a four-note pattern descending by step. The chaconne, although usually minor, sometimes uses a major key.

Clearly, if all the parts simply repeated, then the music would soon become rather dull. What Pachelbel does instead is to write a canon for the three violins with each section becoming increasingly rhythmically complex as the piece goes on (see Chapter 9). Probably the most famous use of ground bass can be found in the aria 'Dido's Lament' from *Dido and Aeneas* by Henry Purcell (1659-1695).

Contemporary use of the ground can be heard in the soundtrack to Peter Greenaway's film *The Draughtsman's Contract*. In creating the score, the English composer Michael Nyman (b. 1944) researched the music of Purcell and set out to use Purcell's grounds as a starting point to 'signify a kind of Baroque prototype'89. 90 The well-known opening sequence, 'Chasing sheep is best left to shepherds' is based on the Prelude to Act III from Purcell's *King Arthur* (see below).

Although Purcell's original was not conceived as a ground, Nyman takes the music and fashions his own ground from the opening bars, creating a series of layers out of the given music, each one being treated differently in terms of rhythm, texture and melody. Nyman's pitches stay more or less the same as those of Purcell, but the rhythm changes. The resulting 15-bar sequence

89 Pwyll ap Siôn. *The Music of Michael Nyman: Texts, Contexts and Intertexts*. (London: Routledge, 2007).

90 In 'An eye for optical theory' Nyman unwittingly chose a composition by William Croft, a lesser-known contemporary of Purcell.

is heard four times each time the repetition is varied with changes in texture and instrumentation.

The work of Minimalist composers such as Steve Reich (b 1936), Philip Glass (b 1937) and John Adams (b 1947) features constantly repeated patterns that are subjected to gradual processes of melodic and rhythmic change (transformation) as the piece unfolds. Such transformations include phase shifting (or phasing) where one part repeats constantly and another gradually shifts out of phase with it (see pages 30-4).

Pop music in general makes much use of repeated chord progressions, drum patterns, and riffs not mention whole sections, such as the verse and chorus. Many pop songs are built on riffs - short, repeated melodic or chordal patterns which may be heard at different pitches. Riffs first came to prominence in jazz, swing and rhythm and blues. They formed the basis of the song's structure where musical variants took place above them. In his definition of riffs, the pop musicologist Richard Middleton notes their structural importance, describing them as 'short rhythmic, melodic, or harmonic figures repeated to form a structural framework'.91 Certain riffs are typical of the blues style. The pattern below, for example, would be repeated through the song with the same shape being built on the different chords of the 12-bar blues.

Amongst songs including the best-known riffs are Led Zeppelin's 'Whole lotta love' (1969), 'Sweet Child O' Mine' (1987) by Guns 'N' Roses and 'Seven Nation Army' by the White Stripes (2003). What they have in common is that they are both memorable and catchy, so much so that the 'Seven Nation Army' riff with its roots in garage blues was transformed into a political anthem sung to the words 'Oh, Jeremy Corbyn' when he was leader of the UK Labour party. Thousands joined in with the chant at the Glastonbury Festival in 2017.

In 1977 Donna Summer and Georgio Moroder collaborated on the disco song 'I feel love'. Combining Moroder's incessant synthesizer bass-line, swirling

91 Middleton, Richard. *Studying Popular Music*. (Philadelphia: Open University Press, 1990). .

chords and programmed drums with Summer's ecstatic vocal floating over it, the song was to serve as a blueprint for house and techno, two of the earliest forms of Electronic Dance Music (EDM).92 A simple one-bar pattern in the synthesizer is repeated over and over, for nearly six minutes (roughly 350 repetitions), helping to create a trance-like state. Here is the synth bass line.

In the 1980s technology opened up different ways of creating music. The short, repeating patterns (or ostinati) of house music were facilitated by the new technology available which meant that they did no longer have to be played live. Most EDM songs are based on loops of short repeated rhythms. DJs have different ways to manipulate these sounds in order to add variety cutting up tracks, mixing, scratching, beat-matching, and pitch-shifting, adding these to the creative mix. The ostinati are then layered on top of each other, dropping in and out of the mix.

The rhythmic foundation of much Afro-Cuban music is provided by the clav*é* – repeated interlocking rhythms with a metronomic pulse underlying the music and keeping all the players together. The clav*é* rhythm is made up of several one- or two-bar patterns creating a layered texture made up of melodic and rhythmic ostinati (see pages 44-5). Here is a typical clav*é* pattern as used in 'Se Quema la chumbamba' played by Familia Valera Miranda (see below).

In many styles of African music the bell functions much like the clavé of Afro-Cuban music. It sets the foundation with all other instruments weaving around the pattern. The Ewe musicians of Ghana, for example, often use the

92 Julia Winterson. *Pop Music The Text Book* (London: Peters Edition, 2013).

following bell pattern (see below). Its role is vital in maintaining the flow, energy and clarity of the entire ensemble. L and S refer to long and short strokes (see pages 41-2 for more detail).

**TYPE 2 STEP Translation and glide symmetry**

The Type 2 frieze pattern uses glide symmetry which means that the figure is reflected (inverted) and then translated by being shifted horizontally to the right.

The next two examples, Types 2 and 3, are both taken from Igor Stravinsky's score *Petrushka*. *Petrushka* is a ballet in four scenes which tells the story of three puppets who are brought to life. It is scored for large orchestra with a prominent piano part and was first performed by the impresario Diaghilev's *Ballets Russes* in Paris in 1911.

The following extract is taken from the first scene in the section titled 'Russian Dance'. For ease of reading, the music below has been written using treble clefs. The top two staves are the Bb clarinets and the lower two staves are the viola parts. In both instruments, the music on the second beat is an inversion of the rising arpeggio figure on the first beat. The Type 2 frieze pattern uses glide symmetry which means that the arpeggio figure is reflected (inverted) and then translated by being shifted horizontally to the right.

© Copyright 1912 by Hawkes & Son (London) Ltd. Revised Version: © Copyright 1948 by Hawkes & Son (London) Ltd. U.S. Copyright Renewed. Reproduced by permission of Boosey & Hawkes Music Publishers Ltd.

There are many other examples of uses of musical symmetry in this score. Did Stravinsky (1882-1971) make a conscious decision to use such musical devices? We know from his writings that the answer is almost certainly yes. As well as being a composer, Stravinsky was an influential thinker about music and he wrote much on the subject. His work *Poetics of Music in the Form of Six Lessons* (1942) is considered to be a landmark of twentieth century musical aesthetics. In this he writes at length about his own compositional process.

The creator's function is to sift the elements he receives from her, for human activity must impose limits upon itself. The more art is controlled, limited, worked over, the more it is free.93

## **TYPE 3 JUMP Translation and reflection along the horizontal line**

93 Igor Stravinsky. *Poetics of Music in the Form of Six Lessons.* (London: Harvard University Press, 1942).

The Type 3 frieze pattern uses translation and reflection along the horizontal line. This means that the descending scale figure in the example below is reflected along the horizontal line (inverted). The bars of music below are taken from the harp part of 'The Shrove – tide fair'.

© Copyright 1912 by Hawkes & Son (London) Ltd. Revised Version: © Copyright 1948 by Hawkes & Son (London) Ltd. U.S. Copyright Renewed. Reproduced by permission of Boosey & Hawkes Music Publishers Ltd

## **TYPE 4 SIDLE Translation and reflection along the vertical line**

The Type 4 frieze pattern uses translation and reflection along the vertical line. This means that in each pair of footprints the motif is reflected along the vertical line (played backwards).

The following examples illustrate the ways in which two composers from different centuries have used this type of frieze pattern in their keyboard music. The first example is from the eighteenth century where Domenico Scarlatti uses the device in his Keyboard Sonata K. 514 in C (see below).

Notice how the second half of the bar in the left hand is a retrograde (backwards) version of the first half of the bar in the right hand.

The second example is taken from the twentieth century. The way that the French composer Claude Debussy (1862–1918 wrote for piano altered the way in which the instrument was perceived; he often used the extreme registers, with widely spaced chords and parallel movement as well as taking a new approach to pedalling which produced subtle blending effects. *Estampes* (1903) is a set of three pieces for piano. The first piece *Pagodes* evokes the Javanese gamelan which Debussy had heard played at the 1889 Exposition Universelle in Paris (for bars 26 – 29, see below).

In this example above, the Type 4 frieze pattern is found in the right hand: the second half of the bar is a retrograde version of the first half with the first quaver of the third triplet acting as the point of reflection.

#### **TYPE 5 SPINNING HOP Translation and 1800 rotation**

The Type 5 frieze pattern combines translation with 1800 rotation (retrograde inversion). The original musical idea is combined with the retrograde inversion. Béla Bartók (1881-1945) uses the Spinning Hop frieze pattern in the fifth and final movement (Allegro molto) of his *String Quartet No. IV* (for bars 329-331, see below).

Here is the first violin part with its 1800 rotation immediately underneath it.

The rotation is played by the viola in the first bar of the example and by the cello in the second and third bars.

#### **TYPE 6 SPINNING JUMP Translation 1800 rotation, horizontal and vertical symmetry**

The Type 6 frieze pattern is one of the most complex in its construction. It combines translation with 1800 rotation along with reflections about both the x and y axis. The upper stave is formed by alternating the original with a retrograde of the original. The lower stave is an inversion of the upper stave.

Musical examples of Type 6 are very difficult to spot. The example below from Beethoven's*Diabelli Variations* is almost, but not quite, there. Conway's footprints shown below correspond with the middle complete bar of the Beethoven in terms of the overall shape. The original motif in Variation XXVI is followed by the retrograde version in both staves, and the lower stave is an inversion of the upper stave (bars 9-11). However, strictly speaking the notes on the second beat in the R H should be C D E rather than C# D E. Similarly the C D E of the L H should be followed by E D C (rather than G F E).

10.5920/mathsandmusic.fulltext 10.5920/mathsandmusic.fulltext

#### **TYPE 7 SPINNING SIDLE Translation 1800 rotation, glide, and reflection along the vertical line**

The Type 7 frieze pattern is similar to Type 4 in that it uses translation and reflection along the vertical line. This means that in each pair of footprints the motif is reflected along the vertical line (played backwards). In Type 7 this is combined with 1800 rotation and glide as can be seen in the middle pair of footprints.

The example below is taken from a keyboard suite by Handel. This is the Prelude, the opening movement of *Suite No. 1* marked *quasi fantasia*. Such improvisatory flourishes were typical of Baroque keyboard music particularly in the opening bars of toccatas and preludes.

This chapter has shown how the symmetries of each of the seven frieze patterns can be found in music. Chapter 9 looks in more detail at the ways in which symmetries are fundamental to the musical structure of two musical forms in particular, the canon and the fugue.

## **Scales**

#### 98 MATHS & MUSIC

# 6 **Diatonic scales and tuning**

this chapter describes various scales, intervals and systems of tuning. Most of the scales are commonly used, some of them less so. It opens by introducing the music theory involved; the diatonic major scale, intervals and the concept of consonance and dissonance. For centuries there has been much debate about different scales and tunings with many eminent mathematicians, scientists and musicians across Europe devising and analysing scales. What follows focuses on the three main systems that evolved; the Pythagorean scale, just intonation and equal temperament. It covers the mathematical and scientific principles involved such as ratios, square roots and irrational numbers, the harmonic series and the circle of 5ths.

## **The diatonic major scale, intervals, consonance and dissonance**

A scale is a pattern of notes arranged in order of pitch from low to high (or vice versa) with specified distances or intervals between them. An interval is the distance between two notes and, in Western music, these are usually measured in numbers of tones or semitones. In this way from C up to D or down to B is a 2nd, another step from C up to E or down to A makes a 3rd, and so on.

Major and minor scales are known as diatonic scales and are built on a pattern of seven notes within an octave span. All major scales are built on the following pattern

The semitones come between the 3rd and 4th degrees of the scale and the 7th and octave. Here is the scale of C major. It uses only the white notes on the keyboard.

The major scale can be reproduced at any pitch i.e. it can start on any note. The following shows the names of all the intervals found between the tonic (key-note) of a major key and the other degrees of the scale. This serves as a useful standard by which other intervals can be worked out.

The character of different intervals is often referred to in terms of consonance and dissonance. Consonant intervals feel relatively stable and do not need to resolve to another interval. Major and minor 3rds and 6ths, perfect 5ths and octaves are all consonant intervals. Two voices producing the same pitch are said to be in unison. Dissonant intervals feel somewhat unstable, as though one of the notes needs to move up or down to resolve into a consonance. The major 7th, for example, feels as though it needs to resolve upwards, and the minor 7th feels as though it needs to resolve downwards. Major and minor 2nds, perfect 4ths, major and minor 7ths and all augmented and diminished intervals are dissonant. Consonance is often loosely defined as being pleasing to the ear, dissonance being the antithesis, unstable and needing resolution. These definitions, however, should be treated with caution: the pleasing/displeasing notion depends on aesthetic preferences and implies a psychoacoustic judgment, whereas the notion of resolution of tension depends upon a familiarity with Western tonal harmony.

## **The Pythagorean scale**

One of the earliest recorded musical scales was the Pythagorean scale. Pythagoras, the Greek mathematician and philosopher lived in the second half of the 6th century BCE. Although the system of construction of this scale existed long before his time, the term Pythagorean scale came about because of his theoretical justification in mathematical terms. Apparently, in passing a blacksmith's forge, Pythagoras heard different musical intervals in the striking of hammers against the anvils. Hammers of different weights struck simultaneously produced different consonant and dissonant intervals, so, for example, a hammer weighing half as much as another, a ratio of 2:1, produces a note an octave higher. Pythagoras went on to deduce from this that there was a relationship between consonant sounds and simple ratios and investigated further by carrying out a series of experiments with other bodies; water-filled glasses, strings, bells and pipes. The results pointed to unchanging relationships between the dimensions of the instruments used and the notes they produced. The ability to express these relationships numerically made it possible to analyse scales.

Using a monochord, a single-string instrument said to have been his invention, Pythagoras discovered that when a string is stopped half way the shorter string vibrates with twice the frequency and sounds an octave higher. Several intervals can be identified in the ways that they correspond to simple ratios of sound wavelengths or frequencies. So, for example, an octave is identified by the ratio 2:1 because the frequency of the upper note is twice that of the lower note, the interval of a 5th can be produced by the ratio 3:2 and a 4th by the ratio 4:3 From this the Pythagorean scale can

be constructed by taking a note and producing others related to it through simple whole number ratios.94

In order to find the ratios of further intervals, Pythagoras multiplied two ratios together. As we have seen, an octave uses the ratio 2:1 and a 5th uses the ratio 3:2. Going up an octave and then down a 5th produces a 4th. We know that the 4th uses the ratio 4:3 and it follows that this can be calculated by 1/2 x 3/2 = 3/4. Using this principle, other degrees of the scale can be determined (see Table 1). So for example, if *W* = 9/8, then 1/1 x *W* = 9/8, 9/8 x *W* = 81/64 and so on. Table 1 shows the ratios used in the Pythagorean scale.

**Table 1 – Ratios of each note to the lowest note used in the Pythagorean scale**

Table 1 gives the ratio of each note to the lowest note, the fundamental. In order to calculate the ratio between the notes, that is the intervals between them, we take the ratio of each note to the one preceding it. The result can be seen in Table 2.

94 This rests on the theory of numerical ratios presented in books 7–9 of Euclid's *Elements*.

There is an elegant simplicity in the way that this use of simple ratios leads to the construction of the diatonic scale. This simplicity accords with the ideals of the philosophy promulgated by Pythagoras and many of his contemporaries which sought to explain the nature of the universe through numbers, ratios and proportions. Music theory too was mathematically based dealing with ratios, proportions and number relations. However, there are several flaws within this Pythagorean method of calculation: not all of the intervals of the scale can be produced accurately by strict Pythagorean methods; they do not satisfy the properties of the circle of 5ths (see Figure 1); and the intervals do not always match the acoustical properties found in the harmonic series.

Taking each of these problems in turn, Table 1 shows us that all five steps of a tone are in the ratio of *W* = 9:8 and the two semitones are in the ratio of *H* = 256:243. However, two half steps in this Pythagorean tuning do not add up to one whole step.95

*H*2 = 256/243 x 256/243 = 2`16/310 ≠ 32 /23 = *W*

95 Gareth Roberts. *From Music to Mathematics {Exploring the connections}.* (Baltimore: John Hopkins University Press, 2016): 121.

This discrepancy is referred to as the 'Pythagorean comma', defined as the gap between one whole step and two half steps. It can be found by dividing the ratio for a whole step by the ratio for two half steps.

*W/H*2 = 312/219 ≈ 1.013643265

This means that raising the pitch by two half steps produces a slightly smaller interval than raising it by one whole step.

The next problem with the Pythagorean scale is related to the circle of 5ths. The circle of 5ths is represented by a circular diagram demonstrating the relationship between different keys (see Figure 1).

## **Circle of 5ths**

It shows a series of chords whose roots are each a 5th higher than the previous chord e.g. C-G-D-A (see Figure 1). From any starting note the pitch is raised repeatedly until the starting point is returned to and the circle closes. This is the equivalent to raising the pitch by seven octaves. However, using the specified Pythagorean ratios, the span of seven octaves is not equivalent to twelve 5ths and there is a need to introduce a new note for each octave to fill the gap. This means that rather than returning to the starting point and

closing the circle, the figure becomes an ever-increasing spiral (see Figure 2). The shaded area denotes the Pythagorean comma.

## **Spiral of 5ths**

## **The harmonic series**

Another problem lies in the correlation of the Pythagorean scale with the harmonic series. When a vibrating object, such as a string, is set in motion, it vibrates both as a whole, with a frequency called the fundamental (the lowest note or the first harmonic) and, with lesser intensity, in other sections as well. The harmonics (or overtones) generated may be represented in an ordered series called the harmonic series, a set of frequencies which are successive integer multiples of the fundamental (see Figure 3).

## **Figure 3 – The harmonic series**

The frequencies are related by simple whole number ratios. In general, the *n*th harmonic of a series has a frequency which is *n* times the fundamental frequency. The harmonic series defines many of our intervals. As we have seen, the calculations used to make the Pythagorean scale were very limited and they missed out some of the most important ratios found in the harmonic series such as the major 3rd (5:4) and the minor 3rd (6:5). In this way the Pythagorean scale was out of sync with the laws of acoustics.96

Nevertheless, use of the Pythagorean scale persisted for hundreds of years. Another system of tuning known as equal temperament avoids the Pythagorean comma, but it was many years before it was in common use. The evolution of polyphonic music from the twelfth century onwards led to experimentation with alternative systems of tunings. Polyphony used several instruments or voices and music became more complicated harmonically; music was no longer largely restricted to the predominant use of three harmonic intervals - the octave, 4th and 5th – and 3rds and 6ths were gradually adopted.

## **Just intonation**

Just intonation is based upon the first six notes of the harmonic series, the word ' just' used because this alignment is considered to be more acoustically pure or true. The ratios used are based on smaller numbers than those of the Pythagorean scale (see Table 3). The just major 3rd uses a ratio of 5:4, rather than 81:64, which is more consonant than its Pythagorean counterpart. Similarly the just major 6th is 5:3 (rather than 27:16) and the major 7th is 15:8 (rather than 243:128).

96 Eli Maor. *Music by the Numbers*. (Princeton: Princeton University Press, 2018): 19.

#### **Table 3 – Ratios used in just intonation**

The idea of using the ratio of 5:4 was first suggested by two medieval British theorists, Theinred of Dover and Walter Odington and in ensuing decades there was much debate about different tunings by leading scientists and mathematicians across Europe, both the German mathematician Johannes Kepler (1571-1630) and the English mathematician Isaac Newton (1642- 1727) devised scales. Kepler devised a scale derived from the ratios of planetary orbit minimum and maximum speeds and Newton devised a seven note diatonic scale based on the seven colour spectrum of the rainbow.97 The French polymath Marin Mersenne (1588-1648) is credited with being the first to formulate rules governing vibrating strings, and the first to discern the nature of harmonics related to a fundamental note. In the search for a more consonant scale, the eighteenth century Swiss mathematician, Leonhard Euler, developed an elaborate mathematical theory which was based upon just-intonation ratios.

Although just intonation conforms more closely to the laws of acoustics than the Pythagorean scale, it is not without problems; the circle of 5ths is still not closed and there are two different ratios for the whole step within the scale. The ratio for the step between the first and 2nd degrees (a tone) is 9:8 whereas the step of a tone between the 2nd and 3rd degrees has a

97 Neil Bibby. *Music and Mathematics. From Pythagoras to Fractals*. (Oxford: Oxford University Press, 2003).

ratio of 10:9. The gap between these two different steps (9/8 x 10/9) is the ratio 81/80 which is known as the 'syntonic comma'.98 Although this is not discernible to all listeners and does not pose a problem for instruments that have a continuous range of notes such as violins, for instruments with fixed keys or holes it makes playing in more than one key often difficult and sometimes impossible. Keyboard instruments were particularly problematic in this respect. The celebrated German/Viennese piano maker Johann Jakob Könnicke (1756-1811) attempted to solve this problem when in 1796 he invented the 'Harmonie-Hammerflügel', a keyboard instrument with six diatonic manuals that divided the normal 12-key octave into 31 notes to enable a purer tuning. Unfortunately it was difficult to build, as well as difficult to keep in tune and to perform on.99

## **Equal temperament**

The answer to this problem was to be found in equal temperament tuning. The word tempered refers to the fact that just intonation has been adjusted or compromised. It is so named because the scale is divided into 12 equal semitones. In this system, the circle of 5ths is closed. This means that there are no discrepancies between different keys, all the intervals across keys are equivalent, making it possible to play music with multiple key changes and chromatic harmony.

Rather than using the whole number ratios found in the Pythagorean scale and just intonation, the Flemish mathematician Simon Stevin (1548-1620) came up with the idea of letting the half step (semitone) equal 12√2, the 12th root of 2 which can also be written as 22/12. In other words, when multiplied by itself 12 times, 22/12= 2. This is an irrational number in contrast with the rational numbers used in the aforementioned systems. The Pythagoreans were strong believers in the importance of rational numbers, those numbers which are either integers or can be written as a ratio (or quotient) of two integers. An irrational number is a real number which cannot be expressed as

98 Roberts, *From Music to Mathematics*, 129.

99 Michael Latcham. 'Könnicke, Johann Jakob' in Grove Music online.

an integer or as a quotient of two integers. Irrational numbers have infinite, non-repeating decimals.

Having established 22/12 as the interval for a half step, Stevin went on to define all the other intervals by multipliers of this step (see Table 4). The numbers in the 3rd column are easily found by calculating the number of half steps in each interval.

#### **Table 4 – Ratios used in Stevin's equal temperament**

Marin Mersenne (1588-1648) made an important contribution to the theory of tuning and was an advocate of equal temperament. He also reassessed the nature of consonance and dissonance. The most famous early work to use the system and exploit all 24 keys is J S Bach's *Well-Tempered Clavier* – the two books were written in 1722 and 1738-1744 each comprising a prelude and fugue in each major and minor key. It took a long time for equal temperament to be adopted across Western music; as late as 1851, not one of the British organs at the Great Exhibition was equally tempered.100 Equal temperament is now widely thought of as the normal tuning of the Western 12-note chromatic scale. Table 5 compares the frequencies of notes in a scale of A major according to the ratios of the Pythagorean scale and the equal tempered scale measured in Hertz.

100 Bibby, Music and Mathematics, 27.

Frequency expresses the number of repetitions during a certain length of time. It is measured in time units, usually seconds, and called Hertz after the German physicist Heinrich Hertz (1857–1894). A tuning fork that vibrates 440 times back and forth in one second, concert A, has a frequency at 440 Hz.101 The set of frequencies 100, 200, 300, 400, 500 Hz … is a harmonic series whose fundamental is 100 Hz and whose 5thharmonic is 500 Hz.

#### **Table 5 – A comparison of frequencies used in equal temperament and the Pythagorean scale**

Table 6 shows the approximate differences in frequencies for a whole step (tone) and half step (semitone) used in the equal tempered scale and the Pythagorean scale.

**Table 6 – A comparison of the frequencies for a tone and a semitone used in equal temperament and the Pythagorean scale**

Another commonly used way of measuring the intervals in different tuning systems is through the use of cents. Cents are based on a logarithmic scale and there are 100 cents in an octave. The system was introduced in the late nineteenth century by the English mathematician Alexander Ellis (1804- 90). In his analysis of the scales used in various European musical traditions, he showed that the diversity of different systems could not be explained by a single physical law.

101 Concert A is the standard tuning note in the UK and USA. Until the nineteenth century musical pitch was not standardized and the levels varied widely across Europe. Nowadays ensembles which specialise in music of the Baroque period have agreed on a standard of A = 415 Hz.

Table 7 gives a comparison of the three tuning systems discussed so far measured in cents. It can be seen that although equal temperament is a close approximation to the perfect 5th is significantly sharper than a major 3rd in equal temperament.

**Table 7 – A comparison of the frequencies used in the Pythagorean scale, equal temperament and just intonation measured in cents**

Moving away from the seven-note diatonic major scale with its use of tones and semitones, the next chapter goes on to look at a variety of other scales using a different number of notes and often including the interval of a microtone.

#### 112 MATHS & MUSIC

# 7 **More scales: whole tone, octatonic, pentatonic, and scales that use microtones**

this chapter opens by describing modes of limited transposition, such as the whole tone scale and the octatonic scale, giving examples of their use in Western classical and pop music. It then moves on to the ubiquitous pentatonic scale, the five-note scale which is found in different forms across the world. The next section looks at some of the scales formed by dividing the octave into more than 12 semitones, made possible through the use of microtones, intervals smaller than a semitone. It discusses the long history of experimentation with equal temperament systems, notably the 19-note, 31-note and 53-note scales, and describes some of the instruments devised to accommodate them. The chapter concludes by looking at two twentieth-century composers who have used microtonal material in their music; the Czech composer Alois Hába was one of the first and went on to establish a department of microtonal music in Prague. Sometime itinerant hobo, the American Harry Partch, was similarly devoted to the cause of microtonal music, he rejected Western scales and invented his own, one of which was made up of 43 notes to the octave.

## **Modes of limited transposition**

Olivier Messiaen was the first to use the term modes of limited transposition to denote scales which can be transposed a limited number of times before the original set of pitches reappears. The symmetry inherent in these modes means that no note can be perceived as the tonic they lend themselves to static harmony or what Messiaen described as containing "the charm of impossibilities".102

102 Messiaen, Olivier. *Technique of My Musical Language* (Paris: Alphonse Leduc, 1944): 58.

The two most common examples of modes of limited transposition are the whole tone scale and the octatonic scale (both of which were already in existence before Messiaen coined the term).

## *Whole tone scale*

The whole tone scale is made up of six whole tones starting on either C or Db (its only transposition).

Messiaen refers to this as his first mode of limited transposition. Another French composer, Claude Debussy (1862-1918), uses the whole tone scale in 'La soirée dans Grenade' from his piano suite *Estampes*. In this languid movement, Debussy evokes images of Spain through strumming guitar sounds and a *habanera* rhythm. From bar 23 onwards he uses notes from the whole tone scale below to evoke the exotic atmosphere.

### 10.5920/mathsandmusic.fulltext 10.5920/mathsandmusic.fulltext

This corresponds with the second whole tone scale shown above, but using enharmonic equivalents for Db (C#) and Eb (D#).103

The whole tone scale is not often found in pop music but Stevie Wonder uses the scale in the Intro to 'You are the sunshine of my life' and it is found relatively frequently in jazz, by performers including John Coltrane, Thelonius Monk and Art Tatum.

## *Octatonic scale*

The octatonic scale is made up of eight notes alternating tones and semitones. Each octatonic scale has two versions or modes; one starting with a whole step (tone) and the other starting with a half step (semitone). So, for example, starting on C the two modes are

The octatonic scale can start on any note, but in all, because the same series of notes appears with different starting notes, it is limited to three different patterns. As well as the two above, the remaining pattern is as follows.

Messiaen refers to this as his second mode of limited transposition. Instances of the octatonic scale are generally fairly rare although, as Allen Forte notes

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

in his article about Debussy's use of the octatonic scale, '…it should be emphasised that '… octatonicism was "in the air" during the early twentieth century'. Along with works by Debussy (including the opera *Pelléas et Mélisande*), Forte goes on to cite Strauss's opera *Elektra* (1908), the second of Webern's *Four Pieces for Violin and Piano* (1910), and Ravel's song 'Surgi de la croupe et du bond' (1913) from the *Trois poèmes de Mallarmé*. 104 Scriabin's *Sixth Sonata, Op. 62* for piano (1911) could be added to this list as could its use in Stravinsky's ballets *Petrushka* and *Rite of Spring* and several pieces in Bartók's *Mikrokosmos* including 'From the island of Bali'. A rare example to be found in pop music is the intro to the Radiohead song 'Just'.

## **Pentatonic scales**

All the scales discussed so far are predominantly found in Western classical music and most have seven notes. In contrast, the most widespread scales, both geographically and historically, are pentatonic scales formed of five pitches. They can be found in traditional music across the world from the British Isles, to China, Japan, Lapland, Hungary, West Africa and in the music of the indigenous people of the Americas. There are also examples in Western classical music through the centuries, from Gregorian chant to the piano music of Debussy and they can be heard in some blues, gospel and rock music. Three well-known examples of songs using pentatonic scales are 'Amazing grace', 'Nobody knows the trouble I've seen' and 'Auld Lang Syne'. Pentatonic scales are useful when improvising because they work well over several chords and have a major/minor ambiguity. They are not uncommon in pop music, 'Shape of you' by Ed Sheeran, is an example as are several songs by The Smiths including 'Heaven knows I'm miserable now' and 'There is a light that never goes out'.

Pentatonic scales can be categorized as either hemitonic (with semitones) or anhemitonic (without semitones). The formation varies from country to country, and region to region. Two of the most well-known forms are found below. The first example uses only the white notes of the piano, whereas the second uses only black notes.

104 Allen Forte. 'Debussy and the Octatonic'. *Music Analysis*, Vol. 10, No. 1/2 (1991): 159.

MORE SCALES 117

Note that each of these contains the circle of 5ths. In the first example, for instance, C-G-D-A-E

Slendro, an anhemitonic pentatonic scale, is one of the two major tunings in Javanese gamelan music, the other is pelog, a seven note scale . The five notes of slendro approximate those shown below.

At first sight the intervals employed in slendro seem to equate to the pentatonic scale on the black keys, a mixture of whole tones and minor thirds which are easily distinguished by ear. However, the five notes of slendro are more or less equally spaced, the resultant intervals lie somewhere between a whole tone and a minor third although this is not standardized and differs from one gamelan to another – the intervallic structure is known as *embat* and this marks out the personality of the gamelan.105

In 1889 Debussy attended the Exposition Universelle in Paris where he was particularly impressed by a Javanese gamelan with its ensemble of tuned percussion, gongs and metallophones of different sizes. In his evocative piece 'Pagodes', from the piano suite *Estampes*, he attempted to capture some of the Javanese gamelan sound partly through his use of the pentatonic scale which he uses to emulate the slendro scale. Here is the melody line in bars 3 - 4.

105 Neil Sorrell. *A Guide to the Gamelan*. (London: Faber and Faber, 1990): 56.

118 MATHS & MUSIC

A preponderance of consonant intervals can be formed from the pentatonic scale. The nineteenth-century physician Hermann von Helmholtz (1821- 1894) explained the relative consonance and dissonance of musical intervals in terms of the extent of the beating between the two notes when they are heard simultaneously. He discovered that notes whose fundamental frequencies are related by small whole number ratios have reduced beating because of the coincidences between the frequencies of the harmonics involved. Thus a perfect 5th, with a frequency ratio of 3:2, is a smooth/ consonant interval with low beating because the 3rd harmonic of the lower note coincides with the 2nd harmonic of the upper note. In his study of the relative number of consonant and dissonant intervals formed by pairing notes in different scales, David Huron discovered that the pentatonic scale exhibited an optimum number of consonant intervals. He argued that the

frequency of occurrence of various interval given pitch-class set can be correlated with corresponding perceived consonance for each interval class. If one of the composer's aesthetic goals is to generate predominantly consonant music, an appropriate choice of palette would maximize the availability of consonant harmonic intervals while minimizing the presence of dissonant harmonic interval'.106

It is difficult to give reasons for the ubiquity of the pentatonic scale, but it may be connected with the fact that it can be generated from the circle of 5ths and what has been described as optimum consonance.107 Another reason may be its harmonic adaptability when improvising. However, the extent of the influence these features account for in terms of ubiquity is merely speculative.

106 David Huron. 'Interval-Class Content in Equally Tempered Pitch-Class Sets: Common Scales Exhibit Optimum Tonal Consonance' Source. *Music Perception: An Interdisciplinary Journal,* Spring, 1994, Vol. 11: 289-305.

107 Huron, 'Interval-Class Content', 301.

## **Scales that divide the octave into more than 12 notes**

Most of the scales described so far divide the octave up into 12 semitones. The next section looks at some of the scales formed by dividing the octave into more than 12 divisions. This is made possible by the use of microtones, intervals smaller than a semitone. The term quarter tone is sometimes used, but this is not always accurate given that the interval may be larger or smaller than a quarter tone.108 Since antiquity mathematicians and musicians have experimented with such equal temperament systems, notably the 19-note, 31 note and 53-note scales. The interval tunings that arise from these divisions are often closer to those found in the harmonic series (just intonation) rather than in 12 note equal temperament.109

## **19-note, 31-note and 53-note systems**

Interest in the 19–note and 31-note systems dates from the sixteenth century when such systems were explored by various mathematicians. Könnicke's 'Harmonie-Hammerflügel' use of six diatonic manuals to divide the normal 12-key octave into 31 notes has already been mentioned (see page 108).110 This was not the first instrument to experiment with this system. In 1637 Mersenne described a keyboard instrument utilising the 31-note scale in his treatise *Harmonie Universelle*, an outline of his theoretical and practical ideas on music. The instrument functions by having alternative keys for some notes, the tunings for these alternatives differing by the syntonic comma.111 The Dutch mathematician and astronomer Christiaan Huygens (1629- 1695) took a different approach; he used logarithmic measurements of intervals to ascertain the equal division of the octave into 31 parts. His

108 Microtones are commonly used in contemporary near Eastern musical traditions such as those found in some Turkish, Arabic, Persian and Greek music.

109 In their study of microtones and projective planes, Wilson and Gamer compare the number of cents found in various intervals across scales using just ratios, 12 notes, 19 notes, 31 and 53 notes. See Robin Wilson and Carlton Gamer 'Microtones and projective planes' in John Fauvel, Raymond Flood, and Robin Wilson. *Music and Mathematics*. (Oxford: Oxford University Press, 2003): 148.

110 Michael Latcham. 'Könnicke, Johann Jakob' in *Grove Music*.

111 Neil Bibby. *Music and Mathematics. From Pythagoras to Fractals*. (Oxford: Oxford University Press, 2003): 22-23.

ideas are summarised in his *Lettre touchant le cycle harmonique* (1691) which includes a description of an imaginary transposing harpsichord, with 31 strings to the octave and a shifting keyboard of 12 notes which used various selections from the 31-note set.

Interest in the 53 note system can be traced back to antiquity. It originated from the fact that a series of 53 perfect 5ths equates very closely to 31 octaves. The difference between the two was calculated as 353/284 by Nicholas Mercator (1620-1687) hence its name, the Mercator comma. Although conventional staff notation can be adapted for the 19-note and 31 note systems, this proves difficult with 53 notes, as does performance.

## **Indian classical music and the 22-note system**

Indian music uses just intonation and the use of equal temperament is unknown. In Hindustani (North Indian) classical music, an octave is called *saptak* and has seven notes called *swara*. These can be further divided into 22 notes known as *sruti* or *shruti*. The seven notes are sa, re, ga, ma, pa, dha, ni (similar to the Western do re mi fa so la ti). The notes of the Indian scale are as follows.

*Shadja* (Sa), *Rishabha* (Re), *Gandhara* (Ga), *Madhyama* (Ma), *Panchama* (Pa), *Dhaivata* (Dha), *Nishada* (Ni).

The 22 note system was developed from early experiments in ancient and medieval times based on the fact that the highly concordant notes *panchama* and *madhyama* have ratios of 3:2 and 4:3 respectively. Further research was undertaken to determine the other ratios.

#### **Table 1 – Ratios used in the Indian scale**

The research examined both the cycle of 4ths and the circle of 5ths (see pages 104-5). It was found that the 12th note of the cycle in one case and the 11th and 12th notes in the other case were higher or lower by a small interval, again this gap was found to be in the ratio 81/80, equivalent to the 'syntonic comma' but known as the *praman* or *pramana sruti* in Indian music theory. Using *srutis*, precisely tuned microtones, the scale can be further divided into 22 notes.112 The first and fifth notes (sa and pa) have only one variant, whereas all the other degrees of the scale (re, ga, ma, dha, and ni) have two variants each. There are ten pairs of notes and these along with the sa and pa give the 22 *srutis* of the Indian musical scale, these are mainly used in ornamentation. The two notes constituting each pair are found to be uniformly separated by the 81/80 interval of a comma or *pramana sruti.* The notes re, ga, dha, and ni have natural and flat variants, while ma has a natural and a sharp variant.

## **Microtonal material used in twentieth century classical music**

During the twentieth century a number of Western classical composers incorporated microtonal material into some of their music, notable examples

112 P. Sambamurthy. *South Indian Music Book V. (Chennai: The Indian Music Publishing House, 1963).*

including Charles Ives 'Three Quarter-Tone Pieces' (1925) for two pianos tuned a quarter tone apart, and Karlheinz Stockhausen, whose electronic piece *Studie I (*1953) uses 25 equal-tempered intervals within each span of two and a half octaves but the most notable pioneers of microtonal music were perhaps Alois Hába and Harry Partch.

## **Alois Hába**

The Czech composer Alois Hába (1893-1973) could be credited as the first to use quarter tones and sixth tones in Western art music. His parents were both folk musicians and his mother, a singer, taught him peasant songs from Moravia. These could be characterised by their use of microtones to heighten the expressive effect, the major mode using quarter-tone and sixthtone sharps, the minor by quarter-tone flats. In 1917 he composed his Suite for string orchestra, his first composition to use quarter-tones and went on to write a great many microtonal works from piano pieces to an opera. In 1923 he established a department of microtonal music at the Prague Conservatory. In order to perform microtonal music Hába constructed special instruments: three types of quarter-tone piano, a quarter-tone and a sixth-tone harmonium, and a quarter-tone clarinet, trumpet and guitar. The opera *Matka* ('The Mother', 1929) is arguably Hába's masterpiece.113

## **Harry Partch,** *U. S. Highball (A Musical Account of a Transcontinental Hobo Trip***)**

The American composer, instrument maker and philosopher Harry Partch (1901 – 1974) rejected Western scales and invented his own, one of which was made up of 43 notes to the octave. His work was inspired by ancient Greek music theory and the work of Herman von Helmholtz.114 Partch describes

113 Jiří Vysloužil. 'Hába,Alois' in *Grove Music*.*.*

114 Herman von Helmholz. *On the Sensations of Tone as a Physiological Basis for the Theory of Music*, 1865 (London: Dover, 1954).

how he arrived at this scale and others in his book *Genesis of Music* (1949).115 The scale is based on a 'tonality diamond'; an arrangement of ratios in a diamond shape where the diagonals sloping in one direction form 'Otonalities' and the diagonals in the other direction form 'Utonalities'. Partch's tonality diamond is similar to the diamond devised by Max Friedrich Meyer some 50 years earlier.116 Having devised the 43-note scale, Partch devoted his time to implementing this system, creating new instruments that could be tuned to this new scale and then training performers to play them.

In 1935 at the height of the Great Depression he began a transient existence in the western states, jumping trains, sometimes living in boxcars in train yards, and picking up work occasionally. For several years he kept a small notepad that he used to transcribe fragments of conversations, writings on the sides of boxcars, names of stations and thoughts. These notes became the basis for *U.S. Highball*, a piece for speaker and Partch's instruments. In 1943 he started work on the first draft using the motley assortment of instruments he had built at that point. These included the Adapted Guitar (a guitar with new frets to accommodate his tuning system), the Chromelodeon (a re-tuned foot-pump harmonium), and the Kithara (his own version of the ancient Greek stringed instrument but now with 72 strings). New instruments were added for the second version including a set of marimba-based instruments; a smaller version of the 72 string Kithara (Surrogate Kithara), and a collection of tuned artillery shells and Pyrex water containers. Partch took the comments he had recorded in his notebook and turned them into melodies including quotes from hoboes such as "Hey, don't sleep with your head against the end of the car! You'll get your neck broke when she jerks!" With the small intervals in his new system of tuning applied to both voice and instruments, sliding steam whistle sounds and occasionally rhythmic percussive sounds, *U. S. Highball* has a surreal atmosphere, but it is oddly reminiscent of a train journey.

This chapter and the preceding one have outlined some of the mathematical and scientific thinking behind the creation and analysis of common and less common scales. This field encompasses a vast area of research which has

115 *Harry Partch. Genesis of a Music: an Account of a Creative Work, its Roots and its Fulfillments* (Madison: WI, 1949, enlarged 2/1974).

116 For more detail of the tonality diamond see Chris Forster. *Musical Mathematics. On the Art and Science of Acoustic Instruments*. (San Francisco: Chronicle Books, 2010).

attracted many leading musicians, mathematicians and thinkers over the centuries including J S Bach and Messiaen, Pythagoras, Newton, Kepler, Mersenne, and Hertz, so that inevitably the surface has only been scratched both historically and geographically. In the twenty-first century, technological advances mean that electronic music facilitates any kind of tuning without the need for retuning or the construction of customised instruments.

# 8 **Serialism and the 12-note scale**

perhaps more than any other technique of composing, serialism is rooted in mathematical concepts. It is algorithmic in its use of a strict set of procedures to generate a note row and it uses symmetrical transformations in the construction of different rows. This chapter looks at the procedures of serialism in the music of its originator, the Austrian composer Arnold Schoenberg (1874 – 1951) focussing on his *Variations for Orchestra*. It goes on to examine the different compositional approaches of the other two members of what is known as the Second Viennese School117 - Alban Berg (1885 – 1935) and Anton Webern (1883 – 1945). The next sections look at extensions of serial approaches: rhythmic serialism where patterns of durations are ordered, and total serialism where algorithms are applied to further elements of the music such as dynamics and attack. The chapter ends with a more detailed exploration of both mathematical and musical considerations with a discussion of whether the two can be reconciled.

The twelve-note system of serialism (sometimes known as dodecaphony) follows strict mathematical rules in the form of an algorithm. First of all a note row (or series) is composed - this uses all of the 12 notes of the chromatic scale and is the basis of the whole composition. None of the notes are repeated in the note row (also known as the prime order or prime series). The note row can exist at each of the transpositional levels, as can each of the inversion, retrograde and retrograde inversions. This means that the row

117 The Second Viennese School is an umbrella term for the composers Schoenberg, Berg and Webern. It is sometimes understood to include other pupils of Schoenberg. It is based on the tenuous notion that there was a First Viennese School which might have included Haydn, Mozart, Beethoven and Schubert.

can be used in any of its 48 forms. It can also be transformed through scale symmetry in two ways: augmentation and diminution. The durations of the original notes can be lengthened (augmentation), or shortened (diminution). The original notes can also be used in another octave (octave displacement). There are four main permutations of the row, each of these is rooted in symmetrical transformation. They are often referred to as the prime order, inversion, retrograde and retrograde inversion.

Here are the four permutations of the note row used in the first movement (Introduction and Theme) of Schoenberg's *Variations for Orchestra, Op. 31*  (1928), his first twelve-tone composition for a large ensemble.

**Prime order – P**

**Inversion – I** uses horizontal reflection, each of the intervals is turned upside down

**Retrograde - R** uses vertical reflection where the note row is played backwards

**Retrograde inversion – RI**. The inversion can be played backwards. In other words it undergoes a 1800 rotation.

All the material of the *Variations for Orchestra* is derived from the four versions of the row. Following the 'Introduction', the second section of the *Variations for Orchestra*, 'Theme' (bars 34-57) presents four linear statements of the theme in the order P, RI, R, I. Schoenberg marks this with the word '*Haupstimme*', a term he used to denote the principal part in a complex texture.

In addition to the symmetrical aspects, the notes of the row can also be used vertically to make chords (sometimes known as verticalisation). Schoenberg excluded the use of consonant chords because of their tonal qualities. It can be seen from the examples below that the Prime row and the Retrograde Inversion (in transposition) are both made up from the same two hexachords – Bb E Gb Eb F A and D Db G Ab B C.

## **Prime order – P**

**Retrograde inversion – RI in transposition**

Together these two hexachords form a 12-note aggregate. The series is therefore said to show the property of 'combinatoriality', a technique found in twelve-note compositions whereby a collection of pitch classes (in this case P) can be combined with a transformation of itself (in this case RI) to form an aggregate of all 12 pitch classes. The term 'combinatorics' has been borrowed from mathematics where it refers to the study of the enumeration, combination and permutation of sets of elements and the mathematical characteristics of their properties.

## **Alban Berg**

Each of the three members of the Second Viennese School took a different approach to serialism. Alban Berg, a pupil of Schoenberg, was quite free in his use of the note row reconciling twelve tone procedures and tonal harmony. His music tends not to keep to a single series, even within a movement. He also incorporated tonal elements into the twelve-tone language, notably in his Violin Concerto (1935) where the original row uses eight ascending thirds followed by a note row and consequently spells out two major and two minor chords

1 + 2 + 3 = G minor, 3 + 4 + 5 = D major, 5 + 6 + 7 = A minor, and 7 + 8 + 9 = E major. 8 – 12 = five notes of a whole tone scale.

The Violin Concerto also uses a Bach chorale taken from his Cantata No. 60 and a Corinthian folk tune, both tonal in nature.

## **Anton Webern**

Webern, who also studied with Schoenberg, took a much stricter approach to the dodecaphonic rules: unlike Berg, he always used a single series for each composition and suppressed all repetition of material. He constantly exploited the possibilities of the note row with much use of concentrated motivic working and symmetrical structures. In his *Concerto Op. 24* (1934) this symmetry can be seen within the row itself where the notes 1 - 6 and similarly 7 – 12 can be reflected along the middle to give a mirror image where the same order of intervals appears forwards as well as backwards (see below).

The order of intervals between the first six notes is as follows

B to Bb (semitone) Bb to D (major 3rd) D to Eb (semitone) Eb to G (major 3rd) G to F# (semitone)

In reverse order (retrograde) the notes F# G Eb D Bb B follow the same pattern - semitone, major 3rd, semitone, major 3rd semitone.

Furthermore if the notes 1- 3 are regarded as a microcosmic Prime version, then 4- 6 is the Retrograde Inversion, 7 – 9 is the Retrograde and 10 – 12 is the Inversion.

Here are the opening bars of Webern's *Concerto Op. 24* which illustrate how Webern has used the row in harmony in his composition. The parts enter in this order – oboe (P), flute (RI), trumpet (R) and then clarinet (I). The piano enters in bar 4 where the overlapping parts follow this order – P (retrograde), RI (retrograde), R (retrograde), I (retrograde). Notice how various forms of the row are unfolded simultaneously, beginning on different beats of the bar and interlocking (see below). This all helps to create a complex rhythmic structure, a characteristic of Webern's compositional style.

In his *Symphony Op. 21* (1928), and all the works that followed, the twelvetone technique was used with unprecedented strictness and the concept was extended to include, not just pitch, but also timbre and rhythm. In Webern's words,

So what has in fact been achieved by this [twelve-note] method of composition?… What territory, what doors have been opened with this secret key? To be very general, it's a matter of creating a means to express the greatest possible unity in music.118

## **Serialism with other pitch-class collections**

Before he arrived at his twelve-tone system, Schoenberg compositions included series containing a number of pitch classes greater or lesser than

10.5920/mathsandmusic.fulltext 10.5920/mathsandmusic.fulltext

118 Letter from Anton Webern to Alban Berg cited in *Die Reihe 2: Anton Webern*. (London: Universal Edition, 1960): 42.

12. In the *Five Piano Pieces Op. 23,* for example, No. 2, uses a series of nine different pitch classes and No. 3 is based on a 14-note series. Similarly, before Stravinsky fully adopted 12-note serialism in *Threni* (1957–8) he used series with fewer than 12 pitch classes; *In memoriam Dylan Thomas* (1954), for example, uses a series of only five pitch classes. There are also examples of series with more than 12 pitch classes which include Berio's *Nones for orchestra* (1954) which is based on a 13-note series made up of two overlapping heptachords, the second being the retrograde inversion of the first.119

## **Rhythmic serialism**

As Paul Griffiths points out 'interpretations of rhythmic serialism are problematic. There is no equivalence class to correspond with the pitch class … there is no analogue for inversion'.120 An early example of the use of ordered patterns of durations can be found in the third movement of Berg's *Lyric Suite* (1926) where he creates a rhythmic series made up of 12 durations, each of one, two or three units. The series is used in exact retrogrades, and is varied by adding rests and 'transposed' by multiplying all the values by the same integer. In 1940 Webern composed his *Variations Op. 30* for orchestra which uses versions of two rhythmic motifs presented in the opening bars and derived from the note row. These are in effect sets of durations.

The American composer Milton Babbitt (1916-2011), a trained mathematician, demonstrated and formalized many aspects of 12-tone compositional technique in several important essays where he introduced algebra and number theory to model 12-note rows and operations. In 'Some Aspects of Twelve-Tone Composition' (1955), 'Twelve-Tone Invariants as Compositional Determinants' (1960), and 'Set Structure as a Compositional Determinant' (1961) he introduced terms derived from mathematics such as 'source set', 'aggregate,' 'secondary set', and 'derived set.' These terms were used in the classification of different types of pitch class set and helped to describe the different procedures used by such sets. He coined the term 'combinatorial' to describe 12-tone sets that yield aggregate and secondary

119 Paul Griffiths in 'Serialism' in *Grove Music*

120 Ibid.

set formations. This nomenclature has since become widely adopted as the basis for theoretical work notably in music set theory.121

In 'Twelve-Tone Rhythmic Structure and the Electronic Medium' (1962) Babbitt demonstrated a variety of methods for interpreting the structures of pitch class sets in the rhythmic domain.122 This included an analogy between the octave (in pitch structure) and the bar (in rhythmic and metrical structure) by dividing the bar into 12 equal units each of which can be articulated by individual points of attack. In this way pitch class sets can be mapped onto 'time-point sets' where a set of integers can be interpreted both as a pitch class set and as a time-point set, defined as the duration which separates the attack from the beginning of the bar. So, for example, 0, 11, 6, 7, 5, 1, 10, 2, 9, 3, 4, 8 can be interpreted as a pitch class set where the numbers are generated by counting up in semitones from the first note (designated as 0) (see below).123

or as a time-point set a particular point of attack is a measure of its position within the bar. In the example below the metrical unit is a demi-semiquaver, a 12th of the whole bar. Here the 12 available points of attack within a bar are ordered according to the numerical set given above. Time-point 0 occurs on the first demi-semiquaver of the bar, time-point 1 on the next, and so on (see below).

121 Set theory in music originated with Milton Babbitt and was then formulated in 1973 by the American music theorist Allan Forte. This led to more rigorous theoretical formulations by John Rahn, Robert Morris and David Lewin who applied mathematical group theory to music.

122 "Twelve-Tone Rhythmic Structure and the Electronic Medium," *PNM*, i/1 (1962), 49–79; repr. in *Perspectives on Contemporary Music Theory*, ed. B. Boretz and E.T. Cone (New York, 1972), 148–79.

123 Elaine Barkin, revised by Martin Brody and Judith Crispin. 'Milton Babbitt' in *Grove Music*

Babbitt used this method in his *Second String Quartet* (1954). The time-point series is manipulated to produce different versions of the rows: inversion by complementation of the time-point numbers to 12; transposition by the addition of a constant value to each time-point number; and retrograde by using the time-point set backwards.124

## **Total serialism**

In the early 1950s, several composers including Boulez, Stockhausen, Maderna and Nono extended the procedures of serialism to the other aspects of the music beyond pitch, such as rhythm, dynamics, tempo, timbre and note attack. This method has become known as total serialism or integral serialism. These composers were much associated with the Darmstadt School, a loose grouping of composers associated with the International Summer Courses for New Music in Darmstadt, West Germany.

An early example of total serialism can be found in Babbitt's *Three Compositions for Piano* (1947). Here points of articulation are determined by a set along with dynamics and pitch set classes. In 1949, Messiaen composed his piano piece *Mode de valeurs et d'intensités* (Mode of Durations and Intensities) which was based on scales, not only of pitch but also of duration, attack and dynamics. Two years later Boulez based his piece *Structures 1a* (1951) entirely on the note row used by Messiaen in *Mode de valeurs et d'intensités*. From this he created two number matrices which represented all 48 versions of the row. An integer was assigned to each pitch class, so for example Eb = 1, D = 2 and so on. From these matrices he derived another system to determine the durations of notes where the integers were read as numbers of demi-semiquavers (sixteenth notes). In addition, each statement of the row was given a particular dynamic

124 Griffiths, 'Serialism', *Grove Music.*

and mode of attack.125 These matrices could be described as a pair of magic squares (see Chapter 14 for more detail).

Another leading figure in the serialist avant garde was Karlheinz Stockhausen (1928-2007), himself a teacher at Darmstadt. In his works from 1953 onwards, serial organization permeates every level of the formal process. His works *Kreuzspiel* (1959) and the *Schlagtrio*, are examples of 'total serialism' where the basic series is permutated and applied to durations, dynamics and articulation as well as pitch. In his *Elektronische Studien* (1954) he applied similar rules to the organization of pitch, rhythm and timbre. His work *Gruppen* (1957) for three orchestras (combining 109 players) and three conductors explores the possibilities of different simultaneous tempos using sets of tempi.

## **Some mathematical considerations**

Serialism draws heavily on mathematical principles as we have seen above: it is algorithmic in its use of a strict set of procedures designed to be applied systematically; it uses symmetrical transformations in the construction of different rows (vertical and horizontal reflection, translation and 1800 rotation) and it uses combinatorics in its formulation of harmonies.

Babbitt's theoretical writings about 12-note music eventually led to the formulation and development of music set theory which has some parallels in mathematical set theory and group theory. In music a set is a group of pitch classes, usually a 12-note set, it may also be a set of other musical elements such as durations or dynamics. Music set theory assigns integers to different pitch classes, categorises musical objects and analyses their relationships through operations such as transpositions and inversions.

Because the series can use transpositions of the 12 notes of the chromatic scale, as can each of the inversion, retrograde and retrograde inversions, the original series can exist in 48 different forms. However it is not always the case that a note row will generate 48 different forms. There are some rows which will generate the same row under certain operations, particularly if

125 Jonathan Cross. 'Composing with numbers' in John Fauvel, Raymond Flood and Robin Wilson. *Music and Mathematics.* (Oxford: Oxford University Press, 2003): 137 and Paul Griffiths in 'Serialism' in *Grove Music*.

the original row demonstrates some kind of symmetry. For example, if we were to take a descending chromatic scale beginning on C as the note row then the Prime order would be as follows.

The Inversion would be

and the Retrograde Inversion would be

which is, of course, the same as the original row. In other words it is invariant. In mathematics, an invariant is a property of a mathematical object which remains unchanged after operations or transformations are applied to the objects.

To find the total possible number of note rows, we use the mathematical symbol *n*! which is referred to as *n* factorial. It is the product of all positive integers less than or equal to *n*.

$$n! = n \ge (n - 1) \ge (n - 2) \ge (n - 3) \ge 3 \ge 2 \ge 1$$

There are 12! possible permutations of the row which is calculated thus

12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 479,001,600

Another, more tenuous, connection to mathematics has been made to Einstein's 1915 Theory of General Relativity. In his book *Music by the Numbers*, Eli Maor argues that a comparison can be made between Schoenberg's serialism and Einstein's theory of relativity where all systems of reference are equal to each other, the democracy of the twelve notes and their lack of hierarchical relationships meant that 'Henceforth only the position of each note relative to its immediate predecessor would matter; you might call it relativistic music'. Maor quotes the words of the composer Pierre Boulez 'With it [the twelve-tone system], music moved out of the world of Newton and into the world of Einstein'126 Maor's argument, however, is lacking in detail. Although it is true to say that at the beginning of the twentieth century, Schoenberg's new system created a profound revolution in music just as Einstein's Theory of Relativity shook the foundations of science, there is little or no evidence to suggest that Schoenberg's twelve-tone system was influenced by the work of Einstein. When speaking about any potential link, Schoenberg pronounced that 'There may be a relationship in the two fields of endeavour but I have no idea what it is. I write music as music without any reference other than to express anything but music'.127 Einstein in turn was baffled by Schoenberg's music which he described as 'crazy'.128

## **Some musical considerations**

When Schoenberg invented his twelve-note system he abandoned the tonal system which had been used in music for the previous 300 years. In comparison with the hierarchies of the tonal system, serialism took a democratic approach where all 12 notes of the chromatic scale were equal and no note was bound to the home key. Tonality emphasised the importance of the tonic and dominant and every note had a specific relationship to the tonic. Put simply, in the words of the composer Christopher Fox, 'The overriding characteristic of the series is that it is non-hierarchical – everything is as

126 Eli Maor. *Music by the Numbers. From Pythagoras to Schoenberg*. (Princeton: Princeton University Press, 2018): 125.

127 https://www.schoenberg.at/index.php/de/1933-r-p-arnold-schoenberg-receives-theamerican-press

128 Andrew May. *The Science of Sci-Fi music*. (London: Springer, 2020): 88.

important as everything else …'129 The algorithmic nature of serialism with its set of mathematical rules led some to see the method as the compositional equivalent of painting by numbers. As Jonathan Cross puts it 'Accusations of lack of artistry, lack of creative imagination, and even lack of musicality have been hurled by critics and music-lovers alike at very many twentieth-century composers not least at the Viennese composer, Arnold Schoenberg'.130

The new system of serialism raised several questions, not least whether it could be perceived within the music by the listener or whether instead it was incoherent with a limited capacity to communicate the composer's thoughts. The musicologist Fred Lehrdahl maintains that 'if the serial construction is so important – and yet it is not an explicit part of the listeners' experience, the listeners' experience must be impoverished'.131 In his book on aesthetics, *Music, the Arts, and Ideas,* the philosopher Leonard B Meyer argues that listeners are conditioned to the perception of tonal music making it more difficult to perceive serial pieces, a situation compounded by the fact that composers often used quite different serial procedures in different pieces of music, and that a satisfactory perception of serialism in one work would not necessarily help the listener in others.132 In tonal music the hierarchical relationships between pitches in the diatonic scale help to establish and define the listener's expectations. The counter arguments from serial composers usually adopt the position that serial procedures are not intended to be perceived by the listener or that although they are not perceived consciously, the music nevertheless gives an effect of coherence which the listener cannot explain.133

Many works using serial techniques were premiered at the Darmstadt International Summer Courses for New Music and were subject to much debate.17 Such discussions tended to foreground the technical analysis of the music rather than the expressive purposes to which it was applied. Perhaps the final word should go to Christopher Fox who, in his essay 'Darmstadt

131 Fred Lehrdahl 'Cognitive constraints on compositional systems'. *Contemporary Music Review*,

129 Christopher Fox. 'Music after Zero Hour' *Contemporary Music Review*, Vol. 26, No. 1, 2007: 16.

130 Cross, 'Composing with numbers', 131.

Vol. 26, No. 1, 1992: 97-121, cited in Christopher Fox. 'Darmstadt and the Institutionalisation of Modernism' *Contemporary Music Review*, Vol. 26, No. 1, 2007: 120.

132 L.B. Meyer: *Music, the Arts, and Ideas* (Chicago and London, 1967)

133 Griffiths, 'Serialism', *Grove Music.*

and the Institutionalisation of Modernism', argues that 'generally the aesthetic purpose of Darmstadt serialism is not that the series should occupy the musical foreground; rather, it is that the serial principle is the method whereby musical transformations can be achieved.'134

10.5920/mathsandmusic.f ulltext 10.5920/mathsandmusic.fulltext

134 Christopher Fox, 'Darmstadt and the Institutionalisation of Modernism', *Contemporary Music Review*, Vol. 26, 2007: 120.

## **Forms and Structure**

#### 140 MATHS & MUSIC

# 9 **Canons and fugues**

this chapter focuses on two musical forms, the canon and the fugue, where geometric transformations and imitation are central to their composition. The word canon is used to refer to both a musical form and a compositional technique where an initial melody is imitated at a specified time interval by one or more parts. A section of a piece is described as canonic if the imitation prevails throughout the section, if the imitation is exact and continues through the whole piece, it is called a canon. As a strict form canon makes use of different types of geometric transformation: reflection, translation, scale and, less frequently, rotation. Canons fall into several discrete categories which are outlined in this chapter. These are distinguished in different ways: by the number of voices used; imitation at different distances; imitation at different upper or lower intervals; and by the geometric transformations used. The fugue, a more complicated form, follows a detailed set of rules; the composition of fugues has often been regarded as one of the most highly-skilled compositional techniques.

One of the earliest examples of the canon as a musical form is 'Sumer is Icumen In' which dates from the mid-thirteenth century. Four voices sing the same melody one after the other, accompanied by two lower voices (the *pedes* meaning 'foot') which themselves alternate and overlap. A culmination of 14th-century canonic technique was reached in the works of Guillaime de Machaut (c1300-1377), his rondeau 'Ma fin est mon commencement' is the earliest known piece based on retrograde procedures (see pages 63- 4). The canon had its heyday during the late fifteenth and early sixteenth centuries with the works of composers such as Guillaume Dufay, Johannes Ockeghem and Josquin des Prez. During this period, mastery of canonic technique was equated with proficiency in composition; practical treatises

#### 141

of the 16th century often included compendia of canonic devices. Johannes Ockeghem (c1410-1497) almost certainly invented the prolation canon where each melodic line uses the same music but played at different speeds and is one of the most difficult types of canon to execute.135

Canon as a learned form of counterpoint lost much of its popularity during the Enlightenment: it was seen more as an outdated academic exercise which placed ingenuity ahead of musical value and freedom of expression. J S Bach (1685-1750), however, used canon throughout his career notably in the *Musical Offering,* the *Goldberg Variations*, the canonic variations of *Vom Himmel hoch* and the *Art of Fugue.* In these works the form hit a new peak and, chiefly through Bach's influence, the canon once again became an important musical device.

The use of canonic principles, although less popular, carried on and examples can be found in the works of Haydn, Mozart, Beethoven, Brahms and Mahler.136 The twentieth century saw a renewed interest in canonic principles: its use of geometric transformations was central to both serialism and minimalism. There are many examples of canons and canonic writing in the music of Schoenberg, Stravinsky and Anton Webern.

## **Rounds and catches**

A straightforward form of canon is the 'catch' or 'round' where each of the voices enters one after the other with the same melody creating a polyphonic texture.137 Because all the voices sing the same tune, rounds are relatively easy to sing; well-known examples are the children's songs 'Row, row, row your boat' and 'Frère Jacques'. A single-line melody is constructed so that a round forms its own harmony when sung at the unison, consequently they are frequently constructed around only one or two chords to make the compositional process easier. During the seventeenth century, rounds,

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

136 A well-known example of a canon can be found in the third movement of Gustav Mahler's *Symphony No. 1* (1888) which is based on a minor version of 'Frère Jacques'

137 In some ways the terms 'round' and 'catch' are interchangeable but 'catch' is often used to refer to comic, sometimes bawdy, English rounds written between roughly 1580 and 1800.

often drinking songs with bawdy words, were largely regarded as light entertainment being sung in clubs and taverns – usually by men, frequently on the topics of drinking and sex, and often with lewd words. Henry Purcell (1659-1695) contributed many such bawdy songs such as 'Once, twice thrice'. In this context, they were usually referred to as catches.

## **The main types of canon**

Canon in unison is one in which the imitating part(s) enters at the same pitch as the leading part. This is the most common form and is often found in folk music around the world, a well-known example being 'Shalom Chaverim'. Canon at the octave is one in which the imitating part(s) enter an octave apart from the leading part. Canon at the fifth is one in which the imitating part(s) enter a fifth apart from the leading part. Entries at any other interval are similarly explained.

The following two examples are taken from the first volume of *Tabulatura nova* (1624) by Samuel Scheidt (1587-1684). In both, the two canonic parts form contrapuntal lines upon a *cantus firmus* taken from the Magnificat.138 The first is an example of canon in unison where the imitating part follows the leader a quarter of a bar later at the same pitch (see below).

138 *Cantus firmus* is a Latin term referring to a pre-existing melody used as the basis of a new polyphonic composition. It is found in Medieval and Renaissance music.

The second example is of canon at the fifth where the imitating part follows the leader a quarter of a bar later a fifth lower (see below).

The thirty harpsichord variations of the *Goldberg Variations* by J S Bach (1685-1750) are grouped in threes where each trio of variations culminates in a canon. The canons themselves progress sequentially according to intervals. The first canon is at the unison, the second canon at the second, the third canon at the third and so on through to variations 25–27, which culminate in a canon at the ninth.

In **canon by inversion** (sometimes referred to as *Al rovescio*) the imitating part plays the melody in inversion: an upward interval becomes a downward one and vice versa. This use of inversion can be seen in the opening bars of Canon No. 2 in *Fünf Canons, Op. 16* (1924) by Anton Webern (1883- 1945). The five pieces are composed as strict canons and the texts themselves could be regarded as 'canonical', given that they are taken from Latin liturgical texts. The work is scored for soprano, clarinet and bass clarinet. Canon No. 2, *Dormi Jesu*, is for soprano and clarinet. Notice how the inverted intervals are precise. The clarinet opens with a leap of an augmented 5th (nine semitones) from Bb up to F# whereas the soprano opens a bar later with a descending minor 6th (nine semitones), E down to G#. The next two clarinet notes rise up a minor 3rd, A to C, whereas the singer follows with a falling minor 3rd F down to D.

**Double canon** uses two melodies which are given out at the same time by two parts and are imitated by two other parts: in other words, two two-part canons proceed simultaneously. The Trio of Wolfgang Amadeus Mozart's *Serenade No. 12 in C minor* (1728) is a double canon using two melodies in inversion for two oboes and two bassoons (see below). It is marked 'Al rovescio'; Oboe II leads with Oboe I entering two bars later in inversion. Bassoon 1 leads the second canon with Bassoon II entering two bars later also in inversion.

*Symphony Op. 21* (1928) by Anton Webern makes much use of mirrors and palindromes through a series of double canons.

**Retrograde canon** is a type in which the imitating part presents the melody backwards. It is also known as canon cancrizans or crab canon (even though crabs move sideways rather than backwards). Unlike most other canons, the entries are not staggered. Instead both parts start at the same time. Crab canons are relatively uncommon. One of the most famous crab canons is included in J S Bach's *Musical Offering* (1747). This is a collection of pieces based on what is known as the Royal Theme, so called because it was presented to Bach by Frederick the Great, himself a composer and flautist, as a subject for an on-the-spot improvisation. 139 Here is the Royal Theme.

The *Musical Offering* includes 10 canons of different types. These were notated as puzzle canons in the original printed edition, and the full parts were not transcribed. A puzzle or riddle canon is one in which the canon is notated in an enigmatic way where the composer gives only the leading voice along with cryptic instructions for its resolution. It is then up to the performer to solve the intellectual exercise in order to discover the correct time and interval distance for the voice(s).140

Three of the canons in the *Musical Offering* have no title or instructions as to when the imitations should be played by the follower. Instead they have the Latin inscription *Quaerendo invenietis* (Seek and ye shall find) with clues provided within the music. So, for example, in No. 1, Canon *a 2 cancrizans*, the title tells us that the canon is for two voices and it should be played backwards. Furthermore at the end of the main melodic line, the clef and key signature were originally written facing backwards. The Royal Theme is presented in the first five bars of the upper part (see below). The lower part is an exact retrograde of the upper part and the upper part is an exact retrograde

139 Webern used this royal theme in his 1935 orchestral arrangement: *Ricercare Fuga (Ricercata) a 6 voci* from J S Bach: *Musical Offering*).

140 Two of Bach's former students, Johann Philipp Kirnberger and Johann Friedrich Agricola are credited with solving these in the *Musical Offering* as cited in http://www.early-music.com/js-bachmusical-offering/ and Gareth E Roberts. *From Music to Mathematics.*(Baltimore: John Hopkins University Press, 2016): 178.

of the lower part. The point of symmetry, a vertical reflection, is between bars 9 and 10. This canon as well as displaying symmetry has another interesting mathematical aspect: it can be visualised on a Möbius strip. A Möbius strip is a 2-dimensional strip where one continuous side is formed by joining the ends of a rectangle after twisting one end through 180°. The canon should be written as one long line of music on a strip of paper and then cut in half at the end of bar 9. It should then be glued together. It could then be played by two instruments travelling in opposite directions around the strip. The twist is the point at which the two parts interchange.

**Mirror canon** usually refers to a canon in which the combined principles of inversion and retrograde are used in the imitating part. In other words, an inverted entry moves in contrary motion – in the opposite direction. Sometimes it is used to refer to canons in inversion. Canon 3 (a 2 *per Motum contrarium*) in J S Bach's *Musical Offering* is a mirror canon at the fifth. Here the Royal Theme is used as an accompaniment and is not treated canonically. It can be found in the upper voice, but using diminution where the original note values are halved (and consequently played twice as fast). The title tells us that the follower sings the leader's theme in contrary motion. It is also played backwards (see below).

**Canon by augmentation** has the imitating parts in longer notes than the leading part. The notes are usually twice as long.141The following example is the taken from the opening bars of the first movement of the Brahms motet *Schaffe in mir, Gott, ein rein Herz, Op. 29 No. 2*, a setting of Psalm 51. Here the bass II part imitates the soprano part, this time at the same pitch but in augmentation. The note values are twice as long. This means that the bass part takes twice as long, consequently the soprano line is repeated. The example below shows the entire canon at the octave in augmentation between the Soprano and Bass II. To make this clearer the other voices have been omitted.

141 Canon by augmentation is sometimes known as a sloth canon.

**Canon by diminution** has the imitating parts in shorter notes than the one that they are imitating. The notes are usually half as long. In his *Miscellanea Musica*, C P E Bach (1714-1788) composed several canons, partly for teaching purposes. These include 'Diminution canon for two voices in C major'. Examples of canon by diminution are rare. This is because the voice with the shorter note values soon overtakes the 'slower' one with its longer note values. C P E Bach avoided this in his 'Diminution canon' by beginning the piece with straightforward imitation at the lower fifth and then from bar 5 imitating the leader (two octaves lower) but with half the note values effectively at twice the speed.

It should be noted that in the above examples the transformational devices used are often unrecognisable to the listener unless they have analysed the score first. Although the main melodic themes can act as a unifying factor, holding the music together, the transformations are often easier to see than to hear.

## **Prolation canons**

A prolation canon combines two or more identical musical lines at different rates; each voice sings or plays the same music, but at different speeds. Voices may either enter successively or simultaneously. The concept is simple but the compositional technique is difficult, hence prolation canons are quite rare. The earliest master of the prolation canon is Johannes Ockeghem. Each movement of his *Missa Prolationum*, dating from the second half of the fifteenth century, uses a different variant of this form where there are different gaps between the entries and different relative speeds of the voices. 'Kyrie Eleison I' is a double prolation canon, where two separate pairs of voices, soprano/alto and tenor/bass, take up a different motif. The ratio of tempos between the soprano and the alto is 3:2. Each of the first six note values of the soprano part is multiplied by 3/2 in the alto part. After the opening passage shown below, the music continues as a regular canon.

Other examples of prolation canons include J S Bach's *Canon a 4 per Augmentationem et Diminutionem,* the final movement of *14 Canons, BWV 1087,* which is built on a repeating bass line taken from his *Goldberg Variations.* The Estonian composer Arvo Pärt (b 1935) uses a prolation canon in his vocal work *Cantus in Memory of Benjamin Britten (*1977) which was used in the 2004 film *Fahrenheit 9/11*. Here a series of falling A minor scales is heard across five voices with each group entering an octave lower and at half the speed. Another relatively recent example can be heard in the first

movement of Shostakovitch*'s Symphony No. 15.* It has been argued that prolation canons are examples of fractals in music (see pages 199-200).

## **Rhythmic canons**

The final type of canon to be explored is the rhythmic canon, one that is relatively rare although used by both Alban Berg and Olivier Messiaen (see page 28). A rhythmic canon is one which is based around rhythm alone. It is again based on the principle of strict imitation, but here it is an initial rhythm that is imitated at a specified time interval by one or more parts. Canons are found throughout the opera *Lulu* (1929-1935) by the Austrian composer Alban Berg (1885-1935). In Act I, scene 2, the death of the Painter is accompanied by a five-part rhythmic canon played on unpitched percussion - drums, a tam-tam and a gong.

A more specialised form of the rhythmic canon is the **rhythmic tiling canon.** This is a type where the canon is crafted so that once all the players

have begun to play or sing, there will be exactly one player striking each beat. In other words the rhythms yield one and only one onset per pulse. A basic form of the rhythmic tiling canon can be found in the vocal genre, *katak* music, which is sometimes referred to as the Balinese monkey chant (see pages 46-7).

## **Fugues**

Unlike the canon, where one theme continues from the beginning to the end, the fugue is a more complicated form which follows a detailed set of rules: two or more voices use the systematic imitation of a principal theme (the subject) an answer and (sometimes) a countersubject (a second theme that accompanies the subject), in simultaneously sounding melodic lines (counterpoint).142 The answer follows the subject in the second voice. It is a statement of the main subject but in a different key, usually the dominant. Statements of the material incorporate contrapuntal devices that alter the subject in some way. These devices are based on symmetry and include those of scale (augmentation and diminution) and horizontal reflection (inversion).

The term 'fugue' has been in continuous use since the 14th century. At first the word held a variety of meanings with imitation being the defining characteristic. However, from the mid-seventeenth century, a theoretical model began to evolve and in 1725, this model was largely set by the Austrian composer and theorist Joseph Fux (1660-1741) in *Gradus ad Parnassum*, a theoretical treatise on composition whose middle sections focus on species counterpoint and fugue. *Gradus ad Parnassum* served as a textbook for several generations of composers and the rules set out were followed by many, Haydn, Mozart and Beethoven amongst them. Central to Fux's treatise are the four rules for voice-leading which are applied to species counterpoint

142 The fugue opens with the subject, the main theme of the fugue. The answer follows the subject in the second voice. It is a statement of the main subject but in a different key, usually the dominant. Some fugues also have a countersubject, a second theme that accompanies the subject. The word episode is used to delineate a passage where no voice states the subject in its entirety. The music in episodes is usually, but not always, developed from the subject or countersubject. Stretto is an overlapping effect where one voice states the subject and is closely followed by another voice before the first one has finished.

in two, three and four voices before sections on imitation and fugue. The method of instruction is through a dialogue between teacher and pupil and is based on exercises using a given *cantus firmus* for which a counterpoint is to be constructed. Several rules are set up in the first exercise which include:

This strict set of procedures has much in common with a mathematical algorithm where a set of precisely described instructions or routine procedures are designed to be applied systematically through to a conclusion in a number of steps.

*Gradus ad Parnassum* was held in high esteem by J S Bach whose keyboard work *The Well-Tempered Clavier,* a set of preludes and fugues in all 24 major and minor keys was completed in 1742. This collection of fugues is commonly viewed as the zenith of the form (see page 77). By the 1750s the fugue began to be seen as outdated and it no longer enjoyed a central role in contemporary compositions. However, the study of fugue and counterpoint was still an important part of a composer's training and fugal imitation was often found in compositions. Haydn wrote fugues as finales to several of his works, as did Mozart, and Beethoven who adopted a free approach to fugues, notably in the finale to the *Hammerklavier Sona*ta Op. 106 (see pages 68-9) and the string quartet the *Grosse Fuge Op. 133*. Tonality was embedded in the strict rules of the structure of fugues therefore twentieth-century developments in atonality did not lend themselves to the form. Between the 1920s and 1940s however, Neo-classicism revived some of the elements of eighteenthcentury musical precepts translated into a modern idiom.143 Later on both Shostakovitch and Hindemith composed collections of fugues inspired by J S Bach's *The Well-Tempered Clavier*: Shostakovich's *24 Preludes and Fugues* (1950–51) and *Ludus Tonalis* (1943) by Paul Hindemith (1895-1963). *Ludus Tonalis,* subtitled 'Studies in Counterpoint, Tonal Organization and Piano

143 Examples of fugues written in this style include the second movement of Stravinsky's *Symphony of Psalms* (1930) and Bartók's *Music for Strings, Percussion and Celesta* (1936).

Playing', includes twelve three-part fugues.144

It should be noted that the formulaic/symmetrical rules of the fugue are sometimes modified to fit the irregular/asymmetrical nature of the major/ minor scale. So, for example, in diatonic music, although strictly speaking the answer in a fugue is a statement of the main subject in the dominant (a real answer), the intervals are sometimes adjusted based on the key signature. The melodic contour remains the same, but the transposition is not exact (a tonal answer). So for example in J S Bach's Fugue No. 16 in G minor (from Book I of the *Well-Tempered Clavier*), the subject opens with a rising semitone from D to Eb. If the answer were an exact transposition of the subject, then it would open with A to Bb, however, it opens with a rising minor 3rd – G to Bb, a tonal answer.

As time went on artistic licence increasingly overrode the formulaic approach to the point where in the fugues of Hindemith and Shostakovitch, twentiethcentury contemporary musical idioms, (harmonic and rhythm techniques, for instance) are integrated into fugal procedures in ways that are impossible to reduce to formulas.

## **Fugues, computers and Artificial Intelligence**

Composers of fugues followed a set of steps and procedures and as well as adhering to the rules of counterpoint, they also had to understand how to handle the entity of thematic, contrapuntal, and harmonic elements so that even the most complex counterpoint had a firm harmonic basis. In

144 Hindemith, living in the United States, was inspired to compose *Ludus Tonalis* following a then much derided broadcast of Shostakovitch's *Leningrad Symphony*. Hindemith condemned the trend toward "despicable rubbish" in orchestral music and hoped to "remind those who have not succumbed what music and composition really are"cited in Alex Ross. *The Rest is Noise.* (London: Fourth Estate, 2008): 299.

conceiving of the original subject, they needed to bear in mind the answer and the countersubject to ensure that they were a satisfactory harmonic and contrapuntal combination; few could disagree that the design of a good fugue subject is a significant part of the process of writing a fugue. It could be argued that the multi-faceted reasoning employed is similar to that used in solving a mathematical problem. As Joel E Cohen writes of the kinship between mathematics and music 'The processes of creation are similar; that is, in both the most beautiful combination is selected from an infinitude of possibilities and its logical potentials are developed consistently'. He goes on to write 'The end products of mathematics and music depend for their meaning upon the successive relationships of their elements and upon the order imposed by the creator'.145 To this could be added that, ideally, the end product should be convincing and musical. Given that the set of procedures which need to be followed in order to compose a fugue could be described as an algorithm, perhaps a computer programme could be designed to fulfil this function. As the computer scientist Andrés Garay Acevedo writes of Fux's *Gradus ad Parnassum*, 'such a well-studied process starts to suit the idea of computer assisted composition.'146 This fact was recognised by the American electronic music composer Laurie Spiegel (b. 1945). In 1981 she delivered a conference paper entitled 'Manipulation of musical patterns' where she argued that a 'working knowledge of all the processes of transformation which can be aesthetically applied' to musical patterns should be important considerations in computer music system design adding that there also needs to be a 'practised awareness of how such materials and operations … relate to and influence each other's' potentials'. She goes on to list relevant materials and operations which include, transposition, retrograde, inversion and scaling.147 In 1986 Spiegel went on to design her algorithmic composition software *Music Mouse* which she labelled as an Intelligent Instrument with reference to the programme's built-in knowledge of chord and scale

145 Joel E Cohen. 'Some Relationships between Music and Mathematics'. *Music Educators Journal*, Vol. 48, No. 1, 1961: 108-109.

146 Andrés Garay Acevedo. 'Fugue Composition with Counterpoint Melody Generation Using Genetic Algorithms'. *Computer Music Modeling and Retrieval*, Volume 3310, 2005.

147 Laurie Spiegel. 'Manipulation of musical patterns'. In *Proceedings of the Symposium o Small comuters and the Arts:* 19-22. Philadelphia, PA: IEEE, 1981.

conventions, and stylistic constraints. She composed several works using *Music Mouse* including *Three Sonic Spaces* (1991).

One of the first to attempt to create computer-generated fugues was the American musicologist and composer, David Cope, who published *The Well Programmed Clavier* a set of keyboard pieces which he had generated through a computer programme 'Experiments in Musical Intelligence' devised to produce new works in the styles of various composers. The works are discussed in his books *Experiments in Musical Intelligence* (1996) and *The Algorithmic Composer* (2000).148

In 2001, an experiment was conducted at Cornell University in New York. The aim of the artificial intelligence project 'Fugue Generation with Genetic Algorithms' was to establish whether the project team could create fugues 'from scratch'.149 The subject was generated first by the programme and then the answer and countersubjects were generated, following 'certain rules and guidelines'. The first part was 'accomplished through the use of a genetic algorithm' using 'random subjects which were slowly improved 'into something decent'. A 'fitness function' was used to evaluate the suitability of the different features of the subject. This looked at, for example, similarity of pitch where each note was discouraged from being 'different from its predecessor', similarity of note length where notes of similar length were encouraged to avoid 'switching around arbitrarily', and key, where notes were awarded for 'being in the key of C' which is followed by the rather naïve explanation that 'This goes a long way toward making the subject sound better. It removes the majority of sharps and flats from the subject, making it much nicer to listen to.' This set of criteria betrays a lack of musical knowledge in its assumptions. Testing the final product on a team of listeners, the research team came to the conclusion that although the system was 'able to successfully generate subjects, and from there create the beginning of a fugue' it was unable to generate complete fugues, due to the complexity involved. This is not surprising given the naïve assumptions of the fitness function. They found that 'while many of the pieces generated by our system have been rated highly by our test listeners, they are still not competition for a human composer'.

148 David Cope. *Experiments in Musical Intelligence (1996)* and *The Algorithmic Composer (2000)*  published by A R Editions, MIT Press.

149 Eric Milkie and Joel Chestnutt, S 473 Artificial Intelligence Project, Fugue Generation with Genetic Algorithms, Cornell University.

In his paper 'Fugue Composition with Counterpoint Melody Generation Using Genetic Algorithms' (2005), Andrés Garay Acevedo presented the results of the implementation and evaluation of a genetic algorithm which was used to help in the task of automatic counterpoint generation. A fugue subject was used as an input for the system and then a genetic algorithm was used in order to find a suitable counterpoint melody that could be arranged in a basic fugue structure with an answer and two counter-subject melodies arranged over time for three voices. As such, the process was more sophisticated than that used by Cornell University; Acevedo recognised that 'if an algorithm is to be used, then it must incorporate higher-level concepts of musical arrangement and coherence' in order to have 'some sort of aesthetic value.' The algorithm was tested with two different input melodies, and evaluated for fitness: the results were deemed to be satisfactory.

In 2014, a further piece of research was undertaken by Yu Yue Yue Yang and Andrew Horner which sought to improve on David Cope's results and those of Cornell University. 'Automated Fugue Generation' worked on the premise that 'A fugue is a musical composition of multiple voices built on a subject (recurring theme) that imitates itself frequently throughout the piece. The objective was to design and build a system that generated three-voice fugues, emulating the style of J. S. Bach. The subject was to be two bars long and then the position of the new entries was determined bearing in mind the voice, bar, key and 'tonal centre progression'. The characteristics of Bach's fugues were taken into account and the results were evaluated in terms of the frequency of repetition, the dominance of stepwise motion, the range, the link between segments and the fitness of the chord and the pitch structure. The evaluation found that their results 'made musical sense in general, occasionally with unnatural transition and were significantly better than the Cornell results but 'pale' compared to David Cope's pieces. They concluded that they had built a system that could generate polyphonic music similar to fugues, but that it had limitations and the results sounded more like pop music than authentic Baroque music.

It would appear that although the composition of fugues is often referred to as one of the most mathematical and formulaic procedures, it cannot be reduced to these two features; understanding the mathematical structure does not tell us anything about the effect on the audience. Although the algorithm is a partner in the creative process it could be argued that it is no more than that.

#### 158 MATHS & MUSIC

# 10 **Proportion: Golden Section and the Fibonacci series**

golden section and the associated Fibonacci series give rise to satisfying natural proportions. This chapter explores their use in music, looking in particular at the music of Mozart, Satie, Bartók, Debussy, and composers associated with the Darmstadt School, questioning whether such ratios are used consciously by composers or otherwise. The final section outlines the theory of proportional parallelism in the music of J S Bach. It asserts that the composer consciously revised his works so that total number of bars was a multiple of 10, 100 and sometimes 1000 resulting in perfect proportions such as, 1:1 or 1:2.

The term Golden Section (GS) refers to the unequal division of a line into two parts such that the ratio of the smaller part to the larger is the same as that of the larger to the original whole. This ratio is approximately 1:1618. It was first documented by the Ancient Greek mathematician Euclid in the *Elements* (c 300 BC) where it was found that this 'division in extreme and mean ratio' appeared frequently in geometry.150

150 The *Elements* is a mathematical treatise, one of the most influential works in the history of mathematics, most notably geometry and for over 2000 years has had a huge influence on scientific thinking. The work is attributed to the ancient Greek mathematician Euclid and is a compilation of mathematical research from the previous two centuries including Pythagoras, Archytas and Eudoxus.

## **Golden Section**

At this time the concept was known simply as the 'section' but during the period of the Renaissance it was taken up by artists, sculptors and architects as a 'divine proportion'; a link was made between the golden ratio and abstract ideas, such as aesthetic beauty and perfection. The educator and composer John Paynter wrote

'The asymmetrical relationship of one-third to two-thirds in measurements of duration or space has immense significance in the history of the world's poetry, literature, visual arts, architecture and music… It is found the world over, in music of every kind, old and new. It is a proportion that is especially satisfying when we observe it in nature - in the shapes and patterning of seashells and fir cones, for example and, perhaps for this reason, it has been consciously emulated in painting and sculpture.… It is discernible in numerous ancient manmade structures and in the `magic' shapes of the pentagon and five-pointed star….'151

Mathematically GS is expressed in the Fibonacci series, a summation series in which each number is the sum of the two which precede it.152 The ratio between the successive terms (0, 1, 1, 2, 3, 5, 8 13, 21, 34, 55 and so on) is an expression of Euclid's golden ratio.

151 John Paynter 'Music in the school curriculum: why bother?' *British Journal of Music Education*. 2002 19:3, 215-226.

152 The Fibonacci series is named after Lionardo Fibonacci the thirteenth- century Italian mathematician who first presented the sequence in his manuscript *Liber abbaci* (1202).

As can be seen, the ratios gradually approach the value known as the golden ratio –

1:<PHONE_NUMBER>749894. This is designated by the Greek letter *phi* – Φ, an irrational number.153 Since the nineteenth century, the number

Φ = 1/2( 1 + √5) = 1.6180339887 ….

has been called the golden ratio, golden section or golden number. A golden rectangle is a rectangle with sides in this ratio.

The golden ratio can also be applied to durations. As Paynter wrote '… the highly satisfying proportion is also perceptible in defined periods of time, such as musical works, which frequently take off in new and unexpected directions around 0.6 or 0.7 of the overall duration'. It is true to say that the proportions of one third to two thirds are frequently found in music, not least in sonata form (see below). In pop and rock verse and chorus songs, for example, this is the approximate point where the middle eight, a contrasting section, is most frequently to be found.154 A clear example of GS in pop music can be found in Lady Gaga's 2016 song 'Perfect Illusion' where there is a dramatic key change at exactly this point; the song is 179 seconds long and the key change happens at 111 seconds (179 x 0.618 = 110.622). Paynter argued that GS can also be found in the 12-bar blues but warns that 'It would be unwise to make too much of this, but it is interesting to note how often a composer's intuitive sense of proportion in time structures coincides with the Golden Section divisions'.155 Many classical pieces feature a dramatic arc somewhere after the midway of the piece, so inevitably this climax will happen close to GS proportions according to the number of bars or the time which has elapsed. Although GS can be viewed and measured in, for example, the façade of a building, Paynter points out that this is more difficult with music; we have to hear the music through. As he wrote, 'Of all mankind's attempts to model perfection, music is, perhaps,

153 An irrational number is a real number which cannot be expressed as an integer or as a quotient of two integers. Irrational numbers have infinite, non-repeating decimals.

154 The middle eight (sometimes referred to as the bridge) is a contrasting section (not necessarily eight bars long) where new material is introduced with, for example, a different arrangement of instruments, and/or different chords.

155 John Paynter. *Sound & Structure* (Cambridge: Cambridge University Press, 1992): 215.

the most subtle. Its meaning is manifest not in objects viewed or touched but in events that can only be experienced in the time it takes to make each one audible'.156 Consequently it can be difficult to gain any accurate purchase on structural plans in music through listening alone and the use of GS can be audibly imperceptible.

In Mozart's time, sonata form was conceived in three parts: the exposition in which the musical themes are introduced; the development and recapitulation sections where the themes are developed and then revisited. In 1995 John Putz set out to determine whether Mozart had composed the movements of his piano sonatas according to GS. In order to establish this he examined the lengths of two sections: the exposition and the combined length of the development and recapitulation across 16 of Mozart's piano sonatas. Putz found that Mozart was interested in mathematics but that although he 'may have known of the golden section and used it' there is 'considerable deviation from it [which] suggests otherwise'. He came to the conclusion that 'Perhaps the golden section does, indeed, represent the most pleasing proportion, and perhaps Mozart, through his consummate sense of form, gravitated to it as the, perfect balance between extremes.157

In some ways this is a chicken and egg situation: Putz points to the argument put forward by the music critic and academic Eduard Hanslick in his 1854 treatise on aesthetics, *The Beautiful in Music (Vom Musikalisch-Schönen)*

… we should be wrong were we to construe … that man framed his musical system according to calculations purposely made, the system having arisen through the unconscious application of pre-existent conceptions of quantity and proportion, through subtle processes of measuring and counting; but the laws by which the latter are governed were demonstrated only subsequently by science.158

Musical analysis by means of GS proportions has aroused much

156 John Paynter, Bernarr Rainbow, *Music Education in Crisis*, Woodbridge, Suffolk: Boydell Press, 2013): 50.

157 John F. Putz. 'The Golden Section and the Piano Sonatas of Mozart'. October 1995 *Mathematics Magazine*

158 Eduard Hanslick, *The Beautiful in Music, translated by Gustav Cohen* (London: Novello, 1891): 151.

controversy. In her study of GS in the music of Erik Satie (1866- 1925), Courtney Adams argued that because the concept itself involves approximation, 'lack of precision provides easy fuel for argument' and consequently 'GS analysis has engendered some scepticism.' As Adams wrote, in compositional analysis it is important to minimize the extent of acceptable deviation from the exact GS ratio because 'GS proportions are perilously close to one-third and two-thirds, fundamental divisions in music.' A useful margin of error is 1 per cent or less. 'Typically, analysts find 2 per cent a reasonable margin, although occasionally a writer will allow a deviation as high as 10 per cent, which I find wholly unacceptable.'159

In order to establish the degree of deviation across Mozart's piano sonatas, a similar table was constructed based on Putz's own table but focusing only on first movements (see Table 1).160 Here the first column identifies the piece by the Köchel cataloguing system, 'a' represents the number of bars in the exposition,161 and 'b' represents the combined length of the development and recapitulation (without including any codas). The column 'c' shows the number of bars that would have been used if the movement had adhered strictly to the proportions of GS and the final column shows the percentage difference from the actual number of bars in the Mozart sonata.

159 Courtney S. Adams. 'Erik Satie and Golden Section Analysis'. *Music & Letters* , May, 1996,

Vol. 77, No. 2: 242-252, Oxford University Press: 243-244.

160 The first movement of K. 331 in A has been omitted because it is not in sonata form, rather it is a set of variations.

161 Although the expositions of Mozart's piano sonatas are normally repeated in performance, Putz gives the number of bars of the exposition without repeats.

#### **Table 1 Golden Section proportions in the first movement of Mozart's piano sonatas**

In terms of adherence to GS proportions, five of the piano sonata movements stand out – K. 333 is exact according to GS proportions, and K. 279, K. 283, K. 309, K. 330 and K. 545 all have less than 1% difference. As was mentioned earlier, Adams considered a difference smaller than 2% to be a reasonable margin. This would mean that K. 282 would also fit into this category giving a total of 6/17 movements - (35%). On the other hand some movements, K. 311, K. 570, and K. 576, for example, show what Putz describes as 'considerable deviation' leading us to the same conclusion as Putz, that is that Mozart lent towards the 'pleasing proportion' of GS and gravitated towards it intuitively.

Before we move onto the survey of in the music of Erik Satie, it is revealing to examine a piece of music which follows GS proportions exactly and was written only 34 years after Mozart completed his final piano sonata. In her

10.5920/mathsandmusic.fulltext 10.5920/mathsandmusic.fulltext

analysis of Schubert's lied 'Du bist die ruh' (You are my calm) (1823), the music theorist Pozzi Escot discovered that, discounting the first seven bars of solo piano introduction, the song falls into two distinct parts. It has a total of 75 bars (82 - 7), Part I (bars 8- 53) is 46 bars long and Part II (bars 54-82) is 29 bars long. This means that Part II starts at exactly at the point of Golden Section.162 Furthermore, at this point the music is marked by a dramatic increase in tension.

75 x 0.618 = 46.35

'Du bist die ruh' is a setting of a love poem by Friedrich Rückert. The first part tells of calm repose, the opening bars (below) have a feeling of stillness reflected in the simple harmony and quiet dynamic. The vocal line opens with the following four bars repeated.

When the point of GS is reached there is a dramatic change as the mood is transformed from one of calm repose to deep yearning. There is a tremendous tonal shift as the harmony modulates from Eb to Cb and we hear the first crescendo as the vocal line rises to the highest note in the song. This is followed by a dramatic silence.

162 Pozzi Escot. *The Poetics of Simple Mathematics in Music*. (Cambridge: Publication Contact International, 1999).

**Erik Satie and Golden Section**

Returning to the GS analyses of Adams, in her survey of this mathematical device in the music of Erik Satie, Adams found four multi-movement sets and six individual movements using GS proportions 'that are exact or within a deviation of one per cent.163 The works belong to both the early (1887-92) and late (1914-19) periods of Satie's output. She set out to find an explanation as to these two clusters, aiming to establish whether the appearance of GS was the result of 'conscious use, coincidence or, as some would argue, instinctive application.' In summary, she found that although it was possible that Satie had learnt about the principle from Debussy in late 1891 and had experimented with it, there was no convincing answer with regard to the early pieces, and that the use of GS in these was largely either instinctive or coincidental. This was reinforced by the 'absence of any calculations in the sketchbooks over this long period' However, she found that in the later period 'there is ample evidence of his close association with a number of people who were demonstrably aware of GS' and that the GS proportion appears regularly in Satie's music between 1914 and 1919, including the

163 The sets are the *Trois Gymnopedies, Les Trois Valses, Avant-dernieres pensees* and the first three *Nocturnes.* The individual movements are *Le Fils I, Sonneries I* and *III* and *Trois melodies.*