Focus on...connecting mathematics and music
Many connections between mathematics and music are interesting to explore. Some have led mathematicians to very significant insights.
It is possible to see similarities in what mathematicians and musicians do. For example, both mathematicians and composers of music reason, discover, create and communicate. Mathematicians do and create mathematics – and ‘bodies’ of knowledge known as branches of mathematics exist at any particular time. Musicians create and play music – and ‘bodies’ of music, such as classical music or rock music, exist at any particular time.
And there is a distinction, shared in both mathematics and music, between the practice of ‘exercises’ and the real activity or creation. As Marcus du Sautoy wrote in The Music of the Primes in Issue 28 of Plus:
“It is one of the failings of our mathematical education that few even realise that there is such wonderful mathematical music out there for them to experience beyond schoolroom arithmetic. In school we spend our time learning the scales and time signatures of this music, without knowing what joys await us if we can master these technical exercises. Very few would have the patience to learn the piano if they were denied the pleasure of hearing Rachmaninov.”
But, as Professor du Sautoy continued:
“It isn't just aesthetic similarities that are shared by mathematics and music. Riemann discovered that the physics of music was the key to unlocking the secrets of the primes. He discovered a mysterious harmonic structure that would explain how Gauss’s prime number dice actually landed when Nature chose the primes.”
The significance of the Riemann Hypothesis, the role played in its formulation by Riemann’s discovery of a link between a mathematical graph and a musical note, and how mathematicians have tried to verify or falsify the hypothesis, are entertainingly sketched for students in the whole article and in Marcus du Sautoy’s book The Music of the Primes.
Students may enjoy exploring for themselves some of the simpler mathematical ideas and relationships that underlie, or are revealed when they investigate, music and its history.
Vibrating air creates sound. An initial vibration creates air-waves – or sound waves – just as a stone thrown into a still pond creates ripples.
The pitch of a note is how high or low it sounds. It is determined by the frequency of the vibration that is producing the sound and is measured in hertz (hz) – a unit giving the number of vibrations, or wave cycles passing a point, per second. The lower the frequency of the sound waves the lower is the pitch of the sound. Listen to sounds with different frequencies.
This can be simply demonstrated. Hold a ruler on a table so that it juts out beyond the edge of the table, and twang the ruler. Then if you make it jut out further and twang it again you get a lower note. The longer the part of the ruler is that’s vibrating, the slower is the vibration, and therefore the lower the note.
Students might enjoy Hearing Subtraction – an interactive sound activity about frequency and beats.
The ancient Greeks in the time of Pythagoras, who took for granted a link between mathematics and music, discovered that musical notes are related to simple ratios of integers. They found that if two strings with lengths in the ratio 1:2 are plucked, the notes sound the same but at different pitches – one is higher than the other. This difference in pitch is what we call an octave.
The note that we call middle C, which can be found more or less in the middle of a piano keyboard, has a frequency of around 256 hz – anything producing sound waves at this frequency will make a sound like that note. In the western scale of musical notes, C, D, E, F, G, A, B, (you can play those seven notes here), notes with a frequency ratio of 1:2 are given the same name.
Piano Keyboard by Jens Egholm
Doubling the frequency creates a note an octave higher. Reversely, halving the frequency creates a note an octave lower.
In this diagram middle C is coloured blue:
Piano Keyboard by Cyndaquazy
The note A that is nine white keys below middle C has a frequency of 440 hz. The 12th root of 2 (1.0594630943593...) is the ratio of the frequencies between half tones. So, the frequency of A# is 440 × 1.059... = 466.16376... hz, the frequency of B is 466.1637 × 1.0594 = 493.8833 hz, and so on. If you do this 12 times you end up with the A that is one octave higher and which has a frequency of 880 hz. By multiplying 12 times by the 12th root of two you have multiplied by (21/12)12. If you then multiply 880 by 21/12 another 12 times you obtain the frequency of the note of the yellow key in the diagram.
The pattern of a sound wave can be displayed using an oscilloscope.
Oscilloscope, photograph by Ilja
If a tuning fork
Tuning Fork, photograph by John Walker
is struck, it produces a pure note, which on an oscilloscope traces a sine curve.
Sine Curve by Geek3
The tuning fork has two tines, which vibrate when the fork is struck. This vibration can be linked to the formation of a sine curve. As each tine vibrates, the distance of its end-point from the mid-point of the distance between the two tines in their starting position can be plotted against time. At first, the tine moves away from the mid-point until the displacement is at a maximum. And then, as the tines get closer together, the displacement decreases, until it reaches a minimum when the tines are closest together. This cycle is repeated many times per second, the exact number depending on the frequency of the tuning fork. A fork tuned to middle C and marked 256 hertz will vibrate 256 times per second, giving a continuous sequence of displacements and producing a sine curve.
If the same note is played on a violin the curve will not fit a sine curve – it will be ‘messy’. This is because when a violin string is stroked, secondary notes called harmonics are produced.
The piano keyboard pattern reveals one of those surprising connections that pervade mathematics.
Piano keyboard by Lanttuloora
The black notes show the pattern 2, 3, 2, 3, 2, …
In any section of the keyboard showing an octave there are five white keys and eight black keys, making 13 keys altogether:
Piano keyboard by Lanttuloora
Where have you seen the sequence 2, 3, 5, 8, 13, … before? It’s the Fibonacci sequence!
Fibonacci Rabbits by MichaelFrey & Sundance Raphael
When you play the notes on all 13 keys you play a chromatic scale.
Chromatic scale by Hyacinth
But if you start with C, and play only the white notes you play the scale of C major.
Diatonic scale by Benjamin D. Esham
If you do the same thing but play the second black key instead of the third white key, Eb (E-flat) instead of E, you play a minor scale.
Some instruments are tuned to scales with numbers of notes other than 13. For example many folk songs use the pentatonic scale with only five notes. Interested students can investigate patterns in, and relationships between, the intervals between notes in different scales.
At Phil Tulga’s Music through the curriculum site, students can play and hear intervals between notes on Virtual Panpipes, or on a Virtual Water Bottle Xylophone, on a Virtual Glockenspiel, or on Musical Fraction Tubes.
This JavaSound applet is a really great resource for students to use to explore the sounds of particular notes.
At the Exploratorium students can play listening memory games on the theme of the quality of sound.
Musical rhythm helps students explore and develop some mathematical ideas. For example, beating half time, quarter time, and eighth time, can help students ‘feel’ fractions…
Solfege subdivision by Christophe Dang Ngoc Chan
…before exploring more complex patterns and their changes.
They can use objects, such as Cuisenaire rods, to represent specific beats.
The students lay out the rods to make a pattern – say, one red and two browns – and this pattern is repeated several times. Then they ‘read’ the pattern by clapping, or beating an instrument. They can be encouraged to create their own version of musical notation to record their patterns.
This is a simple, but very nice, interactive rhythm builder for students to explore.
A teacher describes how she uses rhythm to engage 10, 11 and 12-year old students in mathematical exploration.
By systematically allocating musical notes to represent numbers or digits students can listen to, rather than look at, number patterns and patterns within numbers – sometimes patterns that are overlooked when looking are discerned when listening!
And aesthetic properties of numbers may be revealed. For example on the website of the Mathematical Association of America (MAA) Ivars Peterson wrote:
“The decimal digits of the mathematical constant pi, 3.14159265. . ., ring out an intricate melody that sounds vaguely medieval. Those of the constant e, 2.718281828. . ., progress at a relentless, suspenseful pace. Euler's prime-number-powered phi function bounces about with a semitropical rhythm. Lorenz's butterfly meanders through a ragged soundscape. Pascal's triangle echoes with an eerie beat.”
Professor Chris K. Caldwell of the department of Mathematics and Statistics at the University of Tennessee at Martin has developed a scheme for listening to sequences of primes, in which he represents one number by the musical note middle C, the next by C-sharp, the one after by D, and so on, for a total of 128 numbers. That correspondence is then used to ‘play’ primes. Because there are infinitely many primes, Caldwell’s strategy is to divide each prime by a particular number, then play just the remainder – a novel application of modular arithmetic! For example, if the divisor, or modulus, is 7, then for the primes 2, 3, 5, 7, 11, 13, 17, 19, 23…, the numbers played are 2, 3, 5, 0, 4, 6, 3, 5, 2…. Students can explore these ideas at Chris Caldwell’s Prime Number Listening Guide.
It’s interesting that people agree that some particular musical chords are consonant (pleasing to hear) while other are dissonant (unpleasant). People agree that chords in which the frequencies of the two notes that are played together are in the ratios 1:2, 2:3, 3:4 and 4:5 all sound pleasing. Students could investigate for themselves the ‘pleasantness’ of chords created by notes with frequencies in various different ratios.
The jangly opening chord of The Beatles’ hit A Hard Day's Night is one of the most recognisable chords in pop music.
A mathematician in Canada tried to find out how The Beatles produced that sound back in 1964, without using the synthesisers and studio electronics of today. Stanford University professor Keith Devlin said, “Sounds themselves are very mathematical things. And that was the key to unravelling this particular mystery.”
The Beatles, photograph from the
United States Library of Congress
Professor Jason Brown of the department of Mathematics and Statistics at Dalhousie University used sound-wave analysis based on the work of French scientist Joseph Fourier to deconstruct the opening chord. He found that it isn’t created using only guitar and bass, as previously assumed. He concluded that that Beatles producer George Martin also played a five-note chord on the piano!
In this Open University podcast, Alan Graham, who has worked in Mathematics Education at the Open University since 1977, explores and entertainingly illustrates relationships between mathematics and music.
From links in this news item you can reach articles about how researchers at the University of Plymouth are finding that the use of music technology can improve students' achievements in mathematics.
Page header - piano keys, image by julianrod some rights reserved
Ripples by Mila Zinkova some rights reserved
Piano keyboard by Jens Egholm some rights reserved
Piano Keyboard by Cyndaquazy in the public domain
Oscilloscope by Ilja some rights reserved
Tuning Fork by John Walker in the public domain
Sine Curve by Geek3 some rights reserved
Piano keyboard by Lanttuloora some rights reserved
Piano octave by Lanttuloora some rights reserved
Fibonacci Rabbits by MichaelFrey & Sundance Raphael some rights reserved
Chromatic scale by Hyacinth some rights reserved
Diatonic scale by Benjamin D. Esham in the public domain
Piano diagram by Hyacinth some rights reserved
Solfege subdivision by Christophe Dang Ngoc Chan some rights reserved
The Beatles photograph from United States Library of Congress's Prints and Photographs division in the public domain