The Intersection

of Math, Music,and

PhysicsMike Thayer

Summit High School, Summit, NJGood Ideas in Teaching Precalculus And… Conference

Rutgers University, New Brunswick, NJMarch 19, 2010

Introduction Why is this a good topic to discuss? Who is this meant for? What are the big ideas, and how do they fit

into a mathematics classroom? Big dreams! Recommended readings

Mathematics and Music

Music is integral to so many of our students’ lives – why not try to use it in some of our courses?

There are many paths we can follow to tie music, math, and physics together – too many for a short presentation!

The goal today: to provide a few possible directions to explore (and to explore bigger things…?)

Mathematics and Music

These topics have been connected for thousands of years: Pythagoraeans investigated “consonant”

intervals The “tuning”/”temperament” problem (15th –

18th century) Investigators: Galileo, da Vinci, Descartes,

Kepler, Newton, Huygens, J.S. Bach, among others

More recent: sound and music as objects of research Acoustics Physical and psychological perception of sound Digital music

Mathematics and Music

More recent connections: College-level courses on connection between

math and music (e.g., David Wright’s course at Washington University of St. Louis – see handout)

Topics appropriate for high school/middle school (see The Mathematics Teacher, Sep. 2009 theme issue): Transformations (transpositions, reflections,

dilations) Modular Arithmetic and groups Intervals

Why should we connect them?

Think back to your own high school experiences: If you studied a musical instrument, you learned

how to read music, how to play your instrument, etc. But there are questions left unanswered (and

unasked!): For example, why did you choose a particular

instrument? “I liked its sound” – you were thinking about its timbre

If you played an instrument that could play more than one note simultaneously (e.g., guitar, piano, violin, xylophone), why did some combinations of notes “sound better” than others? Intervals, consonance, and dissonance – this is the

“psychoacoustic” basis of sound

Mathematics and PHYSICS and

Music Keep thinking back to your own high school

experiences: If you took a physics class:

You probably had a unit on sound (pipes with closed ends, wavelength, etc.)

It almost certainly didn’t include music (what is an octave? What is a perfect fifth?)

In your mathematics classes: You had lots on trigonometry

You might have talked about adding sinusoids Probably NOT about Fourier synthesis (which is accessible

to advanced precalculus students) Did you discuss modular arithmetic? Maybe…

Topic in musical composition (intervals, “Circle of Fifths”)

Mathematics and PHYSICS and

Music The moral:

Students’ interest in music might provide natural connections into mathematics (and physics) courses

Is there a good way to do this (within the framework of existing courses)?

Or should we think of something else? Which students would this type of material be

best for?

Who is this material for?

Most of this material requires some degree of mathematical sophistication on the part of the student

For purposes of this presentation, assume a student who has completed Algebra 2 with some trigonometry

In many cases, students will be helped by knowledge of music theory (clefs, key signatures, musical notation), but much can be taught “on the fly”

What is needed will depend on what topics one chooses to investigate!

The Big Ideas Hearing: Why is it so difficult to explain the perception of

sound? Sound generation: Vibration of an elastic medium (usually

air). Why do different instruments generate different “types” of sounds (timbres)? What is resonance?

Why are scales important? What is temperament? What is consonance and dissonance?

Why are intervals important? Why an “octave”? How does one compose music? What manipulations are

available to the composer? What about digital phenomena in music? What is “aliasing”? For each item, where does the math come into play?

The perception of sound (after Loy)

Sounds can be thought of in 6 “dimensions”: Frequency (perceived as pitch) The point at which the sound begins (onset) Amplitude or intensity (perceived as

loudness) The length of time that the sound lasts (the

duration) The change in the sound’s intensity over time

(the envelope of the sound) The quality of the sound – that which

distinguishes a trumpet from an oboe, for example (the wave shape)

These are the most important descriptors of sound

Sound Perception The ear – can be thought of as a receiver –

translates information about incoming sounds into the six “dimensions” we discussed

Objective measures of sound perception and music are difficult, and a major research topic. Examples:

pitch and loudness (as perceived) are not linear functions of frequency and amplitude (and they actually influence each other!)

One will perceive sounds that “aren’t there”: experiment of Seebeck on missing fundamental (Audacity Demo)

The Missing Fundamental

Sound Perception The phenomenon of “beats”

example: y = sin(2πx)+sin(2.1πx) (quite noticeable!):

Sound PerceptionNow we make the frequencies closer together:

y=sin(2πx)+sin(2.01πx) – much slower change in amplitude – the frequencies are getting closer together (heard as a “slower beat”)

Used in tuning instruments!

The Generation of Sound

Understanding of vibrating systems is critical for both generation and detection of sound

Connections between math and physics: Every object that has any elastic properties vibrates at a particular frequency – and this frequency is dependent on a property of the material Examples:

Springs: f α √(1/m) Pendulums: f α √(1/L) Helmholtz resonator: f α √(A/LV)

The frequency of vibration can be related to sinusoidal functions in the usual way (x(t) = sin(2πft), where x is displacement)

Musical Vibrating Systems

Stringed instruments Categorized in several ways:

How they are played (bowed, picked, struck) How they choose pitch (unstopped, stopped

fretted, stopped unfretted) If sound can be continuously produced (e.g.,

plucked vs. bowed)

Percussion instruments 1-dimensional (bars) vs. 2-dimensional

(membranes and plates) Wind instruments (brass, woodwinds, flutes)

Musical Vibrating Systems

These systems each have natural frequencies at which they resonate – that is, they tend to be efficient at producing sounds at that particular frequency.

Musical systems produce multiples of a particular frequency as well; e.g., a clarinet playing a 440 Hz note will also tend to generate frequencies of 1320 Hz, 2200 Hz (“odd harmonics”), etc. The same is true for a flute – except it will also generate EVEN harmonics (88o Hz, 1760 Hz, etc.)

A note played by a particular instrument is therefore a LINEAR COMBINATION of frequencies, each with different amplitudes (using a model of y =sin(2πft)): For example: Clarinet A-440: a*sin(880t) + b*sin(2640t) + c*sin(4400t)…Flute A-440: d*sin(880t) + e*sin(1760t) +f*sin(2640t)+…The presence of even harmonics gives a flute a different characteristic sound – a

different TIMBRE! This is a rich area for mathematics students to explore –what sorts of

sounds are generated when you add together different harmonics with different amplitudes? (This is essentially Fourier analysis!)

Scales, Intervals, Temperament

An interval in music – the “distance” between two pitches An interval of an octave in music – two notes that sound

“identical” at different frequencies - one note is exactly one-half the frequency of the other

Some intervals were classically considered to be consonant (“agreeable”), such as the octave (freq. ratio 2:1), the perfect fifth (freq. ratio 3:2), and the perfect fourth (4:3). Other intervals were considered to be more dissonant.

A scale is a systematic way of dividing up the octave. The question of how to divide up the scale leads to the

idea of temperament.

Scales, Intervals, Temperament

How do you divide up an octave? This is an example of a scale in C major, played ascending by the right hand and descending by the left hand. The first notes are the same (“middle C”), and the last notes are one octave above and one octave below middle C, respectively. The familiar “do-re-mi…”!

Scales, Intervals, Temperament

In Western music, within one octave, there lie twelve notes, represented as

C, C♯ (or D♭), D, D♯ (or E♭), E, F, F♯ (or G♭), G, G♯ (or A♭), A, A♯ (or B♭), B, and then C again.

The interval between any two adjacent notes is a half-step.

A major scale consists of the pattern: W-W-H-W-W-W-H (e.g., C-D-E-F-G-A-B-C). So what is the problem?

Temperament Originally, the octave was divided up using

Pythagorean ratios (dividing a string):

There are several problems with this… Going up or down from C by perfect fifths (powers

of 3/2) and then “adding or subtracting octaves” can actually produce all half-intervals as well, BUT…

Notes that should be the same (G♭,F♯) are slightly different – audibly so!

Note C D E F G A B CLengths Ratio

1:1 9:8 81:64 4:3 (perfect fourth)

3:2 (perfect fifth)

27:16 243:128

2:1 (octave)

Temperament Problems (continued):

This means that scales will sound “off” if they involve that tone – which is many scales!

Mathematically, it’s because there are NO INTEGER SOLUTIONS to the equation (3/2)m=(2/1)n. (excluding m=n=0).

So, one possible solution is to simply deal with it, and build instruments that resonate in a single key – but then no transposition is possible. Not what Western musicians desired…

Another possible solution is to attempt to divide the octave into differently-spaced intervals (this led to other tuning techniques, such as meantone temperament and well temperament). Each have their own problems, again often with transposition.

The “Solution” A third option is to create a system of temperament in which the

spacing between notes is equal: the equal-tempered scale. For a given frequency f, the frequency of the note that is N half-

steps above it is given by the formula

Note that only if N is a multiple of 12 will the frequencies of different notes be integer multiples of each other – the natural resonances of instruments will be somewhat lost….but everything is a trade-off! (Every interval in any key will have the same “sound” – equally good and bad!)

Students with access to music programs (or, preferably synthesizers) can create their own types of scale and temperament.

f (N )= f*2N/12

Transformations in Music

Many of these discussed in The Mathematics Teacher (Sep. 2009) issue (“Listening to Geometry”, Brett D. Cooper & Rita Barger): Translations (repeated structures in music) Reflections (“retrograde” – playing a melody

backwards; “inversions” – flipping the melody over some note)

Rotations (“retrograde inversions”) Dilations (temporal) “Atonal” transformations (modular

arithmetic!)

12-Tone Music Consider a composition using the following

notes:

This contains an example of all 12 notes in the chromatic scale. Each note can be associated with a number (begin from D):

D=0, F♯=4, A=7, C=10, G♯=6, F=3, E=2, G=5, B=9, C♯=11, A♯=8, D♯=1

12-Tone Music (and modular arithmetic)

We create a matrix from these numbers. The first row consists of the note classes from the composition on the previous slide. A note class is the set of all notes that are equivalent, modulo octave (mathematically, modulo 12, since 12-half-steps in an octave).

The first column consists of the note classes that result from the inversions of the notes in the first row.

Each row is then formed from keeping the intervals from the first row, but using the entry in the first column of the particular row you’re filling in.

Our example follows:

The 12-tone matrix

So how do you compose?

The composer may take a particular row or column from this matrix and create a musical piece from it, freely altering rhythm, creating harmony, and adjusting octave.

Example: 3rd row of previous matrix:

Convert these back into notes (D=0, D♯=1, E=2, F=3, F♯=4, G=5, G♯=6, A=7, A♯=8, B=9, C= 10, C♯=11)

):

[5] [9] [0] [3] [11]

[8] [7] [10]

[2] [4] [1] [6]

Digital Music Or, “Why is analog better than digital?” Musical signals are continuous – analog, described

using the 6 “axes” from earlier The capture of musical signals now by technology is

generally digital – that is, an analog waveform is quantized into digital information by sampling. The conversion occurs with a particular frequency (e.g., 44000 samples per second are collected and quantized – this would be a “sample rate of 44 kHz”)

There can be losses (usually are), plus other effects such as aliasing.

Aliasing Complicated phenomenon, but basically

comes down to this: If you wish to “accurately” sample a signal that is oscillating at a frequency of f, you must collect at least 2f samples per second (Nyquist sampling theorem).

An example (a rotating bicycle wheel) will help understand where aliasing can occur.

The Rotating Bicycle Wheel

(after Loy) Paint a spoke of a bike wheel red, set it spinning clockwise at 1

rotation/second (1 Hz) in a dark room Set a strobe light flashing at 1 Hz. Will appear that the spoke doesn’t move. Increase the speed of the wheel to 2 Hz; still will not appear to move! (Any

integer speed – no apparent motion!) Set the wheel to 0.1 Hz , strobe to 1 Hz – will see the spoke move CW 1/10

of a rotation per flash. (Note – this will still happen if wheel spins at 1.1 Hz! Or 2.1, or 3.1, … these are ALIASES of the correct frequency, 0.1 Hz.) If you set the wheel to 0.2 Hz, the spoke will “move correctly”.

Set the wheel to 0.5 Hz – you will see the spoke alternate positions. Note at 0.9 Hz – will appear to see the spoke moving BACKWARDS (at 0.1

Hz!)! For a 1 Hz strobe, the only frequencies you can be confident you’re

measuring correctly are between 0.5 and -0.5 Hz. There are other frequencies that are ALIASED to these frequencies. Big problem in audio!

Why is aliasing a problem?

Example (from Loy): You wish to record a violin playing a 750 Hz note. You sample

at 10000 Hz. The violin plays the note as well as all of the harmonics (multiples) of this frequency that give the note its timbre: 750, 1500, 2250, 3000, 3750, 4500, 5250, 6000, 6750, ...

All frequencies above 5000 Hz are aliased – so, for example, the 5250 Hz harmonic will be perceived (recorded!) as a 4750 Hz harmonic – which is NOT part of the violin’s harmonic sequence! Creates inharmonicities (harmonic distortions).

There are ways around this, but they are complicated and also affect the sound that is recorded! This leads to the entire “digital vs. analog” debate.

To dream…the impossible

dream… Many, many possible topics discussed – so

is this a class in the making? What would an elective in math and music

consist of? Who would it be geared towards? Would there be physics included? Who would teach it?

Recommended Readings

My personal favorites: Isacoff, Stuart. Temperament: The idea that solved music’s greatest

riddle (Alfred A. Knopf, 2001) Rothstein, Edward. Emblems of Mind: The inner life of music and

mathematics (Times Books, 1995) Loy, Gareth. Musimathics: The mathematical foundations of music,

volumes 1 and 2 (MIT Press, 2006 and 2007) Benson, Dave. Music: A mathematical offering (Cambridge University

Press and online at www.maths.abdn.ac.uk/~bensondj) Wright, David. Mathematics and Music (American Mathematical

Society, 2009) The Mathematics Teacher, Sep. 2009 issue:

Listening to Geometry, by Brett D. Cooper and Rita Barger Introducing Group Theory through Music, by Craig M. Johnson