Maths IA · Geometry and trigonometry · topic 31 of 40

Fourier series: what is in a musical note? · IB Maths IA idea: guide, sample graph and step-by-step plan

music + maths: great for engagement

A flute and a violin can play the same note and sound completely different. You're going to find the maths of the difference.

Difficulty●●● Stretch
KitPhone recording, Audacity or Python, GDC
TimeAbout 14 to 15 hours
FitsAA HL
  • ✓ you play an instrument
  • ✓ you like integration and trigonometric identities
  • ✓ you can use software for the heavy numerical work

The big idea

← derive it. don't just quote it!

Any periodic wave can be written as a sum of sines and cosines, and the coefficients say how much of each harmonic is present. Derive the coefficients by integration, test them on waves where you know the answer, then analyse a real recording of an instrument you play. The sound's shape is in its harmonics. Computer help is needed for the real data, but the exploration is about the maths, so derive and justify the coefficients yourself, then ask when a finite sum is good enough. Fourier series are in neither course, so introduce them carefully; the AA HL integration (by parts, trigonometric identities) does the work, so be rigorous.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Derive the coefficients

Use orthogonality of sines and cosines and integration to find a₀, aₙ and bₙ, then check on a square wave.

→ A real proof at the heart of the exploration: every step justified, every integral shown.
Study the partial sums

Plot the sum of the first N terms for a square wave or sawtooth, and look at the error as N grows.

→ You can discuss convergence, and the overshoot near a jump (the Gibbs effect), with numbers.
Analyse a real instrument

Record a sustained note on an instrument you play and extract the strengths of its harmonics.

→ Personal data and decisions (sampling rate, which section of the note); you must explain what software is doing, not just show output.
Compare two instruments

The same note on two instruments (or two ways of playing one).

→ A comparison of coefficient patterns and a judgement, with a clear measure of difference.
Rebuild the sound

Use your estimated coefficients to build the wave back up and compare it to the recording.

→ A test of the model: the residual shows what the series leaves out.
Move from series to the discrete case

Compare a finite Fourier series with the discrete transform used in software.

→ Links two ideas and forces you to be honest about sampling and the limits of the data; keep to one clear comparison.

Your Maths IA research question: run the check

say which wave, which note

Write your own question first. Then make sure it ticks every box:

  • Names the wave or instrument and the note, and says how you will get the data.
  • Says what you will extract or test: the coefficients, the error of partial sums, or a comparison of instruments.
  • Defines the period and the variable of the function so the series is well defined.
  • Says how you will judge success: an error measure between the rebuilt wave and the original.
  • Is narrow enough for 12 to 20 pages, with the integration shown and not hidden in software.
  • Can be answered with numbers and a stated precision.

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The picture to draw

A wave built from sines. Draw yours with a wave you actually study, and mark the harmonics you will keep.

= sum ofharmonics ↓what the mic recordsharmonic 1: fundamentalharmonic 3: smallerharmonic 5: smaller stillf(t) = a₀/2 + Σ ( aₙ cos nωt+ bₙ sin nωt )your job: find aₙ and bₙone period, T = 2π/ω

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Title page, question and rationale~1 page
Your question; the instrument or sound that made you ask it.
Aim and plan~0.5 page
What you will derive and what you will analyse.
Periodic functions and sums of sines~2 pages
Period, frequency, harmonics; why sines and cosines are a good building set (orthogonality).
Deriving the coefficients~3 pages
Integrals for a₀, aₙ and bₙ, with the orthogonality results shown or proved. At HL this is where rigour must show.
A test wave~2 pages
A square or sawtooth wave: coefficients by integration, partial sums, and error against the number of terms.
Real data~2 pages
How you recorded, chose a section, and extracted harmonic strengths; units, sampling and uncertainty.
Comparing model and sound~2 pages
A rebuilt wave against the recording, a measure of the difference, and what is left over.
Reflection throughout~1.5 pages
Limits (is the sound truly periodic? how much does the data hold?), and what software did that you had to trust.
Conclusion~0.5 page
Answer with numbers, and say whether the aim was met.
Bibliography and appendixnot counted
Sources, code and extra plots.

What your Maths IA graph could look like

n is a whole number: points only!

Made-up data, to show you the form of a strong graph, not the answer. Yours will look different. example only

123456789100.00.20.40.60.81.01.21.41.6Harmonic number, nAmplitude bₙ (relative to the wave's height)square-wave model (from your derivation), odd ninvented harmonics from a recording, with error barswhere are the even harmonics?what does a gap tell you?ILLUSTRATIVE · MADE-UP DATA
  • The horizontal axis is the harmonic number, a whole number: points only at n = 1, 3, 5 and so on.
  • A model curve from your derivation, with the data tested against it.
  • Error bars show the uncertainty in the extracted amplitudes: say how you estimated them.
  • The axis says what the amplitude is relative to, so the reader can judge it.
  • Another graph to consider: the error of the partial sum against the number of terms, or the relative power of each harmonic.

Where does the power sit?

Rank the harmonics by their share of the wave's power. This chart is for an idealised square wave. A chart with your instrument's numbers will look different, and the difference is your analysis.

Harmonic 181.1%Harmonic 39%Harmonic 53.2%Harmonic 71.7%Harmonic 91%most of it is the first!% of the wave's power: square-wave model, bₙ² / 2ILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

do the test wave BEFORE real data

Tick steps off as you go. We'll remember on this device.

0 of 11 done

  1. Read and predictFind what a Fourier series says and what the coefficients mean. Before any calculation, predict which harmonics a wave with a jump would need.
  2. Learn the orthogonality resultsUnderstand why integrals of products of sines and cosines over a period vanish. Write the argument in your own words.
  3. Derive for a simple waveFind aₙ and bₙ for a square or sawtooth wave by integration. Check a few partial sums against the wave.
  4. Trial run (one evening)Record a note on your phone. Look at the waveform. Is it clean? Is it close enough to periodic? Fix the recording first.
  5. Choose the data and the methodDecide the instrument, the note, the section of the sound and how you will get coefficients. Explain what the software computes.
  6. Extract the harmonicsFind harmonic strengths from the recording. State sampling rate and uncertainty. Check against the test wave first.
  7. Rebuild and testSum the series with your coefficients and compare it with the data. Choose an error measure and decide what counts as a good fit.
  8. Compare or deepenCompare two instruments, or study the overshoot in partial sums. Show your reasoning, not only a plot.
  9. Reflect as you goWhat did software do that you had to trust? What limits does a finite recording put on the answer?
  10. Write, then cut12 to 20 pages. One worked integral per type, the rest in a table. Proper notation, with no calculator syntax.
  11. Check before your teacher's draftCheck every symbol is defined once and used consistently, every graph has titles and units, and the conclusion answers your aim.

Maths you'll need

Traps to avoid in this Maths IA

  • Letting software do the Fourier analysis and showing only the output, with no sign of understanding.
  • Leaving out the integration: the derivation is the HL-level maths.
  • Treating a real recording as perfectly periodic without saying how you chose the section.
  • Claiming a series converges, or a finite sum is 'close enough', with no error measure.
  • Mixing up the period, the angular frequency and the frequency in the formulas.
  • Copying a derivation from a textbook with no use of your own data (personal engagement).
  • Too many topics: Fourier series, transforms, filters and synthesis. Pick one line and go deep.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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