Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

Why do a flute and a violin sound different on the same note?

AA SLAA HLAI SLAI HL Solid Also in: Modelling

Research question

Can the waveform of a note played on [two instruments] be modelled as a sum of sine waves at integer multiples of the fundamental, and how do the relative amplitudes of the harmonics differ?

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Why it makes a good exploration

Timbre — the colour of a sound — comes from the mix of harmonics. Recording instruments with your phone and modelling their waveforms with sums of sinusoids links trigonometry to something you can hear.

The mathematics you'll need

Sinusoidal functions and superposition
Period and frequency
Fitting amplitudes (reading a spectrum; explain Fourier ideas simply)
Comparing models with residuals

Where the data comes from

Record notes with phyphox's audio oscilloscope and spectrum tools; export data.

phyphox (RWTH Aachen) — Free app that turns your phone's accelerometer, microphone, barometer and gyroscope into data loggers with CSV export.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Explain sound as pressure waves.
Record the same note on two instruments.
Read harmonic amplitudes from the spectrum.
Rebuild each waveform from sinusoids and compare.
Reflect on phase, room acoustics and microphone response.

Pitfalls that cost marks

Presenting spectra without explanation.
Ignoring phase in reconstruction.
Too many harmonics to discuss meaningfully.

Showing personal engagement

Record your own instrument.
Synthesise a sound from your model and compare by ear.
Test how harmonics change with loudness.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Sinusoidal functions and superposition; Period and frequency
AA HL	Good fit	Sinusoidal functions and superposition; Period and frequency
AI SL	Good fit	Sinusoidal functions and superposition; Period and frequency
AI HL	Good fit	Sinusoidal functions and superposition; Period and frequency

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: presenting spectra without explanation; and ignoring phase in reconstruction. Say how each one affects your answer.

Use of mathematics (E)

SL: sinusoidal functions and superposition; period and frequency; fitting amplitudes (reading a spectrum; explain Fourier ideas simply); comparing models with residuals, each explained in your own words and carried out correctly.

How art & music explorations reach the top bands in general: Art, music & design IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Compare odd-harmonic-dominated instruments (clarinet) with others and explain the physics.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL art & music exploration adds.

See a complete IA, marked

Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) is a complete art & music exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Sinusoidal functions and superposition, period and frequency, fitting amplitudes (reading a spectrum; explain Fourier ideas simply) and comparing models with residuals.

Where can I get data for this IA?

Record notes with phyphox's audio oscilloscope and spectrum tools; export data. Checked sources and tools: phyphox (RWTH Aachen) and Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Why do a flute and a violin sound different on the same note?” filled in: the question, the maths and the data sources.

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