Why do two nearly-equal notes 'wobble'? Beats and trigonometric identities

AA SLAA HLAI SLAI HL Solid Also in: Trig & navigation, Modelling

Research question

When two notes of slightly different frequency are played together, does the loudness pulse at the difference of the frequencies as the sum-to-product identity predicts, and how well does a sum of sines model a real recording?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

A trigonometric identity explains something you can hear and record. Fitting the model to a real waveform and checking the beat frequency is a tidy, testable investigation.

The mathematics you'll need

Sum of two sine waves
Sum-to-product identity: sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2)
Envelope and beat frequency
Fitting to recorded data
Effect of unequal amplitudes

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Record two tuning-fork apps, two instruments or two phone tones with phyphox (audio oscilloscope) and export the 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.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Derive the beat formula.
Predict the beat frequency for chosen notes.
Record and measure the real beats.
Fit the model and look at residuals.
Reflect on unequal amplitudes and room echo.

Pitfalls that cost marks

Sampling rate too low to resolve the waveform.
Assuming equal amplitudes without checking.
No comparison between prediction and measurement.

Showing personal engagement

Tune an instrument by ear using beats and explain what you are doing.
Predict the beats before recording.
Compare with a musician's tuning.

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	Sum of two sine waves; Sum-to-product identity: sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2)
AA HL	Good fit	Sum of two sine waves; Sum-to-product identity: sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2)
AI SL	Good fit	Sum of two sine waves; Sum-to-product identity: sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2)
AI HL	Good fit	Sum of two sine waves; Sum-to-product identity: sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2)

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

Personal engagement (C)

Use an artwork, instrument or design you know or make yourself, and measure it yourself. Explain the aesthetic question as well as the mathematical one.

Reflection (D)

Test popular claims honestly (the golden ratio is often not where people say it is) and reflect on how measurement choices affect the result. For this idea, start with: sampling rate too low to resolve the waveform — say how it affects your answer.

Use of mathematics (E)

SL: Geometry, trigonometry, sequences or functions used to model or test a real piece of art or music, with measurements and error discussed.

HL: Complex numbers, parametric curves, transformations as matrices, or Fourier-style sums of sinusoids, used to explain the structure rigorously.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Model three notes at once, or analyse the beats in a slightly out-of-tune chord.

Extending it for HL

Describe the construction with parametric or complex-number methods, or prove a property of the pattern or scale you studied.

See a complete IA, marked

Our annotated exemplar Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) asks a different question, but shows how a complete art & music exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).