Why does a piano have twelve notes per octave?

AA SLAA HLAI SLAI HL Solid Also in: Pure maths

Research question

How closely do the intervals of 12-tone equal temperament approximate the pure frequency ratios 3/2 and 5/4, and would a different number of notes per octave (19, 31, 53) do better?

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

Why it makes a good exploration

Equal temperament uses a geometric sequence with ratio the twelfth root of 2, sacrificing pure harmony for flexibility. Measuring the errors in cents, and testing other systems, shows why 12 is a good compromise.

The mathematics you'll need

Geometric sequences and ratios
Logarithms and the cent (1200 log₂ r)
Rational approximations of log₂(3/2)
Comparing errors across systems

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

No data needed; optionally record your instrument with phyphox's audio spectrum to measure frequencies.

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

Explain pitch, frequency and pure intervals.
Derive equal-temperament frequencies.
Compute errors in cents for the fifth and major third.
Compare 12, 19, 31 and 53 notes per octave.
Reflect on why musicians accept the compromise.

Pitfalls that cost marks

Mixing ratios and differences.
Not explaining the cent scale.
Musical detail that crowds out the maths.

Showing personal engagement

Measure your own instrument's tuning.
Listen to (or play) intervals in different systems and describe the difference.
Link to continued fractions for the choice of 12.

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

Taking it further

Use continued fractions of log₂(3/2) to explain which numbers of notes per octave work well.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).