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

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?

Make it yours, then agree it with your teacher.

Next step: plan this IA free — the planner opens with this question filled in. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.

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

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.

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.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Geometric sequences and ratios; Logarithms and the cent (1200 log₂ r)
AA HL	Good fit	Geometric sequences and ratios; Logarithms and the cent (1200 log₂ r)
AI SL	Good fit	Geometric sequences and ratios; Logarithms and the cent (1200 log₂ r)
AI HL	Good fit	Geometric sequences and ratios; Logarithms and the cent (1200 log₂ r)

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: mixing ratios and differences; and not explaining the cent scale. Say how each one affects your answer.

Use of mathematics (E)

SL: geometric sequences and ratios; logarithms and the cent (1200 log₂ r); rational approximations of log₂(3/2); comparing errors across systems, 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

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

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?

Geometric sequences and ratios, logarithms and the cent (1200 log₂ r), rational approximations of log₂(3/2) and comparing errors across systems.

Where can I get data for this IA?

No data needed; optionally record your instrument with phyphox's audio spectrum to measure frequencies. Checked sources and tools: phyphox (RWTH Aachen) and Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Why does a piano have twelve notes per octave?” filled in: the question, the maths and the data sources.

Plan this IA free

Your plan is kept in this browser. A free account keeps it with your drafts on any device: Join free · Sign in

Also: free IA self-check · A–E checklist (PDF) · have a draft? AI feedback or a review with Pete (€100)

Writing it up: how to write a Maths IA, step by step · structure and format · mistakes that cost marks · annotated exemplars, marked on every criterion