Why do guitar frets get closer together? A geometric sequence on the fingerboard

AA SLAI SLAA HL Accessible Also in: Modelling

Research question

Do the fret positions on a real guitar (or ukulele) follow the geometric sequence that equal temperament predicts, how large are the deviations, and how good was the old luthiers' 'rule of 18' approximation?

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 geometric sequence you can measure with a ruler on an instrument you own. Comparing the exact rule, a historical approximation and the real instrument gives a clear, finishable investigation.

The mathematics you'll need

Geometric sequences with ratio 2^(−1/12)
Distance from the nut as a function of fret number
Comparing measured and predicted positions; residuals
Percentage and cumulative error of an approximation
Logarithms to linearise the model

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

Measure fret positions on your own instrument with a steel rule; repeat to estimate measurement error.

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 fret positions from equal temperament.
Measure the real instrument.
Compare with residuals and a log plot.
Test the 'rule of 18' approximation.
Reflect on compensation, string stretch and measurement error.

Pitfalls that cost marks

Measuring from the wrong reference point.
Ignoring that real necks are compensated.
No estimate of measurement error.

Showing personal engagement

Use your own instrument.
Compare two instruments.
Explain why the 12th fret is half-way.

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 with ratio 2^(−1/12); Distance from the nut as a function of fret number
AA HL	Fits, but add an HL technique	Geometric sequences with ratio 2^(−1/12); Distance from the nut as a function of fret number
AI SL	Good fit	Geometric sequences with ratio 2^(−1/12); Distance from the nut as a function of fret number
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Accessible. A good first extended piece of maths, with room to go deeper. 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: measuring from the wrong reference point — 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

Investigate how pressing a string down raises the pitch slightly, and how builders compensate for it.

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).