Spreading beats evenly: Euclid's algorithm and musical rhythm

AA SLAA HL Solid Also in: Pure maths, Cryptography

Research question

How can k drum hits be spread as evenly as possible across n steps, why does Euclid's algorithm for the highest common factor produce the answer, and which traditional rhythms does it generate?

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

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Why it makes a good exploration

A link between the oldest algorithm in mathematics and rhythms from many musical traditions. You can listen to every result, and the 'evenness' question can be made precise.

The mathematics you'll need

Euclid's algorithm for the highest common factor
Floor functions and even spacing
A measure of evenness (sum of distances between hits)
Rotations and when two rhythms are 'the same' (counting)
Proof that the spacing differs by at most one step

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; play the rhythms in a free drum app or clap them; compare with rhythms from music you know.

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

Define 'as even as possible' precisely.
Generate rhythms with Euclid's algorithm and with floor functions.
Show the two methods agree on examples and explain why.
Compare with traditional rhythms you know.
Reflect on whether 'most even' means 'best sounding'.

Pitfalls that cost marks

Claiming a rhythm comes from a tradition without a source.
No precise definition of evenness.
Listing rhythms without mathematics.

Showing personal engagement

Play the rhythms yourself.
Find a Euclidean rhythm in a song you like.
Invent a measure of evenness and test it.

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	Euclid's algorithm for the highest common factor; Floor functions and even spacing
AA HL	Good fit	Euclid's algorithm for the highest common factor; Floor functions and even spacing
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
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: 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: claiming a rhythm comes from a tradition without a source — 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

Count how many distinct rhythms (up to rotation) have k hits in n steps.

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).