Music & Ratios: A Year 8–9 (13‑year‑old) Interdisciplinary Lesson — Proportional Reasoning, Sound Frequency & Musical Intervals (ACARA v9 linked)

PARTIES & PURPOSE

Plaintiff: Mathematics (Ratios & Proportional Reasoning).
Defendant: Sound & Music (Frequencies & Intervals).
Purpose: Show how numerical ratios describe musical intervals and how to test those relationships with experiments and measurements.

LEARNING OBJECTIVES

• Define ratio and demonstrate simplifying and comparing ratios.
• Use measurements of frequency to calculate ratios between notes.
• Connect frequency ratios to musical intervals (octave, fifth, fourth, major third).
• Design and carry out an experiment to show frequency changes (e.g., changing string length) and analyse results with proportional reasoning.

CURRICULUM LINKS (ACARA v9 — cross‑disciplinary)

• Mathematics: Ratios and proportional reasoning; using measurement and units; calculating and simplifying ratios.
• Science: Sound as vibration; frequency and pitch; experiment design and measurement.
• The Arts (Music): Pitch, intervals and tuning systems; historical context (Pythagoras, monochord).

KEY DEFINITIONS — Quick, in cadence

• Ratio: a comparison of two numbers written a:b or a/b (e.g., 2:1).
• Frequency: how many vibrations per second (Hertz, Hz). Higher frequency → higher pitch.
• Interval: the distance between two pitches (e.g., octave, fifth).
• Proportional reasoning: understanding how one quantity changes when another changes by a constant factor.

FACTS (History & Background — Pythagorean vibe)

Long ago, Pythagoras noticed that simple number ratios produce pleasing intervals. He used a single string (a monochord). Halve the string length, the pitch becomes one octave higher — a 2:1 frequency ratio. That observation links maths (ratios) to music (intervals).

COMMON MUSICAL RATIOS (Just Intonation examples)

• Octave = 2:1 (e.g., 440 Hz → 880 Hz)
• Perfect fifth = 3:2 (e.g., C at ~261.63 Hz → G at ~392.44 Hz, ratio ≈ 392.44/261.63 ≈ 1.5)
• Perfect fourth = 4:3
• Major third (just) = 5:4

Note: On modern keyboards tuned in equal temperament these ratios are approximated, but the ideas still hold.

MATHEMATICS STEP — Using ratios and proportional reasoning

1. To find the ratio of two frequencies f1 and f2 compute f2:f1 or the fraction f2/f1.
2. To check for an octave, test if f2/f1 ≈ 2. For fifth, check if ≈ 1.5 (3/2), etc.
3. Simplify ratios by dividing numerator & denominator by a common factor when possible, or express as decimal and compare to exact fraction.

EXPERIMENTAL INVESTIGATION — Step‑by‑step

Two options: using a string (monochord/guitar) or measuring pure tones with a phone/computer.

Materials

• Stringed instrument (guitar, ukulele) or a DIY monochord (a string stretched over a board) OR a set of tuning forks.
• Smartphone with a frequency analyzer app ("Spectroid", "AudioSpectrum", "Frequencimeter") or a laptop with a tone generator + audio spectrum software.
• Ruler or tape measure, pencil, paper.

Procedure A — String length experiment (demonstrates inverse relation length ↔ frequency)

1. Pluck the open string and measure its frequency f_open using the app. Record the string length L_open (distance between nut and bridge).
2. Mark the midpoint (L_open/2). Press the string at the midpoint (or move bridge to halve vibrating length) and pluck. Measure frequency f_half and record L_half = L_open/2.
3. Calculate the ratio f_half / f_open. Expect ~2. Explain: halving length doubles frequency (f ∝ 1/L for a fixed tension and mass per length).
4. Try other lengths (e.g., 2/3 L_open or 3/4 L_open). Measure frequencies and compute ratios. Compare results to expected inverses of length ratios.

Procedure B — Measuring musical intervals using tones

1. Use a tone generator: set Tone A to 440 Hz (A4). Use app to confirm.
2. Set another tone to 880 Hz. Measure and calculate 880/440 = 2 → octave. Listen: same note name, higher pitch.
3. Set third tone to 660 Hz. Compute 660/440 = 1.5 → perfect fifth (3:2). Listen and compare.
4. Try a tone at 550 Hz. Compute 550/440 = 1.25 → major third (5:4).

Data table (example)

Analysis questions

1. Compare measured ratios to ideal fractions (2:1, 3:2, 4:3, 5:4). How close are they? Explain small differences (tuning system, measurement error).
2. If you halve string length and frequency doubles, what happens to wavelength? (It halves — wavelength and frequency inversely related for fixed wave speed.)
3. Use proportional reasoning: if one string vibrates at 300 Hz, what frequency will it have if you reduce length to 3/4? (Since f ∝ 1/L, f_new = f_old × (L_old / L_new) = 300 × (1 / (3/4)) = 300 × 4/3 = 400 Hz.)

ARGUMENTS (Why this matters — courtroom cadence)

Mathematics gives us the language (ratios) to describe why two notes sound consonant. Science explains the mechanism (vibrating strings/air columns) and how frequency creates pitch. Music uses both to build scales, melodies and harmony. Understanding ratios helps in tuning instruments, constructing electronic music, and modelling sound phenomena (e.g., harmonics, resonance, Doppler effect).

EXTENSIONS & CROSS‑CURRICULAR LINKS

• History: Research Pythagoras and the monochord — write a short paragraph linking ancient discoveries to modern tuning.
• Music technology: Explore equal temperament vs just intonation — why modern instruments use equal temperament (compromise across keys).
• Physics: Investigate harmonic series — frequencies at integer multiples (1×, 2×, 3× ...) create timbre and overtones.
• Real world: Discuss how proportional thinking is used in audio engineering, instrument making, and digital sound synthesis.

ASSESSMENT TASKS (Short & focused)

1. Practical: Measure three pairs of notes using an app and calculate ratios. Identify each interval (octave, fifth, fourth, etc.) and explain any differences from ideal ratios.
2. Problem: A string at length 80 cm vibrates at 220 Hz. If you press it at 60 cm of vibrating length, what is the new frequency? Show your working using proportional reasoning.
3. Written: Explain in three sentences how ratios help musicians tune instruments.

CONCLUSION — The Verdict

Proportional reasoning (ratios) is the legal language between maths and music. Frequencies compare by simple ratios — 2:1 for octaves, 3:2 for fifths — and experiments (strings, tuning forks, apps) let you test these claims. You now have the steps to measure, calculate, explain and extend. Court adjourned — go make noise, measure it, and be precise about the numbers.