The Mathematics of Music – Ratios and Frequencies
Mathematics and music share a deep connection — both rely on patterns, structure, and proportion. When students explore musical notes and harmonies through ratios and frequencies, they see how simple numbers shape the sounds we hear every day.
The Concept
Musical notes are based on frequency, the number of vibrations per second (measured in hertz, Hz). When two notes are played together, the ratio of their frequencies determines how harmonious they sound.
For example:
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Octave: 2:1 ratio (e.g., 440 Hz and 880 Hz)
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Perfect fifth: 3:2 ratio (e.g., 440 Hz and 660 Hz)
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Perfect fourth: 4:3 ratio
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Major third: 5:4 ratio
These simple ratios create consonance — pleasant, stable sounds. More complex ratios produce dissonance, which gives tension and colour to music.
The Experiment
Students can use tuning forks, keyboard apps, or digital synthesiser tools to measure and compare frequencies. By analysing waveforms or using PASCO sound sensors, they can see how frequencies combine to form interference patterns and beats.
Plotting these waves shows visually why harmonious intervals have simple repeating patterns, while dissonant ones do not.
The Maths
If the frequency of a note is , then an octave above is , and a fifth above is .
Modern tuning (equal temperament) divides the octave into twelve semitones, where each note is times the frequency of the previous one — a perfect example of exponential growth in sound.
Skills Highlight
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Exploring ratios and proportional reasoning in a musical context
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Applying logarithmic and exponential relationships to real data
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Visualising wave patterns and frequency combinations
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Linking mathematical precision with creative expression
Why It Works in Teaching
This topic shows students that mathematics isn’t just abstract — it creates harmony, melody, and rhythm. Linking sound, frequency, and ratios provides a powerful and engaging way to explore number patterns and scientific thinking together.
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