Mathematics of Music – Teaching Ratios, Frequencies and Scales

Music may seem like pure art, but beneath the melodies lies mathematics. Ratios and frequencies govern why notes sound good together, why scales exist, and why different musical traditions—Western, Asian, or otherwise—sound distinct.

The Ratio Behind Harmony

When you pluck a string, it vibrates at a fundamental frequency. Divide that string in half, and it vibrates at twice the frequency—an octave higher. The octave ratio of 2:1 is one of the most important in all music. Other pleasing notes arise from simple ratios:

• 3:2 → the perfect fifth

• 4:3 → the perfect fourth

• 5:4 → the major third

These ratios explain why certain intervals sound "in tune"—the waveforms line up regularly, reinforcing each other.

Building Scales

Western music is built on the idea of dividing the octave into 12 equal parts—known as equal temperament. Instead of keeping the "pure" mathematical ratios, each note is adjusted slightly so instruments can play in any key without sounding out of tune. It’s a compromise: mathematically imperfect, but musically versatile.

This is why a piano can play a Bach fugue in C major and then jump to B-flat minor without retuning. The twelfth root of 2 sneaks in here: each semitone is a multiplication of frequency by 2^(1/12).

Different Musical Worlds

Not all traditions make the same compromises. Much Asian music uses different scales and tuning systems. For example:

• Indian classical music uses 22 shrutis (microtones) in an octave, allowing for subtler distinctions in pitch.

• Chinese pentatonic scales divide the octave into five notes, creating the recognisable "open" sound we often associate with Chinese folk music.

• Japanese gagaku music sometimes uses in scales with intervals that don’t exist in Western tuning, producing tones that sound exotic or even dissonant to Western-trained ears.

The difference isn’t just cultural—it’s mathematical. Changing the ratios, or the number of divisions in an octave, changes how we perceive harmony and mood.

Teaching Through Experiment

In the classroom, you can demonstrate this beautifully:

• Stretch a guitar string and measure the frequency at different lengths.

• Use a frequency generator and show how doubling the frequency jumps an octave.

• Compare a Western major scale with a pentatonic one on a keyboard, asking students which feels "familiar" and which feels "different."

Mathematics becomes audible—students hear the ratios in action.

Why It Matters

By exploring the maths of music, students see how numbers connect to the real world. They discover that 2:1 is not just a ratio—it’s the reason lullabies, symphonies, and pop songs make sense. And they learn that cultural variety in music is built not just on history, but on mathematical choices about how to divide sound.