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You don’t have to be a musician to know when music sounds bad. But what is it, exactly, that makes music sound good? Turns out it’s all in the numbers. The combinations of notes we find pleasing are based on simple ratios.

In this lesson, students learn how the notes plucked on stringed instruments are mathematically related. By calculating the ratios of string lengths, students explore the relationship between notes that sound good together — and also those that don’t. Math and music have more in common than you might think!

Students will

  • Measure string lengths and calculate the ratios for different notes in a major scale
  • Apply fraction division to calculate the length ratio between any pair of strings
  • Explore the relationship among string lengths, ratios, and sound quality of simple harmonies

Before you begin

This lesson could be used to introduce the concept of ratios and explore their relationship to fractions, or could serve as an application of ratios. Either way, students should have a strong foundation in fractions. They should be comfortable multiplying fractions by whole numbers and other fractions, and some previous exposure to fraction division would also be helpful. Although the lesson focuses on music, no prior musical knowledge is assumed, either for the student or the teacher!

Common Core Standards

Content Standards
Mathematical Practices

Additional Materials

  • Rulers
  • Colored pencils


The Piano Guys, One Direction