**Author: Joseph Iafrate**

**Editors: Christina Vallianatos, Scott Barolo, and Bryan Moyers**

**Editor’s Note: This post has several sound files to help readers understand the author’s message better. These sound files can be accessed via bolded links.*

Part one of this post explained how physics gave us a new language for talking about musical notes. In part two, we look at combinations of notes. Will two notes sound pleasant together, or will they clash? We can apply what we’ve learned about frequencies to get an answer.

**The Harmony of Ratios**

If you’ve ever used the Pythagorean theorem, you are well-acquainted with one of Pythagoras’ contributions to society. Pythagoras was an ancient Greek philosopher and mathematician dedicated to discovering mathematical principles in the world around him. During his time, the Greeks already had an idea of which notes sounded good together, a pleasant combination of two or more notes that we call a harmony. Pythagoras and his followers could identify harmony by ear, but they wanted to see if the math that permeated the rest of their worldview had anything to say about this phenomenon.

According to legend, they took two taut strings of different lengths and plucked them at the same time. The sounds seemed to **clash** with one another. So the Pythagoreans increased the length of one of the strings and tried again. It was a **bit better**, but the notes still seemed to clash in their ears. So they increased the length again. This kept going until the sounds complemented one another. Eventually they got it just right, and the two notes were in **harmony**.

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