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If we plot a graph of the air pressure as a function of time while a musical note is played, we will get a wave-like shape – called a waveform – that repeats itself, such as these three: The upper waveform – the simplest of the three – was created by a computer, and it can be heard in example 1.

This waveform corresponds to a basic mathematical function called sine, and perhaps this is why it sounds “synthetic” or “boring.” Waveforms of real musical instruments are much more complex – the second waveform is of a guitar (example 2), and the third, of a piano (example 3).

In December 1717 a young musician called Johann Sebastian Bach arrived in the German town of Köthen.

Musical notes are periodic oscillations in air pressure, which we sense through our eardrums.Some note combinations please the human ear, while others don’t.The Greek mathematician Pythagoras is often credited with the discovery that ear-pleasing combinations obey mathematical regularity.A sequence of notes, played one after the other, is called a melody. It is built from three notes, whose frequencies are 440, 660, and 733.3 Hz.Now listen to example 7, whose note frequencies are 550, 825, and 916.6. Even though the two melodies used totally different notes, we easily recognize that this is “actually the same melody.” What made us feel that this is “actually the same melody”? The human ear and brain, it turns out, recognize two note sequences as “the same melody” when the ratios between their note frequencies are kept the same.

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