How did Pythagoras and Ptolemy measure the relative pitch of musical notes?

Question

Both Pythagoras and Ptolemy believed that the intervals between notes in music should be ratios of small integer numbers. This is known as Just Intonation. Pythagoras liked them to derived from octaves (2:1) and perfect fifths (3:2), which is 3-limit tuning. Ptolemy was more flexible, and his scales are in 5-limit tuning.But how did they measure the relative frequency of two notes? How did they even know that the pitch they heard was a result of the frequency of vibrations?Today, I can make a recording of a note on the flute with Audacity (audio software), and then look at the waveform to see that it repeats 440 times per second. Alternatively, I can use an electronic tuner that will instantly tell me the notes frequency.But Pythagoras died in 495 BC, and Ptolemy died in 170 AD, so they obviously didn't have these technologies. So how did they measure the frequency of sounds and the ratios between them?

Answer 1

Hi there! These guys used a device called a monochord to come up with these ratios. It's kind of like a guitar with just one string, but with a sliding bridge wedged under the string. By sliding the bridge around, you can change the pitches on both sides of the string. For example, when the bridge is slid so that it sits smack in the middle, the string is evenly divided into two halves. The resulting pitch will be what we now think of as being an octave higher than the pitch of the same string when it is plucked without the bridge. You are, of course, correct that people from this time did not have a way to measure vibrations per second––they were comparing string lengths, and the relative sizes of things that produced a musical pitch when struck, such as blacksmith's anvils (which, according to legend, is how Pythagoras is said to have noticed these ratios in the first place).