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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?

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    I thought the story was that the ratios were derived from string lengths, not frequencies. – Dekkadeci Feb 15 '18 at 23:02
  • @Dekkadeci indeed. They may have had an idea that the rate of vibration was proportional to the string lengths, but vibration doesn't seem to be a necessary component of the theory, in which case it does not matter whether they had such an idea. Surely they must have recognized that constant tension was a requirement, though, and I suppose they must have had some idea that strings and winds has something in common, and that seems likely to have been vibration. – phoog Oct 12 at 18:37
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To the best of my knowledge there was not a specific known frequency of one pitch that was compared to a specific known frequency of another pitch.

However, it was relatively simple to pluck a string, hear the pitch. Then pluck the string at half of the length (possibly by depressing that string at that point, or touching the string at the half-way point), and hear that the string produced the same pitch, only higher.

To the mathematic mind of one such as Pythagoras, it is not too far of a stretch that he was able to determine the frequency must have been twice as fast. Likewise, when the string was one third of the length, the pitch was 3 times as fast.

This interval, what we now call the 5th, is the closest mathematical relationship that can exist between 2 different notes. And by extrapolating this 5th interval, all twelve tones that comprise our modern tuning system can actually be mathematically derived. Even without knowing the starting pitch as far as Hz is concerned, the relationship between notes came from physical phenomenon.

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They used a monochord, an instrument consisting of a single long string. The string could be shortened (by touching, just as we do now) and the relative lengths of string lengths giving tones measured. Rather than dealing with frequencies, these guys dealt with string lengths.

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