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Take a guitar. Very gently put finger on string. Start sliding finger while continuously strumming with other hand. On some places (usually 3rd 5th 12th fret) the string will produce clean sounds, that's where the frets should be.

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This also results in a per-string just intonation ;)

right, isn't that kind of what i said? so isn't that how they did it back in ancient times?

Those are so called flute sounds or harmonics, ie you don't press the string against the fretboard or fret normally at all. Instead you touch it very gently to force a vibration node there where your finger is. It doesn't work at all on most places along the string, you end up just damping the string. The easiest harmonic is halfway between bridge and nut, or at 12th fret, ie you force the string to vibrate with half the wavelength, or a node in the middle, or double frequency, or one octave higher, compared to free string. Then second easiest is 7th fret and then 5th (which is 2 octaves, ie quadruple frequency) and 9th. So you get less than pentatonic scale if you only use those, but they are useful pointers or anchors anyway.

But this is not exact!

Music harmonics are just about multiples. Ie one note rings at some frequency and another note that sounds nice with it rings at some multiple of it. Or they have some common factor. The twelve note system and the twelvth roots are just approximations for some of these harmonies. There's even very recent innovation like true temperament frets. But I don't think that works with all the different song keys and tunings... Maybe it's another reason why I should buy a guitar for every song key we have!

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