In this system, every note in the scale is birthed through the repeated application of a single mathematical operation: multiplying a frequency by $1.5$.
Pythagoras discovered that if you divide a string in a $3:2$ ratio, you get a note that sounds perfectly harmonious with the original. By repeating this process twelve times, you "gather" all the notes of the chromatic scale.The Math: You start at a base frequency ($1$), then go to $1.5$, then $2.25$ ($1.5 \times 1.5$), and so on.The Re-entry: Because these numbers quickly become huge, you use the Law of Octaves to "fold" them back down into a single octave (dividing by $2$ until the number is between $1$ and $2$).The Pure Fifth: In this system, the fifths are "perfectly" tuned to the physics of the string. They are "purer" than the fifths on a modern piano.
Researcher Note:
The Wolf Tone: To deal with this "extra" bit of sound, tuners have to hide it in one specific interval. That interval becomes shockingly dissonant—the Cacophony that prevents the system from being perfect.
The Wolf Tone: To deal with this "extra" bit of sound, tuners have to hide it in one specific interval. That interval becomes shockingly dissonant—the Cacophony that prevents the system from being perfect.