Swiss Mathematician Leonard Euler...
was trying to resolve a major problem. Pythagorean and Just Intonation prior to Euler had a serious problem. An instrument that played an ascending scale could not return to the exact starting note by descending back down the same scale. (scroll down...)

Music Historians identify five principle scales.
Pythagorean is the root of European music. In Gann's original list of recommended books to read, he listed several books on music theory.

While Euler solved one problem for instruments unable to return to a starting note, he created others in modern Equal Temperament. The only notes in our modern scale that retain perfect harmonic relationships are the beginning of each octave. The notes between are all sharp or flat...meaning they exceed or fall short of perfect harmonic intervals.

Only the ancient scale developed in 560 BC retains perfect harmonic ratios... the scale of Pythagoras. But you will later learn it too had a problem called the Pythagorean comma.

Below is a slide showing an extended Pythagorean scale. The lower numbers are the notes defined in cents. The unit of measurement to define frequencies between notes is called "cents".

The well known Fibonacci numbers are in red boxes.


The musical tuning scales listed below from ancient and modern times all contain Fibonacci ratios if highlighted in pale yellow. The Chinese names are in fact names of instruments...not scales. If you have information in English about Chinese scales and their precise interval construction please email Connie Brown