The theory of the three worlds according to Sir Roger Penrose connects different fields of science and philosophy.

The three worlds are:

• The platonic world contains objects that are non-local and „timeless“.
• The physical world contains objects which can be examined from outside.
• The mental world is how we experience things in our head.

Music in the three worlds

• Musical scales are part of all three worlds
• No musical scale without the octave.
• Resonance plays an important role
• The perception of the octave in the mental world
• Real constraints for musical scales
• The overtone series is not a musical scale
• “Breaking down” the fifth
• Fractions and Resonances govern the scales
• Criteria for attractive musical scales (summary)
• First musical scales found with these criteria
• Standard pentatonic scales
• The major scale introduces tension to the resonances
• Pure and impure temperament
• How does the pythagorean comma come about
• The distribution of tones within the octave
• Tones for the two gaps
• How the tempered scales came into being
• The equal temperament
• Why resonance also works with imprecision
• Advantages of the equal temperament

Appendix

• Calculationg with frequencies and intervals
• Expressions around waves and sine waves