Mathematics of musical scales

Outline

Basis of harmony
- Harmonics
  - Plucking a string
  - Octaves
  - Integer harmonics
  - Consonance = aligned harmonics
Octaves and fifths
- Building a scale (circle of fifths)
- Major chord from 4:5:6 overtones; minor from 1/6:1/5:1/4
- Fifths don't add up: (3/2)^12 = 129.75 != 128 = 2^7
So fix it!
- Keep perfect fifths and octaves, except seriously break one of them (Pythagorean)
- Keep as many "just" intervals as possible, restricting music to one or two keys (just intonation)
- Redefine fifths (meantone temperament / well temperament)
  - Quarter-tone temperament: 4th root of 5 → 4 5ths makes a 3rd
- Complete uniformity (equal temperament)
  - Twelfth root of two
... but why 12?
- define "cents"
  - Compare 12-TET with just intervals
  - 19-TET
    - play "Space Man"
  - 53-TET
    - (3/2)^53 = 2151972563 != 2147483648 = 2^31
Just temperament
- the way we sing, play strings, etc.
- pure musical idealism
Well temperament
- "Well-tempered Clavier"
- 81/80 is the "syntonic comma"
- Quarter-tone tuning: pretend the syntonic comma doesn't exist
- Indian 22-sruti scale: two versions of each note
Equal temperament
- Why octaves?
  - Bohlen-Pierce scale
  - play "Solemn Song for Evening"... but not for too long