Harmonics and Temperament
Harmonic series, the comma
The first 16 harmonics of a low C. Columns 3 and 5 are Danhauser's two “easy to hear” harmonics (§ 107): the twelfth and the seventeenth.
Listening experiments
The harmonics blending into a single tone
386 cents, steady, no beating
400 cents, slightly “shimmering”
702 then 700 cents: almost identical
E as a harmonic (5/4), then E by four fifths (81/64), then together: 21.5 cents apart
- Equal temperament
- The octave divided into 12 equal half steps of 100 cents (ratio 2^(1/12) ≈ 1.0595).
- Syntonic comma
- 81/80 ≈ 21.5 cents, about 1/9 of a whole step, as Danhauser writes.