Table 1.
Beat rate relationships for different piano note pairs (Capleton, 2007)
| Major third (4 st) | Beats (Hz) | Perfect fourth (5 st) | Beats (Hz) | Perfect fifth (7 st) | Beats (Hz) | Major sixth (9 st) | Beats (Hz) | Minor third (3 st) | Beats (Hz) | Minor sixth (8 st) | Beats (Hz) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| F–A | 6.9 | F–Bb | 0.8 | F–C | 0.6 | F–D | 7.9 | F–Ab | 9.4 | F–Db | 11.0 |
| F#–A# | 7.3 | F#–B | 0.8 | F#–C# | 0.6 | F#–D# | 8.4 | F#–A | 10.0 | F#–D | 11.7 |
| G–B | 7.8 | G–C | 0.9 | G–D | 0.7 | G–E | 8.9 | G–Bb | 10.6 | G–Eb | 12.3 |
| Ab–C | 8.2 | G#–C# | 0.9 | G#–D# | 0.7 | Ab–F | 9.4 | G#–B | 11.2 | G#–E | 13.1 |
| A–C# | 8.7 | A–D | 1.0 | A–E | 0.7 | A–F# | 10.0 | A–C | 11.9 | A–F | 13.9 |
| Bb–D# | 9.2 | A#–D# | 1.1 | Bb–F | 0.8 | Bb–G | 10.6 | A#–C# | 12.6 | A#–F# | 14.7 |
| B–D# | 9.8 | B–E | 1.1 | B–F# | 0.8 | B–D | 13.3 | B–G | 15.6 | ||
| C–E | 10.4 | C–F | 1.2 | C–G | 0.9 | C–Eb | 14.1 | ||||
| C#–F | 11.0 | C#–F# | 1.3 | ||||||||
| D–F# | 11.7 | D–G | 1.3 | ||||||||
| Eb–G | 12.3 |
We simulated the process of listening to beats in a frequency window through the use of a five-element harmonic complex where only the middle component is modulated at a rate relevant to the tuning procedure. Table 1 shows the range of beat rates for different note pairs for the scale area F3–G4 (difference in semitones indicated at the top) corresponding to the intervals used in tuning. The major thirds, perfect fourths, perfect fifths, and major sixths are the most useful for tuning, but the minor thirds and sixths that are also useful are included as well. In the experiment we used rates of 2 Hz and 16 Hz that are both within this range. st, Semitones.