4-4 Beat Theory
Who created the beat theory?
Helmholtz is praised for discovering the beat theory.
What happens when two complex tones are played together?
When two tones are played simultaneously, the harmonics of each tone is perceived by the human ear. We have learned that beats occur when the frequencies of the two tones are slightly different.
Helmholtz discovered that in some instances the harmonics aligned for the component tones, which produced a pleasant sound to the human ear. These are the consonant harmonics. On the other hand, other tones do not have harmonics that align, which produce an unpleasant sound. These are the dissonant harmonics. This theory became known as the beat theory.
What would the beat theory say about unison tones?
In unison tones, the harmonic frequencies of each tone matches so they sound consonant and peasant to the ear.
The figure below shows how all frequencies perfectly align:
What would the beat theory say about whole note intervals?
The ratio of frequencies between whole notes is 1.123 making it so no harmonics would align. They produce a dissonant (clashing) sound which is unpleasant to the ear.
The figure below shows the obvious mismatch of all the frequencies:
What would the beat theory say about the perfect fifth interval?
A musical fifth occurs when complex tones with frequencies f and (3/2)f are played simultaneously. The harmonics of these two tones would have some matching frequencies (making a pleasant consonant sound) while other frequencies collides (producing an unpleasant dissonant sound).
The figure below shows the aligned and dis-aligned frequencies:
How do you find the ratio for a musical interval?
(f2/f1) = (n/m) where n/m is the numerical ratio
What are the first pair of matching harmonics for common musical intervals?
| Interval | Frequencies | First Pair of Matching Harmonics |
|---|---|---|
| Unison | f1 and f2 = f1 | f2 with f1 |
| Octave | f1 and f2 = 2f1 | f2 with 2f1 |
| Fifth | f1 and f2 = (3/2)f1 | 2f2 with 3f1 |
| Fourth | f1 and f2 = (4/3)f1 | 3f2 with 4f1 |
| Third (major) | f1 and f2 = (5/4)f1 | 4f2 with 5f1 |
| Sixth (major) | f1 and f2 = (5/3)f1 | 3f2 with 5f1 |
| Third (minor) | f1 and f2 = (6/5)f1 | 5f2 with 6f1 |
| Sixth (minor) | f1 and f2 = (8/5)f1 | 5f2 with 8f1 |
| Whole Tone | f1 and f2 = 1.123f1 | No matches |
| Semitone | f1 and f2 = 1.0595f1 | No matches |
Practice Problems:
1. If the harmonic frequencies of two tones align will it sound peasant to the human ear?
2. If the harmonic frequencies of two tones do not align will it sound pleasant to the human ear?
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