Physics of the 4th
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The harmonic interval of a perfect fourth has traveled through musical history, first as an interval of consonance, then to one of dissonance and then moving back to one of consonance. From an acoustics perspective, the perfect fourth is a sensory consonance, however, in the context of Common Practice Period harmony it is considered a stylistic dissonance, especially when it appears in a two-voice texture above the bass (e.g. 6/4 position of a chord). In this context, it is a tension/suspension device and seeks resolution. The alternating perfect fourth is ubiquitous in Common Practice Period timpani literature and was the bread and butter of early timpani culture for centuries. It was the only interval that Mozart wrote for his timpani parts, but he never wrote it as a double stop, only as alternating notes. Johannes Brahms, on the other hand, made use of the perfect fourth double stop in his monumental Ein deutsches Requiem, nach Worten der heiligen Schrift Op. 45 (A Greman Requiem). In the second movement, Denn alles Fleisch, es ist wie Gras (For all flesh, it is as grass), Brahms expertly uses this device to enhance the harmonic tension.
Johannes Brahms: Denn alles Fleisch, es ist wie Gras
Verdi did write perfect fourth double stops at the end of the Dies Irae in his Requiem, however they are marked pianissimo and are not used as a tension/suspension device. Their only function is to support the tonic chord, acting as a substitution of a perfect fifth. As composers’ harmonic language began changing at the beginning of the twentieth century, the perfect fourth began showing up as a harmonic device in timpani literature. e.g. the very end of Rachmaninoff’s Piano Concerto No. 3 (1909), William Walton Symphony No. 1 (1935), Aaron Copland Fanfare for the Common Man (1942).
Hiding in every timpanists tuning of the perfect fourth is the potential for a strong major second dissonance. This may be why more composers of the Common Practice Period did not write for perfect fourth double stops. When the notes of the interval are struck in succession and both notes are allowed to ring, the ear tends to hear the result as being slightly dissonant. This repeated dominant-tonic pattern, which often occurs at the end of a work in the Common Practice Period, is usually damped between notes and not allowed to blend together. When the tempo is fast enough where damping is not feasible, the notes blend together, but when the tonic is finally reached, the dominant is damped. Many young timpanists find this interval difficult to tune accurately. They find it somewhat discordant, or annoying when it is played harmonically. There is an explanation for this based on acoustics, and the physics of how the instrument generates pitch.
In a harmonic vibrating system, the interval of a perfect fourth corresponds to a pitch ratio of 4:3. From the perspective of coincidental harmonics, this also means that within the overtones series of each note, the fourth partial of the A, A, e, a, series will coincide with the third partial of the D, D, a, d series.
Harmonic Series of a Perfect Fourth Interval
Coincidental Harmonics 4:3
Since a timpano’s spectrum consists of only a quasi harmonic series with a missing fundamental, the first coincidental partials are now modes 3,1 and 2,1 (A, e, a – D, a, d), which by nature sounds like it might be more consonant since 3:2 is the ratio of the perfect fifth. However, a strong dissonance of the major second now exists between modes 2,1 and 1,1 respectively. Adding to this is the fact that the effect of the bowl as a baffle decreases the decay times, especially the modes of the lower frequencies, which means that mode 1,1 will decay faster than mode 2,1 (the fifth) leaving it to dominate the spectrum.
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