In Search of the Missing Fundamental: by Richard K. Jones
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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.

BReq
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. 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 timpanists find this interval somewhat discordant or annoying when trying to tune it or 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.

Overtones-4-3Harmonic 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.

If the “A” drum is struck first and is allowed to ring, the principal tone of that drum will begin to decay leaving mode 2,1 (e) to beat against the principal tone mode 1,1 of the “D” drum.  A major second dissonance is heard as the two pitches ring together. One must also bear in mind that the principal tone of the the “D” drum will also fade quickly as well. If the head in not tempered well, the fifth of the spectrum on that drum will also dominate.

Overtones-Missing-1-1Partial Tones of a P4 on Timpani Struck Consecutively
The Principal Tone Decays Leaving a M2 Dissonance

It is interesting to note that from a voice leading perspective, this e is a tendency tone that wants to resolve downward and does in this case, which only reinforces the Common Practice Period dominant/tonic (tension/resolution) relationship. The use of timpani in CCP literature only adds to the strength of the Dominant/Tonic cadential function, albeit in a very subtle manner when viewed from this perspective.

Due to the effect of the baffle created by the bowl, not only are the lower concentric modes damped (e.g. mode 0,1 the dissonant fundamental), but so also is the principal tone mode 1,1 to some degree. This can lead to what timpanists refer to as the overbearing fifth and pitch creep in the spectrum the longer the pitch sustains. When the frequency of the principal tone is not consistent from lug to lug, the overall strength of perceived pitch is severely diminished and permutations of the more audible mode 2,1 will tend to dominate the spectra. An overbearing fifth generates a pitch shift once the principal tone begins to decay. (see Pleading the Fifth)

Kolberg-Timpano-Spectrum1Waterfall chart (frequency, time and amplitude) of a timpano sound spectrum
(single struck note) highlighting six preferred modes (1,1), (2,1), (3,1), (4,1), (5,1) and (6,1)
(Fleischer & Fastl)

A harmonic P4 on timpani can be difficult to tune when the drums have the predominant fifth in the spectrum. When struck consecutively, the first overtone of the A drum, which is an E is beating against the principal tone D of your D drum. These two frequencies are only a major second apart. This is somewhat unavoidable with the interval of a P4.  A way to make it less noticeable is to temper the heads so that fifth is not so strong. Spend some time tempering to where there is a strong presence of the tenth above the principal tone.  Mode (4,1)

When tempering timpani heads, it is imperative to adjust the frequency of the principal tone at each tuning lug so that it includes many strong  near-harmonic partials. When the tension of the head is adjusted in such a manner that the secondary preferred modes are focused on creating  the virtual pitch of the principal tone (mode 1,1), mode 2,1 then becomes less pronounced/overbearing and the overall pitch will be perceived as being more harmonic in nature. On occasion, the missing fundamental can be perceived.