What’s Lurking in Your D & A?
The Hidden Major Second Inside a Timpani Perfect Fourth
In most traditional timpani literature using D and A, the A is the lower drum and the D is the upper, smaller drum. For that reason, this discussion treats A–D as a perfect fourth: A below, D above.
The harmonic interval of a perfect fourth has traveled through musical history, first as an interval of consonance, then as one of dissonance, and later moving back toward consonance. From an acoustical perspective, the perfect fourth can function as a sensory consonance. In the context of Common Practice Period harmony, however, it is often treated as a stylistic dissonance, especially when it appears above the bass in a two-voice texture or in the 6/4 position of a chord. In that context, it acts as a tension or 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. In Mozart’s timpani writing, the tonic–dominant relationship dominates, usually as alternating notes rather than harmonic double stops. 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 German Requiem). In the second movement, Denn alles Fleisch, es ist wie Gras (“For all flesh 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 function is to support the tonic chord, acting almost as a substitution for a perfect fifth. As composers’ harmonic language began changing at the beginning of the twentieth century, the perfect fourth began appearing more often as a harmonic device in timpani literature, for example at the very end of Rachmaninoff’s Piano Concerto No. 3 (1909), William Walton’s Symphony No. 1 (1935), and Aaron Copland’s Fanfare for the Common Man (1942).
The Perfect Fourth: Consonance, Dissonance, and Context
Hiding in every timpanist’s tuning of the perfect fourth is the potential for a strong major-second dissonance. This may help explain why more composers of the Common Practice Period did not write perfect fourth double stops for timpani. When the notes of the interval are struck in succession and both notes are allowed to ring, the ear may hear the result as slightly dissonant, even when the two principal tones are correctly tuned.
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 that damping is not feasible, the notes may blend temporarily, but when the tonic is finally reached, the dominant is usually damped. Many young timpanists find this interval difficult to tune accurately because it can sound somewhat discordant or annoying when played harmonically. There is an explanation for this based on acoustics and on the way timpani generate pitch.
Why a Harmonic Fourth Works
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 means that within the overtone series of each note, the fourth partial of the lower A series coincides with the third partial of the upper D series. If A is the lower note and D is the upper note, the fourth partial of A coincides with the third partial of D. That shared A helps explain why the interval can sound consonant in a harmonic system.
Harmonic Series of a Perfect Fourth Interval
Coincidental Harmonics 4:3
What Changes on Timpani
A timpano does not produce a true harmonic series in the same way a string or wind instrument does. Its pitch-bearing structure does not begin with a true harmonic fundamental. Instead, the perceived pitch is organized around the principal tone, mode (1,1), and the near-harmonic preferred modes above it. A timpani perfect fourth is therefore not a pure harmonic interval produced by two harmonic instruments. It is an agreement between two near-harmonic, missing-fundamental spectra.
In an A–D perfect fourth, the first strong coincidental upper partials are no longer the same as they would be in a true harmonic system. The A drum’s mode (3,1) may align closely with the D drum’s mode (2,1), which can make the interval seem as if it should be more consonant. However, a strong dissonance can also appear: the A drum’s mode (2,1), its fifth, produces E, which can clash with the D drum’s principal tone, mode (1,1). That creates a hidden major second inside what appears on the page to be a clean perfect fourth.
Adding to this is the fact that the effect of the bowl as a baffle changes the decay behavior of the timpano’s modes. The principal tone, mode (1,1), can decay sooner than mode (2,1), the fifth. When mode (2,1) remains too prominent, the fifth of the lower drum can dominate the decay and collide with the principal tone of the upper drum.
The Hidden Major Second
If the A drum is struck first and 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. The problem is not that A and D are necessarily out of tune. The problem is that the A drum may continue speaking E while the D drum is trying to establish D. The ear hears the major second E–D inside what looked like a consonant perfect fourth.
One must also bear in mind that the principal tone of the D drum will also fade. If the head is not tempered well, the fifth of that drum’s spectrum may also dominate, adding another layer of pitch instability.
Partial Tones of a Perfect Fourth on Timpani Struck Consecutively
The Principal Tone Decays, Leaving a Major-Second Dissonance
In a cadential context where D functions as the point of resolution, this lingering E can even behave like a subtle tendency tone resolving downward into D. From that perspective, timpani tuned to a perfect fourth can add a subtle acoustical reinforcement to the dominant-tonic tension/resolution relationship so common in older timpani writing.
Why the Overbearing Fifth Makes It Worse
Due to the effect of the baffle created by the bowl, not only are the lower concentric modes damped, such as mode (0,1), but the principal tone, mode (1,1), can also be shortened 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 spectrum. An overbearing fifth generates a pitch shift once the principal tone begins to decay. See Pleading the Fifth.
The fifth is not the enemy. It becomes a problem only when it outlives or overpowers the principal tone. In a perfect fourth, that distinction becomes especially important because the fifth of the lower drum can form a major second against the upper drum’s principal tone.
Waterfall 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 perfect fourth played harmonically on timpani can be difficult to tune when the drums have an overbearing fifth in the spectrum. When the lower A drum is struck first, its first strong upper preferred mode, E, can beat against the principal tone D of the upper drum. These two frequencies are only a major second apart. This is somewhat unavoidable with the harmonic interval of a perfect fourth, but it becomes much more noticeable when the lower drum’s fifth is too strong or the upper drum’s principal tone is weak.
If either drum is not behaving as one membrane, the hidden dissonance becomes more obvious. A fourth that sounds acceptable in single strokes may become unstable when both drums are allowed to ring. Worn or poorly tempered synthetic heads may exaggerate this effect because tension history, dimples, and bearing-edge crease fatigue can weaken mode (1,1) and allow mode (2,1) to dominate the decay.
How to Teach and Tune the Fourth
A way to make the hidden dissonance less noticeable is to temper the heads so that the fifth is not so strong. Rather than strengthening the fifth, temper the head so the principal tone is clear and the upper preferred modes, including mode (3,1) and mode (4,1), support the virtual pitch without allowing mode (2,1) to dominate.
When teaching timpani, it is best to teach the tuning of this interval as a melodic interval, not as a harmonic interval. Teach the fourth first as a melodic skip. The player should hear the identity and direction of the two principal tones before becoming distracted by the mixture of their spectra. Tune the interval quickly and efficiently; focus on the melodic motion, not the blended harmonic interference between the drums.
Learning to master your spectral and holistic listening skills will also help you hear and focus on the melody of the sound rather than the harmony of the sound.
When tempering timpani heads, it is imperative to adjust the frequency of the principal tone at each tuning lug so that it is supported by many strong near-harmonic partials. When the tension of the head is adjusted so that the secondary preferred modes reinforce the virtual pitch of the principal tone, mode (2,1) becomes less pronounced and less overbearing. The overall pitch will then be perceived as more harmonic in nature. On occasion, the missing fundamental can be perceived.
Final Practical Takeaway
On timpani, a perfect fourth is not just two notes. It is two near-harmonic spectra interacting. If the lower drum’s fifth lingers too strongly, it can create a hidden major second against the upper drum’s principal tone. The solution is not to eliminate the fifth, but to restore the authority of the principal tone on both drums so the interval is heard melodically and musically, rather than as a clash of lingering partials.