In Search of the Missing Fundamental: by Richard K. Jones
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A classic study on timpani harmonicity was conducted in 1973 by Arthur H. Benade of Case Institute of Technology/Case Western Reserve University, Cleveland, Ohio. In Cleveland’s Severance Hall, Benade measured the frequency ratios of the first ten components of a timpano (a personal instrument) owned by Cloyd Duff, timpanist of the Cleveland Orchestra. Duff owned Dresdner Apparatebau Anheier/Jähne & Boruvka timpani with calf heads. Benade did not indicate which timpano was used for the measurement.

Figure 3g is the set of frequencies that was reported by Benade for Duff’s instrument which was tuned to C3 130.8 Hz. The letters (beginning with P for the principal tone) represent the sequence of modes starting with P=mode (1,1). Q=mode (2,1), R=mode (0,2), S=mode (3,1), T=mode (1,2), U=mode (4,1), V=mode (2,2), W=mode (0,3), X=mode (5,1) and Y=mode (6,1). The preferred modes are P, Q, S, U, X and Y. Modes R, V, and W are the more damped non-preferred modes and so their respective theoretical ratios are not shown.

Fig. 3g

The missing fundamental effect might then give you the pitch C2 for the instrument under certain conditions and dynamic levels. 15

Benade did not comment on the amplitude or decay time of each of the preferred modes but it is clear that the principal tone, the fifth, the octave and the tenth (modes (1,1), (2,1), (3,1) and (4,1)) were very close to being harmonic. The near perfect harmonicity of modes 3,1 and 4,1 (the octave and the tenth above the principal tone) would strongly contribute the virtual pitch of the spectrum as well. A consistent clarity and harmonicity of the first few overtones is reported by many timpanists who prefer to use calf heads. Note: Comments on the consistent harmonicity throughout the range of a drum of modes (1,1), (2,1), (3,1) and (4,1) of calf heads (as compared to synthetic) is also mentioned by Fleischer & Fastl in their studies. 16

Figure 3h (click to enlarge) is a pitch-class equivalent chart of the preferred modes of Duff’s timpano after he had cleared the head. The sound file includes the frequencies of the preferred modes only.

Fig. 3h

Benade goes on to further explain Duff’s clearing process for achieving timpani harmonicity.

It is not sufficient merely to get the overall skin tension correct for the desired pitch of the kettledrum, one must also make small additional changes in the tension produced by the various screws around the periphery of the drum. Cloyd Duff has a particularly apt word to describe this process of subsidiary adjustment which compensates for the inherent irregularity of the skin and for the possible eccentricity of the kettle rim. When everything is in perfect adjustment, the drum is said to have been “cleared.” It is a revelation to listen to an expert such as Duff clearing a good drum, making the tone ring with smoothness and clarity. This clearing in fact is a process of persuading the partials to more closely match the ideal. As a matter of fact, Duff apologized for his drum’s lack of tonal clarity–a season’s hard use had battered the skin to a point where he no longer considered it possible to bring it into proper adjustment. 17

Cloyd Duff was an extraordinary musician who was known for his wonderful tone and impeccable intonation. His legacy continues through the students he taught at the Cleveland Institute of Music; many of which hold positions in the world’s finest orchestras. His term clearing is has now become synonyms with fine tuning the instrument. (see The Duff Clearing Process)

Cloyd Duff’s Jähne & Boruvka timpani now owned by Peter Kogan (retired)  of the Minnesota Orchestra