Defining A Standard

Can a timpano produce a genuinely pitch-centered, near-harmonic spectrum?

Yes, but the word harmonic requires some precision.

An ideal harmonic series consists of exact integer multiples of one fundamental frequency. A real timpano does not normally produce an exact harmonic series. Instead, several of its important sustained modes can lie remarkably close to the frequency relationships of a harmonic series.

For that reason, this WEBook uses the terms near-harmonic or quasi-harmonic when describing the preferred timpani modes.

One of the most influential historical examples comes from measurements reported by musical-acoustics researcher Arthur H. Benade on a timpano belonging to Cloyd Duff, principal timpanist of the Cleveland Orchestra.

Benade’s Measurement of Cloyd Duff’s Timpano

Benade reported a set of frequencies measured from Duff’s timpano when Mode (1,1), the principal-tone mode, was tuned to approximately C3 at 130.8 Hz.

Historical documentation associated with Duff’s instruments identifies his set with Jähne & Boruvka/Dresdner Apparatebau timpani fitted with natural-skin heads. Benade’s published frequency table does not identify which individual drum from the set supplied the measurement.

Using the mode identifications associated with Benade’s data, the important preferred diametric modes were:

Label	Mode	Measured Ratio to Mode (1,1)	Nearby Harmonic-Series Relationship
P	(1,1)	1.000	2nd harmonic of an implied lower fundamental
Q	(2,1)	1.504	Near 3:2
S	(3,1)	2.000	Near 2:1
U	(4,1)	2.494	Near 5:2
X	(5,1)	2.979	Near 3:1
Y	(6,1)	3.462	Near 7:2

The sequence is striking. Relative to an implied pitch one octave below Mode (1,1), these modes fall close to harmonics 2, 3, 4, 5, 6, and 7.

This does not mean that every component of the timpano spectrum is harmonic. Other membrane modes occur between these preferred modes and contribute to the attack and tone color. Their amplitudes and decay rates help determine how strongly they influence the sustained pitch impression.

Preferred Modes and the Sense of Pitch

The important result is not that every resonance becomes an integer multiple. It is that several strong and musically significant modes can organize themselves into a sufficiently regular frequency pattern for the ear to perceive a convincing pitch center.

Modes (1,1), (2,1), (3,1), and (4,1) are especially important because their frequencies in Duff’s measured spectrum fall close to the relationships:

1 : 1.5 : 2 : 2.5

Modes (5,1) and (6,1) extend the sequence approximately toward:

3 : 3.5

The result is not an exact harmonic series, but it is sufficiently close to support strong musical pitch perception.

Missing Fundamental and Principal Tone

Benade also pointed out that the preferred-mode sequence could support a missing-fundamental interpretation one octave below the Mode (1,1) principal tone:

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

This is one useful auditory interpretation of the spectrum, but it should not obscure the importance of Mode (1,1) itself.

Mode (1,1) is an important audible principal-tone anchor in normal timpani sound. Missing-fundamental perception can strengthen the organization of the spectrum, particularly as relative modal amplitudes change during the decay, but it is not necessary to imagine that the ear always replaces the audible principal tone with a completely absent lower fundamental.

The timpano’s sense of pitch arises from the combined organization of its preferred modes over time.

What Physics Produces the Near-Harmonic Sequence?

The frequency ratios of an isolated circular membrane are strongly inharmonic.

A timpano becomes much more pitch-like because the membrane is coupled to the surrounding air and to the air enclosed by the kettle. This head-air-bowl coupling shifts different modal frequencies by different amounts.

Mode (1,1) is shifted proportionally more strongly than several of the higher preferred diametric modes. The result can move the preferred-mode sequence toward the approximate relationship:

f₁₁ : f₂₁ : f₃₁ : f₄₁ ≈ 2 : 3 : 4 : 5

or, normalized to Mode (1,1):

1 : 1.5 : 2 : 2.5

This effect is a property of the coupled timpano system and is not unique to Duff’s instrument.

Benade’s Results in a Broader Experimental Context

Later experimental research demonstrated similar quasi-harmonic behavior on other timpani.

Christian, Davis, Tubis, Anderson, Mills, and Rossing found that for typical kettle enclosures, the ratios

f₁₁ : f₂₁ : f₃₁ : f₄₁

remain close to

2 : 3 : 4 : 5

over a substantial normal playing range. ^{Christian et al.}

Helmut Fleischer later obtained closely related results from experimental modal analysis of a large Kolberg timpano fitted with a synthetic head. Depending on tuning, the first preferred-mode ratios were measured near:

1 : 1.5 : 2 : 2.5

and remained approximately stable over a broad pitch range. ^Fleischer

Taken together, these measurements show that quasi-harmonic preferred-mode organization is a general feature that well-designed, properly functioning timpani can exhibit.

Head Material Matters—but It Does Not Create Harmonicity by Itself

Duff’s measured instrument used a natural-skin head, and many timpanists have strong musical preferences regarding natural and synthetic heads.

Head material can affect:

damping,
material uniformity,
environmental stability,
tactile response,
and the amplitudes and decay behavior of different spectral components.

These properties can strongly influence how a timpano sounds and feels.

Near-harmonic frequency placement, however, cannot be attributed simply to calfskin compliance, natural fibers, or one particular head material.

The experimental observation of comparable quasi-harmonic ratios on synthetic-head timpani shows that the central mechanism is the behavior of the complete coupled system: membrane tension and geometry, air loading, kettle geometry, damping, and structural condition all contribute.

Duff’s Clearing Process in Benade’s Account

Benade’s description of Duff’s clearing process is especially valuable because it documents what the player actually adjusted:

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. ^Benade

Several important points follow directly from this account.

Duff distinguished overall tuning from fine circumferential adjustment.
He used individual tension screws to compensate for real irregularities in the head and instrument.
The musical criterion was a tone that rang with smoothness and clarity.
The final result was limited by the physical condition of the head and drum.

Modern modal physics gives this empirical practice additional context. Nonuniform circumferential tension can break rotational symmetry, alter preferred modal orientations, and lift the degeneracy of modes with m > 0, producing frequency splitting. Reducing such asymmetry is therefore one physically plausible consequence of careful clearing.

Benade’s account does not identify lifted degeneracy as Duff’s explicit theoretical model. It documents the craft; modern vibration theory helps explain mechanisms through which that craft can improve the acoustic result.

Harmonicity and Degeneracy Do Different Jobs

Two ideas used throughout this WEBook should remain distinct.

Head-air-bowl coupling helps determine where the preferred-mode frequencies lie.

Degeneracy determines whether symmetry-related realizations within a modal family share the same frequency.

A drum can therefore have reasonably quasi-harmonic modal spacing while still containing acoustically significant splitting within one of those modal families.

Conversely, a modal family can remain nearly degenerate without its frequency being positioned at an exact harmonic ratio relative to the other families.

A musically convincing timpano benefits from both:

useful placement of the preferred modal frequencies, and
sufficient circumferential symmetry that those modal families remain stable.

What About Differences Among Historical Measurements?

Different experimental studies of timpani have produced somewhat different frequency ratios, amplitudes, and decay behaviors.

This is expected because the measured result depends on many variables, including:

drum diameter and kettle geometry,
head type and condition,
membrane tension,
circumferential uniformity,
strike location and excitation method,
measurement technique,
and the particular instrument being studied.

Benade later wrote to physicist W. E. Baylis that he stood by the accuracy of his Duff measurements and believed that tuning skill contributed to differences among measured instruments. The correspondence is valuable historical evidence of Benade’s interpretation of the discrepancy. Read Benade’s letter to Baylis.

Modern measurements show that near-harmonic preferred-mode relationships do not depend on one player, one manufacturer, or one head material. They are reproducible physical features of well-functioning timpani, although the exact ratios and acoustic prominence of the modes vary from instrument to instrument.

Defining the Standard

Benade’s Duff measurement remains an exceptionally useful historical example because it captures a professionally prepared timpano in the hands of one of the twentieth century’s most influential timpanists.

But a useful acoustical standard should describe properties of the sound, not require every instrument to reproduce one historical spectrum exactly.

For the purposes of this WEBook, a well-tempered and well-cleared timpano should aim for the following:

Near-harmonic preferred-mode placement: the important sustained modal families lie near relationships such as 1 : 1.5 : 2 : 2.5, with higher preferred modes continuing the pattern approximately.
A convincing principal-tone identity: Mode (1,1) and the surrounding modal spectrum support a clear musical pitch.
Circumferential stability: changing strike position does not produce distracting changes in pitch identity.
Minimal acoustically significant splitting: symmetry-related components do not produce objectionable beating, shimmer, or pitch drift.
Coherent temporal behavior: although the relative strengths of the modes change during the decay, the note maintains a convincing musical identity throughout its useful life.
Robustness across useful dynamics: stronger excitation may reveal more of the modal spectrum without causing the perceived pitch to lose focus.

This standard is not mathematical perfection.

It is a physically informed definition of musical stability.

Benade showed how close an expertly prepared real instrument could come to a compelling quasi-harmonic spectrum. Later experiments show that the underlying physics is broader than one drum or one player. Duff’s contribution was to develop an empirical way of hearing and adjusting the instrument until that physics served the music.

To discuss that process precisely, we need a shared vocabulary. Terms such as mode, node, preferred mode, degeneracy, lifted degeneracy, and missing fundamental allow us to distinguish what is measured from what is heard—and what is heard from what the player physically changes.

That vocabulary gives us the bridge from Benade’s measurements to the Duff Clearing Process and the modal physics that can help explain it.

Pitch Without Harmonics

Explaining Some Jargon