Frequency, Pitch and Timbre
Complex tones, or composite waveforms (both harmonic and inharmonic), consist of many sine waves of different frequencies added together. These individual sine waves are called frequency components. Musical sounds generally contain more than just one or two components; sounds with only a few frequency components tend to lack richness and are usually not perceived as musically interesting. They have little or no timbre (tone colour). Conversely, sounds with an extremely large number of frequency components—such as the noise of a strong windstorm with rain—may be interesting or even pleasant, but because they lack a definite pitch, they are rarely considered musical “notes.”
When someone sings a note or plays a note on an instrument, a very specific set of frequencies is produced. You can visualize each note as a smooth mixture of many sine‑wave components. These components are called overtones or partials. Ideally, they are harmonic (integer multiples of a fundamental frequency), but they may also be inharmonic. The human auditory system blends these components together so effectively that they are not heard as separate tones. Instead, they determine the timbre or tone colour of the sound. Notes containing many inharmonic overtones exhibit what is known as inharmonicity.
Inharmonicity refers to the degree to which the frequencies of overtones deviate from whole‑number (integer) multiples of a fundamental. Inharmonic overtones are often distinguished from harmonic ones by calling them partials, although the term “partial” technically includes both. Whether we perceive a sound as pitched or unpitched depends strongly on the harmonicity of its overtones: the greater the inharmonicity, the less definite the perceived pitch. Many percussion instruments—such as cymbals, tam‑tams, and drums—produce strongly inharmonic spectra, yet these contribute desirable colour and texture to music. By contrast, modern wind, brass, and string instruments are designed to minimize inharmonicity and enhance harmonicity. In the hands of a skilled performer, the timpani can bridge the gap and exhibit traits of both.
If an oboe plays middle C (C4, 261.63 Hz) and a clarinet plays the same note at the same loudness, it is still easy to distinguish one from the other because each instrument has a unique timbre. Timbre arises from each instrument’s characteristic spectrum of overtones—its harmonic recipe. This recipe includes the number of harmonic and inharmonic partials and their amplitude ratios relative to the fundamental, along with transient features such as attack and decay. These spectral differences allow us to recognize one instrument from another even when they play identical pitches.
The chart below compares the first sixteen harmonic partials (and their amplitude proportions) of a clarinet and an oboe playing the same pitch. The clarinet produces a strong fundamental and emphasizes the odd harmonics, whereas the oboe exhibits relatively weak energy at the fundamental but strong fourth, fifth, and sixth partials.
The oboe has very little acoustic energy at the actual fundamental frequency, yet we still perceive the correct pitch. The same is true for the bassoon. Even when the fundamental frequency is absent, the ear reconstructs the pitch based on the spacing of the overtones. This phenomenon is known as the missing fundamental or virtual pitch.2
A similar psychoacoustic effect plays an important role in timpani pitch perception. For timpani, the perceived “fundamental” is often the second partial of a non-harmonic overtone series. Thus, when a timpanist tempers or “clears” the drumhead, the goal is to align certain partials so that the auditory system constructs a virtual fundamental that is not physically present.
Chapter 2 will explore the physics of circular vibrating membranes and explain how their vibrational modes contribute to timpani sound production. Before that, the next section reviews key terms introduced in Chapter 1. A solid understanding of these concepts will enhance the reader’s ability to analyze and appreciate timpani tone.