Timbre and spectra

How string instruments work: 8

Timbre

The specifics of each string instrument make its sound distinct from that of other string instruments. This unique quality of sound of each instrument or voice is known as timbre. Timbre incorporates all acoustic characteristics of sound that do not directly correspond to loudness and pitch. It is the final piece of the puzzle that builds on the previous articles about how string instruments work.

The strings and even the entire instruments are of different sizes and are made from a huge range of materials, ranging from wood, steel, and brass to pumpkin shells, plastic, and nylon. Many instruments have a resonator that acts like a filter, amplifying some frequencies and attenuating others. The characteristics of such a filter depend on its size, shape, and material. The quality of sound of an instrument is also dependent on the craftsmanship of the manufacturer.

Apart from the construction of instruments, even the method of playing contributes to the timbre. For example, some string instruments are bowed (e.g., sarangi and violin), some are plucked (e.g., veena and guitar), and some are struck (e.g., santoor and piano). This also gives rise to different quality of sound in each instrument. Even the presence or absence of frets on an instrument, and whether the style of music freely uses ornamentation, affect some higher-level qualities of sound.

Harmonics are partially responsible for what makes an instrument’s sound unique. The relative amplitudes (or volume, ‘strength’) of each harmonic are different for each string instrument, and this makes the sounds diverse and unique.

Discrete spectrum

The relative strength of each of these harmonics can be visualised on a histogram like this one.

The amplitude of discrete frequencies (usually only frequencies corresponding to the harmonics) is plotted in a discrete spectrum. Image source: Simon Fraser University

The x-axis represents frequency, and the y-axis represents the amplitude (or relative ‘strength’ or loudness of each frequency component). Remember that when a string vibrates, several wavelengths/frequencies are present in it simultaneously. The amplitude of each of these frequencies is represented on the spectrum. This is comparable to splitting sunlight into a rainbow through a prism.

Continuous spectrum

A real sound doesn’t have just the harmonics. Because numerous waves are generated on plucking, striking, or bowing a string (and many of them die out in a short while), and because the instrument’s design may allow various non-harmonic frequencies to resonate, the sound of a real instrument has much more than just the harmonics. A more real picture would be obtained by visualising the relative strengths of all frequency components. This is known as a spectrum. It represents a snapshot of a sound at an instant in time.

The amplitude of all frequencies are plotted in a continuous spectrum. Usually, for pitched sounds such as a plucked string of an instrument, the peaks in the spectrum correspond to the harmonics, whose frequencies are integer multiples of the fundamental frequency. Adapted from the University of New South Wales

Spectra and timbre

Usually, instruments have distinct spectral features, and this directly contributes to the distinct timbres. For example, a clarinet’s sound is rich in odd harmonics, whereas a saxophone’s sound has both odd and even harmonics. Some instruments have the fundamental as the most prominent frequency, whereas some have the second or the third harmonic as the most prominent one. Sometimes, the fundamental frequency is absent, and is only inferred by our brains while listening. The presence of inharmonic partials (frequencies that do not correspond to the ‘expected’ series of harmonic frequencies) also lends unique qualities to any instrument. The following spectra illustrate this distinction.

Spectra of different instruments playing the same note, A4. Image source: University of Colorado Boulder

Time evolution of spectra

The above spectra tell us how much of each frequency is present in a sound at a given instant in time, but doesn’t give any information about how this changes over time. For this, we can look at a series of spectra across time, which lets us see how different frequency components evolve over time.

Evolution of spectrum with time, as we pluck a guitar string. The vertical axis shows intensity (relative strength) of the harmonics. All plucked instruments such as guitars, pianos, and harps show a similar response. Image source: Florida Atlantic University

In the above time-evolving spectrum, we not only see overall spectral characteristics (e.g., the relative strength of harmonics beyond the fifth one are low), we also see that most of the higher frequencies do not sustain for long, whereas the first one (the fundamental frequency) persists the longest. We also see some non-harmonic components in the beginning, which quickly disappear with time.

Spectrogram

Another way to visualise this in a 2-d format is to represent intensity using colour. This is called a spectrogram. In the spectrogram below, the horizontal axis represents time, and the vertical axis represents frequency. The intensity of any frequency at a given time is given by the intensity of the yellow colour. The parallel yellow streaks usually indicate the harmonics, which are all separated by a constant value f₀. We can see how each of these frequencies evolve over time. We can also see if the note being played changes over time. In this image, for example, there is some upward movement in the beginning where all harmonics slide upwards, followed by a vibrato where the harmonics oscillate slightly.

Spectrogram of a single pluck of a digital guitar (played using GeoShred). Point a marks the note onset. The pitch changes from a to b, and it remains stable until c but with a vibrato. Higher harmonics fade away after a while, before point c.

Envelope

Another way to visualise sound is to plot the vibrations directly (as opposed to plotting its spectrum). The vibrating string transmits its vibrations to the surrounding air, which then travel to the listener in the form of sound waves. This sound represents a superposition of all the modes of vibrations. The x-axis represents time, and the y-axis represents amplitude. In this context, amplitude represents the degree to which air pressure at a given point varies because of the wave.

A ‘ting’ sound is produced on plucking or striking a string. This corresponds to the burst of all frequencies, most of which quickly die out and only the harmonics survive, which we identify with a clear pitch that follows the initial ‘ting’. The characteristic evolution of the sound through time represents another important aspect of timbre, known as envelope. This is the outer shape of the sound signal as visualised in this illustration. Notice the variety of envelopes displayed by different instruments.

The audio signals at the top (blue) help us visualise the envelope of the sound. The spectrograms at the bottom (black and white) give information about the frequency components of the sound through time. The sound of a piano decays quickly with time, whereas that of a flute, trumpet, or violin is sustained for longer. Source: Fundamentals of Music Processing; accessed from International Audio Laboratories, Erlangen

The envelope of a sound signal can be understood in terms of attack, decay, sustain, and release.

Envelope characteristics. Image source: zytrax

Modulation: vibrato and tremolo

Other factors such as modulation also affect the timbre of a sound. Frequency modulation in the right frequency range gives rise to vibrato, and amplitude modulation in the right frequency range is perceived as tremolo. These features of sound are part of timbre because they don’t directly affect the pitch or loudness of a sound. However, these are more relevant for musical sounds than for all sounds in general.

In summary, timbre is a complex quality of sound and is affected by several factors, including spectral components, envelope, and modulation.