Musical sounds, musical instruments and musical signals

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Sound, vibrations and resonance

Sound is a longitudinal wave, which means that the vibration of the air is in the direction of travel of the sound wave.



Sound is invisible, so we have to animate to see it. (For film clips of visible examples, see Travelling Waves). In the animation below, the top graph shows a sine wave y(x'), where x' is a coordinate moving with the wave. On the axis below, we turn this into a longitudinal wave by rotating the arrows clockwise, so positive displacement is now to the right and negative to the left. We then apply these displacements to the 'particles'. This gives regions of high density (particles close together) and low density, which is plotted on the next axis. In the final graph, the density is represented in colour, to give an exaggerated picture of a sound wave at a moment in time.

This animation is one of several from Sound, pressure and density. See also this multimedia introduction to sound.

Resonance refers to the capacity of some systems to store energy in vibrations at a particular frequency. The picture at right is a Chladni pattern of one of the resonances of a violin back. The resonances of the violin body are vital to the operation and timbre of the instrument. Oscillations and resonance are a huge topic in physics, engineering and music acoustics. Here are an Introduction to resonance and a multimedia introduction to oscillations.

Frequency and pitch

To increase the pitch by the same musical interval, we must increase the freqeuncy by the same factor. Mathematically, this means that the pitch depends on the logarithm of the frequency. In the film clip below, from Sound: frequency and pitch, we first double the frequency from 500 to 1000 Hz (one octave), then from 1000 to 2000 Hz (also one octave). For nearly all listeners, the pitch interval between 1000 and 2000 Hz is same as that from 500 to 1000 Hz. Any doubling of frequency corresponds to an octave increase in pitch, no matter what the initial frequency.

More explanations, sound files and film clips are given on Quantifying Sound and Sound Frequency and Pitch. A table of frequencies, note names and MIDI numbers is given below. Conversion in either direction is available from this site.



Before leaving this section, it is worth noting that frequency and pitch can only be determined precisely for sustained, steady tones. Musicians know that they must be more careful with the tuning of long notes and chords than the notes in rapid passages. This effect is a fundamental property of waves, which is explained using sound files in The Musician's Uncertainty Principle. (That page also relates it to Heisenberg's Uncertainty Principle.)

Rhythm, time and frequency; muscles, nerves and hearing

Pitch depends on frequency, measured in vibrations per second. Tempo depends on frequency, too, although it is usually measured in beats per minute, which is the marking on a metronome. So, pulses of air arriving at the ear 440 times per second is the note A4, while pulses of air arriving at the ear 3 times per second is presto. Why is there such a huge subjective difference? Ultimately, the distinction comes from our nerves. Above a few tens of vibrations per second, we can no longer distinguish individual vibrations and our sense of pitch takes over. Explanation with sound files and animations is given in Human Sound.

Intensity, pressure and loudness

Sound levels are reported in decibels. The decibel (dB) is a logarithmic unit. The sound pressure level (L_p) in decibels is defined as
L_p = 20 log₁₀ (p/p₀),
where the reference level is chosen as p₀ = 20 μPa. Consequently, a sound pressure of 20 μPa has a sound level of 0 dB (close to the limit of hearing). At the other extreme, a sound pressure of 20 Pa has a level of 120 dB, which is painfully loud.

In this recording (fromWhat is a decibel? ), the first sample of sound is white noise (a mix of all audible frequencies, just as white light is a mix of all visible frequencies). The second sample is the same signal, but with the voltage reduced by a factor of the square root of 2. The reciprocal of the square root of 2 is approximately 0.7, so -3 dB corresponds to reducing the voltage or the pressure to 70% of its original value. The green line shows the voltage as a function of time. The red line shows a continuous exponential decay with time. Note that the voltage falls by 50% for every second sample. What is a decibel? relates sound intensity and pressure and frequency to phons, sones, loudness and the frequency response of the ear. You can also measure your own hearing response using this Hearing Test on-line.

Pure tones, harmonics, periodic and non-periodic sounds; Timbre, spectrum, envelope and transients

A tuning fork produces a pure tone: it produces a sound pressure that varies sinusoidally with time. A more complex sound may be considered as the sum of many sine waves, and the sound spectrum may be thought of as a 'recipe' for creating that sound: each point on the graph of the spectrum tells you how much sine wave at that frequency is present. This graph shows that process for a harmonic or periodic wave, comprising six harmonics.

The first six harmonics of a sawtooth wave, sounded one at a time.

Added one at a time, we can hear the individual 'notes' in a 'chord'. If, however, we hear a series of notes, each containing several harmonics, it is much more difficult to distinguish the individual harmonics. This is demonstrated in the example below: all notes have six harmonics in the same proportions.

These examples are among those given on What is a sound spectrum? and on Spectrum, harmonics and timbre.

Timbre depends somewhat on spectrum. But it depends very strongly on how a note varies in time. For a demonstration, see Timbre and envelope. See also the multimedia presentation on Quantifying Sound.

Harmony, scales and temperaments

Major thirds in just intonation, with frequency ratio 5:4, are very pure and their Tartini tones are exactly in tune. But what happens with modulation into different keys? Begin with a major scale in G major: G A B. Then we use that B to play the first three notes in the scale of B major: B C# D#. Then we use that D# to play the first three notes in the scale of D# major: D# E# Fx, where Fx means F double sharp.

The sound file will convince you that, with this just intonation, F double sharp is not the same note as G. Alternatively, if you said (quite reasonably) that there is no key of D# major, then we could have used E flat, F and G as the final third, and proved that E flat and D# are not the same note in just tuning.

Tartini tones and temperament: an introduction for musicians is an easy introduction. For more detailed explanation and examples, see Tartini tones and temperament, and the multimedia presentation Interference and consonance.

Music as a signal

We classify signals into analog and digital. An analog quantity varies smoothly. Examples are the position of a loudspeaker cone, the vibrating bridge of a string instrument and the voltage produced by a microphone. Today, microphone voltages are digitised on recordings: at regular time intervals the voltage is recorded as a number. (Typically, it is recorded 44100 times per second, with 16 or 24 bit precision, i.e. in 16 or 24 binary digit numbers. 16 bits gives a dynamic range of 96 dB. 44100 samples per second allows it to reproduce frequencies up to 22.05 kHz, and so include nearly all the human hearing range, except perhaps for the highest frequencies hear by children.)

Music notation is a digital signal, in that pitches have specific values and notes have specific lengths. In note processors such as Sibelius, both are stored as numbers. This coding is extremely efficient: a written score contains much less information than the recording of a performance. This photo, from a paper by the author about information creation and storage in music, shows four digital storage media: a pianola roll, a music box drum, the score of Stravinsky's Le Sacré du Printemps and a compact disc.

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An overview of musical instruments

Many musical instruments depend for their operation on standing waves. This simple animation (from Strings, standing waves and harmonics) represents one dimensional waves that are used to model both string and wind instruments. The green wave is travelling to the right, the blue to the left. Together, they add to form the red standing wave, with nodes (points of no motion) at the vertical lines.

The next two animations (from Open vs closed pipes) show standing waves caused by reflections at the ends of two different pipe configurations that are used as very simple approximations of a flute (open at both ends) or clarinet (open at one end).

But there is very much more to musical instruments than these very simple models would suggest. For an introduction, try our web site: Music Acoustics and its pages introducing different families of instruments.

DC-AC conversion Bowed strings, wind instruments and the voice can produce steady sustained tones, because they can convert a steady ('DC') supply of energy from the arms or breath into that of oscillatory ('AC') motion. This observation is of fundamental importance in music acoustics because it gives rise to harmonic spectra and thus to harmony. Nearly harmonic spectra can be produced by plucked and struck strings, but this is, comparatively, a very recent invention. The animation at right shows how a bow, travelling with a steady speed (DC), excites a vibration (AC) in a string. (From Bows and strings.) The lateral force between bow and string is related to their relative speed and other variables by equations that are nonlinear: the force is not proportional to the speed (or position). Similarly, the relations between the vibration of a brass player's lips, a woodwind reed, a flute's air jet or a singer's vocal folds and the air flow past or through them are nonlinear (meaning that the change in air flow is not proportional to the change in the other variable). Without going into mathematical details, we note that this nonlinearity has the effect of producing periodic vibration with high harmonics, and the presence of harmonics has the important musical consequences noted above. The graph below is a schematic of air flow vs time past a vibrating reed for successively higher dynamic levels. Vibration up and down the bold line converts DC flow into AC flow in reed instruments. More details at Introduction to clarinet acoustics.

Strings

String instruments use highly uniform strings that, usually, a very flexible. A consequence is standing wave resonances that fall very close to the harmonic series.

The first twelve harmonics on a C string. In the sound file, notice that the seventh and eleventh harmonics fall about halfway between notes on the equal tempered scale, and so have been notated with half sharps.

A sketch of the first four modes of vibration of an idealised stretched string with a fixed length. The vertical axis has been exaggerated.

These graphics and sound files come from Strings, standing waves and harmonics. See also Bows and strings and Violin acoustics and Guitar acoustics for an introduction to those families.

Wind

In woodwind and brass, the reed, airjet or player's lips provide the non-linear element (somewhat analogous to the bow in the string family). Instead of a vibrating string, the resonator is a tube of air, which has a number of different configurations and geometries.

For an introduction to woodwind and brass, see the various different instrument families on Music Acoustics.

Impedance matching: getting sound power out

A vibrating string on its own tends to slip through the air without transmitting much power: that's what the bodies of violin, guitar etc are for. Technically, this is called impedance matching. Standing sound waves in a narrow pipe also don't radiate well: that's what the bells of brass instruments and some woodwinds are for. Listen to the differences between these 'instruments'.

    A cylindrical pipe, 110 cm long. (No mouthpiece.)

    A 110 cm pipe, including flare and bell. (No mouthpiece.)

See Bells for more information.

Voice

The vocal folds in the larynx vibrate a little like a trumpet player's lips. The big difference, however, is that the pitch of the brass instrument is controlled by the instrument, whereas the singer's vocal tract never determines the pitch in normal playing. Rather, the tract resonances preferentially 'boost' some bands of frequencies, called formants, in the output sound.



A schematic of the source-filter model, from Wolfe et al (2009). The periodic spectrum at left corresponds to normal speech and singing. See also practical examples of the source-filter model, with measurements and sound files.

For a multimedia introduction to the voice, see Human sound and more technical details in An introduction to voice acoustics.

More resources

• Music acoustics with many pages devoted to instruments and to diverse topics in music science.
  
• Physclips: Waves and sound: a multi-level, multimedia introduction.
• FAQ in music acoustics
  
• Measure your own hearing response
  
• A list of other educational web sites from this author
  

© 2011, JoeWolfe