Hot Pans - Stockholm Steelband
© Ulf Kronman, The Pan Page. Publisher: Musikmuseet, Stockholm, Sweden.

Acoustic Function of the Steel Pan

This section presents measurements and theoretical models from my research on the steel pan. The presentation is complemented by some speculations on the relation of the measured data and the models to the practical tuning work and the design of the steel pan. Since my research only has been going on for a short time, this section is still a bit thin, raising many questions that have to be left without answers.

It is not necessary to read this section to use the handbook for steel pan making and tuning. But if you are curious and want to know why you have to perform the various steps during the making of a pan, this part may provide some explanations. For the experimenting panmaker, this section is intended to provide some theoretical insights to guide the further development of the tuning techniques and the instrument.

To be able to discuss the theoretical aspects of the steel pan later, the section begins with a chapter on music acoustics.

19. Music acoustics

Sound

Physically, sound is defined as small, periodical variations - vibrations - in the air pressure. The pressure variations are produced by small differences in the density of the air. The vibrations are transferred to the air when a vibrating body (such as a musical instrument) pushes the molecules of the air.

A musical sound can be described in terms of three fundamental characteristics: loudness, pitch and timbre.

The loudness represents the perceived strength of a sound and it is denoted by the sound pressure level, which is measured in decibels (dB).

The pitch is related to the physical measure frequency. The frequency of a sound describes how many times the air molecules are pushed back and forth each second. Frequency is measured in Hertz (Hz).

If the frequency of a single tone is related to its perceived place on a musical scale, the result is called the pitch of the tone. The pitch of a tone is denoted as C, C#, D, D#, etc. If more accuracy is needed, the octave position of the pitch is also included as a number; C1, F#3, G6, etc. The usable range of musical pitches extends from C0 (16 Hz) to C9 (8372 Hz).

The timbre of a sound is a much more complicated characteristic than the loudness and the pitch. It describes the tone "colour" - the character of the sound, and there is no single physical entity that can describe it fully. The timbre can roughly be described by the wave-form, as seen when analyzing the sound with an oscilloscope, see fig. 19.1. The more rugged the wave-form, the more brilliance and treble in the sound.

Photo

Fig. 19.1 Wave-form of a steel pan tone.

Musical instruments usually produce simultaneous vibrations of several different frequencies, which are added together to form what is called a complex tone. The parts of a complex tone are called partials. Each partial is emanating from a vibrational mode of the instrument (see below). The partials are numbered one, two, three, etc., from bottom to top. Fig. 19.2. shows the resulting wave-form when several partials are added together to form a complex tone.

Photo

Fig. 19.2 Adding of partials in a complex tone.

The process of dividing a complex tone into its individual partials is called a spectral analysis. Nowadays, computers can do a spectral analysis in a matter of seconds. The result of a spectral analysis is a diagram, showing the frequency and the relative intensity of the partials. This is called a frequency spectrum, or a spectrogram. The spectrogram can reveal a lot about the timbre of the sound. Fig. 19.3 shows a spectrogram of a typical steel pan note.

A complex sound that is perceived as a tone is usually built up of a number of partials that are equally spaced on a frequency scale. This means that the frequencies of the higher partials are multiples of the lowest one. This is called a harmonic series, and the tone is therefore called a harmonic tone. Most musical instruments generate harmonic tones. As an example, a harmonic steel pan tone with the lowest partial at 200 Hz has upper partials at 400, 600, 800, 1000, 1200 Hz, and so forth.

When we listen to a complex tone, we unconsciously analyse the sound and split it into its partials. The lowest partial of a harmonic sound is what we perceive as the pitch of the tone, i.e., the note's position on the musical scale. Therefore, the lowest partial is called the fundamental. The higher partials are often called overtones to the fundamental.

It is important that the overtones have a harmonic relationship to the fundamental, because we use them to perceptually define the pitch of a complex tone. A sound with many non-harmonic partials will be perceived as dissonant and vague in pitch. Partials that do not fall within the harmonic series are called disharmonic, and do not contribute to the sense of pitch in the tone. Disharmonic partials instead add more character to the timbre of the sound.

Photo

Fig. 19.3 Frequency spectrum of a steel pan tone.

Vibrations in bodies

The vibrations in a body are generated by two fundamental properties: the mass of the material and a tension that tends to restore the material to its rest position if it is disturbed. When an instrument is exited (hit), the interaction between the mass and the tension will generate sound waves passing through the body.

When the sound waves reach a point of difference in the density of the medium they are travelling in, they are reflected. When the reflected sound wave is travelling back and forth in the body, the sound energy will add at some points and cancel at some other points. This phenomenon is called a standing wave. In the points where the wave energy cancels, the surface of the body remains at rest. These points are called nodes. At the points where the energy adds, the body is vibrating fiercely. These parts are called anti-nodes.

The resonance of a body makes it respond with certain standing-wave vibrations when hit. These vibrations are called the normal modes (or eigenmodes) of the body. Each normal mode has its corresponding resonance frequency, producing a specific partial in the frequency spectrum of the note emitted from the vibrating body.

Partials

As seen from above, the normal modes of a vibrating body generate partials. The distribution of these partials determines whether the sound can be considered to be harmonic or not; if the partials are evenly spaced, the tone sounds harmonic. To be able to discuss the importance of harmonic partials later, we need to take a first look at their musical significance.

Fig. 19.4 shows an example of the partial distribution in a harmonic tone with a fundamental at 200 Hz.

Photo

Fig. 19.4 Harmonic partials of a complex tone.

The table below shows how the same partials can be looked upon in a musical situation. The first row shows the partial number, the second its frequency, the third shows the step between two successive partials, and the fourth shows their musical interval related to the fundamental. The table shows that the harmonic partials all have a relationship to the fundamental that can be derived from the musical scale. For instance, each doubling in frequency represents an octave step.

Partial

Frequency

Step interval

Interval relation to fundamental

1

200 Hz

Fundamental

2

400 Hz

Octave

One octave above the fundamental

3

600 Hz

Fifth

One octave and a fifth above the fundamental

4

800 Hz

Fourth

Two octaves above the fundamental

5

1000 Hz

Third

Two octaves and a third above the fundamental

6

1200 Hz

Minor third

Two octaves and a fifth above the fundamental

Musically, the intervals that sound most harmonic are, in falling order, the octave, the fifth and the third. If another note that has a pitch that is an octave or a fifth above the fundamental sounds together with the hit note, it will fall into its harmonic spectrum and support it. This has implications for the design of steel pans, which will be seen later in the chapter about layout.

Musical instruments

All musical instruments have in common that they convert kinetic energy from the player into acoustic energy, i.e., to vibrations it the air. The player can transfer the energy to the instrument by blowing, bowing, plucking or hitting. In the search for a model of the tone generation in the steel pan, only instruments that are hit - percussion instruments - are of further interest.

Percussion instruments may employ strings, bars, membranes, plates or shells as sources of sound. The listing of instrument types below has been made according to the shape of the vibrating part of the instrument. It represents a scale of increasing complexity in tone generation, leading from the simple string of a guitar to the complex shell of a steel pan note.

String and bar instruments

The string is the simplest example of a vibrating body. The restoring force of the vibrations is supplied by the tension of the string. The frequency of the string vibrations is determined by three factors: The length of the string - a longer string yields a lower tone. The tension - higher tension yields a higher tone. The mass - a heavier string gives a lower tone. Strings generate harmonic overtones automatically. Fig. 19.5 shows the standing wave patterns of the lowest normal modes in a string.

Photo

Fig. 19.5 Vibrational modes in a string.

For musical purposes, a bar may be seen as a string with one new property; its stiffness. The stiffer the material of the bar, the higher the frequency will be. In principle, the bar is acting like a string, but the restoring force in the vibration is supplied by the stiffness of the bar instead of the tension. The partials of bar instruments are not harmonic, but the lower partials can be tuned by thinning the bar at appropriate places. Examples of bar instruments are xylophones, vibraphones or triangles.

Membrane and plate instruments

A membrane may be thought of as a two-dimensional string. The vibration of a membrane is basically the same as the string, but here the vibrations can extend in several directions. In principle, a rectangular membrane acts as two independent strings; one along the length of the membrane and one across it. This means that there will be independent vibrational modes in both the directions, see fig. 20.2.

The partials of a vibrating membrane are seldom harmonic - they depend on the tension and the relative length of the sides of the membrane. Drums are the best examples of membrane instruments.

The next step in complexity is to make the membrane stiff. This yields a stiff plate. Vibrating plates bear the same relationship to membranes that vibrating bars do to strings. The restoring force is produced by the stiffness of the material, instead of the tension. The normal modes for a rectangular plate are approximately the same as for the rectangular membrane. The partials of a plate are not harmonic.

Shell instruments

Finally, if we make the plate slightly curved, we arrive at a physical shape that resembles a steel pan note. Theoretically, the note dent in a steel pan may be seen as a shallow shell. "Shallow" here means that the height of the arch is small compared to the size of the dent. The new acoustic factors introduced by the curving of the shell are the rise and the shape of the dent. Further, the fastening of the note in the pan surface makes it possible to introduce a tension in the dent.

The factors affecting the frequencies of the vibrations in a steel pan note represent the sum of all the factors of the less complicated resonators mentioned above, plus the two extra introduced by the arching:

The tension in a steel pan note is presumably not produced by stretching, as in the string. It is rather of a suppressive type, forced into the note when it is lowered during the tuning. According to acoustic theory, a suppressive tension applied on a vibrating body lowers the frequency of the tone. The existence of a tension in the steel pan note is is still to be proved.

The steel pan is classified as belonging to the instrument class of idiophones, which means that the pitch is determined by the shape and the internal state of the resonator. There are not any ready-made formulas for the calculation of the vibrational modes in such a complex shape as the steel pan note. A formula would have to include all the variables (factors) mentioned above and would be very complicated.

To entangle the situation further, the tone generation in a steel pan note seems to be very non-linear in its nature. This means that the variables involved in the tone generation affect each other in a way that makes it impossible to express the mechanism in a simple formula. For more about the non-linear tone generation, see below.

But some relief is to be found; my studies show that, as a first approximation, the steel pan note may be considered as an almost rectangular plate, disregarding the arching and the tension in the note. This means that the vibrational patterns of a rectangular plate (or a membrane) are to be found in a steel pan note, and that the study of the vibrational modes can be simplified accordingly.