20.Tone generation in steel pans
The shallow dent of a steel pan note is indeed a complicated resonator. No one has so far been able to explain how the steel pan works and why it generates such harmonic tones. As a physicist and a steel pan-player, I got curious and decided to try to find an explanation to the tone generation in the steel pan. Therefore, I started to do some part-time research at the Royal Institute of Technology in Stockholm.
During the three years the research has been going on, I have made some measurements and I also have some preliminary results. In this chapter I want to present the findings from my work and what ideas and thoughts that will govern my future research on the steel pan.
My research started off with two simple questions:
(A) How is the unique tone of the steel pan generated?
(B) How does the choice of material, design and crafting methods affect the tone?
To make progress with these two main questions, I needed to know more about both the tone and the instrument. Therefore, I decided to focus my study to three fields:
(1) To register and understand the acoustic properties of the steel pan tone.
(2) To register and understand the construction of the instrument.
(3) To reveal the relationship between the acoustic properties of the tone (1) and the physical properties of the instrument (2).
The discussions with the tuners and the practical section of this book can be seen as main actors in the part of understanding the instrument. The laboratory work that is to be presented in this chapter represents the understanding of the tone. The rest of this theoretical section is a first attempt to explain the relationship between the construction and the tone.
Modes of vibration
A good steel pan tone has many harmonic partials; at least five or six that are strong relative to the fundamental, see fig. 19.3. The question is, how are these partials generated? If we disregard arching and thickness of the note and look at the note as a membrane, we get some clues to the normal modes generating the partials.
Fig. 20.1 Research on the steel pan. Experimental set-up with (left to right) frequency analyser, computer, oscilloscope, frequency counter and microphone.
Fig. 20.2 shows a graphic view of standing wave patterns of the lowest three normal modes of a rectangular membrane. The lowest three modes of typical steel pan notes have been measured to be the same as these. Fig. 20.3 shows the lowest three modes of a steel pan note, together with cross-sections of the motions along and across the note.
Fig. 20.2 The lowest three normal modes of a rectangular membrane, which are equal to the standing wave patterns of normal modes in a typical steel pan note.
The lowest mode, the fundamental, is easy to understand - the whole note is vibrating up and down like the head of a drum. This is equivalent to the situation in which two hypothetical strings stretched along and across the note would be moving up and down in their whole length, see fig. 20.3.
Fig. 20.3 The three lowest modes of vibration in a steel pan note.
For the second mode, the octave, the note is vibrating up and down twofold along its length. A node line - a part where the surface is standing still - can be found in the middle, see the dotted line in the second note in fig. 20.3. The existence of the octave mode is easy to prove by putting a finger in the middle of the note and striking it at one end. This damps the fundamental and generates a flageolet - the octave sounding alone.
The third partial is generated in the same way as the octave, but here the note is vibrating up and down twofold across. A nodal line can be found lengthwise in the note. This can be proved in the same way as with the octave, but is a bit harder to hear. Put two fingers along the dotted line of note three in fig. 20.3 and hit the note near the side, preferably with the hard end of the stick.
I have measured the frequency of the lowest three modes of several steel pan notes and examined their relation to the lowest three partials in the steel pan spectrum. This is easily done by driving the note at the measured partial frequencies with a tone from a loudspeaker and looking at its mode response. The result can be seen in fig. 20.4.
Fig. 20.4 Relation between normal modes and partials in a steel pan note. Nodal lines of the modes indicated above corresponding partial peaks in the diagram.
From the discussion above, we see that the lowest two harmonic partials in the steel pan tone can be explained as generated by the two lowest vibrational modes of the note. But what about the higher partials? According to the measurements they are present in the tone, but the modal theory does not seem to be able to explain their existence. In fact, the third mode of a steel pan note often generates a partial that not is harmonic, see fig. 20.4. It often sounds with a tone that is one octave plus a third or a fourth above the fundamental.
The theoretical value for the third partial in a harmonic series should be one octave and a fifth above the fundamental. As seen from fig. 20.4, the tone of a steel pan has a partial that matches this harmonic interval perfectly, but it does not come from the third mode of vibration. A suggested explanation of the generation of higher partials can be found in the chapter about non-linearity.
When looking at the onset (the beginning) of a steel pan tone, one notices that the higher partials tend to arrive a bit after the beginning of the tone. My measurements show that they arrive about 20-30 milliseconds later then the onset of the fundamental, see fig. 20.5. In ordinary percussion instruments, the higher partials usually arrive at the onset, together with the fundamental.
The notion that the higher partials tend to arrive later in the steel pan tone suggests that they are not initiated by the stroke. It rather seems that they are generated by some other mechanism that needs a few milliseconds to start working.
Fig. 20.5 Relative strength of the partials during the first 80 milliseconds of a steel pan tone.
The contradiction that the frequencies of the normal modes of the steel pan note do not have a harmonic relationship but the frequency spectrum still shows a large number of harmonically spaced partials, reveals that the higher partials must emanate from some other source than the vibrational modes. My hypothesis at this stage is that the harmonic partials are generated by a non-linear "distortion" process of the vibrations in the note, but this still has to be fully explained and proved.
The non-linearities may be introduced in the tone generation by the shape of the note and its tension. These properties will make the note move asymmetrically when it is vibrating, probably moving a bit easier and farther upwards than downwards. If the note moves asymmetrical, the vibration of the fundamental will also start to generate overtones.
This would explain why the higher partials arrive later: Only the lower normal modes are exited by the stroke and the higher partials are then generated later by the shifting of energy from the fundamental mode to higher frequencies.
This non-linear "distortion" process may be compared to the situation where a pure sine wave is the input to an amplifier that tries to output it beyond its range of power. The output tone will then exhibit harmonic overtones. You might say that the fundamental mode of the note acts like a sine wave, but it is forced to generate overtones by the asymmetric shape and tension of the note.
To speculate further, it seems as if the harmonic spectrum of the steel pan tone is generated by some intricate interaction between the non-linearities of the fundamental and the octave mode. The octave mode of the note also generates a series of harmonic overtones when it is vibrating. These overtones will be equally spaced over the frequency axis, but with an interval that is equal to the frequency of the octave. If the fundamental has a frequency of 200 Hz, the octave will be at 400 Hz. Then the partials generated by the octave mode will be found at 800 Hz, 1200 Hz, 1600 Hz, etc. This means that they will coincide with every second overtone of the fundamental. The cooperation between the harmonic spectra of the fundamental and the octave would explain why the partials with even numbers (2, 4, 6, etc.) are stronger than the odd numbered partials, see figures 19.3 and 20.4.
If a proper harmonic spectrum is to be generated, the octave has to be tuned to a perfect two-to-one relationship to the fundamental. If the octave is only a few cents (100 cents equals a half-tone interval) away from its exact value, the sound of the note will be dissonant. This can also be seen in the spectrogram, which will show a tone with double partial peaks, see fig. 20.6
My thoughts about the generation of higher partials have eventually led me to form a working-hypothesis. The following is a conclusion of the hypothesis that will govern my future research on the steel pan, together with some measurements that support it:
Hypothesis regarding the mechanism for generation of harmonic partials in steel pans
"The harmonic partials above the octave in a steel pan tone are not generated by vibrations of the higher normal modes of the note, but by a non-linear process involved in the motion of the fundamental and the octave modes."
The measurements supporting the hypothesis are:
1. The frequencies for measured normal modes of order three and four do not fall into the harmonic spectrum of the tone. Mode three is sometimes too high, sometimes too low, seldom in the harmonic spectrum. Mode four seems to be most often too high. Measured harmonic partials, on the contrary, are all spaced with exactly equidistant intervals.
2. After striking the note, the fundamental reaches its maximum within a few milliseconds (ms) and the octave within 15 ms. The higher partials reach their peaks after 20-30 ms. This indicates that the higher partials are not generated by the stroke.
3. A note driven by a pure sine wave at the frequency of the fundamental responds by emitting a tone containing a large number of harmonically spaced partials.
4. If the frequency of the driving sine wave is varied, the higher partials are more sensitive to these variations than the fundamental and the octave. A variation of +/- 2 Hz gives as an average a 30 dB difference in the higher partial level, whereas the corresponding difference for the fundamental and the octave is about 10 dB.
More focused measurements in the future and the development of a refined model will have to reveal if this hypothesis can be considered to be valid for the tone generation in steel pans.
Non-linear effects are present in all instruments but so far the acousticans have paid little attention to them because the tone generation of regular string and wind instruments can be explained fairly well by linear models. But to the steel pan, the non-linearities seem to be the foundation of the tone generation. Therefore, my belief is that a future theoretical model of the tone generation in steel pans has to be non-linear.
Fig. 20.6 Spectrogram of a dissonant steel pan tone. The octave is tuned a few cents below the first overtone of the fundamental, resulting in split partial peaks.
The final notion will be that - whatever the mechanism - the tone generation in the steel pan does not work like any other tonal instrument. The only instruments that seem to have a similar tone generation are cymbals and gongs.