July 8, 2012
Vol. 7, Num. 3
Kalimba Magic NEWS
This article is somewhat technical, but I have tried to present the explanations so that a high school junior will be able to grasp the important points. Stick with it, you just might find this stuff to be fascinating. Comments from musical instrument maker and scientist Bart Hopkin follow at the end of the article.
Anything that vibrates will vibrate at more than one frequency (they make more than one note at a time), but usually the lowest frequency (what we percieve to be the pitch of the sound if the vibration is audible) will dominate, and we will hear the sound as that one note. All the other higher frequencies vibrating - often refered to as "overtones" or "harmonics" or "partials" - will not usually be perceived as separate notes, but they will change the character of the sound, i.e., the exact frequencies and the relative strengths of those overtones are what permit the listener to distinguish the sound of a flute from the sound of a kalimba.
Vibrating strings (on guitars or violins for example) and vibrating air columns (flutes, pipe organs, or the resonators on marimbas) have simple mathematical relationships between the frequencies of the fundamental vibrational tones and overtones. On a guitar string, the nth overtone will have a frequency which is n times the fundamental frequency. When n = 1, well, that is just the fundamental. For n = 2, the "first harmonic", the frequency is twice the fundamental - in other words, it is an octave above the fundamental tone. For n = 3, we get a pitch that is an octave and a fifth above the fundamental. For n = 4, well, that is 2 x 2 - in other words, two octaves above the fundamental. Octaves and 5ths - these all harmonize with each other, and on a guitar or piano, the overtones are called "harmonics" because they harmonize and work with the fundamental tone.
Vibrating bars (on marimbas) or vibrating kalimba tines fixed at one end and free on the other have overtones that do not follow such simple mathematical relationships with the fundamental tone as the strings and air columns do. In other words, the overtones on the marimbas and kalimba tines don't harmonize with the fundamentals. Which means these instruments don't sound so good. But of course, you know that these instruments do sound pretty good. Why?
On the marimbas, each bar is sculpted so that it has the perfect shape to vibrate at the fundamental plus a note two octaves above the fundamental. Concert marimbas are often made such that three, four, or even five of the overtones are tuned to notes that sound good with the fundamental. And to stack the deck further, resonators (those long tubes underneath each bar) are added to further emphasize the fundamental note. Add those together - bars with one or more tuned overtone and a resonator that is also tuned to the fundamental - and you have a beautiful-sounding instrument.
What about kalimbas? Most kalimbas are made with constant-thickness spring steel, and these will produce overtones that are out of tune with the fundamental. The first overtone vibrates at a frequency about a factor of 6.3 higher than the fundamental's frequency - a note more than two octaves above the fundamental, but less than three octaves - an oddball note that doesn't harmonize with the fundamental. But why doesn't the kalimba sound bad to us? I believe that the anharmonic overtones of the kalimba are actually part of what makes the kalimba sound like a kalimba. David Chapman was one of the first people to publish this idea, that the anharmonic overtones of the kalimba make for its unique sound.
My own contribution to this debate is the fact that the overtones have relatively little energy in them, so the fundamental note dominates... but perhaps more important, the overtones die out very quickly, and after a fraction of a second, the fundamental tone represents 99% of the energy in the vibrating kalimba tine, and we hear a very pure sound. Furthermore, those high overtones can be several octaves above the fundamental, so unless you are playing a bass kalimba with very low notes, those overtones may very well be out of your hearing range. Very young listeners who have not lost any of their hearing range are sometimes put off by the sound of the kalimba - perhaps because they hear more of those high overtones. Older people who have lost some high frequency range typically find the kalimba's sound to be charming and lovely.
Recently, a friend engaged with me in a discussion about tuning the overtones of kalimba tines. His idea was that you should be able to change the composition of the metal to get harmonic overtones. My thinking: it doesn't matter what kind of wood your marimba bar is, the overtones still need to be explicitly tuned to get them to harmonize with the fundamental, and you do that by sanding or cutting away material from the middle of the back side of the bar. To get kalimba tines which have tuned overtones, you need to change the shape of the kalimba tine - either distribute the mass in some non-uniform manner, or distribute the spring strength in some nonuniform manner. You can do this by filing tine material off the tine, as suggested by Bart Hopkin in has fantastic book "Musical Instrument Design." Or you could do it by flattening out the tine with a hammer in a particular way, or by adding some mass to the tine in specific places.
David Bellinger is a kalimba maker known for his interesting innovations. Someone once sent me a Bellinger kalimba for me to retune, and as I was retuning it, I noticed that each tine has a small bead of solder on the underside near the tip. The main effect of this extra weight at the end of the tines is to lower the vibrating pitch of the fundamental - i.e., make the pitch each tine plays lower. But it doesn't have as strong an effect on the overtones as on the fundamental, I noticed that the first overtone seemed to be about two octaves above the fundamental. I think that David Bellinger can tune his first overtone in this way.
But only a day after I was discussing with my friend about tuned overtones, I pulled out an old rustic karimba made in the style of the Kwanongoma College of African Music karimbas introduced by Jege Tapera and brought to the USA by Paul F. Berliner and Dumisani Maraire. I bought it on EBay for about $40 5 years ago. It is in the key of F, and I recently fine-tuned it so it could play with my Hugh Tracey Alto in F (more on that one later). Anyway, when I started to play the rustic karimba, I noticed that I could hear the overtones sustaining for much more than one second. If you pluck a kalimba tine but put your soft thumb flesh on the tip of the tine, you will damp out the fundamental vibrations, but often you can still hear the overtones. I was shocked to hear that the overtones seemed to be two octaves above the fundamental. I checked every tine, and it seemed that every tine had tuned overtones. Wow! My opinion of this $40 instrument just went way up!
Andrew Tracey and Bart Hopkin both state that some African makers of traditional instruments - karimbas and marimbas - and quite probably some of the ancient makers of traditional instruments as well, understood about tuning overtones. That said, I have also read scholarly papers that concluded that non-modern mbiras did not have tuned overtones. My work on this rustic karimba comes down firmly on the side of ancient African genius - somehow the people who made this karimba, and probably the people who made similar instruments centuries ago, could hear the overtones and know what they were doing.
There is a great piece of free software that is perfect for studying overtones - Audacity. (OK, just like Pharma-city in Boston, I actually sometimes call it Auda-city instead of audacity.) I record samples from each note on the rustic karimba with my Handy H4 Zoom Recorder, select a few seconds of the note, and form a power spectrum plot... and cool, the overtones are tuned!
Power spectrum of the karimba's F4 note. The fundamental is at 350 Hz, the highest peak. On this log-log plot, each light vertical line is another octave. The first overtone is one octave above F4, the second overtone is two octaves above F4. The broad feature between 3Hz and 22 Hz is an artifact. I think the feature at around 100 Hz is my air conditioning. Smaller peaks at high frequencies may be overtones or may be noise.
Power spectrum of the karimba's A3 note. There may be an overtone less than an octave above A3, and there may be a smaller overtone at A4 an octave above, but there is clearly an overtone two octaves up at A5.
How are the overtones tuned on this rustic karimba? I do not see any filings or any extra mass soldered onto the tines, but the tines have a characteristic shape made by taking a long steel bar of uniform rectangular cross-section and pounding the end flat. The flattening cannot be seen in this face-on photograph, but it can be inferred from the widening of the tine as you go towards the tine end. At the bridge, there is very little widening (flattening), but the width increases towards the end of the tine. It is my hypothesis that this detailed shape tunes the overtone.
I have a bit more work to do, but this is my thinking: to first order, this flattened end doesn't change the mass distribution on the tine very much, only the strength distribution. Because the spring strength is minimally affected at the bridge, I think the fundamental pitch is not affected much by the flattened tine shape. However, halfway between the bridge and the tip of the tine, the tine is significantly flattened, and the spring strength is significantly decreased, and decreasing the spring strength in this region should bring the overtone pitch down. Iin other words, down from 6.3 times the fundamental to about 4.0 times the fundamental to be 2 octaves above fundamental.
I believe the overtone at one octave above the fundamental can be explained by another vibrational mode caused by the tine sliding over the bridge at twice the fundamental frequency.
So, all you kalimba makers of the world: how much do tuned overtones matter to you?
I learned about tuning overtones from musical instrument maker and designer Bart Hopkin, and we are lucky enough to have a comment on this article from him:
Hammering the tines affects the overtone relationships as your paper makes clear, so traditional African techniques which involve hammering are in the realm of overtone tuning, whether deliberate or not. So at this point I'd make a distinction between (1) overtune tuning as it's commonly practiced on western marimbas, in which the maker deliberately seeks certain uniform overtone relationships of mostly octaves and maybe fifths, and (2) overtone tuning in a slightly different spirit, in which one seeks out an overall sound that is pleasing to the ear without specifically tuning the overtones in all tines to certain prescribed intervals. It would make sense to me to think that the latter approach wouldn't be unusual in traditional African kalimba making. This approach can be seen as both more sophisticated and less sophisticated than the more western approach. Less sophisticated because it may be more intuitive, less mathematical- techy (no Auda-city involved). More sophisticated in that, if my imaginings on the process are on target, it may involve overtone tuning with an ear to the pitches of the other tines, not only the sound of each tine as an individual. Also more sophisticated in the end result because with some variation from tine to tine, it may offer a richer overall palette than would be the case with uniform tuning of all tines.
Remember, of course, that these thoughts are entirely speculative. Mostly they just flow in a non-rational way from the heavenly feeling I get sometimes listening to the cascading rainbows of tones and overtones from traditional mbiras and kalimbas.