Overtone-Tuning in Lamellaphones

Last modified 8-6-2010

by Bart Hopkin

This article is about mbiras, kalimbas, sansas, and similar musical instruments with vibrating tines or tongues, including non-traditional or home-made variations. These vibrating-tine instruments make up the family known as lamellaphones. Within that field, the article’s focus is on overtone-tuning – the practice of adjusting the overtones within an instrument’s sound to bring them into better alignment with the fundamental tone.

Overtone-tuning is often done with other sorts of instruments. For instance, people who are serious about marimba-making know that overtone-tuning, or its absence, is central to the character of that instrument. The same applies for some other tuned idiophones such as steel drums and carillon bells. But the question of overtone tuning isn’t often considered when it comes to making lamellaphones. That’s too bad, because lamellaphones can benefit from overtone-tuning just those other instruments can, and the tuning work is at least as easy to do.

This article starts with foundational information for the process of overtone-tuning in lamellaphones. Then it provides how-to for two possible overtone tunings.


Overtone Tuning – the basic idea

While we think of a typical musical tone as a single note, most musical tones actually have several frequencies present in the sound. Usually the lowest of these frequencies, commonly called the fundamental, defines the pitch. In other words, the pitch of the overall sound, as perceived by the listener, is the pitch of the fundamental. Other frequencies present in the sound can be thought of as overtones. They contribute a lot to sound quality or tone color, while playing a lesser role in the perceived pitch.

The musical relationships between the fundamental and its overtones vary from one type of instrument to the next. For many musical instruments, the relationship is harmonic. In this context, “harmonic” does not simply mean “harmonious;” it refers to a set of specific mathematical relationships between the frequencies. In sounds containing harmonic overtones, the overtones seem to blend imperceptibly into the fundamental. The pitch-sense is very clear, and the tone seems coherent and integrated. Instruments with harmonic overtones include string instruments (assuming the strings are well made and under sufficient tension), and wind instruments (assuming the bore shape is suitably designed for the purpose).

Then there are instruments that produce inharmonic overtones. For these, the simple relationships of the harmonic series don’t apply, and overtones appear at various quirky pitches relative to the fundamental. This is the case for most drums and idiophones. With these inharmonic tone qualities, the pitch-sense tends to be less clear and the tone usually seems, subjectively speaking, less integrated.  With the instruments mentioned above – marimbas and xylophones, steel pan and similar instruments, bells, and the instruments of the lamellaphone family - it’s possible to realign at least some of the overtones, moving them from potentially confusing inharmonic relationships to much clearer harmonic relationships.

Before going all out on the overtone-tuning is idea, however, let’s look at the other side. The presence of inharmonic overtones in lamellaphones and other muscal sounds isn’t necessarily a bad thing at all. Inharmonic tone qualities can be piquant and colorful. The world would be a boring place if all musical sounds were somehow required to be perfectly harmonic. In listening to some rather beautiful mbira music recently, I noticed how I was able to follow the musical relationships inherent in the fundamental tones, even as the constant play of high, mostly inharmonic overtones could be heard, almost as a separate dance, coloring the texture and feel of the music from above.  This is  part of the richness of the music. Furthermore, I’ve been told that some African makers do pay a lot of attention to the overtones as they shaped their instruments’ tines. Their approach is quite different from that described in this article, and they don’t aim for the specific tunings described here.  So let me be clear about this: I don’t mean to suggest that lamellaphone overtones always should be deliberately tuned, or that if they are to be tuned it should necessarily be done in the manner described here.  

But I am saying that, if you choose to do it – shifting those unruly inharmonic overtones into harmonic agreement with the fundamental – the clearer tone quality that results is often worth the effort. This is especially true when the instrument plays in dense and harmonically complex musical surroundings. Its greatest value may be in western musical contexts, particularly when lamellaphones play in ensemble with other western instruments.  Prominent inharmonic overtones in such contexts can be musically confusing. In some cases, the listener’s ear may even tend to track the overtones rather than focusing on the intended fundamental pitches (definitely a confusing musical situation).

Further Specifics

In lamellaphones, as in marimbas and other instruments, overtone tuning is especially valuable for notes in lower registers. In lower notes the overtones tend to come through strongly because they are near the heart of the audible musical range. Sometimes they even dominate the fundamental. Overtone tuning is less important in for high notes because the overtones in high notes, being much higher still, are less conspicuous. For this reason, it’s usually not necessary to tune the overtones for high notes. This is convenient, because the task of overtone tuning is more difficult and time-consuming for higher notes. How high before overtone tuning is unnecessary? There’s no single answer to this question; it’s best decided on a case-by-case basis, as each instrument and each maker’s preferecces are different.

With marimba bars, overtone tuning is done by reshaping the bars, and the same approach works for the vibrating tines of lamellaphones. The underlying idea is as follows:

The fundamental and the overtones in the sound correspond to different patterns of vibratory movement, or modes of vibration, in the tine. These different patterns have different regions of maximum flex over the length of the tine. If you thin the tine in the regions of maximum flex for any given mode of vibration, the resulting weakening will lower the frequency of that mode. It will have much less effect on other modes not sharing the same flex points. So if you can figure out where those regions are and do your selective thinning accordingly, you can adjust the relationships between the overtones and the fundamental, bringing them more in line with the ideal of harmonicity.

This process of selective thinning is potentially a complex and difficult task. It certainly is in marimba bars, where the tuning of multiple overtones can reach the level of high art. But in lamellaphone tines, it’s possible to achieve very nice results with a simple approach. The approach I’m going to suggest involves tuning just one overtone relative to the fundamental. The process is systematic, predictable and not labor intensive.

Why is it that you can get away with tuning just the fundamental plus one overtone? There are several reasons, but the main one is that in vibrating tongues the overtones are typically very widely spaced. In a uniformly shaped, rigidly mounted tine, for instance, the first overtone above the fundamental typically appears around two octaves and a minor sixth above the fundamental. That first overtone, despite being so high, is often conspicuous in the tone. For this reason, it makes sense to tune it, especially for notes in the lower registers. Additional overtones are arrayed higher still. These additional very high overtones are usually less conspicuous and tend not to affect the pitch-sense as much as the lower overtone. The ear tends to hear them just as added color. For that reason, tuning those very high overtones is less important.

With these things in mind, we’d like to tune the tine so that that first overtone over the fundamental – the one that stands out most significantly in the tone – ends up in some coherent, harmonic musical relationship to the fundamental. In this article I’ll suggest two tunings. One positions the first overtone three octaves above the fundamental, and the other positions it two octaves and a fifth above. Both of these fall within the harmonic series and provide the kind of musical coherence we seek. My preference is for the octave tuning because it’s clearer and more lucid – I love the sound of it. On the other hand, the quint tuning (the one using the fifth) offers a bit more personality, and has the advantage of requiring less thinning.


Two Tunings

You can find sound clips of these tunings near the end of this article.

Octave Tuning: In this tuning, the first overtone is positioned three octaves above the fundamental. This gives a pleasing and lucid tone with excellent pitch definition due to the octave agreement. The higher inharmonic overtones are left untuned. Being very high in pitch and usually not prominent in the overall tone, they add a little spice without distracting too much from the pitch-sense.
This is an excellent tuning throughout the range. It’s especially valuable in the lower ranges where the quint tuning described below isn’t as effective. It has one potential drawback: it requires a lot of thinning near the base of the tine. If the tines you’re working with are not fairly hefty to begin with, the tine may be left rather thin near the base, subject to bending or breaking. For this reason, this tuning isn’t good for very thin tines.

Quint Tuning: In this tuning the first overtone is positioned two octaves and a fifth above the fundamental. Higher overtones are left untuned. The dominant-tonic relationship between the first overtone and the fundamental points to the fundamental as the defining tone, helping to clarify the pitch sense, but the tone quality is more complex and not as transparent as an octave tuning would be.
This tuning is most effective in the mid ranges and up, where the prominent fifth in the tone is high enough not to stand out too much. (In the lower registers, that fifth falling nearer the heart of the musical range makes the pitch sense less clear.) On the plus side, this tuning requires relatively little thinning of the tines, meaning both less work and less danger of weakening the tines too much. Thus, it’s practical with thin tines, for which the deeper thinning of an octave tuning wouldn’t be feasible.

 In the following paragraphs, I’ll jump right into the procedure for doing the octave tuning. With that procedure understood, I can then detail a few more underlying ideas more fully. Finally I’ll describe the quint-tuning procedure, as it differs from octave tuning in a few particulars.

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 The Octave Tuning

 Preliminary #1: Temporarily mounting the tine for tuning and filing purposes.
The tuning will be much easier to do if you can temporarily mount the tine in a bench vise prior to the final mounting and tuning on the instrument. If you don’t have easy access to a bench vise or something similar, a C-clamp on the edge of a table will do. Be sure that the edge of the mounting is well defined: double check to be sure that the vise or clamp is grabbing the tine very firmly at the very point where the free vibrating portion of the tine is to begin. If this edge point is a little loose, it confuses the tuning process and compromises the results.

 Preliminary # 2: Learning to hear the overtone.
To do the tuning, you need to be able to hear separately the tones of the fundamental and of the first overtone. Both are present when you pluck the tine normally, but your ear will tend to focus on the pitch of the fundamental. Without practice, it’s difficult to mentally separate out the tone of the overtone.
To isolate the sound of the first overtone, use these tricks. Pluck at a point about a third of the tine’s length from the bridge. Pluck not with the fleshy part of the finger but with a fingernail or with something edgy like a guitar pick. As you do this, try touching the tine lightly with the fleshy part of a finger of the other hand at a point about three-fourths of the tine’s length from the bridge. (This is similar to the string player’s technique for isolating string harmonics.) With luck, this will isolate the overtone almost completely, allowing you to hear it perfectly. It may not work well with smaller tines, in which case you can skip the soft finger touching and just depend on sharp plucking near the one-third point.
You’ll also want to be able to hear the fundamental with minimal distraction from the overtone. To do this, pluck at the end of the tine with the fleshy part of the finger for a soft, round sort of tone.
If you have electronic tuner, you can use it. With the soft end-plucking technique just described, the tuner should register the pitch of the fundamental. With the three-fourths finger-touching technique, it should be able to recognize the pitch of the overtone.

Preliminary #3: Understanding the strategy for the octave tuning; knowing where to thin the tine.
In the octave tuning, the first overtone in each tine is to be set three octaves above the fundamental. Before tuning, in the great majority of cases, this overtone islocated somewhat less than three octaves above the fundamental. To do the tuning, you will need to increase that interval to three octaves. Counter-intuitively, you will do this not by raising the overtone, but by lowering the fundamental. This will be done by thinning the tine in a location that lowers the fundamental while having much less effect on the overtone. The thinning will be done over a region starting at the base of the tine and extending over about one fourth of its length.
Important note: If you find that, for the tines you are using, the overtone is more than three octaves above before tuning, read the notes following these steps to know how to proceed.

 Preliminary #4: Thin one side only, or thin all around?
If your tine is flat and wide in shape, as in rectangular bar stock, you don’t need to thin uniformly all around the bar. You can thin what is to be the underside of the bar only. But if your bar is as thick as it is wide, as in square stock or cylindrical rod, you’ll need to thin uniformly all around. This is discussed more fully later in this article.

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 Step 1) Determining the approximate vibrating length and the region to be thinned:
Before doing any thinning of the tine, you need to set the tine at about the right length for the note it is ultimately to produce. Place the tine in the bench vise (or C clamp or whatever you’re using) and secure it firmly with what seems like a reasonable portion of its length protruding, ready for plucking. Pluck it, and listen for the pitch of the first overtone (review Preliminary #2 above for information on how to do this). Adjust the playing length (unclamp and re-clamp as needed) so that the first overtone is a little more than three octaves above the intended fundamental pitch for this tine. At this length, the fundamental will initially be too high. The idea is that when you thin near the base to drop the fundamental to three octaves below the overtone, it will end up at about the right pitch. Use a marker to mark the start of the tine’s sounding length where it meets the jaws of the vise. This is (very approximately) where it will cross the bridge when mounted on the instrument. Also mark a quarter of the sounding length from there, to indicate how far along the tine the thinned portion will extend.

Step 2: Thinning Near the Base
The thinning should result in a smoothly curved dip extending over roughly the first quarter of the tine’s sounding length. To do the thinning you can use a grinder, file, or any other method that does the job. To drop the fundamental pitch to three octaves below the first overtone, you’ll typically need to thin the tine to about half of its total thickness at the narrowest point, but this is very approximate and will vary depending on the shape of the tine and the shape of the arch you make.
The procedure is: thin just a bit, then check the pitches of the fundamental and the first overtone. Unless you overdid it, the fundamental won’t have dropped enough yet. Thin some more, check again, and repeat until you’ve got the desired three-octave relationship. Don’t worry about whether the fundamental ends up on the exact right pitch; this will be adjusted later. Instead, focus on the correct tuning of the three-octave relationship between the fundamental and overtone.

Step 3: Mounting on the Instrument and Final Tuning
The tine can now be mounted on the instrument. Position it with the length mark you made earlier at the bridge. It should then produce a tone fairly close to the intended pitch but probably not perfect. Tune in the usual fashion by adjusting the tine length. If the final bridge location is reasonably close to your original mark, the three-octave relationship will remain acceptably close. If it turns out that a major length adjustment is needed to get the pitch right, you’ll probably need to go back and retune the three-octave relationship.

Step 4: Repeat
The fourth and final step in the process is to repeat steps 1 – 3 for the remaining tines that are to be overtone tuned. Remember that, as discussed in the first half of this article, you may not need to do overtone tuning for higher pitched tines; it’s most valuable for those in mid or lower registers.

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Further Underlying Information

And now, here’s additional information to deepen the understanding of what’s going on in these procedures. Knowing these things will allow you to customise the procedure as needed for different circumstances and different tunings.

 Where to Thin the Tine for Tuning – Generalized Rule

Lamellaphone tines can be made in different shapes. For instance, they may be uniform in cross sectional shape over their length, as with as the cylindrical steel rod or rectangular bar stock often used by western instrument makers. In contrast, many traditional African instruments have tines of cold-hammered steel with a flared shape, broadening toward the end. In the case of uniformly shaped tines, the interval between the between the fundamental and the first overtone typically appears at a little less than two octaves and a minor sixth above the fundamental. In other words, if the tine hasn’t been reshaped to adjust the overtone tuning, the first overtone naturally appears near that interval. In non-uniformly shaped tines, the musical interval between the fundamental and the first overtone is less predictable. Typically it will still be less than three octaves above, but for tines heavily weighted toward the end it may be more.

Our goal in overtone tuning is to take whatever that interval happens to be, and shift it to a preferred interval. If you’re doing one of the two tunings suggested in this article, the preferred interval will be three octaves or two octaves and a fifth. If you’re doing some other tuning of your choosing, it could be something different. In any case, before tuning, the existing interval is either too small and needs to be made bigger, or it’s or too big and needs to be made smaller. Here’s what to do in each case:

 If the untuned interval is too small (i.e., the overtone is not far enough above the fundamental),
then make it larger by lowering the fundamental.
To lower the fundamental with minimal effect on the overtone, make the tine thinner near the base in a gradual cutaway starting near where the bridge is to be and extending over about a fourth of the length.

 If the untuned interval is too big (i.e., the overtone is too far above the fundamental),
then make it smaller by lowering the first overtone.
To lower the overtone with minimal effect on the fundamental, make the tine thinner over a fairly narrow region located about 3/5 of the bar’s length away from the base of the tine.

 

Thin One Side Only or Thin All Around? Rectangular tines vs. cylindrical or square tines

There’s an important consideration having to do with the direction in which a tine vibrates. Some tines have a round shape — for example, those made from cylindrical rods. These tines will normally vibrate in an up-and-down direction, because that’s the direction in which they are normally plucked. But they are also capable of vibrating side to side. In theory, cylindrical tines should produce the same frequency regardless of what direction they vibrate, because the round shape has the same rigidity in all directions. But for various reasons this isn’t always entirely true. In particular it’s not true if you thin the tine for tuning purposes on one side only. Then the tine is significantly less rigid for one direction of vibration than for another. You end up with a dual-pitch tine. You could hope that the dual pitch effect won’t be a problem if you consistently pluck from the same direction, exciting the tine to vibrate in a certain direction that gives the desired pitch only. There’s some validity to this hope, but not quite enough. The dual pitch still tends to come through at least some of the time, and more in some tines than others. To avoid this problem, when thinning cylindrical tines for overtone-tuning purposes, it’s a good idea to thin them not just on one side, but uniformly all around the circumference. If, after doing that, you still notice dual pitch effects, you can correct them by further thinning on the side that weakens the tine for the direction of vibration that is too high in pitch.

Here’s a trick that may be useful for uniform circumferential thinning of cylindrical tines: as long as the tine remains straight, you can get good results by putting it in a drill chuck. Thin it as it turns using either a hand file or a grinder bit in a separate hand-held drill.

What was just said for cylindrical tines applies as well to ones which are square or near-square in cross section.

Contrast this with flat tines, wider than they are thick, such as those made from rectangular bar stock. Due to their shape these tines don’t as readily vibrate side-to-side. They’re much more inclined to vibrate in the up-and-down direction only. Dual pitches, as a result, don’t arise (or at least aren’t audible) and you can confidently thin on one face only. For the sake of appearance, you’d probably choose to do this on the side that is to be the underside of the tine.


 Shaping the arch (how long should the thinned area be?)

When you thin the tine, you create a kind of arch-shaped cutaway in the surface of the tine. You can make the thinned region relatively long, extending over a large part of the tine’s length, or you can make it short. If you make it short, it will have to be made deeper to achieve the same effect on pitch as a longer thinned region. Making a shorter, deeper thinned region is less work than making one that is shallower but more spread out.  But the greater depth may then weaken the tine too much, making it prone to bending or breakage. Thus, for tunings that require substantial thinning, it’s best to thin the tine over a longer part of the tine’s length, make a rather spread-out thinned region. That’s why I suggested thinning over a quarter of the tine’s length for the octave tuning described above: since it requires fairly substantial thinning, a longer thinned regionis better. But with other tunings that require minimal thinning, it’s ok to make the thinned region shorter. This is true for the quint tuning described below.


 And now here’s the procedure for the quint tuning. To reduce repetition, this description is less detailed than the description given above for the octave tuning.

 
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 Quint Tuning

 Preliminaries: Review the preliminary notes for the octave tuning described above. The same considerations apply here, with the following exception:

In this tuning, the first overtone in each tine is to be set two octaves and a fifth above the fundamental.
In most cases the first overtone, appears a little more than this interval above the fundamental prior to tuning. You can achieve the desired interval by lowering the overtone. You will do this by thinning the tine over a short region located about 3/5 of the tine’s length from the base.
In other cases, the overtone initially appears at a little less than the desired interval of two-octaves and a fifth. In these cases, you can achieve the desired interval by lowering the fundamental. Do this by thinning the tine near the base, much as was described above for the octave tuning but with less thinning required.

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 Step 1) Determining which way the tuning needs to go.
Place the tine in the bench vise (or C clamp or whatever you’re using) and secure it firmly with what seems like a reasonable portion of its length protruding, ready for plucking. Adjust the playing length (unclamp and re-clamp as needed) so that that the fundamental is at the intended pitch for this tine. Use the plucking and listening techniques described in Preliminary #2 to determine whether the first overtone is appearing above or below the desired interval of two octaves and a fifth above the fundamental. 

If the interval is less two octaves and a fifth, then ignore the steps below. Follow the procedure given for the octave tuning above, but with this difference: wherever that procedure refers to the interval of three octaves, substitute the interval of two octaves and a fifth. The amount of thinning required to do this tuning will be much less than that described for the three-octave tuning.

If the interval prior to tuning already happens to match the desired two octaves and a fifth, be happy!  No more overtone tuning is needed for this tine.

If the initial interval is more than two octaves and a fifth, follow the steps below.

Step 2: Establishing the approximate vibrating length and the region to be thinned:
With the tine positioned so that the fundamental is already at its intended pitch, use a marker to mark the start of the tine’s sounding length where it meets the jaws of the vise. This is approximately where it will cross the bridge when mounted on the instrument. Also mark a point three-fifths of the length of the tine from the base, indicating where the thinning is to take place.

Step 3: Thinning Near the Three-Fifths Point
The thinning should result in a smoothly curved dip in the region the three-fifths mark. The thinned region will typically cover about 10% of the chime’s sounding length, but it’s not crucial to be precise about that. To drop the overtone to the intended pitch two octaves and a fifth above the fundamental pitch, you’ll typically need only a small amount of thinning. Proceed carefully so as not to overshoot. Thin a tin bit at a time, frequently checking the pitches of the overtone and the fundamental, until the overtone has dropped to the intended interval. Don’t worry about whether the fundamental remains on the exact right pitch; focus on the correct tuning of the relationship between the fundamental and overtone.

Step 4: Mounting on the Instrument and Final Tuning
The tine can now be mounted on the instrument. Position it with the length mark you made earlier at the bridge. It should then produce a tone fairly close to the intended pitch but probably not perfect. Tune in the usual fashion by adjusting the tine length. If the final bridge location is reasonably close to your original mark, the overtone interval at two octaves and a fifth will remain acceptably close. If it turns out that a major length adjustment is needed to get the pitch right, you may need to go back and retune the overtone.

Step 5: Repeat
To comlete the job, repeat the process for the remaining tines that are to be overtone tuned. Remember that you may not need to do overtone tuning for higher pitched tines; it’s most valuable for those in mid or lower registers.

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Sound Samples

The tines in the three short samples below are tuned to the same fundamental pitch.  Each sample features two tines pitched a fourth apart.

TWO TINES WITH OCTAVE TUNING
In this sample, listeners with practiced ears will hear clearly
the overtone three octaves above the fundamental.

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TWO TINES WITH QUINT TUNING
Quint tuning works best with tines in the mid-range and higher.
Because the tines in this sample are fairly low in pitch, the tuning isn’t
shown to its best advantage, but you can get some sense of the sound of it.

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TWO TINES WITHOUT OVERTONE TUNING
In the higher of these two tones you can hear the typical prominent inharmonic
first overtone close to two octaves and a minor sixth above the fundamental. 
In the lower of the two, the overtone happens to come in closer to two octaves
and a fifth, reminiscent of an imperfect version of quint tuning.

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Hearing these pairs of tines in isolation does not tell the whole story. One of the reasons for overtone tuning is to clarify the sound of the instrument as a whole. The final sample is from a marimbula with the octave tuning. (The name marimbula is used for various large, low-pitched lamellaphones in the the Spanish Caribbean.)

MARIMBULA IN OCTAVE TUNING.
If you listen closely you can here the first overtone at three octaves above the fundamental,
sounding as a brief plink at the start of each tone.

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