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Why we use 1/4 Wavelength Resonators - (Discussions with Luís Henrique)

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-----Original Message-----
From: "Luís Henrique"
To: jim@percussionclinic.com
Sent: Wednesday, October 02, 2002 12:24 AM
Subject: marimba bars
Dear Jim McCarthy,

I am a lecturer in musical acoustics in the High School of Music and Performing Arts (ESMAE) of the Polytechnic Institute of Porto (IPP), and I have a particular interest in percussion instruments. I have reed your paper "Marimbas: Exploring the Depths" which I found very interesting and useful for makers of mallets instruments. When you said:(.) Firstly that for a given frequency, the minimum length of open ended tube that will resonate is twice that of a closed tube. Secondly, that the open-ended tube is capable of vibrating at all of the harmonics of its fundamental frequency, whilst the closed tube can only produce the on numbered harmonics.
A marimba generally uses closed resonators because of both these considerations. In the lower register, where the resonators are particularly important, the wavelength is such that the space and length of tube saved with a 1/4 wavelength resonator is considerable. A closed tube also means that the harmonics of the marimba bar can be tuned to suitable notes without being amplified along with the fundamental. You mean that the closed tube for resonator is appropriated because it doesn't amplify the higher modes (4 and 10). Have I correctly interpreted?
If so, why do you think like that?

Waiting your reply,

Regards,

Luís Henrique


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Dear Luis..

Good to know that there is one or two people actually reading that stuff - mostly people want to know about the latest catalogue item/gimmick from Pearl or Zildjian!

To answer your question about resonators:

To a certain extent you have correctly interpreted - although perhaps the reasons of which upper partials are resonant is secondary. I will explain later...

More important, are the issues of efficiency and physical space. Firstly it seems that a tube with a single open end is simply a more efficient resonator then a tube with two open ends - ie it produces more of the sound you want - the air vibration inside the tube. I guess this is most likely because the energy of the vibrating bar is coupled with the only possible antinodal position in the air column. An analogy would be a skipping rope - you don't actually need a person at either end of the rope to keep it moving, you can tie one end to a pole and it works fine - but if you hold one end and the other is just left free - for every bit you move your arm one way, the other end of the rope goes the same amount the other - because the energy is split between two directions it is half the amplitude - ok so not a great analogy, but you get the idea.

The most important issue, for low pitch instruments in particular, is purely one of the physical length of tube required. Imagine an A natural in the bottom space of the bass stave - frequency 110Hz. (This is not a particularly low note by today's standards - a modern concert marimba will go to the E or even the C below this - let alone a purpose built bass instrument.) Once end corrections and such have been calculated for the length of tube required for a 1/4 wavelength resonator is typically pretty close to 0.74m. With a classical marimba design - 0.74m plus a bit of clearance underneath so the instrument is on wheels, and a bit on top for the gap to the wooden bar, and the thickness of the bar, and your instrument is about 0.8m to 0.9m tall. That's about what it needs to be for people under 6ft tall to be able to play it comfortably. If you were to use a 1/2 wavelength resonator, the tube would have to be pretty much double the length for the same frequency. Your only choice would be to put in a full U bend at the bottom, and get the pipe going back up again. This would be a big hassle, a big extra cost, and it wouldn't sound as effective! As for notes any lower than this, the instrument would simply have to become taller or have strange and tricky resonator design. As it is with 1/4 wavelength tube, we already see the U bend at the bottom of the lower notes. Once you go lower than A 55Hz the tricky design is inevitable. If you get the time to read the "Jim McCarthy's Marimbas - Lowest bass marimbas in the world?" article, you will see some pics of "Boris" the bass marimba where these kinds of things are done.

In terms of those vibrational modes which are amplified... I'll include attached here the picture of modes in closed and open tubes which I drew for the essay, as a reference.

If we take first the open tube model down the left hand column: The first possible mode is with 1/2 the wavelength in the tube, the second with the whole wavelength, the third with 1 and 1/2 the wavelength, the fourth with 4x the wavelength etc. This means that the frequency of the note of the second mode is 2x that of the 1st - the 3rd mode 3x - the 4th mode 4x the frequency etc.

If we take next the closed tube model down the right hand column: The first possible mode is with 1/4 the wavelength in the tube, the second with 3/4 the wavelength, the third with 5/4 the wavelength, the fourth with 7/4 the wavelength etc. This means that the frequency of the note of the second mode is 3x that of the 1st - the 3rd mode 5x - the 4th mode 7x the frequency etc.

This is laid out in the table below which indicates the standard harmonic series:

               OPEN TUBES                    CLOSED TUBES
               ----------                    ------------
1ST MODE       fundamental                   fundamental
2ND MODE       8ve                           ------------
3RD MODE       8ve + 5th                     8ve + 5th
4TH MODE       2 8ves                        ------------
5TH MODE       2 8ves + M3rd                 2 8ves + M3rd
6TH MODE       2 8ves + 5th                  ------------
7TH MODE       2 8ves + m7th                 2 8ves + m7th
8TH MODE       3 8ves                        ------------
9TH MODE       3 8ves + 2nd                  3 8ves + 2nd
10TH MODE      3 8ves + M3rd                 ------------
11TH MODE      3 8ves + 4th                  3 8ves + 4th

etc etc...

So from this we see that open tubes can resonate at every possible mode along the harmonic series, but closed tubes can only manage the odd numbered modes.

Now sorry for the tedious explanations of this - I'm sure this is all very familiar territory to you. You are correct in suggesting that closed resonators are chosen partly because of this inability to function in the even numbered modes - yes in particular modes 4 and 10. I think the other reasons of physical space and efficiency are more important ones and were probably the only reasons considered in the early day of marimba building. As marimba building progresses though - particularly as it develops into the lower pitched regions - the concept of odd numbered modes in the resonator becomes more important.

To explain this we have to consider the characteristics of a bar's vibrational modes, and in particular the tendencies of bars made in the classic way from timber. In much the same way as a tube of air, or a string - a bar of wood can of course vibrate in many modes at once. We take the frequency of the notes produced from these modes as f=v/wavelength as we would for the air column, BUT v, is the velocity in the wooden bar rather than in air. The wooden bar is tuned (simply put) by carving an arch in the centre - so the velocity is not a constant figure as it will depend on the thickness of the timber at any given point. The point of this, is that tuners of marimba bars have noticed over the years that the 2nd and 3rd vibrational modes tend to produce notes roughly, 2 8ves, and 3 8ves plus a bit, above the fundamental. The exact placement can of course be tuned along with the fundamental by carving the timber in the correct spots. In average registers of marimbas the placement of these secondary notes may be more a matter of fussiness - producing a more refined sound - we do not hear them that strongly, as they have way less energy than the fundamental note. As the notes get really low however, we can hear them better as they are much better positioned frequencies according to the loudness level curves. So while we still get a low relative percentage of sound pressure at these secondary frequencies to that at the fundamental - we get a much higher percentage of SUBJECTIVE LOUDNESS relative to the fundamental. So what notes are the best exact places to position these secondary modes at then? The second modes is easy - its roughly 2 octaves up anyway, so that's where its placed. This is a good placement to our ears because diatonically speaking its the same note as the fundamental. The third mode is placed at 3 8ves and a Major 3rd above the fundamental. Once again - it is a better position diatonically speaking than the notes either side of it, and has the advantage of becoming more able to be heard as secondary to the fundamental by not being amplified in the resonator.

And this is the crux of the matter - that what we want to hear from a marimba note is that note - not its various partials because they would not be at the frequencies we expect them to be at anyway according to the harmonic series. Because of psycho acoustic effects, it would just confuse our subjective ear. This is one of the reasons why constructing marimba notes way down low is so difficult. The lowest note on a standard piano has a fundamental of 27.5Hz. There is no way a piano string produces enough sound pressure at that frequency for us to really hear it much if at all - but we do hear it, because our brain reconstructs the fundamental from listening to all the upper partials, and the relationships between them. For a marimba bar - these relationships do not exist - therefore the note has to produce an enormous amount of sound pressure for the fundamental to be heard.

Hope this has been some help.
Please don't hesitate to email if there are any other things that come up.

Jim McCarthy

Jim Reccomends www.makeamarimba.com for comprehensive blueprints and building guides to make your own marimbas.

answers by Jim MCCarthy - 21/06/01

For more help on marimba building you can email Jim.

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