The case of the missing fundamental

THE CASE OF THE MISSING FUNDAMENTAL

In our daily lives we take for granted that experience of being directly in touch with the world: seeing, hearing, touching, tasting, smelling. We are there, present as the world unfolds around us.

But recent discoveries in neuroscience have shown that no matter how convincingly our senses tell us we are in direct touch with the outside world, experience is in fact produced via a complex mental construct we have built over a lifetime of trial and error. The data coming in through our senses has to be processed by our brains using this model before it can become our consciousness perception.

When as children we learn to catch a ball, part of the difficulty we have is that by the time visual data has been processed and recognized by our brain, the ball has already moved on from where we see it. The brain picture is out of date by nearly a quarter of a second, so to catch successfully we have to learn by experience how to project ahead in time to where the ball will be.

We are not directly in touch with the world at all.

As Duke University neuroscientist Dale Purves [1] points out, our experience of sound is also a mental construct. Audition works on the same basic principle as vision: it takes in data through a pair of sensory organs and processes it through a complex audition model before it becomes conscious experience.

The physics of vibration, resonance, and soundwaves happens in the real world.

Our perception of that reality is an experience inside our minds.

As successful organisms (that is, still alive!), we can assume that our auditory model accords with physical reality accurately enough for us to survive in the real world.

The relationship between our model and the real world is the relationship between a map and the terrain it sets out to represent. The map is not the terrain.

This has some very interesting consequences. The difference between frequency and pitch is a good example.

 

FREQUENCY AND PITCH

FREQUENCY

Frequency is a measure of how rapidly something vibrates.

Vibration is cyclic. Think of a kid on a swing – the swing seat moves backwards and forwards, passing through its central rest position at the beginning of each cycle and again at the end, on its way back to start the next cycle.

The frequency is the number of cycles completed in one second. The unit cycles per second is given the name Hertz (Hz). 10Hz is 10 cycles/sec. 0.1Hz is a slow one cycle every 10 seconds.

Frequency is a property of the soundwaves that reach your ear. Pitch is the result of your brain reconstructing frequency data received by the ear into a sensory experience.

 

PITCH

Pitch is the sensation of “highness” or “lowness” we have when we hear a tone. Pitch is perception, not an external physical reality.

You could compare pitch in audition to colour in vision. The perception of colour is the way our brains represent data from the outside world that relates to the frequency of the light taken in by our eyes. “Red” is our perceptual response to lower light frequencies and “violet” to higher frequencies.

 

The pitch you’ll hear on this audio clip is a low A from a piano:

Audio file 1: Piano low A note

(Use good quality headphones if possible.)

You can hum this pitch, and we all experience tones like it as single notes. We even name notes: this one, for example, is called A2.

 

MORE ABOUT FREQUENCY

The diagram below shows a frequency analysis of the piano note you just listened to.

PIANO A NOTE

Diagram 1: The power spectrum of a piano low A string

This diagram is called a frequency power spectrum, and what it shows is how energy is being radiated from the vibrating piano string, amplified by the instrument’s soundboard.

A single pure tone would be just one of the spikes on the spectrum.

So here is the first indication of the difference between frequency – a measurable quantity – and pitch, which is a sensation manufactured by the brain. We hear a single pitch, and we experience the other frequency spikes as the timbre of the note, making it easily identifiable as a piano.

At some frequencies the piano radiates no energy at all, while pumping out audio energy at narrow discrete frequency bands. This series of spectrum spikes is called a harmonic seriesand it stems from the particular ways a string can and cannot vibrate.

I’m now going to switch from piano notes to guitar notes [2].

The next audio file is what the open guitar string 5 sounds like when plucked:

 

Audio file 2: Guitar string 5 (A)

 

This is the spectrum of the note from a guitar A 110Hz string:

SPECTRUM OF GUITAR A STRING

Diagram 2: Spectrum of a guitar A 110Hz string

The frequency peaks that make up this tone are at 110, 220, 330, 440, 550, 660, 770, 880, and 990Hz.

These are all multiples of the fundamental, 110Hz.  Mathematically the peaks can be predicted using:

fn= nf1

where n is called the harmonic number (1, 2, 3, etc).

Notice, by the way, that the guitar string harmonic mix is not as rich as the piano note. You hear the difference between the notes in terms of the timbre, which stems from the harmonic mix that make up the notes.

 

THE OCTAVE INTERVAL

The next diagram shows the spectrum of a guitar G string played on Fret 2, giving a second A note.

Have a listen to the note first:

Audio file 3: Guitar G string played on 2ndfret

 

Here’s what the spectrum looks like:

SPECTRUM OF GUITAR G STRING FRET 2

Diagram 3: Spectrum of a guitar G string played at the second fret

The peaks here are at 220, 440, 660, and 880Hz [3].

You’d expect this to sound different from the original A note with the fundamental at 110Hz, and it does. Most people recognize the pitch difference as a leap of one octave.

 

THE MISSING FUNDAMENTAL

You’re about to experience something strange and intriguing. There are two audio files below, the first of which is simply the guitar open A string (string 5) note.

 

Audio file 4: Guitar A note

We know that its fundamental is 110Hz, and that its harmonic series is given by fn= n ×110.

The second audio file is the same note, but I have used a filter to remove the first frequency spike at 110Hz, leaving the rest of the note untouched.

 

Audio file 5: Doctored guitar A note with the fundamental removed

 

Here is the spectrum of that artificially altered note:

SPECTRUM OF A STRING MISSING FUNDAMENTAL

Diagram 4: Spectrum of a guitar A string with the fundamental (110Hz) removed

Here we have something completely unnatural – a harmonic series 220, 330, 440, 550Hz. A natural series starting on 220Hz, as shown in Diagram 3, would go 220, 440, 660, 880Hz.

Compare the sounds of the two tones (Audio files 4 and 5). Given that the doctored tone now has its lowest peak at 220Hz, you might expect to hear the pitch as an octave above – but you don’t!

Why not?

I had to artificially remove the fundamental using Audacity’s notch filter. There is no natural sound in the world corresponding to my doctored note, so your brain obligingly puts the missing frequency back in.

Pitch recognition is to some extent hard-wired into the brain, there are centres dedicated to pitch, and the ear certainly provides the brain with excellent pitch data. We “know” that, except for that missing fundamental, the frequencies present in the doctored note form a harmonic series identical to A 110Hz.

Our brain doesn’t waste any time puzzling over why the fundamental isn’t there – instead, it just fills in the gap.

 

WHAT CAN WE LEARN FROM THIS?

Aside from pointing out the complex relationship between physics and perception, the main point is the importance of psychoacoustics in determining what we hear. For example, we can focus on a single conversation in a noisy room by filtering out the background.

Psychoacoustics draws in much more than just our self-constructed audio model of perception. Our wider knowledge and assumptions feed into the process as well.

For example, everybody knows the Stradivarius violin as the best of all, despite the fact that a number of blind tests have shown that expert violinists often fail to pick the Strad from an array of other high-quality instruments. Maybe the experience of those who play and listen to Stradivarius instruments stems from our assumptions as much as the genuinely superior quality of the instruments themselves.

For those interested in guitars, I can recommend particularly Gore And Gilet’s monumental Contemporary Acoustic Guitar Volumes 1 and 2. Sections 1.1.2 to 1.1.3 delve into the how the ear works in terms of attack and decay, roughness, and masking in particular.

These volumes are brilliant and I can’t recommend them highly enough.

 

 

REFERENCES

Music as Biology Purves, Dale; Harvard University Press 2017

Contemporary Acoustic Guitar Vol 1 & 2 Gore, Trevor and Gilet, Gerard; Trevor Gore publishing 2011

 

https://goreguitars.com.au

There is also a video on YouTube of Trevor sharing his knowledge of the physics of acoustic guitars:

https://www.youtube.com/watch?v=DdXEzIo6IDc

 

[1] Music as Biology Purves, Dale Harvard University Press 2017

[2] The reason for the switch is that the piano has a large number of undamped strings that can resonate freely when you hit a particular key. The spectrum of the A octave 220Hz showed a spike at 110Hz because that string resonated when the 220Hz string sounded.

[3] For those wondering about the very sharp peak at 50Hz, this comes from the 50 cycle electrical power supply in the room.

What is a “live back”?

When guitar makers refer to a “live back”, they mean that a soundbox has been created in such a way that as you play the back plate vibrates and adds to the sound being produced by the soundboard. A live back instrument has a more complex timbre.

A useful way to understand guitar acoustics is to think of the soundbox as a resonator with three connected parts, each of which has an effect on the other two. It’s an example of a complex system in which you can’t change one element without affecting all the others – this is known as coupling.

The three coupled resonators in an acoustic guitar soundbox are the top plate (or soundboard), the back and sides assembly, and the air contained in the box. How these three react together when you play the guitar determines the sound (timbre) of the instrument.

This, by the way, opens up the possibility of changing the overall sound by messing with any one of the three resonators – if you know what you’re doing. For example, you can vary the size of the sound hole to change the air resonance, which will in turn affect how the other two resonators respond.

You’ll notice I carefully used the words “back and sides assembly” rather than just the word “back”.  For the three element model to work there’s no need for the back and sides to do anything other than vibrate up and down for them to play their part. The soundboard, we know, behaves in a more complicated way and is the most important element in creating a good guitar sound.

But what if the back plate developed ambitions of its own and didn’t want to limit itself to going up and down as a rigid block with the sides? What if it had been watching Mr Soundboard doing his smartypants modal tricks just across the way and wanted to get in on the action? Maybe they could make beautiful music together.

This needs a different design approach, of course. Traditionally guitar backs are reinforced by four crosswise spruce braces laid like the rungs in a ladder. Cleverly, we call this ladder bracing – but that’s just the kind of guys we guitar makers are.

So how does the traditional method work out for poor old under-appreciated Mr Back?

Here is  the spectral signature of a a Martin 000-18 made in the early seventies. I recorded the tones produced first by tapping the soundboard at the bridge and then again by tapping the back plate underneath the bridge. Here’s the response of the back plate;

MARTIN BACK

You can see that the back response has one main peak at 202Hz followed closely by another at 221Hz.

The rest of the signature doesn’t really mean much – remember that the loudness is being measured on the dB scale which is not exactly intuitive. The 202Hz peak drops away by over 10dB into the trough to its right, which means the sound level at the trough is less than one tenth of the peak.

If we take the -40dB level running through the centre of the graph, the 202Hz peak is about 1,000 times more intense. So anything less than -40dB isn’t very significant.

If the Martin has a live back, we’d expect the back to contribute to the overall tonal signature. In other words, some peaks in the back signature should imprint themselves onto the overall signature produced by tapping the top to simulate the impetus given by the strings when you play.

In the chart below the back signature is red and the overall top signature in blue.

MARTIN TOP AND BACK

There’s no really strong  imprinting of the back signal onto the overall signature in the Martin. The peak at 100Hz is the resonance of the air body (the coupled Helmholtz resonance), so you would expect that to appear on both signals. The back plate peak at 202Hz is echoed rather reluctantly in the top. Better is the next peak at 221Hz which clearly reinforces the main top peak – that’s one for Mr Martin. At 258Hz there’s another increase in the overall response that matches a back peak as well.

And my live back score for the Martin 000-18 is……(drum roll)…..three and a half!

I have also done the same test with one of my earlier Jumbo models. Here’s the back signal:

JUMBO BACK SIGNAL

The first thing you’ll notice is that this back signal is similar to but a little more complex than the Martin’s – there is quite a peak at 412Hz that isn’t there in the Martin. I can’t claim any credit for this at that stage in my building career. Both instruments have the same four rung ladder back brace system, so I don’t quite know how I managed it except by perhaps making the back plate a little thinner.

Anyway, how well does this imprint itself onto the overall tonal signature?

JUMBO TOP AND BACK

Well, I give this maybe five out of ten. There’s a reasonable imprinting visible at the main back response of 210Hz, a small one at 258Hz, another at 275Hz, not a bad one at 412Hz.

The overall tonal signature of the Jumbo is more complex and interesting than the Martin, and you can hear a difference between the two. How much of the difference is due to the live back effect is hard to tell.

This is all very fine, and I’m excited about the response of my latest guitar, a bamboo classical, that uses Greg Smallman-style lattice bracing on the top and a new experiment of mine for the back bracing. The initial tests for live back are very encouraging – I’m giving it seven out of ten so far, but it doesn’t have its neck on yet.

But…

…is there any evidence that a live back guitar actually sounds better than any other? Is having a live back necessarily a good thing?

And anyway, don’t you have your body in contact with the back when you’re playing? Doesn’t that damp out any back vibration anyway?

I’ll tackle these questions in a later blog. At least now you know what “live back” means.

 

 

 

Cocky chorus with a crow solo

Here’s a recording Wendy made on her phone on an autumn morning in our back yard. The chorus is made up of about ten Sulphur-crested Cockatoos, who politely allowed a few blow-ins (Australian ravens, aka crows) to take a couple of solos. I think you’ll agree the the results are worth listening to:

And because I know you want another spectral analysis, here it is with the cockies in blue and the crows in red:

Cocky chorus

Cocky chorus

So which is the more musical out of a cocky and a crow? Hmmm. The spectrum points to the crow because the cockies, admirable birds though they are, put out a blast right across the spectrum and therefore qualify more as noise than music.

Sulphur-crested cockatoo

Sulphur-crested cockatoo

 

Cicadas

Early summer days in this part of Australia ring with sound of cicadas:

These extraordinary little creatures spend years underground before popping up to find a mate  by climbing into the nearest tree. (Good David Attenborough dialogue, don’t you think?)

It’s the males that call – the females are silent but respond to a male by flicking their wings. For such a small critter the noise level they produce is incredible – over 100dB from around a metre away from just one insect.

I decided to record the sound and analyse it. To my surprise, this is what the spectrum looked like:

Cicada sound spectrum

Cicada sound spectrum

What’s surprising is that no natural oscillator I have ever seen has a gap in the spectrum, as you see here (between 600 and 800Hz). The only conclusion possible is that there there are two quite different species calling at the same time, most likely a bigger one in the lower frequency and a smaller one in the higher.

The two likely types are what are commonly called greengrocers and black princes. I have no idea which call is which…

This is a black prince:

BLACK PRINCE

Link

As I do with every component of a guitar, I want to understand the purpose of the back plate so I can build it in the best way I possibly can. Gore and Gilet’s book Contemporary Acoustic Guitar has been very useful to me (as always) by raising the possibility of building what they call a “live back” – in other words, a back plate that contributes to the overall tone of the instrument by vibrating along with the top when the strings are sounded. This adds a complexity to the tone that wouldn’t be there with a non-live back, so should be desirable.

I’m not convinced I have succeeded in this yet. In this page I’m going to show how I make and brace the back plate, and evaluate my current success in the quest for the live back. I have adopted the back brace plan used in Trevor Gore’s instruments as a starting point, as you’ll see. At the end of this entry I’ve put some evidence that indicates I haven’t breathed life into my backs just yet. Perhaps I need a thunderstorm and lots of electric arcs and an assistant called Igor.

But now I want to show you some test results that have got me thinking more about this concept.

What is the purpose of the back plate?

“Traditional” steel string guitar backs use a ladder bracing system that I also used in my early attempts. It looks like this:

Traditional ladder bracing

Traditional ladder bracing

The purpose of the back braces has always been seen as mechanical rather than tonal. Thin panels tend not just to flex under load but to distort as they absorb and lose water. Wood always expands much more across the grain than along it, so this can lead to weird shape chances unless stiffly braced. The back centreline join can also be pulled apart by dimensional changes in the wood if the guitar is moved into a dry climate after being built in a humid one.

Many people argue that when you play the guitar your body is in contact with the back and damps out all the vibration anyway, so attempting a live back is pointless. This isn’t at all true, as I think I can show you.

The T(1,1)1 “breathing mode

A guitar’s simplest mode of vibration is called the “breathing” or T(1,1)1 mode. You can see it in this animation as Mode #2:

http://www.acs.psu.edu/drussell/guitars/hummingbird.html

This simple mode would not exist if the back were not there. It’s as simple as that. The “Breathing” mode is low frequency (usually around 100Hz for a guitar) and contributes bass boom and depth to the sound.

But can the back contribute higher modes to the overall guitar sound as well? This is what we mean by “live back”.

Can I convincingly show you a live back in action? Sadly, no. But I live in hope…maybe this guitar will achieve it. Stay tuned!

What does the back actually do when a guitar is played?

This isn’t as simple a question as it sounds. Gore and Gilet’s analysis shows that a soundbox is made up of three separate resonators: the top, the back/sides, and the air contained in the box.

Each of these has its own fundamental frequency and an associated string of harmonics or partials, as does every resonator. Here, for example, is the frequency spectrum of a singing wineglass – the highest peak is the fundamental, and the other peaks represent the inevitable series of higher frequency partials produced by different modes of vibration.

WINEGLASS


Here is the string of a guitar string ringing. It behaves in a very similar way, although in this case we can call the partials “harmonics” because they have a simple mathematical relationship with the fundamental, which partials do not:

JUMBO B STRING

Where it all becomes complicated is when you start to join simple resonators together, as you do when assembling a guitar. Because they’re connected, each one influences the others, and not in a simple way.

Okay, enough physics for now. Let me show you what happens when you start to mess with one out of the three joined soundbox resonators. I’m going to use some data from tapping one of my Jumbo guitars to illustrate.

Firstly, here’s the spectrum produced by tapping the top with the guitar held up by the neck and no contact with the soundbox:

Jumbo top response held free from body

Jumbo top response held free from body

Notice that it isn’t a simple spectrum like that for the wineglass, although there are a number of peaks that stand out. Notice particularly the first one at around 100Hz, which is the total resonance of the top, back/sides and airbody (the T(1,1)1 or “breathing” mode we’ve already seen). Now look at what happens when I hold  the guitar in my normal playing position resting lightly against me:

Jumbo top response in playing position

Jumbo top response in playing position

I have kept the original signature (the red line) so you can see the effect of body contact (the black line). Very little at all, wouldn’t you agree?

Okay – one last graph before we get on with construction details. This one shows what happens when I aggressively damp the back by pressing my hand firmly against it while tapping the top:

Jumbo top, back heavily damped

Jumbo top, back heavily damped

See the difference? Again, I have left the first freely-held response (red line) for comparison. The most striking effect of heavy back damping is the complete disappearance of the first peak in the response, the “breathing” mode. Immobilising the back has broken the link between the three resonators in a dramatic way. The tap sound was an anaemic thunk.

What does this tell us? Well, it makes clear a number of vital things about how a guitar works:

1. Messing with one of the three coupled resonators affects the other two, sometimes dramatically. This understanding opens up possibilities for tuning the soundbox if you know how.

2. The response with the guitar held in the playing position shows very little difference to the guitar held freely – the T(1,1)1 airbody peak is still strongly there. This suggests that the back continues to have a strong influence on sound production when playing, and that the common wisdom (that the player’s body contact completely damps the back and makes it useless to strive for a live back) is emphatically wrong.

3. The “breathing” mode is important to the production of sound by a guitar, and it isn’t there unless there is a back to contain it and importantly that it is free to respond to the strings. When tapping  with the back aggressively damped, the tap sound lost its depth and became a rather pathetic thud. This is because with the back immobilised, the box can’t “breathe”.

To sum up:

a) the guitar back is a very important factor in determining the overall sound of the guitar, and deserves close design attention;

b) all guitar backs are “live” in the sense that they are essential for the guitar to produce a sound with any depth (damp it aggressively and the “breathing” mode dies);

c) I haven’t yet achieved the extra Gore/Gilet step of building a back that further enhances the sound by adding complexity through its own resonant response on top of the “breathing” effect.

Building the back

This is the bamboo guitar back nearly finished, showing the Gore-pattern ladder/radial bracing. Here I have been cutting the “gable” with a brace chisel (just visible bottom left). The back is resting is the 20ft radius dish to preserve the curvature as the braces are fitted.

The Gore back bracing design

The Gore back bracing design

These are the spruce brace blanks machined to size:

Brace blanks (back braces on the right)

Brace blanks (back braces on the right)

The are the steps in putting a back together are:

1. thinning the panels to the right thickness;

2. joining the two panels to get enough width for a back plate;

3. bracing the back plate;

4. attaching the back plate to the sides.

Thinning the panels

I thin the back panels using a drum sander before joining them:

Drum sander

Drum sander

I check the thickness often with vernier calipers. A trick I have learned in using a drum sander is that you don’t need to adjust the drum height for each pass, especially when you’re nearing the thickness you want.

Vernier calipers

Vernier calipers

This way I can control the thickness to an accuracy of 0.1mm. In another page (The bamboo guitar – Part 3) I have explained how I chose what thickness I would work to for the bamboo. My aim was to produce a back with properties as close to my usual blackwood backs as I could.

Once the panels are the right thickness it’s time to join them together. Though initially it daunted me, it’s surprisingly easy to get a good invisible joint. Here’s a top I’ve joined using my method (the chalk marks help keep the two panels in the right relationship while I’m working on them):

A top joint

A top joint

These are actually spruce top panels, but the principle is the same and I forgot to take photos for the bamboo back.

The first step is to plane the joining edges straight on a shooting board, then fitting them together on a flat surface to look for gaps. Using an old plane with sandpaper stuck to the sole,  with care you can remove the high spots and get an invisible joint. The trick is to get one edge as straight as possible, then adjust the other to fit it. The shooting board allows you to keep the edges square as you work on them.

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Joining the panels

Once you’ve got the edges fitting perfectly, it’s time to join them together. I use a simple but effective setup that I got from somewhere I can’t remember. I have a flat table with a batten fixed along each side, one of them adjustable so I can clamp different size pieces.

The jointing table

The jointing table

I slip a small batten under the the panels where they join so the edges are slightly lifted, and adjust the edge battens to just hold them in place:

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To make sure the panels stay in vertical alignment, I hold the joint down with cork-backed blocks and go-bars.

When the batten is slipped out the edges are gently forced down to make a beautifully tight joint. Oh, and don’t forget to put glue on first, and put wax paper down so you don’t glue the panels to the table…

Once the glue is set, you can take the joined back plate out of the clamps, clean up any dried glue, and cut out the correct shape.

Adding the back braces

Time to get out the old 20ft sanding dish again, last seem while profiling the side assembly. This time it will be used for two things:

a) sanding the braces to the right curve;

b) acting as a mould to make sure the back conforms to the right curvature.

The first bracing piece is the marriage strip, a piece of 20 x 4mm spruce kept from past top panel production. It is glued along the centre joint to reinforce it, so it’s important to cut it so that its grain crosses the joint at right angles – you can see the strip across the bottom of this photo:

Marriage strip

Marriage strip

The marriage strip is glued on and the whole back pressed into the dish using go-bars. Glue is slippery, so keep a sharp eye on the strip to make sure it doesn’t absent- mindedly wander off before the glue grabs. I also make sure that I know exactly where the marriage strip must end to meet the end blocks neatly once the back is glued to the sides.

The back brace blanks have already been machined to 20 x 10mm, keeping the grain vertical. I cut each one to length, mark out the scallops, then sand the bottom surfaces to the 20ft curve by rubbing them in the sanding dish, I the cut out the scalloping on my bandsaw and finish them with a small sanding drum in my drill press.

I use a sharp X-acto knife to cut through the marriage strip and a sharp chisel to remove wood so the braces can be glued in.

Here’s how it looks with the go-bars in place:

Braces held in place with go-bars

Braces held in place with go-bars

Next to go in are the radial tone bars:

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The scalloping of the braces reduces the mass of the back, but importantly also allows some later tuning of the back resonances once the guitar is finished. (Remember we’ve found that messing with one of the three resonators will affect both the others.) The general principle is that the thicker the braces the higher the fundamental resonance of the back plate will be. If I want to lower the back resonances later, I can carefully thin down the central scallops working through the soundhole.

By the way, the idea that you can tune each brace separately doesn’t work in practice (or even in theory, for that matter – sorry Roger Siminoff) because the back responds as a whole, not a simple sum of its individual parts. Maybe there’s a saying in that somewhere.

Finally, I use a very sharp chisel to carve the top edge of each brace at an angle so it comes to a point like a gabled roof. This reduces their mass but pretty much preserves their elasticity.

Joining the back to the sides

The sides of  the soundbox have already been shaped to match the curvature of the back plate (see The bamboo guitar – Part 6) so it’s now a matter of joining the two assemblies. Because I don’t use an edge binding to hide the joint, I need it to be a perfect fit and firmly clamped:

Joining the back to the sides

Joining the back to the sides

I use cylindrical screw clamps from Stewart-MacDonald (www.stewmac.com).  Once the glue has set, I can carefully trim the overlap down and sand the edge so it is smooth and slightly rounded off.

I now put on a coat or two of epoxy to seal and harden up the grain, leaving the guitar looking like this:

Epoxy sealing coats on

Epoxy sealing coats on

Not bad for bamboo, wouldn’t you say?

And now we return to the live back issue…

So how will I know if I succeed in producing a live back? The evidence would be there if the spectral signature of the back, or at least parts of it, showed up clearly in the overall signature of  the guitar. This sounds simple, but actually isn’t.

We’ve seen that the three resonators, the airbody, the top, and the sides/back that make up the soundbox of a guitar, link together as a whole but not in a simple way – each one changes the others in subtle ways. So if they’re linked so intimately, how can we sort out if any part of the overall sound (other than the breathing mode) comes from the back specifically? Can we isolate what the back is up to and compare it to the overall tonal signature?

There is a way to decouple the top signature from the back signature: block up the soundhole. This takes the airbody out of the mix because it can’t breath, leaving the top and back much less intimately connected.

Here is a comparison of the overall tonal signature of the guitar (the top tapped with the soundhole open) – the red line –  in the normal playing position. The blue line is as close as we’ll ever get to knowing how the back would respond on its own. What I’m looking for is a convincing overlap between the two, showing that the back is contributing to the overall sound.

No evidence of a live back here

No evidence of a live back here

The outstanding feature of the back response is the peak at 210Hz, and maybe the overall response is higher there as a result. Maybe. The airbody response isn’t present because the soundhole is blocked.

Except for a teensy little peak at 175Hz and a fat-looking feature between 210 and 230Hz, I can’t see any convincing evidence that the back is having any influence on the overall sound. Oh, well.

And for my next trick…

Next I’ll describe making and fitting the top panel, the most crucial element in any guitar.

The bamboo guitar Part 6 – the wedge, blocks and splints

A strangely painful-sounding title, wouldn’t you say? I’m a bit worried how this will turn out, but I promise you won’t end up with a wedgie or a splint as a result of reading it.

Top and tail blocks

The end blocks hold the two halves of the soundbox together. They are important for the structural integrity of the guitar, particularly the top block that has to absorb the stress of the neck joint under string tension of around 70kg weight. It has to be a substantial chunk of wood accurately fitted to the soundboard, sides and back. That 70kg weight is another reason I like laminated top linings – imagine an adult man standing on the end of the guitar.

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I make mine from two blocks of Niugini Rosewood joined at right angles and pinned vertically with two 9mm dowels, all held with Titebond glue. The reason for the shape is that the guitar will have a bolt-on-bolt-off neck, so there needs to be space for horizontal and vertical bolts to land. (In this picture the guitar is resting in sanding dish because I am profiling the bottom edge to the right curvature before fitting the side splints and the linings.)

The front face of the block is shaped to fit by temporarily taking the sides out of the construction mould and sticking some adhesive backed sandpaper where the block will land. The block then gets sanded to an exact fit – and I do the same with the tail block before putting the two half sides back into the mould.

Both blocks then need to have a channel cut into them to fit the 18x6mm top linings. Some careful shaping is needed to get a good fit, because of course the linings have a subtle curve to them. When there’s a good fit I glue them both in place  with firm clamping.

The tail block will have to support the strap button and the jack for the pickup, so again I laminate it up out of four 4mm thick pieces with grain crossed to stop it splitting if the guitar is ever dropped on its end.

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In this picture you can clearly see the notches in the top lining waiting for the side splints.

The tail wedge 

It’s customary to cut away the sides where they join at the tail and replace them with a nicely-contrasting wedge of wood – here I’ve used blackwood. To get a good fit. I make the wedge first, clamp it to the guitar and carefully cut into the sides, and then remove the waste with a chisel. The subtle shape means the when you tap the wedge into place it fits very tightly for a hardly-visible joint.

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Profiling the back edge

Once the tail wedge is in and trimmed off, I profile the bottom edge of the soundbox before I glue in the side splints (in all the pictures so far, the guitar is resting in the 20′ radius sanding dish). No matter how hard I try, somehow the sides never match the designed profile and I have to adjust them.

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Side splints

The side splints in a guitar are small strips of wood glued vertically to the sides between the soundboard and the back. They reinforce the side panels by crossing the grain to stop splitting, and they add extra rigidity. They are quite important, as Gore and Gilet point out, because when a guitar is played, at the simplest level of analysis the soundboard and the soundbox necessarily move out of phase with each other. The more rigid the soundbox back and sides are – that is, the more the behave like a single unit – the more efficiently the soundboard can vibrate.

So I take my splints seriously now, where before I used to slap in a few little matchsticks made from scrap spruce and move on. I now laminate them from hardwood and notch them carefully into the top and bottom linings. I make them from three 2mm laminations of leftover Blackwood, so they’ll be the same 6mm thickness as the top linings. They’re 9mm wide, and carefully shaped to fit the  curvature of the sides.

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It’s much easier to fit the splints a bit over-long, then trim them with a saw to match the depth of the side panels – otherwise you’re left sanding end grain when doing the final profiling of the back.

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FITTING THE KERFED LININGS

The last task before fitting the back panel is to install the kerfed lings. I fit them by bending a length of the lining around the curve between each side splint, mark the length, and cut the length off with a bandsaw. If they’re cut slightly overlength you can fit them exactly by sanding the ends.

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Each piece is glued in, and they need to be clamped. I use spring clothes pegs, with the odd more powerful small spring clamp when needed.

The thing to look for here is that the lining fits well against the sides. Because you’ll be working with the soundbox face down, you can sometimes get an unpleasant shock when it’s all finished – you proudly turn it the right way up and find the top edges of the linings are standing slightly away from the sides.

With a bit more work in the sanding dish the linings and the ends of the side splints will be profiled exactly to mate up with the back plate when it’s ready to fit.

The bamboo guitar Part 5 – laminating the top linings

OLYMPUS DIGITAL CAMERAThe top linings form the joint between the soundboard and the sides of the guitar. Generally guitar makers use kerfed linings like the one above. They’re easy to bend around the curves, and they act like a series of small reinforcing blocks to tie the top and sides together. They work well, and I use them for the back-to-side joint.

But I have a theory that solid top linings are better at transferring vibrational energy from the soundboard to the soundbox body. They also add greatly to the overall strength and structural rigidity of the guitar. So I laminate them out of thin strips of Australian oak, a wood that takes to heatbending well. In future I think I’ll use bamboo instead, because the trees the “oak” comes from, Eucalyptus regnans or Mountain Ash, are precious. They’re one of the tallest tree species in the world:

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I cut the strips on my table saw from a blank that I have thicknessed to 18mm in my drum sander. I then thickness the strips to 2mm in the other dimension (actually, I stay in this dimension to do it) by feeding the cut faces through the drum sander. Using good old Titebond  glue, I laminate three 2mm thicknesses in a mould to end up with a pre-shaped lining of 18x6mm.

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To make it easier I prebend the strips using the side bender, otherwise they put up a hell of a fight.

Here’s what it looks like out of the mould:

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What I’m doing here is cutting notches to house the top of the side-splints that will reinforce the sides. The splints also add rigidity to the sides, and a bit of useful mass as well.

I then glue the linings onto the sides, after which I can saw the ends flush with the construction mould. After that I bolt the two sides of the mould together ready to take the top and tail blocks that unite the sides. Notice the very clever placement of baking paper to avoid gluing the guitar to the mould. Yes, we guitar makers are a cunning lot.

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