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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 The brain picture is out of date by nearly a quarter of a second, so to catch successfully we have to learn by experience how to project ahead in time to where the ball will be.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