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 The brain picture is out of date by nearly a quarter of a second, so to catch successfully we have to learn by experience how to project ahead in time to where the ball will be.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  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 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 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.
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.
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:
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:
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:
Diagram 3: Spectrum of a guitar G string played at the second fret
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:
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!
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.
Music as Biology Purves, Dale; Harvard University Press 2017
Contemporary Acoustic Guitar Vol 1 & 2 Gore, Trevor and Gilet, Gerard; Trevor Gore publishing 2011
There is also a video on YouTube of Trevor sharing his knowledge of the physics of acoustic guitars:
 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.