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The Missing Fundamental | Electronic Ramblings

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Before We Start

This blog post utilizes WebAudio to dynamically create audio demos to demonstrate the subject matter.

It is highly recommended you check your volume because things might get a little loud.

Finally, it is highly recommended that you use high quality headphones or speakers. Enjoy!

Introduction

Just as the human vision system can be fooled with the help of special edge cases that we call optical illusions, so can the human acoustic system in the form of acoustic illusions. The study of the acoustic processing and how we perceive sound (as opposed to the physical audio signal) is called psychoacoustics.

I will explain (and demonstrate!) the psychoacoustic phenomena called the Missing Fundamental in this blog post.

Establishing Fundamentals

Before we discuss the missing fundamental phenomena, we should learn a bit about the basics of psychoacoustics. Specifically, lets talk about pitch.

The Pure Tone

Lets first have a listen at a pure tone, otherwise known as a sinusoid.

Play Sound

As the name implies, the sinusoid is made up of a repeating sine wave at a specific frequency, which in this demo is 440Hz, or A4 in the musical scale.

That sounds pretty uninteresting. But do take note about the pitch of the sound you just heard. I cant really describe what pitch is in words, but its how high or low it sounds, and is the human perception of the frequency of the signal.

Adding Harmonics

Now try listening to this sound.

Play Sound

That sounded a little different! Feel free to play the two tones back and forth.

The part thats different about those two signals is called timbre in musical terms. Its why a guitar sounds different to what a piano sounds like even if they play the same note.

However, the two sounds are not completely different. You should be able to tell that the pitch of the two sounds is the same. The timbre is different, but it somehow feels like the same pitch, and youre right.

The sound you heard is actually made up of not only the sinusoid at 440Hz (just like the pure tone) but also a sinusoid at 880Hz and a sinusoid at 1320Hz all added up together.

You dont hear the constituant sinusoids all on their own, but rather a single sound that seems to have a well defined pitch of the fundamental.

Note that the two other tones (880Hz and 1320Hz) have frequencies that are integer multiples of the lowest tone (440Hz)two times and three times respectively.

We call the lowest tone the fundamental and the two other tones the harmonics. The harmonic that has a frequency double that of the fundamental is called the second harmonic, and the harmonic thats triple is called the third harmonic, and so on.

The Missing Fundamental

The previous sound example was made up of three different sinusoids, but the harmonics were not the same amplitude as the fundamental. The fundamental had an amplitude of 1.0, the second harmonic had an amplitude of 0.5, and the third harmonic had an amplitude of 0.2.

These numbers are totally arbitrarypretty much any set of amplitudes will make you perceive a pitch of the fundamental.

Try playing with the next bit of codeit lets you modify the relative strength of the harmonics. I also added a few more harmonics for you to play with.

For now, keep the fundamental fixed at maximum.

Play Sound

Youll quickly see that they all seem to have the same pitch as the original pure 440Hz tone.

It is as if the brain knows that the harmonics are related to the fundamental in integer multiples, and calculates the fundamental all subconsciously.

Heres the kicker: it turns out you can actually get rid of the fundamental and still hear the missing fundamental.

Try it! Try cranking the harmonics up, and crank the fundamental down to 0. You can still hear the missing fundamental as the pitch!

Missing Fundamental in the Wild

The missing fundamental is actually used behind the curtains in a lot of applications, but I think the coolest application of this psychoacoustic phenomena is using it to seemingly enhance the bass response of speakers.

Small speakers tend to have poor bass responselower frequencies are not reproduced as well as they should. This is pretty much unavoidable because of just plain old physics. Small speaker diaphragms just cant create enough low frequency air waves.

One can exploit the the missing fundamental phenomena by intentionally creating a lot of harmonics of the bass portions of the audio. The listener would then perceive the missing fundamental of the bass harmonics, which is the bass itself.

Stay tuned for a future blog post in which a bass boost demo utilizing the missing fundamental will be presented!

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