What Are Waveforms And How Do They Work?

What is an Audio Waveform?

A waveform is a graph that displays amplitude or level changes over time. Amplitude is measured in a bipolar manner, with positive and negative values, not to be confused with level, which can be the absolute value of amplitude changes or an average. This concept is abstract because waveforms typically contain tens of thousands of discrete changes within an unimaginably short period, crammed into a short block in a sequencer. As you probably already know, as you zoom in on a waveform, its contour becomes more and more visible.

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PCM and Nyquist Frequency

A waveform is a digitized recreation of very dynamic voltage changes over time. Here is how they are typically generated…. The discrete changes in an input signal are rectified instantly through a process called "Pulse Code Modulation" (PCM). Simply put, PCM assigns a bit value to each sample at whatever sampling rate you're running. Furthermore, the higher the bit depth, the more values the computer has to choose from, and the more accurate the rectification is. The sampling Rate for recording purposes is pretty standard - 44100 Hz, the Nyquist Frequency for audio. Here is why it is standard…

The highest audible frequency for humans is 20,000 Hz.
Every frequency has a positive and negative half (compression and rarefaction).
So, as long as we sample (analyze and generate) rapidly enough to catch both the positive and negative portions of the highest (most rapidly oscillating) audible frequency - we can confidently rectify just about every audible frequency.
This means at a sampling rate of 40000 Hz, we will catch both the positive and negative portions at 20,000 Hz. However, since sine waves are infinitely smooth and gradual, just one sample of each portion will not produce a very accurate waveform at very high and very low frequencies - but this gets pretty close. This is why the extra 4100 Hz in Nyquist Frequency exists. It accounts for any aliasing when sampling very low and high frequencies.

Of course, at this point, you may ask yourself, "Why not just sample as frequently as possible? Why not just go to 50,000 Hz or 100,000 Hz?" The answer is negligence. We cannot hear the improvement past 44100 Hz. BUT THIS DOES NOT MEAN OTHER SAMPLE RATES ARE NOT USED. For example, when producing sound or music for film, the sample rate of the audio should match the video resolution rate - often over 90,000 Hz.

Synthetic Waveforms

Synthetic waveforms are MILES less complex than audio from an acoustic or electrical source. This is their beauty - they allow us to create sounds from scratch without getting ahead of ourselves. After all, every frequency part of a sound - harmonic or not - can potentially send you down a path of unintentional, unfocused ideas that will do more harm than good in context. These are the four basic synthetic waveforms.

Sine

One harmonic, one frequency. It is so straightforward that it cannot technically exist acoustically or electrically. Overall, even the purest-sounding oscillators and self-resonating filters have a little noise in their output. Y= Asinx describes it mathematically.

Square

Contains odd harmonics (odd whole number multiples of the fundamental). This means that if the fundamental frequency of a square wave is 200 Hz, it will also generate 600 Hz (3rd harmonic), 1000 Hz (5th harmonic), 1400 Hz (7th harmonic), and so on…..

Triangle

In short, the triangle is like a square wave containing odd harmonics, except the harmonic content is lower in amplitude than in a square wave. Meaning the harmonics have less influence on the overall shape of the wave.

Sawtooth

Contains all harmonics. This is the most complex of the four basic synthetic waveforms - but nowhere near as complex as natural sound. In addition, if the fundamental of a sawtooth is 100 Hz, it also contains 200 Hz, 300 Hz, 400 Hz ….etc. Each harmonic is a little lower in amplitude than the previous one.

To conclude, IT IS IMPORTANT TO REALIZE that triangle, square, and sawtooth waves ARE MADE UP OF SINE WAVES - AS IS EVERY SOUND IN THE UNIVERSE. Essentially, one can synthesize a somewhat accurate square wave by adding odd multiples of the fundamental frequency to a sine wave. In the audio below, I have done just that! Listen to the sine wave gradually form into a square wave as I sweep in odd harmonics. The synthesized square wave plays in isolation at the end. I have isolated each partial on a different track and, in doing so, created a crude "spectrum" analysis of the sound.