Waves, Frequencies, and Pitch

The Sine Wave: Music's Atom

The Foundation of Sound

At its core, music is the mathematical organization of air pressure variations. The sine wave is the simplest periodic sound, defined by the function: y(t) = A sin(2πft + φ).

Amplitude (A): Perceived as loudness.
Frequency (f): Perceived as pitch.
Phase (φ): The starting point of the cycle.

To understand how we hear and create music, we must first understand the physics of the sine wave—the simplest building block of sound. Every note you hear is essentially a variation in air pressure over time. By adjusting the amplitude, we change the volume. By adjusting the frequency, we change the pitch. And the phase determines exactly where that wave begins its journey through space.

Sound is modeled as air pressure changes over time.
The sine wave is the fundamental building block.
Mathematical variables map directly to musical perceptions.

Interactive Wave Lab

Experiment with the wave parameters to see and hear how they change. Try to create a high-pitched, quiet sound.

Now, it's your turn. Use the sliders to manipulate the wave. Notice how increasing the frequency adds more cycles per second, while increasing the amplitude makes the wave peaks taller. See if you can find the settings for a high-pitched but soft tone. Changing the amplitude alters the wave's height. In the physical world, this corresponds to the intensity of the pressure wave. As the frequency increases, the cycles pack closer together. Physically, this means more vibrations per second hitting your eardrum.

Frequency is measured in Hertz (Hz).
Higher amplitude creates a taller wave visualization.

Frequency vs. Pitch

Human hearing is logarithmic, not linear. This means we perceive musical intervals based on ratios of frequencies rather than fixed additions.

The Octave: A 2:1 frequency ratio.
Standard Reference: A4 = 440 Hz.

While frequency is a physical measurement, pitch is a human perception. Our ears don't hear in a straight line; they hear logarithmically. If we add 440 Hertz to a note at 440 Hertz, we get an octave. But if we add that same 440 Hertz to a high note at 4000 Hertz, we barely hear a difference. To maintain the same musical interval, we must multiply the frequency, usually by a factor of two to go up an octave.

Doubling frequency results in a pitch increase of one octave.
Adding a fixed amount of Hz does not result in the same musical interval.

The Semitone Calculator

Calculate the frequency of any note using the formula:
f_n = f₀ × 2^(n/12)

Where n is the number of semitones away from A440.

In modern music, we divide the octave into twelve equal parts. To find the frequency of a specific note, we use this exponential formula. Try calculating the frequency for a note 12 semitones above A440. Since twelve divided by twelve is one, you're simply multiplying 440 by two.

12 semitones make up one octave.
The 12th root of 2 is the ratio for a single semitone.

Fourier Synthesis: Building Timbre

Fourier’s Theorem states that any periodic wave can be constructed by summing simple sine waves.

Square Wave: Odd harmonics only (1, 3, 5...).
Sawtooth Wave: All harmonics (1, 2, 3...).

Pure sine waves sound clinical, like a flute or a tuning fork. But most instruments produce complex waveforms. We start with a fundamental frequency. By adding the third harmonic at one-third volume, and then the fifth, we begin to build a Square Wave. This process of stacking sines is called Fourier Synthesis.

Complex waves are sums of sine waves.
The 'Fundamental' is the lowest frequency (the perceived pitch).
Harmonics are integer multiples of the fundamental.

The Oscilloscope Challenge

Examine the waveform and the harmonic spectrum. Identify the wave type based on its visual and mathematical properties.

Correct! The Sawtooth wave is known for its 'buzzy' timbre because it contains all integer harmonics, making it perfect for string-like synthesizer sounds. Look at the oscilloscope. This wave has a buzzy, bright sound and contains every single harmonic—1, 2, 3, 4, and so on. Based on what we've discussed, is this a Square wave or a Sawtooth wave? Click the correct label. Not quite. Remember, Square waves only contain odd harmonics and have a 'hollow' sound. This wave has a more complex, jagged shape.

Visual identification of Square and Sawtooth waves.
Understanding harmonic content.

The Silence of Phase

If two identical waves are 180° out of phase, they cancel each other out. This is known as destructive interference.

Phase is often overlooked, but it's critical. Here are two identical sine waves. If I shift the second wave by 180 degrees, the peak of one meets the trough of the other. The result is total silence. Use the slider to find the point of maximum cancellation.

Phase is the starting position of a wave.
Phase cancellation leads to silence (destructive interference).