Sound frequency

Introduction

The signal frequency is the "pitch" of the sound. Some facts about sound frequencies you might encounter in a pub quiz...
The Python notebook is a convenient playground to generate sounds of those frequencies and listen to the sounds. https://github.com/yasumori/blog/blob/main/2025/2025_12_21_signal2.ipynb.

An example code snipet to generate a 2,000 Hz sound is also below:
import numpy as np
from IPython import display

def gen_audio(frequency, duration, sample_rate):
    t = np.linspace(0, duration, duration * sample_rate)
    return np.sin(2 * np.pi * frequency * t)

hz_2000 = gen_audio(2000, 3, 44100)
display.Audio(hz_2000, rate=44100)
The nice thing about speech and audio processing is to generate sounds and we can actually listen to it. The 2,000 Hz signal sounds like this:

Then, the 8,000 Hz sound has a lot higher pitch.

The 16,000 Hz sound is very annoying or maybe we can't here it. This sound can scare mice.

The music note C sounds like this.

Then, the note E sounds like this.

We can also add C and E sounds that form a chord. 

Time Domain to Frequency Domain 

In digital signal processing, the x-axis is time and the y-axis is amplitude (loudness) in a typical figure of a waveform:

The figure shows a simple sine wave of 1 Hz (1 cycle per second). We can still see this is only one cycle per second.

Tracking cycles however becomes very difficult as the signal frequency increases. 
This figure shows the first 5 cycles of 261 Hz sine wave. 5 cycles are finished within 0.02 seconds, meaning that we need to count cycles for samples of additional 0.98 seconds to ensure that this is a 261 Hz signal. 
We can add two signals of C (261 Hz) and E (329 Hz) to form a chord. The result is the blue line at the bottom.

Ideally, we want to have a way to know frequency components of the signal at once. Thankfully, there's a magic called Discrete Fourier Transform (DFT) to obtain a figure below:

This figure illustrates that the blue signal in the previous figure consists of two peak frequencies of 261 Hz and 329 Hz. 

To summarise this post:
  • The frequency characterises the "pitch" of a sound.
  • We can generate a sound of a certain frequency using Python and listen to it.
  • A signal can consist of multiple frequencies and the Discrete Fourier Transform can show "strenghts" of certain frequencies in the signal.

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