Fourier Series

Introduction

Can circles do more for us than draw these lovely patterns? Can they give us an alphabet, a universal way of generating and representing many forms of interest? Can we treat them like a bunch of kitchen ingredients, that we throw into a recipe to conjure up new dishes that look different?

Inspiration

Take a look at these paintings:

https://x.com/jagarikin/status/962449509782495232

https://www.jezzamon.com/fourier/index.html

What is the Fourier Series?

Videos

Rolling Circles and the Fourier Series

So, if we choose number of circles $M$ and their complex amplitudes $a_j$, $j=1..M$ relying on our (hopefully growing) intuition, we can systematically generate symmetric patterns based on the idea of rolling circles. By trial and error, we can design both the value of $M$ and the values for $a_j$, $j=1..M$. So far, so good.

But how about the other way around? What if we had a “pattern” in mind, and wanted to compute the circles, their number and amplitudes, that would generate that pattern? This is where the Fourier Series comes in.

Wait, But Why?

• Think of the Fourier Series as a set of sinewaves that are derived by decomposing an original waveform
• How are these components related? As integer multiples of a fundamental frequency.
• How are their amplitudes calculated? By taking a correlation between the original waveform and the given sinewave component (unit amplitude)
• How is this accurate? By minimizing a “least square error” between the original waveform and the sum of sinusoids.

A Sound Vocabulary

Some terms will show up repeatedly in our work and we should be clear what they mean:

1. Frequency
2. Amplitude
3. Phase
4. Sinusoid
5. Oscillation: https://natureofcode.com/oscillation/
6. Harmonic
7. In-harmonic
8. Partials
9. Waveform
10. Transient
11. Alias

References

1. Alex Miller. (2018). Fourier Series and Spinning Circles. https://alex.miller.im/posts/fourier-series-spinning-circles-visualization/
2. Aatish Bhatia (November 6, 2013). The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face. https://nautil.us/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-face-234629/
3. Better Explained. An Interactive Guide to the Fourier Transform. http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/
4. Amid Fish.(May 2017). Karplus Strong String Synthesis. http://amid.fish/karplus-strong
5. Karplus, K., & Strong, A. (1983). Digital Synthesis of Plucked-String and Drum Timbres. Computer Music Journal, 7(2), 43. doi:10.2307/3680062. https://sci-hub.se/https://doi.org/10.2307/3680062

Resources

1. https://mathlets.org/mathlets/fourier-coefficients/
2. Working with Audio in p5.js. https://pdm.lsupathways.org/3_audio/
3. Violet Whitney. (Sep 28, 2023) Sounds: Working with sounds and speech in P5.js. https://medium.spatialpixel.com/sounds-bd05429aba38
4. Mister Bomb. p5.Sound project tutorials. https://www.youtube.com/playlist?list=PLIsdHp2z9wFl7A1wWb2VmQUUojEGsKELE
5. https://musiclab.chromeexperiments.com/oscillators
6. https://www.electronicbeats.net/the-feed/excel-drum-machine/
7. https://junshern.github.io/algorithmic-music-tutorial/
8. https://blackwhiskercult.com/visual-music-in-p5-js-i/

Other tools to explore

1. Strudel REPL https://strudel.cc
2. Introducing Jukebox, a neural net that generates music, including rudimentary singing, as raw audio in a variety of genres and artist styles. We’re releasing a tool for everyone to explore the generated samples, as well as the model and code: https://openai.com/index/jukebox/ (OpenAI ((OpenAI?)) April 30, 2020,via Twitter https://twitter.com/OpenAI)
3. https://algorithmicpattern.org/2023/05/15/strudel-live-coding-patterns-on-the-web/
4. https://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/
5. Freesound: Find Any Sound you Like. https://freesound.org
6. WebSpeech API. https://developer.chrome.com/blog/voice-driven-web-apps-introduction-to-the-web-speech-api/
7. https://dogbotic.com

R Package Citations

Package	Version	Citation
ambient	1.0.2	Pedersen and Peck (2022)
mosaicCalc	0.6.4	Kaplan, Pruim, and Horton (2024)
plot3D	1.4.1	Soetaert (2024)

Kaplan, Daniel T., Randall Pruim, and Nicholas J. Horton. 2024. mosaicCalc: R-Language Based Calculus Operations for Teaching. https://CRAN.R-project.org/package=mosaicCalc.

Pedersen, Thomas Lin, and Jordan Peck. 2022. ambient: A Generator of Multidimensional Noise. https://CRAN.R-project.org/package=ambient.

Soetaert, Karline. 2024. plot3D: Plotting Multi-Dimensional Data. https://CRAN.R-project.org/package=plot3D.