Sounds of the Collatz Conjecture: Generating Music from the 3x + 1 Problem

Using sequences from the famous unsolved Collatz conjecture to generate musical passages as MIDI notes.

The Collatz conjecture is also known as the 3n + 1 problem, the 3x + 1 problem, the Ulam conjecture, Kakutani's problem, the Thwaites conjecture, Hasse's algorithm, or the Syracuse problem.

Different strategies are used to map the "hailstone sequences" into sequences of MIDI note numbers, including a straightforward "additive" numerical mapping, "directional" mappings using fixed jump sizes, and mappings based on pitch class.

These visualizations were written in Java using a graphical library called Processing (https://processing.org/), and Java's built-in MIDI library for generating MIDI data (package javax.sound.midi).

0:00 The Collatz Conjecture
1:27 Mapping to MIDI Notes
2:07 Strategy No. 1
4:03 Strategy No. 2
4:45 Strategy No. 3
5:21 Strategy No. 4
5:54 Strategy No. 5
7:18 Strategy No. 6
8:04 Strategy No. 7
9:12 Extra Long Example

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Interested in learning more about algorithms, math, and how to program? Here are some useful and/or classic textbooks that I recommend (these are affiliate links, if you buy one, I get a small commission):

▶ "The Ultimate Challenge: The 3x+1 Problem" by Jeffrey C. Lagarias: https://amzn.to/4aVejxH
▶ “Algorithms” (4th Edition) by Robert Sedgewick & Kevin Wayne: https://amzn.to/3uo25xR
▶ “Effective Java” (3rd Edition) by Joshua Bloch: https://amzn.to/3HOnYJL
▶ “Design Patterns: Elements of Reusable Object-Oriented Software” by Erich Gamma, Richard Helm, Ralph Johnson, & John Vlissides: https://amzn.to/49fpr7R
▶ “Discrete Algorithmic Mathematics” by Stephen B. Maurer & Anthony Ralston: https://amzn.to/4bmsOvG

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