Sonification of prime factors and prime numbers

This applet allows you to hear a sequence of notes or combinations of simultaneous notes whose frequencies depend on the prime factors of each natural number n in sequence. According to the option selected, a formula determines the fundamental frequency f0 of each note in relation to the base frequency f, the prime factor p, and the position i of that prime factor in the list of all primes (so for 2, i = 1, for 3, i = 2, etc.).

• Divide base by prime factors: f0 = f / p
• Multiply base by prime factors: f0 = f * p
• Nth harmonic of base = nth prime: f0 = f * (i + 1)
• Nth semitone above base = nth prime: f0 = f + i semitones
• Nth semitone below base = nth prime: f0 = f - i semitones

For each n there are as many simultaneous notes as there are different prime factors. The amplitude of each note is m / l, where m is the multiplicity of p in the factorisation of n, and l is the sum of m for each prime factor of n (i.e., the length of the factorisation of n).

Notes may be sine tones or complex tones. Other controls are self explanatory. After the first few primes, prime numbers tend to stand out of the sequence one hears, making audible the pattern (or non-pattern) of their occurrence.

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