Every track is assembled from mathematics, not samples. This is the physics and engineering behind the 40 Hz pulse — from a single sine wave to a validated 60-minute audio file.
Every track is built from two independent audio streams mixed, shaped, validated, and exported. Layer 1 is synthesised entirely from equations — nothing borrowed from any recording.
Everything starts here. A sine wave is the simplest possible oscillation. It describes a pendulum, a guitar string, a tuning fork. And the 40 Hz pulse at the heart of every Gamma40 track.
At 40 Hz, the wave completes 40 full cycles per second, each lasting 25 ms. At 1,000 Hz (the carrier inside each pulse) a full cycle takes just 1 ms. The 1 kHz wave oscillates 25× faster than the pulse rate.
A computer cannot store a continuous wave. Instead it takes 44,100 measurements per second, each called a sample, and records the amplitude as a number. This is the CD standard.
Why 44,100? The Nyquist theorem states you must sample at least twice the highest frequency you want to capture. Humans hear up to ~20,000 Hz, so 44,100 provides a safe margin above the 40,000 Hz minimum.
The 40 Hz stimulus is not a continuous 40 Hz hum. Instead it is a pulse train: a rapid series of brief sound bursts, each just 1 ms long, repeating 40 times per second.
This matches the Wang et al. (PNAS 2026) stimulus exactly: "1 ms duration at a frequency of 40 Hz."
At 44,100 samples/second, one 40 Hz cycle spans exactly 1,102.5 samples. Not a whole number. Rounding to 1,102 or 1,103 accumulates timing error. Over 144,000 pulses this becomes measurable.
A running counter tracks position with sub-sample precision. When it reaches 1,102.5 it resets, but the 0.5 carries forward. Over an entire 60-minute session, zero phase error accumulates.
Switching a sine wave on and off abruptly creates a click. Mathematically this is a rectangular envelope applied to the carrier. A rectangle has infinite-bandwidth harmonics in the frequency domain.
A cosine fade is applied to the first and last 4 samples (≈0.09 ms) of each burst, smoothing amplitude from 0→1 at onset and 1→0 at offset.
The Brown Noise track contains no recording. It is synthesised from mathematics. Starting from white noise (pure randomness at every sample) and integrating it into something warm and natural.
Integration in the time domain produces a −6 dB/octave rolloff in the frequency spectrum: low frequencies dominate and highs fade. The result sounds like distant thunder or a low waterfall: deep, warm, enveloping.
RMS (Root Mean Square) is the standard measure of average signal power. It represents what the ear hears as sustained loudness, independent of momentary peaks.
The pulse is scaled so its RMS sits 2 dB below the ambient. It carries about 79% of the ambient's average power. Clearly audible, always present, but the ambient stays dominant for comfortable 60-minute listening.
Before any track is exported, the pulse train is analysed by a Fast Fourier Transform. The FFT decomposes the signal into its frequency components — a spectrum showing exactly how much energy is present at each frequency. If either check fails, the track is not exported.
The pulse train — a 1 kHz sine burst repeating at 40 Hz — produces a characteristic fingerprint:
40 Hz — the fundamental repetition rate of the pulse
80, 120, 160 Hz — harmonics from the 4% duty cycle shape
1,000 Hz — the carrier sine wave inside each burst
960, 1,040 Hz — sidebands at 1 kHz ± 40 Hz
A common concern: does MP3 or AAC encoding alter the 40 Hz timing? No — and here is why. Lossy codecs operate by removing spectral content the ear cannot perceive given simultaneous louder sounds. They do not alter timing. The 40 Hz pulse envelope (the on/off pattern) survives encoding intact.
This has been confirmed empirically: FFT analysis of exported MP3 and M4A files shows the same 40.0000 Hz peak as the source WAV.
Streaming platforms apply loudness normalisation that alters the carefully calibrated pulse-to-ambient ratio. Direct download and local playback is always recommended for Gamma40 tracks. The −18 LUFS master is calibrated precisely — streaming adjusts this without your control.