Mathematics

Every signal is a stack of pure waves

Any sound your phone records is one wiggling line: air pressure over time. Fourier decomposition rewrites that line as a sum of pure tones, and both descriptions hold the same information. A 1965 shortcut, the fast Fourier transform, made that split cheap enough for a phone to run it live.

Sources & further reading

W3K Publishing / Stanford Center for Computer Research in Music and Acoustics (CCRMA)Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications, Second EditionA free Stanford textbook that works out, from scratch, why any recorded sound can be written as a sum of pure tones.Mathematics of Computation (American Mathematical Society)An Algorithm for the Machine Calculation of Complex Fourier SeriesThe 1965 paper with the shortcut that made computers fast enough to break signals into waves in real time.International Telecommunication Union (CCITT / ITU-T)ITU-T Recommendation T.81, Digital Compression and Coding of Continuous-tone Still Images (JPEG)The official 1992 international standard for JPEG images, which stores photos as frequency recipes in 8 by 8 blocks.Proceedings of the 4th International Conference on Music Information Retrieval (ISMIR 2003)An Industrial-Strength Audio Search AlgorithmThe paper describing how Shazam recognises a song from a noisy phone recording by reading its frequency pattern.NobelPrize.org (Nobel Foundation)The Nobel Prize in Physiology or Medicine 2003, Paul C. Lauterbur and Sir Peter MansfieldThe Nobel Foundation's record of the 2003 prize for inventing MRI, a scanner that builds pictures out of frequencies.Questions and Answers in MRIWhat is k-space?A radiologist's reference explaining that an MRI scanner records frequencies first and only turns them into a picture afterwards.NobelPrize.org (Nobel Foundation)The Nobel Prize in Physiology or Medicine 1961, Georg von BekesyThe Nobel record for the scientist who showed the inner ear physically sorts sound by pitch along its length.Sinauer Associates / NCBI Bookshelf (NBK10946)Neuroscience (2nd ed.), chapter: The Inner EarA standard neuroscience textbook chapter describing how the inner ear lays out pitch along its length like a keyboard.