The other night we finally watched Hidden Figures, which is a really great movie. The scenes which depicted the IBM computer being installed reminded us of this 10″ box set, which includes a book and record on the subject of the relationship between mathematics and music.

The book includes photographs of the computer used at Bell Laboratories to compose the music heard on the record. It’s an IBM 7090, the same $2.9 million machine that was used by NASA at the Langley Research Center to calculate trajectories for the Mercury and Gemini space flights.

Music from Mathematics begins with a history of scientific inquiries into the nature of musical composition, from Pythagoras to Hermann von Helmholtz, who designed a resonator to identify the frequencies in music (an invention which indirectly lead Alexander Graham Bell towards his work on the invention of the telephone). The book also breaks down a composer’s work in strictly mathematical terms, noting for instance that even in Schoenberg’s restrictive twelve-tone technique, a sequence of twelve notes offers 479,001,600 possibilities. A factorial such as this is expressed “12!” because mathematics is exciting!

Another part of the book points to the appeal of the unexpected, using Mozart’s Musikalisches Würfelspiel as an example. A popular 18th century game, dice compositions feature sets of alternate sequences of notes depending on the numbers shown when the “composer” throws a pair of dice. The book perpetuates an uncertainty by attributing the work to Mozart, for though published in 1792 and included in the Köchel catalog, it has never been verified as Mozart’s work. Musikalisches Würfelspiel is capable of producing 11¹⁶ similar but distinct waltzes.

The book and record contains a number of experiments beginning at this point with Music by Chance, produced at Bell Telephone Laboratories. The second side of the record opens with a remarkable piece composed on ILLIAC computer at the University of Illinois in 1955.

The process began by assigning numbers to notes of the scale from low C upwards. In the beginning sharps and flats were omitted, but in later experiments a full chromatic scale of two and a half octaves was used. The computer then generated random numbers. The numbers were screened through a series of tests representing the various rules of musical composition such as tonality and the standard of counterpoint formalized in the 16th century. If the next number did not conform to the rules it was rejected and a new random number was generated and tested. The numbers which passed the testing were stored in the computer until a short melody was created, and it was printed out and translated into notation for a human performer.

The Illiac Suite produced by this experiment is regarded as the first musical score composed by a computer. The record inside Music from Mathematics contains only a two minute sampling from its fourth movement, but you can hear a performance of the entire work on Youtube here. Although this is certainly the sort of music which gave John Hartford the “Steamboat Whistle Blues,” you’ll find the Illiac Suite no less accessible than Bela Bartok’s quartets, although hardly as rewarding.

The following year a second Music from Mathematics was released on the Decca label. Not a documentary like this set, it presented eighteen performances by the IBM 7090 recorded at the Bell Laboratories. Fans of this album most famously include author Arthur C. Clarke, who later had HAL 9000 the computer sing “Bicycle Built for Two” (ie “Daisy Bell”) as he fades away in 2001: A Space Odyssey. This was the first song sung by a computer, and appeared on that record.