 ‘Music charms us, although its beauty consists only in the agreement of numbers and in the counting, which we do not perceive but which the soul nevertheless continues to carry out, of the beats or vibrations of sounding bodies which coincide at certain intervals.’ (Leibniz, 1714)
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| In this section, we will begin to examine the relationship between numbers and music. Such an association is often manifest in the ways in which music is rhythmically organised, of course – think of the relationship between note values as numerical proportions, or as fractions – but many composers have also sought to apply number relationships to other components of music, such as pitch, dynamic and articulation organisation and, on the macro level, to formal structuring. |
| The topic will examine the theories and works of a number of composers who have extended the Pythagorean notions of proportion, ratio and the harmonic series, first realised in the music of the medieval and renaissance periods. Such approaches characterise the growing tendency for composers to utilise ‘pre-compositional’ systems, often based on number relationships, in the creation of musical works. Building on relatively simple structural ideas such as the canon, through more complex systems, such as the prolation canon, composers have transferred the observable patterns of ratio from the sonic domain to the temporal domain. Twentieth-century composers who have proposed new ways of working with musical material in these ways include: Henry Cowell, who sought to apply the principles of the harmonic series to elements of rhythmic organisation; Arnold Schoenberg and Anton von Webern, who focused on a quasi-numerical approach to pitch organisation; and Olivier Messiaen, who heralded the much-maligned ‘total serialist’ approach in the 1950s through works such as the Etudes de Rythme. |
Class Documents
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Articles & Books
- Cowell, Henry, 1996. New Musical Resources. Cambridge: CUP. Available via Google Books or complete download here.
- Smith, Leland, 1973. ‘Henry Cowell’s Rhythmicana’ in Anuario Interamericano de Investigacion Musical, Vol. 9, (1973), pp. 134-147. Also at JSTOR.
- Cohen, David, 1974. ‘Anton Webern and the Magic Square’ in Perspectives of New Music, Vol. 13, No. 1 (Autumn – Winter, 1974), pp. 213-215. Also at JSTOR.
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Audio & Visual
- Canon, by the acclaimed Canadian animator Norman McLaren, is a humorous visualisation of musical canon
- Henry Cowell Musical Autobiography (April 6th, 1961). In this audio stream from the Internet Archive, Henry Cowell talks about his life and plays examples of his music from his entire career as a composer—a fascinating document. He mentions the diversity interests from his childhood, giving up violin at age 8, hearing Irish tunes hummed by his father, early American Ozark mountain tunes sung by his mother. Living in San Francisco, he was exposed to international influences, playing and humming along with his Japanese, Chinese & Tahitian playmates.
- A demonstration of the third version of Rhythmicom built by Leon Theremin at Moscow State Conservatory in early 1960s. The first Rhythmicon was developed by Leon Theremin for Henry Cowell in 1932.
- Albrecht Dürer’s famous Melencolia I features a magic square, the subject of much discussion and analysis.
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Internet Sites
- The Online Rhythmicon at the American Mavericks site, produce in association with the San Francisco Symphony, allows you to play a virtual Rhythmicon and to hear compositions by others.
- Antique Puzzles – On this page you’ll find a collection of interesting latin rebuses and riddles, pangrams, a vanish puzzle, magic ROTAS squares, Greek and Latin palindromes, chronograms, tongue twisters, famous double-meaning sentences, anagrams, a verbal labyrinth, some jokes, and finally the Archimedes’ puzzle (aka ‘Stomachion’ or ‘Ostomachion’). Specta, lege atque delecteris. Vales!
- The n-ISM (Network for Interdisciplinary Studies in Science, Technology, and Music) Microtonalism website includes a discussion of 19-TET.
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