Proportions; music and science; the role of numbers The relationship between music, mathematics and science has a long history. Pythagoras (580 - 572 BC) who studied the vibration of strings and determined the ratios between overtones thought everything is governed by numbers and mathematical relations: music as well as the entire universe. Interestingly, the beginning of the Baroque music coincides with the birth of modern science and its peak parallels the advent of classical mechanics as exemplified by the lifespan of: Monteverdi 1567 – 1643 Galileo Galilei 1564 – 1642 René Descartes 1596 – 1650 and J. S. Bach 1685 – 1750 G. F. Händel 1685 – 1759 Sir Isaac Newton 1643 – 1727 More than a superficial coincidence, both the music and the science of this period describe a world which is complex and intricate but rational and deterministic, in accord with the ideas of the Enlightenment. Another example of an encounter between science (acoustics) and music is Jean-Philippe Rameau's Traité de l'harmonie (1722), the book that established the basis of tonal harmony. Later on, French philosopher Henri Bergson who dealt extensively with the relativity, time perception and memory, influenced the music of Debussy while the DADA movement (1916) is presumed to have some bearing over the advent of Quantum Physics (1920s). Could it be that all this was the effect of zeitgeist (the ghost of time) ?

*** Vibrating strings and air columns show the existence of nodes and anti-nodes as well as multiple modes of vibration

http://en.wikipedia.org/wiki/Normal_mode and

http://www.glenbrook.k12.il.us/GBSSCI/PHYS/CLASS/waves/u10l4c.html (you may skip the math). The presence of such modes of vibration determines the existence of what musicians call overtones and scientists call partials. The fundamental frequency or pitch is partial #1, first overtone – partial # 2, etc. Partials, or harmonics, or overtones are integer multiples of the frequency of the fundamental pitch:

http://en.wikipedia.org/wiki/Harmonic_series_%28music%29 In class we talked about the clarinet and the fact that it produces only odd-numbered partials:

http://www.phys.unsw.edu.au/jw/woodwind.html

*** Traditional Western rhythmic notation (divisionary system) uses ratios/proportions to divide a large value (whole note) into subdivisions of 2, 4, 8.., 3, 6, 9,... 5,... 7,... etc. By contrast, the Aksak (additive system) common to non-Western traditions uses a fast pulse in usually uneven groupings: (2+2+3)/8. Bartok, among other composers, have used them (see Six Dances in Bulgarian Rhythm at the end of Mikrokosmos). The 1950s saw the emergence of proportional notation, where the length of a line is proportional to the duration of a sound and its height on the page corresponds to its pitch. The score will usually contain an indication of the type: 1 inch = 1 sec. Here is an impossibly long link to a few pages from Reginald Smith Brindle's book The New Music: http://books.google.com/books?id=VxQ4OO2fjPYC&pg=PA63&lpg=PA63&dq=proportional+notation&source=bl&ots=tuG-W_i9PK&sig=r01JrPOoPeDxAXIIFHu16_PmzZk&hl=en&ei=jh6YSrvlJpHcNqD_4LIF&sa=X&oi=book_result&ct=result&resnum=4#v=onepage&q=proportional%20notation&f=false

*** Among the most common devices used by artists and musicians are the Fibonacci series of numbers and the Golden Section. The Fibonacci series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... is produced by adding two adjacent numbers in order to obtain the following one. For more information than you might want, ake a look at:

http://goldennumber.net/goldsect.htm http://en.wikipedia.org/wiki/Golden_ratio

In music, on can find such proportions in many compositions from Renaissance to present days. In most cases they are (probably) the result of a conscious attempt but just proof of a keen sense of proportions. However, two composers, Batok and Debussy have used these devices consistently as documented in Roy Horwath – Debussy in Proportion and Ernö Lendvai - Béla Bartók: An Analysis of his Music. The pitch intervals on the first page of Bartok's Music for Strings, percussion and celesta contain exclusively Fibonacci numbers of semitones. We also looked in class at Debussy's first piano prelude, book 1 ...Danseuses de Delphes

http://ems.music.uiuc.edu/courses/tipei/M202/Notes/danseuses.pdf

For more about Bartok go to http://solomonsmusic.net/diss7.htm.