From Ancient India to Silicon Valley: Music, the Binary System & More

Author: Edel Sanders, M.A., EdM

Position: Instructor of Music and the Mind

Music is the pleasure the human mind experiences from counting without being aware that it is counting.

Gottfried Leibniz – mathematician and philosopher

My doctoral thesis at the University of Cambridge examines the relationship between musical and mathematical cognition.  I explore mathematical elements, such as number, sequence and proportion, existing within music in order to support the hypothesis that there is a link between both domains of thinking.  This can be observed when analyzing the physics of music for instance.  One can also see the connection between music and number by looking at the history of mathematics. 

This association in ancient India is now widely recognized by scholars (van Nooten, 1993; du Sautoy, 2007). For example, the use of binary numbers and a noted numerical sequence found in nature and prominent in art are said to have origins in India (Kaplan, 1999; van Hall, 2005; Brown, 2008).

The binary numeral system (base-2 numeral system) is the foundation for computer science.  Yet, in the earliest known Sanskrit treatise on prosody, Pingala first used it in the analysis of musical and poetic meters in his Chandahsutra (Science of Meters) text (c. 200 BC, in Bag, 1966).  Meters were arranged in differing patterns of long and short, with the temporal ratio of 2:1.  His study of meter led him to describe the addition rules defining rows of the triangle later known as Pascal Triangle, widely used in the mathematical branches of geometry, algebra and probability. 

Ancient African bush tribes communicated via intricate messages using two-toned drums (Shectman, 2003; Finnegan, 2012).  They used a form of binary system that included an error-correcting code.  Retransmission messages were sent to neighboring villages, possibly with protocol similar to what is now used by Ethernet.  This would have ensured a clear channel for retransmission (Hellman, 2013).

Additionally, use of long-short binary patterns identifies the temporal organization of dance music in various cultures.  For example, clave rhythmic cycles in Salsa music, which most likely originated in sub-Saharan Africa before moving to Cuba and other regions, utilizes multiple variations of 2 and 1 (Hall, 2005).  Argentine tango, influenced both by ancient African rhythms and music from Europe (Davis, 1995), is another case of mathematical patterns used within dance structure, often revealing mathematical sequences such as the Fibonacci sequence, explored below.

In 12th century India, Hemacandra also studied meter, as did his countryman many centuries earlier.  After creating a matrix of all of the rhythmic possibilities of a binary pattern with a ratio of 2:1[1] he discovered a sequence in which each subsequent number - following the first two - is created by adding the sum of the two preceding numbers (Koshy, 2001).  This series later became known as the Fibonacci sequence, after the Italian mathematician, Leonardo Fibonacci, who introduced it to Europe in the 13th century.  Educated in North Africa, Fibonacci then travelled widely with his father for business.  After visiting India, he spoke of the advantages of the Hindu-Arabic system of mathematics.  He later introduced this system, including the now-famous sequence, to Europe.  Thus he became prominent as a result (Brown, 2008).

This sequence is related closely with the Golden Ratio.  The Golden Ratio, also known as the Golden Section or Golden Mean, has been described as a universal law found in natural phenomena.  It parallels the cycle of growth and decline, and is reflected in the microcosmic spirals of a seashell and a pinecone as well as the macrocosmic spirals of a hurricane and the solar system.  It has also been considered the essence of beauty and is found in the proportions of the Parthenon, the paintings of Leonardo da Vinci and even the doorway to your room.

Numerous composers have employed this sequence or its related Golden Ratio, whether consciously or unconsciously.  Beethoven utilized the Golden Ratio at key points in his Fifth Symphony and use of this ratio also appears in works of Bach, Bartok, Debussy, Satie and Schubert (Knott, 2010).  These examples further demonstrate a link between mathematics and music.  Therefore, the next time you listen to your iPod, take note that more might be going on than you think!

 

[1] This ratio exists in harmonic and overtone structures within musical compositions and tones as well.

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