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Music and Mathematics: From Abogi to
Arithmetic The Duke of Illyria opens Shakespeare’s “The Twelfth Night” by ordering “If music
The fact, for instance, that such disparate performers as Ravi Shankar, Yehudi Menuhin and Andre Previn could produce a great concerto with the London Philharmonic Orchestra in 1971 is just one manifestation of this essentially transcendental nature of music. In essence, as noted by the eminent neurologist Oliver Sacks, music can “calm us, animate us, comfort us, thrill us…”[1] Music that is pleasing to the ears has a certain cadence and tonal regularity to it. Music with simple notes and uncomplicated beats is immediately accessible to a wider audience. Music may hence have evolved from its simplest forms – lullabies and the chanting of prayers, for example - via uncomplicated folk songs to more complex pieces like ragas with their rigorous structures. Whatever the level of complexity, we appear to exhibit a primeval nexus with music because the lilt and structure of a tonal arrangement reflect the underlying pulse of life. As Sir Thomas Browne (English writer and physician, 1605-82) wrote, “… there is music wherever there is a harmony, order or proportion; and thus far we may maintain the music of the spheres; … and … they strike a note most full of harmony.” Music’s main components are melody, harmony and rhythm and all three may be related to mathematics to some extent. Melody is a mellifluous series of tones or pitches perceived as a unity. Harmony is the relation of notes to notes – and measures their synchronicity - as rendered by the singers and musical instruments. As there can be both consonant and dissonant harmonies, the acceptance of dissonance varies from one musical genre to another. (For instance, there is more tolerance comparatively for dissonant harmony in Western classical music as compared to Hindustani or Carnatic music.) Rhythm, the third component, is an expression of the temporal structure of music. It attributes time to the melody. Rhythmic structures (or talas) in Indian classical traditions tend to be relatively a lot more complex or elaborate as compared to their Western counterpart. There is no disputing the fact that in all cultures and in all musical traditions tonal structures and rhythm (reflecting their “number, measure and proportion as likeness of the heavens,” as the Renaissance intellectual Agrippa of Nettesheim noted)[2] are considered equally important in the final composition. In his article on “Music, Mathematics and Bach”,[3] Rahul Siddharthan from the Indian Institute of Science, Bangalore, highlights certain musical features (incorporating a mathematical base) which appear to be common among many cultures of the world.
The above series – which in mathematics is called a Pythagorean scale – represents also the ‘pentatonic scale’ which, Siddharthan observes, is a very commonly used sequence among many cultures. In Carnatic music, it is the scale of Raga Mohanam (Sa, Ri, Ga, Pa, Da, and Sa – the ascending notes) and in Hindustani music it is Bhopali. The relationship between harmony and mathematics has in fact been widely discussed since the time of Pythagoras. Legend has it that the great Greek mathematician once ran into a forge to investigate the harmony of hammers. He noticed that most of the hammers generated a harmonious sound when struck simultaneously whereas a combination containing one particular hammer always generated an unpleasant noise. He analysed and found that those hammers which together created a consonance had a simple mathematical relationship – their masses were simple ratios or fractions of each other. Pythagoras applied his new theory of musical ratios to the lyre by examining the property of a single string. He was to discover that harmonious tones only occur when the string is plucked at very specific points. For example, by fixing the string at a point exactly half-way along it, plucking generates a tone which is one octave higher and in harmony with the original tone.[4] The Pythagoreans found also that the ratio 1:2 between the lengths of two vibrating strings would produce an octave, the fifth in the ratio of 2:3, the fourth in 3:4, and so on. Certain famous composers in the West were later to be so enchanted with this observation (of association between music and numbers) that they attempted to incorporate its allure in their musical structures. Claude Debussy, for example, was fascinated with the mathematical concept of the Golden Ratio. The Golden Ratio which is approximately equal to 1.618033989 is the decimal value that the fractions of adjacent Fibonacci numbers head towards. The Fibonacci sequence is a famous unending series of numbers in which each number, except for the first two, is the sum of the preceding two. This sequence which may have been well known in ancient India was studied in the West by Leonardo de Pisa (alias Fibonacci) who made it famous in 1202 A.D. when he posed the following question in his mathematical treatise: How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on? The number of rabbits produced during each month obeys the following series of Fibonacci numbers:
In Debussy’s “Prelude to the Afternoon of a Faun”, the Golden Ratio can be perceived in pitch, rhythm and dynamics. For instance, its musical pulses (also called quaver units) incorporate fortissimos (or ff, very loud) at bars 19 and 28, the next fortissimo is then at bar 47 (the sum of 19 and 28). The piece contains also a number of similar patterns which exhibit Fibonacci characteristics.[5] More than 2000 years later, Johannes Kepler was to conclude that musical harmony could be better explained using geometry instead of numbers. He imagined a string to be bent into the form of a circle which can be divided in equal parts by drawing symmetrical figures with varying numbers of sides inside it. The drawing of a pentagon will thus divide the circumference into parts of 1/5 and 4/5 – and these will correspond to consonant chords. But a heptagon’s sides will produce ratios of 1/7 and 6/7 which on the contrary equate to discords. This, according to Kepler, was because the pentagon can be constructed using a compass and ruler, but not the heptagon. In Kepler’s opinion, geometry is the only language which enables man to understand the working of the divine mind.[6] At a more spiritual level, as Gottfried Leibnitz envisioned it, music is “a kind of counting performed by the mind without knowing that it is counting”.[7] Oliver Sacks has noted that while anatomists cannot identify the brain of a visual artist, writer or mathematician, they can recognise the brain of a professional musician.[8] It appears that something pleasurable happens in our neuronal circuits when we hear music and it is akin to the pleasure we derive from symmetry in architecture. And, this symmetry can be explained by the logic of mathematics. Whether this relationship is immediately apparent or not, in the end it perhaps does not matter how we become aware of nature’s heartbeat: it can be either directly via the numbers themselves, their geometry - or simply the music itself at an elemental level with out the need for a rather cerebral awareness of any association between abogi and arithmetic. Venkat Ramanan References 1. In “Musicophilia”, Picador, 2007, p: xii. 2. From “The Transformation of Medicine by the Magic of Music in the Romances of Shakespeare” by Henry D Janowitz, MD, in the “Journal of the Royal Society of Medicine”, 3. www.ias.ac.in/resonance/Mar1999/pdf/Mar1999p06-15.pdf 4. From “Fermat’s Last Theorem”, Simon Singh, Harper Perennial, UK, 2007, p: 15-17. 5. In “Coincidences, Chaos, and All that Math Jazz: Making Light of Weighty Ideas” by Edward D Burger and Michael Starbird, W W Norton & Company, New York, 2005, p: 129-130. 6. Arthur Koestler, in “The Sleepwalkers: A History of Man’s Changing Vision of the Universe”, Penguin Books, UK, 1982, p: 394-6. 7. From “Dictionary of Quotable Definitions”, ed: EE Brussell, Prentice-Hall Inc, NJ, 1984, p: 388. 8. www.pbs.org/wgbh/nova/musicminds/
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Comments on this Article Romesh Sen [Boston, MA. Nov 11, 2009 3:13:04 PM]
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