Music, at its very core, is mathematical. It’s often said that mathematics is a universal language that could be used as a first means of communication between intelligent species on different worlds. But that claim of universality might also apply to music and, in fact, we’ve already sent some of our music towards the stars in the hope that beings out there might hear it and so come to understand something of the creatures who made it.
Voyager 1, launched on September 5, 1977, recently became the first human-made object to enter interstellar space. Having flown past Jupiter and Saturn, it headed out of the solar system and in 2012 passed beyond the heliopause, the boundary where the influence of the Sun’s magnetic field ends and that of the rest of the Galaxy begins. Its sister craft, Voyager 2, launched in the same year, is also heading for the void between stars but in a different direction. Both remain in contact with Earth, sending back data from a handful of science experiments that their dwindling power reserves can supply, but neither is destined for any close encounters with another star system in the foreseeable future. Their speed is so low compared with the immensity of interstellar distances that it would take them tens of thousands of years to reach even the nearest star, Proxima Centauri – assuming they were heading directly towards it (which they aren’t).
According to NASA’s current estimates, Voyager 1 will come within 1.6 light-years of the star Gliese 445 and Voyager 2 within 1.7 light-years of Ross 248 about 40,000 years from now. By the time of these very remote flybys both probes will be long dead. But structurally the Voyagers could remain intact for millions of years, drifting through the Milky Way galaxy and – who knows – possibly be found by some advanced race that would be curious about the probes’ origins and creators. In that unlikely event each spacecraft carries a message in the form of a gold-plated copper phonograph record containing sounds and images intended to portray the variety of life, environments, and human cultures on Earth. As well as 116 pictures, a variety of natural sounds, and spoken greetings in 57 different languages, the Voyager Golden Record features 90 minutes of music from different ages and regions of the world, including passages from Stravinsky’s The Rite of Spring, a gamelan piece from Indonesia, Bach’s Brandenburg Concerto No. 2, and Chuck Berry’s ‘Johnny B. Goode’. Thoughtfully, a stylus and coded instructions are provided by which to play the record. But, assuming aliens ever found one of the Golden Records and managed to play the music as intended, the question is whether they’d recognise it for what it is. And, similarly, if alien music somehow reached our ears, would we appreciate it as being musical?
One of us (David) is a singer and songwriter whose album Songs of the Cosmos combines science with music in tunes such as ‘Dark Energy’. But as well as songs with a scientific flavour, there’s also science in the making of music, and mathematics, deeply rooted, in the relationship between notes and the construction of scales. It was the ancient Greeks who first discovered that there’s a strong link between music and mathematics. Pythagoras and his followers, in the sixth century bc, built an entire cult around their belief that ‘all is number’ and that the whole numbers were especially significant. Each of the numbers 1 to 10, they held, had a unique significance and meaning – 1 was the generator of all other numbers, 2 stood for opinion, 3 for harmony, and so on, all the way up 10, which was the most important and known as tetraktys because it is the triangular number made from the sum of the first four numbers, 1, 2, 3, and 4. Even numbers were considered to be female, and odd numbers male. In music, the Pythagoreans delighted in their discovery that the most harmonious sounding intervals corresponded with whole number ratios. The very same numbers they held in such high esteem on an intellectual level determined, as simple fractions, what set of notes was most satisfying to the ear. A vibrating string held down at its halfway point (2:1) sounds an octave higher than when open. Held down and played so that the length of the vibrating section to the entire string is in the ratio 3:2 gives a perfect fifth (so called because it is the fifth note in the scale and highly consonant with the root note). Likewise 4:3 produces a perfect fourth and 5:4 a major third. Since frequency depends on one divided by the string length, these ratios also give a relationship between the frequencies of the notes.
The simplest of the ratios (apart from the octave) – the perfect fifth – is the basis for what’s become known as Pythagorean tuning because modern musicologists ascribe its origins to Pythagoras and his brotherhood. Start with a note such as D and move up by a perfect fifth and down by a perfect fifth to produce other notes of the scale, A and G, respectively. Now move up another perfect fifth from A and down another perfect fifth from G to generate the next notes, and so on. Eventually, we end up with an 11-note scale centred on D like this:
E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯
Without some adjustments this would span a wide frequency range, equivalent to 77 notes on a piano. To make the scale more compact, low notes are shifted to a higher octave, by having their frequencies doubled or quadrupled, and higher notes are similarly shifted downwards by an octave or two.
The result of bringing the notes closer together in this way is what’s called the basic octave. Pythagorean tuning was used by musicians in the West until about the end of the fifteenth century, when its limitations for playing a wider variety of pieces became apparent. So enamoured were the Pythagoreans with their discovery – that simple ratios of vibrating strings equated to harmonious musical intervals – and their belief that the universe was based on whole numbers, that they saw a perfect marriage of music and mathematics in the heavens. At the centre of physical space, according to their cosmology, was a great fire. Around this, carried on transparent celestial spheres and moving in circular paths were 10 objects, in order from the centre: a counter-Earth, Earth itself, the Moon, the Sun, the five known planets or ‘wandering stars’ (Mercury, Venus, Mars, Jupiter, and Saturn), and, finally, the fixed stars. The separations between these spheres, they taught, corresponded to the harmonic lengths of strings, so that the movement of the spheres gave rise to a sound (inaudible to human ears) known as the ‘harmony of the spheres’. Both the Greek words harmonia (meaning ‘joint’ or ‘agreement’) and arithmos (‘number’) come from the same Indo-European root ari, which also crops up in English words such as ‘rhythm’ and ‘rite’. Harmonia was also the Greek goddess of peace and harmony – fittingly, since her parents were Aphrodite (goddess of love) and Ares (god of war). The Pythagorean idea that musical harmonies were inherent in the spacing of heavenly bodies persisted throughout the Middle Ages. The philosophy of Musica Universalis (‘universal music’) found its way into the quadrivium, a quartet of academic subjects, including arithmetic, geometry, music, and astronomy, that was taught after the trivium (grammar, logic, and rhetoric) in medieval European universities and was based on Plato’s curriculum for higher education. At the heart of the quadrivium was the study of number in various forms: pure number (arithmetic), number in abstract space (geometry), number in time (music), and number in both space and time (astronomy). Following the lead of Pythagoras, Plato saw an intimate connection between music and astronomy: music expressing the beauty of simple numerical proportions to the ears, and astronomy to the eyes. Through different senses, they expressed the same underlying unity based on mathematics.
About the book:
A couple of years ago, David and his genius student Agnijo decided to write a book that would explain, in clear language to non-mathematicians, some of the strangest and most fascinating of mathematical ideas – especially those that had been developed recently or that had a bearing on the world around us. Their main aim was tackle any subject in maths, no matter how technically difficult or abstract, and present it in a form that anyone could appreciate. Their mantra was “if we can’t explain something in everyday language then we don’t properly understand it ourselves”.
Excerpted with permission from Weird Maths: At the Edge of Infinity and Beyond, HarperCollins India. Pages 288, Price: ₹ 499. The book is available on Amazon- https://amzn.to/2IXYLfZ
Cover photograph: Voyager Golden Record