Models of harmony

from Conrad Shawcross: The Steady States,
Jenny Uglow and Andrea Bellini, (Cornerhouse Gallery Walsall, 2005)

Ring out ye Crystall sphears,
Once bless our human ears,
(If ye have power to touch our senses so)
And let your silver chime
Move in melodious time:
And let the base of Heaven’s deep organ blow,
And with your ninefold harmony

Make up full consort to th’Angelicke symphonyMilton’s ‘Hymn’, one of his earliest poems, makes one think of infinity and time, of our smallness in the cosmos and the greatness of our ambition. It leads us to puzzle over our desire to order and model, as if we are always searching for a lost pattern, a key to proportion and harmony at one with the great rhythms that surround us like the ocean tides.

In the Tempest, as soon as Ferdinand is cast ashore from the chaos of the sea, he notices the ubiquity of sound. But he is bewildered by the source of Ariel’s song:

Where should this music be? I’the air, or the earth?
It sounds no more: - and sure it waits upon
Some god of the island. Sitting on a bank,
Weeping again the king my father’s wrack
This music crept by me upon the waters
Allaying both their fury and their passion,
With its sweet air: thence have I follow’d it,
Or it hath drawn me rather:-

From the myths of Orpheus onwards music has seduced us with its promise, its offer of a charm to soothe the wild beast and the savage heart, to calm the stormy waters and appease - but not quite - the horrors of hell. We distinguish between harmony, Ariel’s ‘sweet air’, and unwanted noise. And the search for pattern in sound is as ancient as Western culture. At the centre of the Greek scientific revolution of the 6th century BC was the idea that the world is regulated by a hidden order that can be perceived by human reason if only we try hard enough: indeed the word Kosmos itself means a kind of supreme rational order and beauty.

Building on earlier teachings from Egypt, Pythagoras and his followers, the mathematikoi, sought this answer in numbers. They were especially intrigued by the numbers that seemed to govern music: the different tones Pythagoras distinguished – according to an ancient story - when he heard a blacksmith striking the anvil with blows of different weight. When a string of a certain length was plucked, and then one of half that length, the difference was an octave. Every time a string was halved it produced a new dominant note: a harmonic series. And if the string was divided by 2/3 you could create an unending, spiralling cycle of harmonious notes. The most pleasing sounds came, the Pythagoreans thought, from the simplest ratios 2:1, 3:2, 4:3 - intervals which were named the diapason (octave), the diapente (fifth), the diatesseron (fourth).

This was the first known translation of an experienced quality - sound – into quantity. In Aristotle’s words, once the Pythagoreans saw that the ratios of musical scales could be expressed in numbers, then ‘all things seemed to be modelled on numbers…they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number.’ The world was composed of invisible particles of matter, each with its resonance, sounding at a level inaudible to the human ear. The only person who could hear this universal music was Pythagoras himself, a demi-god, an Apollonian being.

Numbers as ‘elements of all things’ flowed into the personal and social spheres: even numbers were masculine, odd ones feminine and their relationships infused all human institutions whether it be marriage, or justice, or medicine. Numbers are the wires that hold the material and spiritual world together, fighting the destroying desert of infinity. Inherent in matter they merge into solid mathematics, geometric beauty - and we recognise today that life is indeed inscribed in geometry, from the spirals of DNA to the wondrously varied angles of atoms within molecules, or the growing, starry lattice of crystals. And in geometry the Pythagoreans also discovered that there were five ‘regular solids’ – and only five - the tetrahedron (4), cube (6), octahedron (8), dodecahedron (12) and icosahedron (20). Now comes the magic: if you fit any regular solid just inside a sphere all of its points will touch the inside of the sphere: if you fit a sphere inside a solid, it will touch all the faces. These shapes were counterparts to the four elements Earth, Water, Air and Fire, and to the summation, Quintessence.

The Pythagorean universe was thus modelled on the harmonious relationships of sounds and geometrical forms. At its heart was the unmoving earth, a still sphere, surrounded by one of the regular solids, in turn enclosed in a crystalline sphere bounded by another solid, and so on. Attached to these concentric spheres were the moon, sun and planets, creating exquisite sounds as they spun and whooshed through the air, as if whirled on a string varying in length with its distance from earth. The closer spheres hummed lower tones while those further away, whirling faster, were sweeter and higher pitched. The correspondences were not fixed, but fluid and harmonious, governing the seasons, the tides and all the rhythms of earthly life.

The five known planets of the ancient world, plus the sun and moon, balanced the seven notes of the ancient scale, and philosophers from Plato to Ptolemy brooded on their relationship. In the first century AD Pliny the Elder defined the ‘scale’ of the solar system in relation to the strings of the lyre, with the shortest distance – from Earth to Moon – representing the shortest string. In the 1601 translation of his Historia Naturalis, this appears as follows. Pythagoras, writes Pliny, ‘ calleth the space between the earth and the Moone a Tonus [tone], saying that from here to Mercurius is halfe a tone [ semi-tone], and from him to Venus in manner the same space. But from her to the Sunne as much and halfe againe’ [a minor third]. And so it flows on: Sun to Mars, a tone; Mars to Jupiter, semi tone; Jupiter to Saturn, semi-tone; Saturn to the fixed stars – the Signifier Sphaere or Zodiake’ a minor third. ‘ Thus are composed seven tunes, which harmonie they call Diapason, that is to say the Generalitie or whole state of concent and accord, which is perfect musicke’.

To some people a mathematical equation is a form so rich that you can return to it time and time again and find something new. ‘Much like a work of art, writes Graham Farmelo, a beautiful equation, ‘has among its attributes much more than attractiveness – it will have universality, simplicity, inevitability and an elemental power’. Physicists and mathematicians see elegance as an indicator. Graham de Sautoy, who works on ‘group theory and symmetry’ finds mathematical patterns in palindromes. He is, he says, searching for ‘some deep and subtle structure at the heart of my subject which I don’t yet understand.’ If he can understand the internal symmetry of a zeta function, he writes, ‘ I am convinced it will go hand in hand with revealing a huge vista of structure that we are currently too blind to see’. But for those like me, who find abstract thought difficult, the real excitement comes when these equations and patterns are conveyed in other ways. I understand a palindrome, but not the maths. I can ‘see’ the crystalline spheres and solids, but cannot do the calculations.

Sometimes, too, complex theories suddenly make sense in relation to poetic metre, or musical notation. To the medieval mind there was a strange, mystical power in musical modes, where some dominant notes exert a ‘pull’ on the others, like planets in orbit. But there was no clearly visible musical hierarchy, no map of sound that explained this pull until Guido of Arezzo invented his system of written notation at the start of the eleventh century, with his red and yellow lines foreshadowing the modern clef. This was supplemented by his technique of hand signals and his enduring ‘sol-fa’ mnemonic, its indicators (with the ‘ut’ later replaced by ‘sol’) aptly taken from the first syllables of the hymn to John the Baptist :

Ut queant laxis
Resonare fibris
Mira gestorum
Famuli thorum
Solve polluti
Labii reatum, Sancte Johannes

(That your servants may with relaxed throats sing the wonder of your deeds; take away sin from their unclean lips, O Saint John).

We feel similar leaps and links in architecture. The great Gothic cathedrals were designed in relation to musical and geometrical proportions, and in High Renaissance Italy, Alberti and Palladio both applied Pythagorean mathematics to building, hunting for the secrets of proportion, harmonies of space. ‘ I conclude that the same numbers, by means of which the Agreement of Sound affects our ears with delight’, wrote Alberti, ‘are the very same which please our eyes and mind. We shall therefore borrow all our Rules for the Finishing our Proportions, from the Musicians, who are the greatest Masters of this Sort of Numbers, and from those Things wherein Nature shows herself most excellent and compleat’. At the same time, Alberti longed to make an engine, beautifully geared, 'some unheard of machine to move and carry weights, making it possible to create great and wonderful things', levering his dreams into three dimensions.

In the next generation Copernicus dismantled the Aristotelian universe. Instead of revolving around the earth, the kingly sun governed his family of wheeling stars. But Copernicus still believed that this ‘government’ could be understood in harmonious, transcendent mathematical terms and when Johannes Kepler discovered that the planets orbited in ellipses rather than circles, he still placed them in a musical progression, related to the dimensions of the regular solids. Each orbital plane, he declared in his Harmonia Mundi, created the great harmonic chords: ‘Henceforth it is no longer a harmony made for the benefit of our planet, but the song which the cosmos sings to its Lord and centre, the Solar Logos’.

Kepler designed a model, a scooped sphere whose clumsiness belies the fluid beauty of his thinking. This was the dawn of a great age of model-making in Europe, as instrument makers and clock makers reached new levels of precision. Charting space, the flurry of exploration led to finer astrolabes, quadrants and compasses. Measuring time, Galileo suggested using a pendulum, and in1656 Christian Huygens built the first clock with a freely suspended pendulum, transmitting its regular movement to the mechanism. Huygens- a devout Copernican - also built a planetarium, driven by clockwork: the six planets were moved simultaneously by a central slanting axle, and the orbiting times ( determined by the new mathematics of continued fractions) were governed by pairs of cog-wheels.

After the publication of Newton’s In Principia the new cosmic mechanics seemed to impose a different order, a modality of gravity and weight. But to many these new laws still seemed to compose a social as well as a universal melody, as J. T Desagulier’s verse proclaims :

What made the planets in such Order move
He said, was harmony and mutual Love
The Music of his Spheres did represent
The ancient Harmony of Government.

Travelling lecturers were soon displaying the wonders of electricity, magnetism and astronomy to an enthralled public, while aristocratic patrons requested orreries or armillary spheres, beautiful objects in brass and silver, models of the solar system in which the planets rotated round the sun within a ring engraved with signs of the zodiac.

But was the world really so orderly? In the 1760s Joseph Wright painted a famous pair of pictures, displaying instruments that could clarify the workings of the universe to old and young, men and women. But they also showed two sides to forces and rhythms of nature. In the first painting, of 1766, A Philosopher giving that Lecture on the Orrery, in which a lamp is put in place of the sun, the lecturer uses the lamp to explain an eclipse: events that were once thought frightening, mysterious, supernatural, are now explained. The orrery presents a system that is rational, harmonious, serene. Yet in the companion picture the mood is different. An Experiment on a Bird in an Air pump, 1768, is shocking, violent, uncertain: as the experimenter sucks the air, creating a vacuum, so the bird flutters near to death – instead of experiencing delight, the children now recoil in terror. The music of the spheres has been replaced by a rasp, a rattle of breath. The pale moon still floats on high, glimpsed through the window. But Wright seems to say that art must recognise tumult and chaos as well as the mathematical proportion of natural forces.

In Wright’s day science and technology had not yet become divorced from the arts. Galileo worked on sound and vibrating strings; Huygens experimented with the layout of the keyboard; Benjamin Franklin invented a ‘glassychord’, a mechanised version of musical glasses. On his arrival in England in 1757 Franklin apparently heard an Irishman playing Gluck’s Concerto upon Twenty-six Drinking-Glasses': fascinated, he redesigned the instrument by attaching the glasses to an horizontal spindle in a trough of water in which the glasses were half-submerged. When the spindle was turned (by a pedal or treddle) their wet edges would come up and the player would draw sounds from them by the friction of his fingers on the wet rims, the note varying with the size of the glass. Franklin’s scribbled directions to the glassmaker are minutely precise. And his invention worked - allegedly the young Mozart wanted one, but his father couldn't afford it.

At the same time, James Watt, the improver of the steam engine, was working as an instrument maker in Glasgow, and although he was tone deaf he proudly built a perfect organ for a local Masonic Lodge simply through brilliant calculations. ‘Though we all knew he could not tell one note from another’, remembered his friend John Robison, he went ahead, ‘noting a thousand things no Organ builder would have dreamt of’, delving into books on harmony, following the new science of Equal Temperament, worked out by Bach in his Well-Tempered Clavier of 1722.

It may seem, literally, a plunge from the sublime to the ridiculous to move from the music of the spheres to a home-made organ in Glasgow. But the harmony of numbers knows no bounds. Indeed a good mathematician and engineer - or sculptor - can create music without sound. I felt this when I first saw Charles Babbage’s Difference Engine in the Science Museum in London. Babbage had become obsessed with building a machine to assist in the complex compiling of tables, based on the principle of finite differences. Invented in the 1830s and 40s, his machine was not finally built until the 1990s: on one side are the stacks of intricate, balanced gear wheels and cogs, of bronze, steel and cast iron. But on the other, as the cogs move and lock according to the numbers entered, the ‘fairground whirl’ of levers makes the keys spiral up steel poles, a sculptural ballet of numbers.

It is tempting, but fanciful, to think that Babbage was engineering an harmonic series all his own. The translation of numerical proportion creates a ripple of excitement – a wave. In the 1880s the poet Gerald Manley Hopkins responded with intense interest to the work of Hermann von Helmholtz on vision and acoustics, and the way the human eye and ear grasp order intuitively. Hopkins wrote vigorous letters to Nature, and planned works of his own on the science of music and metre and light. He was fascinated by the tension between stability and flux and in reconciling scientific insights with his passionate religious faith. His own metrical experiments with ‘sprung rhythm’, ‘inscape’ and ‘instress’ reflect a belief that identity can be conveyed in sound as well as image:

As kingfishers catch fire, dragonflies draw flame
As tumbled over rim in roundy wells
Stones ring; like each tucked string tells, each hung bell’s
Bow swung finds tongue to fling out broad its name;
Each mortal thing does one thing and the same;
Deals out that being indoors each one dwells.

It is not surprising that as well as the clattering stone and plucked string, Hopkins chooses the bell’s great note, the sound where we most clearly hear a dominant note amid an aura of others: every note contains a spectrum of sounds, just as there is a range of colour within light, and indeed sounds share the properties of light waves shimmering outwards from a star. We can tell if sound waves are moving towards us if there seems to be an increase in frequency and a decrease in wavelengths, as the Austrian physicist Christian Doppler demonstrated in the early 1800s with an orchestra on a moving railroad car, the music changing pitch as it whizzed past its audience. Within a decade physicists proved that the principle also applied to light waves. Now astronomers measure ‘red shift’ to see if stars and galaxies are moving away from us, as the lengthening of the wavelengths causes the light to move toward the red end of the spectrum.

Contemporary science still finds the links made in the past between music and the cosmos, notably in String Theory, which also ties into the notion of quintessence, since it treats the essence of matter as a loop of energy/(light) rather than the original ‘atomistic’ particle. And in its balance of number and resonance, this provides a Theory of Everything, a single explanatory model for forces and matter, stands as a modern, and equally poetic version of the Pythagorean quest for a universal harmony.

String theorists argue that the fundamental particles—electrons, neutrinos, quarks, and so on— are not point-like but infinitely small, one-dimensional loops, each containing a single ‘vibrating, oscillating, dancing filament’. As explained by the mathematician and physicist Brian Greene:

Just as the strings on a violin or on a piano have resonant frequencies at which they prefer to vibrate—patterns that our ears sense as various musical notes and their higher harmonics—the same holds true for the loops of string theory. But rather than producing musical notes, each of the preferred mass and force charges are determined by the string's oscillatory pattern. The electron is a string vibrating one way, the up-quark is a string vibrating another way, and so on...hence everything, all matter and all forces, is unified under the same rubric of microscopic string oscillations—the ‘notes’ that strings can play.

If electrons vibrate on this almost unimaginably infinitesimal plane, radio signals can reach seventy light years beyond the solar system. Through them we can hear the stars. The sun gives off waves of radiation in a flaring range of frequencies: ‘chords’ with different resonance depict the leaping and flying of sunspots and storms. Pulsars spin with such regular pulses of radiation that a radio telescope can detect a synchronized rhythm as planets orbit them. To return to The Tempest, in Caliban’s words ‘the isle is full of noises/ Sounds and sweet airs’, a thousand instruments humming around our ears. Sounds in every wavelength ripple out from the heart of the cosmos, scribbling their tracks into infinity. We need ears to hear them, and pulses to feel their time-bound, timeless rhythm - and art to show their ravelled mystery.


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