Waves and Numbers

Baghdad, in the ninth century. A scholar is handed a letter written in a secret alphabet. He cannot read a word of it, and he does not try. He counts. How often does each symbol occur? Which is the most frequent, which the second, which the third?

His method rests on a fact about language. In a long enough stretch of any language, some letters turn up far more often than others, in much the same proportions from one text to the next. Rank the cipher’s symbols by how often they occur, rank the letters of an ordinary text the same way, match the two lists, and the secret alphabet gives itself away.

His name was Abū Yūsuf Yaʿqūb al-Kindī, and his method is still called frequency analysis. The word is the one this course uses for waves — how often something happens — and the idea is the one the course is built on. Do not stare at a complicated thing as a whole. Break it into simple parts, count how much of each part it holds, and the counts tell you what the whole is doing.

For a letter, the parts are the alphabet. For a sound, a radio signal, a tide or the shaking of a bridge, they are sine waves. This prologue follows the idea through twelve centuries: from the House of Wisdom in Baghdad, where letters were counted and the ratios of musical strings were written down, through a quarrel over a vibrating string and a bold claim about heat, to brass machines that predicted the sea, and an algorithm fast enough to do the counting as the signal arrives.

Press play and watch the centuries run: the cities lit, in order, where the ideas of this prologue were written down or built, with the empires of the time on the timeline beneath. Drag the timeline to stop at any year.

Al-Kindī was born in Kufa around 801, into a noble family of the Arab tribe of Kinda — his father was governor of Basra — and was educated in Baghdad. There he became a leading figure at the House of Wisdom, and a number of Abbasid caliphs set him to oversee the translation of Greek science and philosophy into Arabic. The bibliographer Ibn al-Nadīm credited him with at least 260 books, on geometry, medicine, philosophy, logic and physics; most are lost (al-Kindī).

One that survives is a short treatise, On Deciphering Cryptographic Messages: the first known recorded explanation of frequency analysis — indeed, of any kind of cryptanalysis (frequency analysis). He built on al-Khalīl of Basra (717–786), whose Book of Cryptographic Messages had used permutations and combinations to list every possible Arabic word, with and without vowels. Al-Kindī’s instructions read like an algorithm:

Al-Kindī was not the last scholar of that world to treat sound as a matter of number. Abū Naṣr al-Fārābī was born around 870 somewhere in Central Asia — the sources name a Fārāb, a common place-name there — and spent most of his scholarly life in Baghdad, before Damascus and Egypt; he died in Damascus in 950 or 951. Whether his family was Persian or Turkic is disputed (al-Fārābī). Later writers called him the Second Master, after Aristotle.

He was a practising musician as well as a theorist. His Great Book of Music, the Kitāb al-Mūsīqā al-Kabīr, opens with the principles of the subject and an investigation of how sound is generated, then applies them to the instruments of his day, their intervals and their melodies (Kitāb al-Mūsīqā al-Kabīr). Behind it stands the Pythagorean theory of harmonic ratios: stop a string at half its length and it sounds an octave higher; at two-thirds, a fifth; at three-quarters, a fourth. Al-Fārābī held that the ear should judge such propositions, even where its verdict disagreed with the arithmetic.

Three centuries later Ṣafī al-Dīn al-Urmawī, born around 1216, probably in Urmia in north-western Iran, came to Baghdad as a young man. He made his name as a calligrapher, became a copyist at the new library of the caliph al-Mustaʿṣim, and there wrote the Kitāb al-Adwār, the first surviving work of scientific music theory since Avicenna’s. It records the final stage in the division of the octave into seventeen steps. When the Mongols took Baghdad in 1258, his music saved him: Hülegü, the sources say, was impressed, and doubled his income (Ṣafī al-Dīn al-Urmawī).

What do the simple ratios have in common? The figure shows the answer in the language of this course. Two tones whose frequencies stand in a ratio of small whole numbers, p : q, add up to a wave that repeats exactly, after q cycles of the lower tone. Small numbers, a short repeat. Make the ratio irrational and the sum never repeats at all — Lesson 1-1 proves it.

The rivals who helped bring al-Kindī down were three brothers, the Banū Mūsā — Muḥammad, Aḥmad and al-Ḥasan, sons of Mūsā ibn Shākir, a former highwayman turned astronomer from Khorasan, whom al-Maʾmūn had taken into his service while governor there. After their father’s death the boys were raised at court and enrolled in the House of Wisdom. They sent to Byzantium for Greek manuscripts, recruited the translator Thābit ibn Qurra, measured a degree of latitude in the desert near Sinjar, and wrote some twenty books, of which three survive (Banū Mūsā). Under al-Mutawakkil they ridiculed al-Kindī’s treatise on the astrolabe and had him beaten and his library taken; it came back to him through the Persian Jewish scholar Sanad ibn ʿAlī.

Their Book of Ingenious Devices, of 850, describes a hundred machines, many of them trick vessels to amuse party guests; some of their valves and controls were not surpassed until modern times. Another treatise describes “the instrument which sounds by itself”. Its tune was set by pins on a cylinder turned by water, as in a music box: each pin, as it came round, sounded one note, and the cylinders could be exchanged. The brothers are credited with the first music sequencer, and their automatic flute player has been called the first programmable machine (music sequencer).

Unroll a pinned cylinder and you are looking at a picture this course draws many times: time running across, pitch running up, a mark wherever a note sounds. Lesson 6-4 computes the same picture from a recording and calls it a spectrogram. The cylinder makes sound from the picture; the spectrogram makes the picture from the sound.

Europe took up the mathematics of sound in the eighteenth century, with the vibrating string itself. Jean le Rond d’Alembert, in Paris, sent the Berlin Academy a memoir on the curve that a stretched string forms when set into vibration, printed in its volume for 1747; some accounts date the discovery to 1746. In it he wrote down the equation of the string — the one-dimensional wave equation — and solved it: every motion of the string is the sum of two shapes, one sliding to the right and one to the left, at the same speed (wave equation).

Which shapes were allowed? Leonhard Euler, then at the Berlin Academy, disputed it with d’Alembert, and their quarrel did much to widen what mathematicians meant by a “function” (history of the function concept). Daniel Bernoulli, in Basel, took the physicist’s view. A string sounds its fundamental and its overtones at once, and its motion is a sum of simple vibrations, each a sine curve with a whole number of half-waves. In 1753 he stated it as a principle — the general motion of a vibrating system is a superposition of its proper vibrations — which the physicist Léon Brillouin later counted as the first statement of the principle of superposition (Daniel Bernoulli).

D’Alembert and Euler would not accept that every shape could be built that way. A sum of smooth sine curves, it seemed, could never make the sharp corner of a plucked string. Each side was partly right, and the mathematics needed to say which part did not yet exist. The figure computes the same motion both ways.

Joseph Fourier was born in Auxerre in 1768, the son of a tailor, and was orphaned at nine. In 1798 he sailed with Napoleon’s expedition to Egypt as a scientific adviser and became secretary of the Institut d’Égypte in Cairo. Back in France, Napoleon made him prefect of the department of Isère, at Grenoble, in 1801; and it was there, between the road-building and the paperwork, that he began to experiment on the propagation of heat (Joseph Fourier).

On 21 December 1807 he presented his memoir On the Propagation of Heat in Solid Bodies to the Institut in Paris. In it he made the claim d’Alembert and Euler had refused: that a function — any function, smooth or not, even one with jumps — could be written as a sum of sines and cosines, each coefficient found by a single integral. Looking back, he wrote that d’Alembert and Euler had both been persuaded that an arbitrary and discontinuous function could never be resolved into such a series (Fourier, letter). The examiners were not convinced either. When he entered a revised essay for the Institut’s prize in 1811, the committee — Lagrange, Laplace, Malus and Legendre among them — concluded:

In the 1860s William Thomson — professor of natural philosophy at Glasgow, later Lord Kelvin — brought Fourier’s method to the tides. A port’s tide is not one rhythm but many: the Moon raises its bulges of water twice a day, the Sun twice a day on a slightly different beat, and the tilts and ellipses of their orbits add rhythms of about a day. The rhythms are astronomy; how much of each a particular port has, Thomson’s harmonic analysis drew out of its tide-gauge records (tide-predicting machine).

Predicting the tide then means adding the rhythms up, hour by hour, for a year: a vast and error-prone labour. In 1872–73, with Edward Roberts of the Nautical Almanac Office and the instrument maker Alexander Légé, Thomson built a machine to do the adding. The engineer Beauchamp Tower had suggested the trick in August 1872: a cord running alternately over and under a row of pulleys, each pulley moved up and down by a crank at the rate of one rhythm, so that the cord’s free end moves by their sum. Larger machines followed in 1876 and 1879, and could draw a year’s tides for a port in about four hours. Machines on Thomson’s pattern are credited with the tide predictions for the Normandy landings of 1944. Lesson 3-1 opens with his machine.

The twentieth century turned the subject from analysis into engineering. Telegraph and telephone companies needed to know how fast they could signal down a line of limited bandwidth. Harry Nyquist’s papers of 1924 and 1928 tied the speed of signalling to the band of frequencies; Vladimir Kotelnikovproved in 1933 that a signal with no frequencies above B is fixed completely by samples taken more often than 2B times a second; and Claude Shannon’s “Communication in the Presence of Noise”, in 1949, made the theorem common property among engineers. Others found it too — E. T. Whittaker, Kinnosuke Ogura, Herbert Raabe — and Lesson 5-1 tells how it came to carry the names it does (sampling theorem).

Sampling turns a signal into a list of numbers, and counting how much of each frequency a list holds is a sum — the discrete Fourier transform — that for N samples takes about N² multiplications. In 1965 James Cooley of IBM and John Tukey published a way to do it in about N log N: for a million samples, the difference between a trillion operations and twenty million. Gauss had found much the same method around 1805, working out the orbits of the asteroids Pallas and Juno, but it was published only after his death, in Latin, and went unnoticed (Cooley–Tukey algorithm). Lesson 6-2 opens with why Cooley and Tukey were asked, and by whom.

None of these people set out to found a subject called signals and systems. Between them they supplied its working ideas: count the parts; let simple ratios fix a period; write a program as a pattern in time and pitch; build any motion out of simple vibrations, and let each vibration evolve by itself; add sines by machine; take samples; compute fast.

Every figure in this prologue has been a picture of a signal: a string’s shape, a column of letter counts, a ring’s temperature, the line a tide machine draws. The oldest picture of a human voice was made in Paris. In 1857 a typographer at a scientific publisher, Édouard-Léon Scott de Martinville, patented an instrument that let a voice draw itself — a horn, a membrane, a bristle scratching a wavy line into soot. In 1860 he sang “Au clair de la lune” into it: the tune on the first cylinder of Figure 5.

He meant his tracings to be read by eye, like writing. It seems not to have occurred to anyone before the 1870s that they held enough to recreate the sound, and no one did so until 2008 (phonautograph). That is where Lesson 1-1 begins: with a signal, what it is, and the few simple operations that change it.