Where Music, Patterns, Math and Chemistry Meet

A pattern of steps

On a piano, the distance from one key to the very next one, black or white, is a half step. Two half steps make a whole step. A scale is a set of notes chosen by a pattern of these steps. The major scale goes whole, whole, half, whole, whole, whole, half, and lands on the same letter one octave up.

Start the same pattern on any of the 12 keys and you get a major scale in a new key. The notes change, but the pattern of steps (and the feel of the tune) stays the same. That is why a song can be sung higher or lower and still be the same song.

Five-note pentatonic scales, with no half steps at all, turn up in folk music around the world. Try playing only the black keys: they form one.

Curves hiding in a times table

Space 200 points evenly around a circle and number them 0 to 199. Now join each point to its double: 1 to 2, 2 to 4, 3 to 6, and so on. When the numbers pass 199, keep counting around the circle, so 150 joins 300, which is point 100. That is clock arithmetic, or arithmetic mod 200.

No single line is curved, but together they outline a smooth heart-shaped curve called a cardioid. The 3 times table outlines a nephroid with two points. In general the $k$ times table makes a curve with $k - 1$ points, all of the same family (epicycloids, the paths traced by a point on a wheel rolling around another wheel).

Stars from polygons

For more than a thousand years, artists across the Islamic world have covered walls, ceilings and doors with interlaced stars. Many of these designs can be rebuilt from a hidden grid of ordinary polygons. In the 1920s the British scientist E. H. Hankin, studying patterns in India, described a simple way to do it, often called polygons in contact:

1. Start with a tiling, say octagons and squares.
2. From the middle of every edge, draw two lines that cross each other at the edge, each leaning in at the same angle.
3. Stop each line where it meets a line from the next edge. Then erase the tiles.

Change the one angle and the whole design changes, from fat flowers to thin, sharp stars. Octagons give eight-pointed stars, and 12-sided tiles give twelve-pointed ones.

Four blocks

The periodic table has a strange shape: tall towers on the sides, a dip in the middle and two loose rows below. Color each element by the kind of subshell its newest electron goes into, and the shape falls into four neat blocks:

• the s block, 2 columns wide, on the left (plus helium);
• the p block, 6 wide, on the right;
• the d block, 10 wide, in the middle (the transition metals);
• the f block, 14 wide, usually printed underneath so the table fits on a page.

Elements in the same column have their outer electrons arranged alike, which is why they behave alike. Sodium and potassium both have one lonely outer s electron, and both react fiercely with water.

Strings and simple ratios

Pluck a string, then stop it at half its length and pluck again. The note jumps up an octave: the shorter string vibrates twice as fast. Stop it at two thirds of its length and you get a fifth, the next-sweetest sound. Three quarters gives a fourth.

Ancient Greek writers credited this to Pythagoras and his followers (6th and 5th centuries BC), who tested it on a one-string instrument, the monochord. The pattern is real: two notes blend smoothly when their frequencies make a simple ratio such as 2:1, 3:2 or 4:3. (A later legend says Pythagoras heard it in a blacksmith’s hammers. The physics of that story doesn’t work, but the string version does.)

The harmonic series: the notes hidden inside one note

A real string doesn’t vibrate in just one hump. It also vibrates in 2, 3, 4 and more equal loops at the same time, at 2, 3, 4 … times the lowest frequency. These overtones form the harmonic series. You rarely hear them one by one, but their mix is a big part of why a violin and a flute playing the same note sound different.

The simple ratios hide in the series: harmonics 1 and 2 make an octave, 2 and 3 a fifth, 3 and 4 a fourth, 4 and 5 a major third. Two notes that share many overtones sound smooth together.

The circle of fifths is a clock

Western music splits the octave into 12 half steps. After 12 steps you are back where you started, one octave higher, so note names work like hours on a clock: arithmetic modulo 12. Stepping up a fifth means adding 7 half steps. Because 7 and 12 share no common factor, stepping by 7 visits all 12 notes before coming home. That walk is the circle of fifths. Try stepping by 4 or 3 instead: you get stuck in a small loop.

There is a catch. Twelve pure 3:2 fifths should land exactly seven octaves up, but $(3/2)^{12} \approx 129.75$ while $2^7 = 128$. The small miss, about a quarter of a half step, is called the Pythagorean comma. Modern equal temperament shares it out by making every half step the same ratio, $2^{1/12} \approx 1.0595$. Its fifth is $2^{7/12} \approx 1.4983$, a hair under 1.5, and the circle closes.

Euclidean rhythms

In 2005 the computer scientist Godfried Toussaint noticed something about rhythms from around the world. Take $n$ beats in a bar and ask for $k$ accents spread as evenly as possible. The answer, found with steps like Euclid’s algorithm for the greatest common divisor (about 300 BC), is often a traditional rhythm. Three accents in eight steps gives the tresillo, heard across Latin American and African music. Five in eight gives the Cuban cinquillo. The same procedure had been used a couple of years earlier by the physicist E. Bjorklund to time pulses in a particle accelerator.

Chladni figures

In 1787 the German scientist Ernst Chladni published a book of strange drawings. He had clamped thin metal plates at the center, sprinkled sand on them, and stroked the edge with a violin bow. Each time the plate rang with a clear note, the sand leapt off the shaking parts and piled up along lines that stayed still. A higher note made a busier pattern.

Those still lines are nodes, just like the still points on a vibrating string, but now in two dimensions. The plate vibrates in a standing wave, and each note it can ring at has its own pattern, called a mode. For a square plate a good approximate formula for a mode is

$$z = \cos(n\pi x)\cos(m\pi y) \pm \cos(m\pi x)\cos(n\pi y),$$

where $x$ and $y$ run from 0 to 1 across the plate. The sand gathers where $z = 0$. Below, each grain takes random hops whose size grows with how hard the plate is shaking under it, so it wanders until it reaches a node and stays there.

Myth vs fact. Chladni patterns are sometimes shown online as proof that sound “shapes matter” in mystical ways, or that certain frequencies heal. The real explanation is plain physics: the sand is simply thrown off the moving parts and left on the still ones. The patterns depend on the plate’s size, shape, thickness and material, not on any special frequency.

A tune has mirror images

Composers treat a melody a lot like a pattern on a tile. Draw it on a grid with time running left to right and pitch going up, and four classic moves appear:

• Transposition slides the whole tune up or down. It still sounds like the same tune.
• Inversion flips it upside down: every step up becomes the same step down.
• Retrograde plays it backwards, a mirror across a vertical line.
• Retrograde inversion does both, a half turn.

Bach wrote a “crab canon” in The Musical Offering (1747), in which one voice plays the tune while the other plays it backwards at the same time. In the 1920s Arnold Schoenberg built his twelve-tone method on exactly these four forms of a row of notes. In math, these moves are the symmetries of the grid: slides, reflections and a rotation.

Newlands’ law of octaves

By the 1860s chemists knew about sixty elements and their atomic weights, but no one had a good way to arrange them. In 1865 the English chemist John Newlands listed the elements in order of atomic weight and noticed that similar ones turned up at regular intervals. Lithium, sodium and potassium, all soft metals that fizz in water, came seven places apart, and so did fluorine and chlorine. Counting lithium as the first, sodium is the eighth.

Newlands compared this to music: the eighth element from any starting point repeats the first, just as the eighth note of a scale repeats the first one an octave higher. Count lithium as do, and sodium lands on the next do. He called it the law of octaves.

When he presented it to the Chemical Society of London on March 1, 1866, the idea was mocked. One listener asked whether he had tried arranging the elements alphabetically, and the society declined to publish his paper. The pattern did break down after calcium, where heavier elements refused to fit.

Three years later Dmitri Mendeleev published a periodic table that left gaps for elements not yet found, and it won the day. But the Royal Society later gave Newlands its Davy Medal (1887) for discovering the periodic law. Today we know why his octaves stopped: the noble gases (found in the 1890s) add one more element to each early row, and from the fourth row on the transition metals stretch the rows to 18.

Myth vs fact. You may read that “each element has its own musical note.” Newlands’ octave was a way of counting, not a sound: elements don’t hum. What is real is that atoms and molecules have exact frequencies at which they absorb and give off light. Those are a hundred billion times higher or more than any note you can hear, so any “element music” is an artist’s choice of how to scale them down.

Molecules vibrate: their “notes”

The atoms in a molecule are held together by bonds that act a bit like springs. A molecule can stretch, squeeze and bend, and like a guitar string, each way of moving has its own natural frequency. A light atom on a stiff bond vibrates fast; heavy atoms on a loose bond vibrate slowly:

$$f = \frac{1}{2\pi}\sqrt{\frac{k}{\mu}},$$

where $k$ is the stiffness of the bond and $\mu$ is a kind of average mass of the two atoms.

A water molecule has three ways to vibrate. Shine infrared light on it and it soaks up exactly the frequencies that match those vibrations, much as a wine glass rings when a singer hits its note. Chemists measure these frequencies as wavenumbers (waves per centimeter of light); an infrared spectrum works like a molecule’s fingerprint.

Only 17 ways to repeat

Wallpaper, floor tiles and woven cloth repeat in two directions. Look closely and each design also has some mix of turns, mirrors and glides (a mirror followed by a slide, like footprints in snow) that leave it looking the same.

In 1891 the Russian crystallographer Evgraf Fedorov proved a surprising fact: there are exactly 17 different mixes. Every repeating flat pattern ever made, from Roman mosaics to modern wrapping paper, uses one of them. George Pólya found the same 17 again in 1924, and the artist M. C. Escher learned them from his paper.

The turns are limited too. A repeating pattern can only have turns of a half, a third, a quarter or a sixth. Five-fold turns can’t tile a repeating floor, any more than regular pentagons can.

A rose window from a compass

Gothic builders in the 12th and 13th centuries filled the round windows of their cathedrals with stone tracery and colored glass. The geometry came from a compass and a straightedge, the same two tools Euclid used.

The key fact is that a compass set to a circle’s radius steps around it exactly 6 times. Bisecting those arcs gives 12, and two diameters at right angles give 4, then 8, then 16. Each petal circle must touch its neighbors and the rim. If the window has radius $R$ and $n$ petals, each petal has radius

$$r = \frac{R \sin(180°/n)}{1 + \sin(180°/n)}.$$

Myth vs fact. You may read that rose windows and other “sacred geometry” have healing powers or encode secret messages. There is no evidence for that. What is true is beautiful enough: medieval builders used simple, exact constructions to make huge designs that fit together.

Why 2, 8, 18, 32?

An electron in an atom is described by whole numbers. The shell number $n$ is 1, 2, 3, … Inside shell $n$, the subshell number $l$ runs from 0 to $n-1$ (chemists call these s, p, d and f). Subshell $l$ has $2l + 1$ orbitals: 1, 3, 5, 7. Each orbital holds two electrons, one of each spin.

So shell $n$ holds two times a sum of odd numbers, and odd numbers always add up to a square:

$$1 + 3 + 5 + \dots + (2n-1) = n^2, \qquad \text{so shell } n \text{ holds } 2n^2 \text{ electrons}.$$

That gives 2, 8, 18 and 32. The same numbers set the widths of the table’s blocks: the s block is 2 columns wide, p is 6, d is 10 and f is 14.

Why do the rows go 2, 8, 8, 18, 18, 32, 32, with repeats? Because subshells don’t fill shell by shell. They fill in order of $n + l$, smallest first, so 4s fills before 3d. A new row of the table starts each time a new s subshell begins.

Balancing an equation is linear algebra

When propane burns, $a\,\mathrm{C_3H_8} + b\,\mathrm{O_2} \rightarrow c\,\mathrm{CO_2} + d\,\mathrm{H_2O}$. Atoms are never created or destroyed, so each element gives one equation:

$$\text{C: } 3a = c, \qquad \text{H: } 8a = 2d, \qquad \text{O: } 2b = 2c + d.$$

That is a matrix equation $M\mathbf{x} = \mathbf{0}$. Its solutions (the matrix’s null space) form a line through zero, and the smallest whole-number point on it, $a = 1, b = 5, c = 3, d = 4$, is the balanced equation. Big reactions with dozens of molecules are balanced this way by computer.

The same math shows up in the nucleus. A radioactive sample loses half of what is left in each half-life, so the amount after $t$ half-lives is $N = N_0 \left(\tfrac{1}{2}\right)^t$, an exponential decay.

Seven shapes of a crystal

A crystal is atoms stacked in a pattern that repeats in three directions, like wallpaper in 3D. The smallest box that repeats is the unit cell. Its shape is set by three edge lengths, $a$, $b$ and $c$, and the three angles between them.

Sorted by the symmetry they allow, all crystals fall into seven crystal systems. Add the ways atoms can sit inside the box, with every turn, mirror and screw, and you get exactly 230 space groups. Fedorov and Arthur Schoenflies each finished the count in 1891.

Snowflakes. Ice is hexagonal: its water molecules link into rings of six. As a snow crystal grows, its six corners stick out into moist air and grow fastest, so it sprouts six arms. Each arm feels nearly the same air, which is why the arms look alike. Johannes Kepler wondered about the six-fold shape in a little book in 1611, long before anyone knew about molecules.

Quasicrystals. In 1982 Dan Shechtman shone electrons through an aluminum–manganese alloy and saw a pattern with ten-fold symmetry, which ordinary crystals can’t have. Many scientists doubted him for years. The pattern is ordered but never repeats, like a Penrose tiling. He won the Nobel Prize in Chemistry in 2011.

The chemistry of stained glass

Plain glass is mostly melted sand. Medieval glassmakers colored the whole pot by stirring in metal compounds, so it was called pot-metal glass. Two kinds of chemistry make the colors:

• Metal ions dissolved in the glass soak up some colors of light and let the rest through. Cobalt gives blue; copper in an airy fire gives blue-green; manganese gives purple; iron gives green and yellow.
• Tiny specks of metal, far too small to see, make other colors. Copper specks give red, gold specks give rose-red, and silver gives yellow.

Myth vs fact. Old windows are often thicker at the bottom, and people say the glass has “flowed” over the centuries. It hasn’t: at room temperature glass would need far longer than the age of the universe to sag that much. Hand-made panes were uneven, and glaziers usually set the thick edge at the bottom.

Building a wave from harmonics

In 1807 the French mathematician Joseph Fourier made a bold claim while studying how heat spreads: any repeating pattern, however jagged, can be built by adding up simple sine waves whose frequencies are whole-number multiples of one base frequency. Mathematicians argued about it for decades, but it holds for every wave a musician cares about.

For sound this is the harmonic series again. A note’s pitch is set by the base frequency. Its color, or timbre, is set by how much of each harmonic is in the mix:

$$y(t) = a_1 \sin(\omega t) + a_2 \sin(2\omega t) + a_3 \sin(3\omega t) + \cdots$$

A pure tone has only $a_1$. A clarinet’s low notes lean on the odd harmonics (1, 3, 5 …), which gives them their hollow sound. A bowed violin string has all of them, falling off roughly like $1/n$, which sounds bright and buzzy. The pattern of the wave on the screen and the sound in your ear are two views of the same list of numbers.

Myth vs fact. Some websites claim that tuning to A = 432 Hz is “natural” or healing, or that special “Solfeggio frequencies” repair the body. There is no good evidence for either. The usual standard, A = 440 Hz, was agreed for convenience so that orchestras and instrument makers match (it became an international standard, ISO 16, in 1975). Any choice of A works musically; what makes notes blend is the ratios between them, not one magic number.

Standing waves are patterns in time and space

Each harmonic is a standing wave: a shape that stays in place and simply swells up and down. On a string the shapes are sine curves with 1, 2, 3 … loops. On a drum or a plate they become the patterns you can see elsewhere on this page. The same idea, splitting a complicated motion into simple standing patterns, runs through all four corners of the diagram.

Balmer’s formula

Heat hydrogen gas until it glows and pass the light through a prism. Instead of a rainbow you see four thin lines: red, blue-green, blue and violet. The Swedish physicist Anders Ångström had measured their wavelengths very precisely. In 1885 Johann Jakob Balmer, a Swiss mathematics teacher, found a simple rule that fit all four:

$$\lambda = B\,\frac{n^2}{n^2 - 4}, \qquad B \approx 364.56\ \text{nm},\quad n = 3, 4, 5, 6.$$

Put in $n = 3$ and you get 656.2 nm, the red line; $n = 4$ gives 486.1 nm, and so on. The same rule also fit fainter violet and ultraviolet lines for $n = 7, 8, \ldots$, seen in the light of stars, and it says they crowd together toward 364.6 nm. In 1888 Johannes Rydberg wrote the rule in a more general form, and in 1913 Niels Bohr explained it with a new model of the atom.

Like harmonics, but not the same

It is tempting to call hydrogen’s lines its “notes”, and the comparison is useful as long as we are careful about the difference.

• Alike: both are lists made from whole numbers, and both come from standing waves. A string’s notes come from waves that fit along its length; hydrogen’s lines come from the electron’s standing waves around the nucleus.
• Different: a string’s harmonics are added in equal steps, $f, 2f, 3f, \ldots$, and go on forever. Hydrogen’s line frequencies are differences of $1/n^2$: proportional to $\tfrac{1}{4} - \tfrac{1}{n^2}$. They bunch up toward a limit instead of spreading out, and they don’t form chords. Played as sound, they don’t sound like a scale.

So hydrogen does not play music. But the hunt for whole-number patterns, the same habit of mind that found the harmonic series, led straight to quantum physics.

Sand on the still lines, electrons off them

On a ringing plate, sand collects on the nodes, the lines and circles that never move. An electron in an atom is also described by a standing wave, found by Erwin Schrödinger’s equation in 1926. Where that wave is zero, there is a node too, and the electron is never found there. So the two pictures are opposites in one way: the sand marks the nodes, while the electron’s dots avoid them. But the patterns of nodes are close cousins.

An orbital is labeled by two numbers we need here: $n$ (the shell) and $l$ (the shape: $l = 0, 1, 2$ for s, p, d). It has $l$ flat nodes (planes through the nucleus) and $n - l - 1$ round nodes (spheres). A round plate’s modes are counted the same way: some lines through the center and some circles. Line up the counts and the two pictures match.

Where the analogy stops

A plate is flat and an atom is three-dimensional, so the orbital’s nodes are planes, cones and spheres, not lines. The sizes don’t match either: the orbital’s round nodes aren’t spaced like the plate’s circles. And an orbital is not a picture of a moving electron; it shows where the electron is likely to be found. What carries over is the deep idea: a wave trapped in a space can only settle into certain patterns, and each pattern has its own frequency, or for an atom, its own energy.

Moves that change nothing

A symmetry operation is a move that leaves a molecule looking exactly as it did: a turn about an axis, a reflection in a mirror plane, or an inversion through the center (every atom goes straight through the middle to the opposite side). The atoms do move, but each lands where an identical atom used to be.

The set of all such moves for one molecule is a group in the mathematician’s sense: do two moves in a row and you get another move in the set, and every move can be undone. Chemists name these groups with short labels. Water’s group, $C_{2v}$, has 4 moves. Benzene’s, $D_{6h}$, has 24, the same symmetries as a flat hexagonal tile.

This is more than tidy bookkeeping. A molecule’s group predicts which of its vibrations can absorb infrared light, how its orbitals can combine into bonds, and whether it can be polar. A molecule with a center of inversion, like benzene or carbon dioxide, can never be polar, because every push one way is matched by a push the other way.

An atom is a tiny drum

Hit a drum and its skin vibrates in a mix of patterns, or modes. Each mode is a standing wave with its own still lines (nodes), its own symmetry and its own frequency. Unlike a string, a drum’s modes are not whole-number multiples of each other: their frequencies follow the zeros of special functions called Bessel functions, in ratios like 1 : 1.59 : 2.14 : 2.30. That is why most drums sound like a thump rather than a clear note.

An electron in an atom is a standing wave too, but in three dimensions. Its shapes, the orbitals, come in families by symmetry: s orbitals are round with no node through the middle, p orbitals have one node plane, d orbitals two, f orbitals three, just like drum modes with 0, 1, 2 and 3 node lines.

How those shapes build the periodic table

Each orbital holds at most two electrons, and each shape comes in $2l + 1$ versions: 1 s orbital, 3 p, 5 d, 7 f. So the s, p, d and f families hold 2, 6, 10 and 14 electrons. Electrons fill them in a set order (1s, 2s, 2p, 3s, 3p, 4s, 3d …), and that order is the shape of the periodic table: two columns for s, six for p, ten for d and fourteen for f. Rows come out 2, 8, 8, 18, 18, 32 and 32 elements long.

So the four subjects meet in the middle. Music gives us standing waves and their frequencies. Math describes them exactly. Their symmetry makes the patterns. And chemistry is what you get when nature fills those patterns with electrons.