The mathematical and musical harmony of chess
When I was a child I dreamed of becoming a great scientist. I loved mathematics and physics.
I read when I was just a teenager the essay "A brief history of time. From the Big Bang to Black Holes" by the great Steven Hawking, modern heir of Isaac Newton.
For scientists, the prospect of understanding and describing the relationship between phenomena with mathematical Laws and reproducible experiments is exciting. Ideally, a Law that can explain everything.
Growing up I discovered myself stronger on a cognitive level but more fragile on an emotional level. Wisely, I preferred to cultivate what most needed to grow and mature, namely the relational and emotional sphere. So the choice to become a medical doctor won.
I found the Law I was looking for through Buddhism.
NAM MYO HO RENGE KYO is the wonderful mystical Law of the Universe. Chanting it makes me feel in rhythm with myself and others.
It is a vibration, a rhythm, a music of the spheres as Pythagoras would say.
By the way my love for mathematics and physics remained alive in me like a burning ember.
In the same way my love for music.
I discovered chess late! What a shame, but better late than never!
Did you know that I used to play checkers when I was a kid?
Calling a chess player a "checkers player" is an unforgivable insult, in my opinion 😆
So, what do math and music have to do with chess?
Let's find out together!
- The Knight's tour problem and the seven bridges of Königsberg
- Chess champion and mathematician: Adolf Anderssen
- Math again? 🥴 The Fibonacci sequence and the Golden Ratio
- Chess games: complex systems approachable with statistical physics
- Chess champion and musician: Francois Andre' Philidor
PRACTICAL DEMONSTRATIONS
- Turning a melody into a chess game: Ode to Joy
- Turning a chess game into music: Bebop/Jazz fusion
Mathematics is a science that studies abstract entities (numbers, geometric figures...).
In mathematics calculations and measurements are performed in order to create models that allow us to develop a theory to explain phenomena and solve problems.
So math is an abstract science with very concrete and useful applications in every area of daily life (agriculture, mechanics, construction, engineering, even electronics and IT).
Mathematics is fundamental for cosmologic speculations too, trying to deduce how the Universe works.
There is a lot of mathematics in chess! There is logical-analytical thinking, there are problems of strategy and tactics to face, the need for concentration, theorems and rules on endgames...
Just to mention a few...
Imagine a square that has a pawn on one of its vertices. If the opponent's king is inside the square or can enter the square with its move, then it can capture it and prevent its promotion to queen.
- Concept of direct, distant and diagonal opposition (for winning endgames)
i.e. the king and pawn endgames...
- Techniques to create a passed pawn
In order to win you need to be mathematically precise
Remember Rooks belong behind passed Pawns!
The Knight's tour problem and the seven bridges of Königsberg
Math buffs will love the famous "knight's tour problem": finding a path that allows the piece to jump over all 64 squares on the board without stopping on the same square a second time.
It's a mathematical problem.
A knight's path is said to be "closed" if the last square on which the knight is positioned is close to the square from which it started, so that the knight, from its final position, can complete the same path all over again.
Otherwise the knight's path is said to be "open".
Well, even today it is not known exactly how many possible paths are open to the knight.
The "Knight's tour problem" can be schematized within the Theory of Graphs, discovered by Euler analyzing the famous problem of the 7 bridges of Königsberg, a city in today's Russia.
There are 7 bridges connecting the opposite banks of the city.
Someone posed the challenge of crossing them all but passing over them no more than once. An impossible challenge given their position in space, but the mathematical demonstration of this impossibility was far from easy.
When playing chess, there are infinite combinations of positions of the pieces on the board. Chess has been played for at least 1200 years and every game is always different.
Numbers are infinite too. There are natural numbers (1,2,3,4...), relative integers (...-3, -2, -1, 0, 1, 2, 3...), rational numbers (expressed as fractions), irrational numbers (for example √2 or π) and algebraic numbers which all belong to the category of real numbers. But there are also imaginary complex numbers.
Zero was formalized by the ancient Greeks. It derives from the Greek letter omicron (O) that was systematically found in the mathematical tables of Ptolemy. The Greeks, however, were unable to internalize the concept of infinity, which was absolutely inconceivable for them, as stated by Aristotle in his "Metaphysics".
Nevertheless since the time of the Pythagorean school (500 BC), more than a hundred years before the birth of the great Socrates, infinity in mathematical terms had actually been found. The Greeks had stumbled upon it and could not figure it out.
Greece was in fact the cradle of art, beauty, culture, "logos" or philosophical thought and reason.
The irrational numbers, with their infinite and non-recurring decimals, frightened and fascinated them at the same time... the esoteric secret not to be revealed of the Pythagorean school was probably all there.
How much beauty and mathematical precision in these irrational numbers!
courtesy of Archimedes of Syracuse (287-212 BC)
Chess champion and mathematician: ADOLF ANDERSSEN
Anderssen deserves to be remembered as an illustrious mathematician as well as a great chess champion.
Born in Breslau – Prussia (odiern Poland) in 1818 and died in the same place in 1879 at the age of 61. Professor of mathematics. European champion.
He beat Howard Staunton (London, 1810-1874), until then the strongest player in Europe.
The Staunton pieces, among the most valuable and used chess pieces, take their name from Howard Staunton. Staunton was the man who played (and won) against Evans, inventor of the Evans Gambit.
Adolf Anderssen lost against Paul Morphy when he came to Europe from the United States; his "Immortal" and "Evergreen" games have gone down in history.
Steinitz himself, the first official World Champion named that games this way, as a compliment for his fresh and brilliant game style.
Here the Immortal game. It was an un official match played in London in 1851 during the interval between rounds of the first International Chess Tournament in history.
Adolf Anderssen opponent was Lionel Kieseritzky, a strong player, Estonian naturalized French. At the end of the game he was so impressed that he decided to telegraph the moves to his chess club in Paris.
It was a King's gambit game. The game is so famous for the astonishing double Rook sacrifice during the middle game (moves 18 and 19) and the brilliant (!!) Queen sacrifice at the end of the game as well (move 22). A fantastic checkmating pattern.
Game explanations by NM Sam Copeland found on chess.com.
A legend tells of an ancient Hindu King who won a great battle to defend his Kingdom. To defeat the enemy he had to take a strategic action in which his son unfortunately lost his life.
From that day on, the King had no peace. He felt guilty for his son's death. He kept thinking about how he could win without sacrificing his son's life.
He was inconsolable.
One day a Brahmin appeared at the palace and, to distract the King, proposed a game he had invented: the game of chess. The King became fond of this game.
By playing a lot he realized that there was no way to win without sacrificing a piece (i.e. his son).
The King was finally relieved. He wanted to reward the monk, who refused.
Having insisted several times, the monk finally looked at the chessboard and asked for "one grain of wheat for the first square, two for the second, four for the third, eight for the fourth, and so on, always doubling".
It seemed like such a modest request...a few grains of wheat.
The day after the court mathematicians went to the King and informed him that the harvests of the entire Kingdom for eight hundred years would not be enough to fulfill the monk's request.
In this way the monk taught the King that a seemingly modest request can hide an enormous cost.
Calculating the Brahmin asked for 18,446,744,073,709,551,615 (18 trillion 446 billion 744 billion 73 billion 709 million 551 thousand 615) grains of wheat.
The King understood. The Brahmin naturally withdrew his request. He was appointed by the King as his advisor and Governor of one of the Provinces of the Kingdom.
This legend was very well known during the Middle Ages with the name of “Duplicatio scacherii”, so much so that there is a reference to it even in Dante Alighieri’s Divine Comedy, where it is used by the great poet to give the reader an idea of the number of Angels present in the heavens:
"L’incendio suo seguiva ogni scentilla
ed erano tante, che’l numero loro
più che ‘l doppiar de li scacchi s’inmilla"
Paradiso, XXVIII, 91-93
Dante Alighieri
Mathematics and chess have a close connection with music too.
In music there are 7 notes... 12 in total if we consider sharps (♯) and flats (♭). With 12 notes, musical compositions are as infinite as chess games.
Pythagoras, a mathematician and philosopher of ancient Greece, was the first to understand that the pitch of a note is inversely proportional to the length of the string that produces it.
Sound is vibration, a physical phenomenon that always occurs, as long as there is no vacuum. The intervals between sound frequencies are simply numerical ratios.
Math again? 🥴 The Fibonacci sequence and the Golden Ratio
The ratio, the perfect proportion (also called divine proportion) is expressed mathematically by the Fibonacci sequence while geometrically by the Golden Ratio.
The Fibonacci sequence is a sequence of numbers that starts with 0 and 1 and where each number is the result of the sum of the previous two. So you will have 0, 1, 1, 2, 3, 5, 8, 13… and it can continue to infinity always following the same principle.
The peculiarity of the sequence is the value obtained by performing the ratio between two numbers placed side by side in the sequence.
In fact, each Fibonacci number divided by the previous one provides a result that tends towards 1.6103 as one proceeds towards higher values. The golden number.
Let's try starting from the beginning: 2 : 1 = 1, then 3 : 2 = 1.5, followed by 5 : 3 = 1.666, then 8 : 5 = 1.6…and so on.
We can then say that the golden number represents the limit to which this series of ratios tends.
In geometry, the "golden proportion" is obtained if the result of the ratio between the entire segment and the larger segment is equal to that between the larger segment and the smaller segment.
It means that AC is mean proportional between CB and AB.
So the Golden Ratio is the ratio between two numbers and results in a constant called φ (phi = 1.6180339887 ...), which has in common with π the fact of being an irrational number, a number having the decimal part infinite and not periodic.
In the same way, the golden angle is the one that with its complement always divides the full angle in a mean proportional way.
The upper Is the Golden angle
In plane geometry it is interesting to consider the golden rectangle, that is, the rectangle that has its two sides that are mean proportional to each other. It's divided in a square and a smaller golden rectangle again.
If we further applied the Golden Ratio to the smaller rectangle A and continued in the same way to infinity, we would have a fractal and obtain what is seen in the figure below, the golden spiral. The curve that joins the subdivision of the golden rectangles is a logarithmic spiral.
We have already said that sound (and therefore music) is a physical phenomenon due to the vibration of a body that emits sound waves and that the pitch of a musical note is inversely proportional to the length of the string that produces it.
The fundamental notes of the musical scale are seven:
The octave is the interval between the two successive Cs. The higher one has double the frequency of the lower one.
For example, in Western music, middle A has a frequency of 440 Hz, so the A one octave above has a frequency of 880 Hz (exactly double), and the A one octave below has a frequency of 220 Hz (exactly half).
The octave can be divided into 12 semitones equidistant from each other. In music, the circle of fifths is used by professionals to compose music (it indicates the chords that when played go well together) and also to transpose the chords, for example if you want to play the same song in C major or A major.
Well, if we consider the natural harmonics produced by a vibrating string (but this is true for any other type of acoustic generation), the circle of fifths does not close at all but becomes a golden spiral, like the one in the figure already shown.
We can therefore conclude that acoustics, in the mathematical relationship that exists between the natural harmonics in the different octaves, has a Golden Ratio in itself, regardless of any compositional structural form chosen.
Because of nature's predilection for efficiency and economy, this mean proportional relationship is commonly found almost everywhere not only in acoustic phenomena: in botany (the arrangement of flower petals, seeds, leaves on the stem...etc.), in the harmony of the proportions between anatomical segments in humans and in the animal kingdom (in the spiral growth phenomena of the shells of snails, land molluscs and shellfish - and in the growth of many other organisms).
The Golden Ratio is present in the helix structure of DNA and in the distribution of the planets in the solar system.
It is aesthetically pleasing precisely because it is commonly present in nature. For this reason it was a model of beauty for Renaissance artists. In painting, architecture and sculpture many artists reproduce the proportions of the golden ratio (the Parthenon of ancient Athens, most of the cathedrals, the painting of the Mona Lisa, the designs of Le Corbusier...).
Musical compositions can be made taking into account the Golden Ratio.
And what does chess have to do with it?
(Not much actually... 😂
...but let's find out together)
Many religious buildings have their plan in the shape of a golden rectangle, many Christian cathedrals for example.
The first architect to take into account the Golden Ratio was probably the Greek Phidias with his Parthenon.
The temples of the Masonic Lodges (esoteric initiatory associations) have the shape of a golden rectangle and an 8x8 checkboard floor.
The logarithmic spiral hidden in the floor symbolically reproduces the hidden spirals of the galaxies in the sky.
The chess game should represent, always symbolically, our Life, the mission we have chosen, the path of spiritual growth, facing the adversities of Life.
It is important to remember, however, that the Golden Ratio, in itself, is a physical phenomenon that has nothing supernatural.
Soon we will see that it is possible to set a chess game to music and, vice versa, transpose a melody into a game!
Chess games: complex systems approachable with statistical physics
A study by Marc Barthelemy, a physicist of complex systems at the University of Paris Saclay, in which he analyzed about 20,000 chess games, was recently published in the "Physical Review Journal" (on 23/01/2025).
Thee study was later disclosed to "non-experts" by science communicator Simone Valentini in a nice article on the Italian page of WIRED.
A chess game can be considered as a complex system composed of multiple components that interact with each other in a non-linear way, to be approached with the methods of statistical physics.
Other examples of complex systems are economic systems and climate systems, which require the use of statistical techniques to be analyzed, given their enormous complexity.
The chess game has a simpler decision tree, in which the variables are discrete (and therefore smaller and more identifiable than economic and climate systems) with only three possible outcomes – victory, defeat or draw.
There are specific moments in a game when the positions on the board become unstable, and when the smallest error tends to have dramatic consequences on the trajectory of the match.
The importance of each piece on the board, at any given moment, depends on the number of interactions it has with the other pieces, on how many it can attack and defend, and how many it is attacked and defended itself.
The speed of change of the situation on the chessboard depends on the ease with which the most important pieces are eliminated, a variable that the physicist defines as “fragility score”. The maximum value of fragility would coincide with key moments, in which a brilliant move by one of the two players changes the outcome of the game.
The study's findings only confirm what is common knowledge among chess players, namely that games tend to be resolved in the transition between the middlegame and the endgame, when even small mistakes begin to cost a lot.
Interestingly, when he applied his analysis to games played by Stockfish, one of the most powerful and widely used open-source chess engines, the patterns that emerged from the study of games between human grandmasters were hard to see. And this suggests that the optimal playing strategies of humans and artificial intelligences are somehow different.
In short, the study offers food for thought, but is still a work in progress.
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.111.014314
Chess champion and musician: Francois Andrè Philidor
Francois Andrè Philidor (1726 - 1795) was a great chess champion, probably the strongest chess player of the eighteenth century and also a great musician.
He joined the choir of the Royal Chapel of Versailles when he was only ten years old. He learned to move the pieces of chess by watching some choristers playing in the intervals.
At about 14 years old, he began to frequent the Cafés where chess was played, neglecting his musical studies. Philidor was soon notice by Legall, became his pupil and in less than three years surpassed his mentor.
The young man was also able to play "blindfold", without looking at the chessboard.
Philidor amazed everyone by playing two blindfold games at the same time and winning them!
In 1748 Philidor wrote his famous Analyse du Jeu des Echecs, a true milestone in the history of the game, the first attempt at a scientific systematization of all the chess knowledge of the time; it had a hundred editions, was translated into over ten languages and until the mid-twentieth century it remained the basic text on which all chess generations and various champions were trained.
Friend of Denis Diderot and Jean Jacques Rousseau. Contemporary of Amadeus Mozart. Lived between Paris and London.
A Don Juan to the point that he had to leave London because he was hounded by the husbands of women who apparently couldn't resist him.
He composed for the theater. He is credited with the birth and success of the Opéra-Comique, or Opera Buffa.
In 1760 Philidor married a singer, the beautiful Angelique-Henriette Elisabeth Richer (1736-1806), herself a member of a family of great musicians.
It was a happy marriage, from which seven children were born. Philidor was an admirable husband, completely devoted to his family.
Philidor was a very generous man; he accepted the collaboration of librettists of little talent to help them, even though he knew that this would harm him.
Compared to other chess champions, who were often (and are) unbearable, to say the least, he was kind and extremely polite.
As a musical legacy it should be remembered that both Mozart and Haydn were inspired by him, who was the inventor of Italian music in France and, at the same time, the most German of all French musicians.
Here is a chess game played by Philidor at the Parsloes Chess Club in London, near Piccadilly. From 1775 onwards Philidor was always there for six months of the year. He often played blindfolded.
Philidor vs Kotter (1-0)
Philidor played with the Whites, but started the game without one of the Rooks.
It was a king's gambit game. Kotter answered with the Falkbeer counter gambit, an uncommon continuation at the time.
The game is commented by Adolivio Capece, italian Journalist and NM (chess National Master) from 1972. He is also the writer of the beatutiful book about the history of chess, the Grand Masters and the memorable games: "Scacchi - I grandi maestri, le partite memorabili" Demetra Ed.
The magic flute is the last Opera composed by Amadeus Mozart. A beautiful and profound story, but with the light tones of the Opera Buffa.
Who knows if Mozart was inspired, even in a small part, by Philidor's Opera Buffa?
This is the plot, set in ancient Egypt:
A prince named Tamino, chased by a ferocious snake, is saved by three young witches in the service of the Queen of the Night, the powerful lady of lunar spells, whose daughter Pamina seems to be held prisoner by the terrible magician Sarastro!
Tamino sets out in search of the girl, with the intention of freeing her, armed with a magic flute, his love for her and escorted by the faithful Papageno, a strange bird catcher in a feathered suit and a slightly beaked nose.
Poor Pamina is watched over by Monostatos, an evil Moor in the service of the magician Sarastro. Through ridiculous threats and flattery he also tries to court her.
The great Sarastro will instead reveal himself to be a good and wise man, capable of saving the girl from the clutches of the wicked Queen of the Night, who is in reality a real witch, to give her to the brave Tamino, but not before he demonstrates that he deserves her, by overcoming three tests (silence, water and fire, or Nature, Reason and Wisdom).
Tamino, after a thousand adventures, and Papageno will succeed in crowning their dreams of love!
The Opera has several interpretations, that of the fairy tale, of the enlightened story and clear mystical-spiritual references, of initiation to a path of interior elevation.
It speaks of enlightenment of wisdom, but also of earthly love. The simplest, nicest and almost clownish, but sincere characters, are those who have remained most in the hearts of the people.
The moral of the Magic Flute is the truth of "rebirth" in Love. Through virtue, harmony in reason, man can be reborn in love. A difficult path, perhaps reserved for a few, but to the others, the simplest, all weaknesses and naivety are forgiven.
The Opera had an overwhelming success perhaps because of its universal character.
It is Love that sets us free ☺️
The scene of the meeting between Papageno and Papagena, who recognize each other as soul mates is among the most beautiful of the Opera.
PRACTICAL DEMONSTRATIONS
Converting a melody into a chess game: Ode to Joy
There is a beautiful tale written by Pablo Despeyroux and Lexy Ortega, which talks about a little girl who was passionate about chess.
One evening she was at home playing chess online while a terrible storm was raging.
A lightning struck the house, short-circuiting the electrical grid and giving her an electric shock.
She lost her consciousness and when she woke up, she found herself in the fantastic world of chess!!
Coming back home would be difficult. She would have to face several challenges.
The most exciting and challenging one would be converting a melody into a chess game. She had to play chess following the notes of a song.
The chessboard is a 8x8 square. Each file is identified from the left by the letter A to the letter H. The ranks are numbered from 1 to 8.
Now we know that there is a musical alphabet (from A to H).
The little girl used the German musical alphabet to decide the destination square of her pieces on each move. In casting the notes could be determined either by the King, or the Rook, or both.
She converted into a chess game none other than Ludwig Van Beethoven's Ode to Joy!
And here is the game. The little girl's moves faithfully followed the musical score, so she won the challenge and was able to come back home.
Turning a chess game into music: Bebop/Jazz fusion
Here is a very famous game, so famous that is called "The Immortal Game" and is considered one of the best ever played.
Kasparov against Veselin Topalov in January 1999 at the famous "Hoogovens Wijk aan Zee Steel Chess Tournament" held in the Netherlands.
Kasparov had the Whites and Topalov the Blacks and chose the Pirc defence.
On move 24 a stunning rook sacrifice that Topalov should have refused, because Kasparov in response started an unstoppable checkmating attack.
Kasparov won the tournament, Anand took second and Kramnik the third place.
Game explained by NM Sam Copeland:
In the previous paragraph we saw how to turn a melody into a chess game. In the same way we can decide to turn a chess game into music.
There are no absolute rules. We just need to establish some criteria and apply them.
A first nice attempt is due to Daniele Trucco, who is an Italian musician with a diploma in piano and a teacher of literature.
I found the musical transposition of the so-called "Immortal Game" by Kasparov on the Italian site "Messaggerie scacchistiche" - October 2, 2024.
The first thing to do is to associate a note with each square on the board.
The note ‘sounds’ whenever a chess piece, while moving, ends up on that square.
Now we need to deal with the lengths of the notes: music develops over time and needs beats, durations and speed of execution.
Ok, we transform each move and its relative countermove into a measure.
The chess game lasted 44 moves,
so there will also be 44 measures in the melody.
The conversion process must finally continue by assigning a length to each piece that moves on the board.
The author, Daniele Trucco, chose to do it like this:
Pawn: semiquaver or sixteenth note (1/16)
Rook: quaver or eighth note (1/8)
Knight: quarter note (1/4)
Bishop: minim or half note (2/4)
Queen: dotted minim (3/4)
King: semibreve or while note (4/4)
And that's the result: the musical score and the melody!
Probably not a great tune... but still better than TRAP music 😂
A very successful method, instead, the one of @BadXinton alias Charles Harrison.
In his blog "Turning chess into music" he explains everything very clearly.
In short it's the same system of using the letters of the chessboard files. He used the English notation, which lacks the letter H.
He chose the game "Magnus Carlsen vs Hiraku Nakamura", played on August 20th, 2020 at the Chess Tour grand final. It ended in a draw.
He decided that each file would correspond to that same note in the A Minor or C Major scale (in any octave) and that the H file or a ‘castle’ could count as a ‘wildcard’ for a note/notes of his choice.
With this approach he was able to give a most beautiful intonation to the melody.
The destination of each move would ‘activate’ these notes. For example:
if a piece moves from the A file to the E file, he would play an E.
To make the music more catchy, he used a jazz chord progression to create a harmonically rich accompaniment in different keys and rhythms, adding to the momentum of the pieces and giving a sense of progression.
The results is FANTASTIC !!
A lovely composition, a SO BEAUTIFUL TUNE and perfectly consistent with the game played.
Enjoy!!!
This is the link to the original article of @BadXinton alias Charles Harrison. I'm impressed.
For the record, the first attempt to turn a chess game into music was the one of John Cage (American composer) and Marcel Duchamp (French painter, sculptor and chess master). It certainly deserves to be remembered.
Actually, they arranged a public game as a pretext for a musical performance called "Reunion", in Toronto with Duchamp and his wife Teeny.
John Cage & Marcel Duchamp's performance
Everything is interconnected.
After all, interconnectivity is also at the heart of how artificial intelligence works.
Chess is no exception ...
We saw together how many useful applications mathematics has and how similar it is to chess, whose games can become real musical harmonies.
I hope you enjoyed the blog and that it helped you to appreciate more the world of science, mathematics and physics.
It was hard this time to document and create the blog, but I am satisfied.
As always every comment is welcome.
See you soon.
DocSimooo
