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The process he chose is quite straightforward. He noticed that there are eight columns of squares on a chess board and also eight notes in a musical octave.
With some rather clever adjustments and using the values of the chess pieces to determine how long each note should be, he transcribed the chess notation into musical scores and then played them on a piano.
Here's my favourite - 'The Immortal Game' played in June 1851, and this is how it sounds on the piano.
Rather delightful!
The full details and a further two examples of the music are available on Jonathan's blog. And if you enjoyed Chess Music you might also like Jonathan's Fibonacci Music!