Is there an infinite number of melodies?

We won't run out of new melodies any time soon. Although the number of possible melodies is finite, it is so very large that for all practical purposes, the supply of new tunes is infinite.

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Will composers run out of new combinations of musical notes to create original melodies? Or are there infinite combinations?

Sandeep Bhagwati (composer)

Concordia University, Montreal, Canada

A melody is more than just a sequence of notes. Rhythm, variations in loudness and the length of musical phrases all enhance the sequence of notes to make a melody “original”. Each of these adds a huge layer of possible combinations. Moreover, the notes that Western composers use are meagre abstractions that represent culturally selected frequencies from the entire, continuous human hearing range. Most music around the world uses frequencies in between Western notes. With these “microtones”, the range of possible notes extends considerably, and so does their combinatorial potential. If you are willing to accept or even invent new musical styles and aesthetics, the potential for composing and relishing new melodies is infinite. “There must be several googols (10100) of different possible melodies that can be composed without repeating yourself”

Richard Ellam

Bristol, UK

We won’t run out of new melodies any time soon. Although the number of possible melodies is finite, it is so very large that for all practical purposes, the supply of new tunes is infinite. I play the bagpipes (smallpipes, not Highland) and my pipe chanter only plays nine notes, whereas most orchestral instruments can cover about three chromatic octaves, totalling approximately 36 notes. So if anyone is going to run out of new tunes, it will be the pipers. We pipers play a lot of marches, and one of the simplest march rhythms is just four quarter-note beats in every bar. If you only have nine notes to play with and four places in a bar to fill, you can have a total of 94 or 6561 distinctly different bars. If you assemble these into 16 bars of music, there are 656116 possible arrangements – a very big number. Lots of these tunes would be awful or boring, or wouldn’t contain harmonious intervals. But even if only one tune in a billion billion (1018) follows all the other rules for composing pipe marches, you still have something like 1.2 × 1043 possible tunes. And that’s only pipe marches – we also play waltzes, jigs, reels, hornpipes, slow airs and the rest. If you do a similar exercise for instruments with a wider range of notes, and for longer and more complex melodies, you realise that there must be several googols (10100) of possible melodies that can be composed without repeating yourself.

Bryn Glover

Ripon, North Yorkshire, UK

It depends what is meant by “notes”. A piano has a finite range of predetermined notes, depending on the fixed tuning of the open strings, but a violin has a theoretically infinite range of possibilities, depending on the variable placing of fingers. But considering the range of notes represented on a piano, and assuming that the questioner is referring to the standard Western notion of seven-note octaves (A to G), then there is a finite number of notes and therefore a finite number of combinations, however large that number may be. Even if one adds in the scales and ranges of, for example, music from China, India, Bali and elsewhere, the answer will still be the same.

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Richard Bridge

London, UK

Composer Sergei Prokofiev, famed for his melodies, gave an answer to this question, which was posed by a reader of Pioneer magazine in 1939. He started with the analogy of chess, calculating that by the white player’s fourth move, there are 60 million possible variants. There are also 257 variations to choose from in a short tune of eight notes, he said, which offers around 6 billion possibilities. Out of these, a composer might be able to find something melodious. Add in different note lengths, rhythm and harmony, and the 6 billion are multiplied still more. He added that some melodies once thought to be appealing are not considered so today, and vice versa. He concluded that “we need not be afraid that there will come a time when all melody will have been exhausted and we shall be obliged to repeat old tunes”. Prokofiev is a good source on this: his body of work is full of melodies in a great Russian tradition. I think Peter’s Theme in Peter and the Wolf is one of the most affecting melodies I have ever heard. He was also a chess master, so he knew about options. To answer this question – or ask a new one – email [email protected]. Questions should be scientific enquiries about everyday phenomena, and both questions and answers should be concise. We reserve the right to edit items for clarity and style. Please include a postal address, daytime telephone number and email address. New Scientist Ltd retains total editorial control over the published content and reserves all rights to reuse question and answer material that has been submitted by readers in any medium or in any format. You can also submit answers by post to: The Last Word, New Scientist, 25 Bedford Street, London WC2E 9ES.

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