Applications and curiosities
It would be quite reductive to think that iterative processes only contribute to solving mathematics or computer science problems. The idea of iterating a process is part of our daily life! Pythagoras himself followed an iterative process (the repetition of an interval, the fifth) to construct the notes of the musical scale, a scale used for centuries. The connections with music don't stop there. Two thousand five hundred years later, the notion of iteration creeps in, to the great delight of our ears, into modern composition techniques. Minimalist music, prime numbers... all inspirations are welcome. These recursive mise en abyme can even inspire writers and artists, or give rise to particularly aesthetic self-referential constructions. Indeed, even the most abstract mathematics invites itself everywhere to amaze us!
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The Pythagorean scale
Pythagoras is often credited with the idea that everything is number. Less well known is that his view of mathematics was closely bound up with musical harmony—not only the "harmony of the spheres," but the actual construction of a musical scale using numbers.
The online encyclopedia of integer sequences | Tangente
Founded in 1995 by British-American Neil Sloane, the OEIS (On-Line Encyclopedia of Integer Sequences) is a collaborative website where anyone can add a new way of generating a sequence of numbers, together with its known properties and the questions it raises. Nearly 400,000 sequences can be browsed there!
Musical iterations | Tangente
The first iterative process in music gave rise to the Pythagorean scale. Since then, recursive processes have been widely used in musical composition, from Johann Sebastian Bach to contemporary music. Their use is limited only by composers’ imaginations!
The thousand-and-first tale | Tangente
Mise en abyme is a classic literary device that masterfully evokes an iterative process. One example is Olivier Salon's 2009 text Les Gens de légende, reproduced by kind permission of Éditions du Castor Astral.
A spiral of digits | Tangente
An entertaining blend of arithmetic and graphics lets us display the successive terms of an arithmetic sequence visually. This calls for some explanation—and decoding!