The mathematical architecture of Bach's "The Art of Fugue"

The Art of Fugue (BWV 1080) is one of the last works of Johann Sebastian Bach, and one of the greatest masterpieces of western music. It can be considered as the supreme achievement of the composer’s contrapuntal abilities. It is composed of 14 fugues and 4 canons, whose order was not established by Bach. The work was published posthumously (1751) with the pieces ordered according to pedagogical criteria of increasing musical complexity. By examining the ratio between pieces’ lengths in bars, a mathematical architecture that fits the whole work based on Fibonacci numbers is reported. A proportional parallelism in the mathematical relationships for different levels of analysis is also reported. This emphatic mathematical architecture of The Art of Fugue is probably owing to Bach’s participation, in the final years of his life, in the Societät der musikalischen Wissenschaften which strongly supported the Pythagorean philosophy, encouraging in particular the intimate association between music and mathematics. The perfect mathematical architecture of the work was probably intended as a ‘secret’ manifesto of the Pythagorean philosophy, confirming the thesis of Dentler (2000).