Article ID: CBB718056038

L'antanairesi e la teoria armonica greca (2011)


"Antanairesis," literally "successive subtractions," is the method today known as Euclidean algorithm. In the "Elements" it is applied to numbers, for computing the GCD, and also to generic magnitudes, to determine if they are commensurable or not. The oldest example of an antanairetic procedure, described in a fragment of Philolaus, regards the construction of the musical intervals in the Pythagorean scale. These intervals - fourth, tone, diesis, comma - are in fact obtained from octave and fifth by successive subtractions. Since octave and fifth are incommensurable, such antanairesis is infinite; therefore does not exist an interval by which all the others can be measured (a passage from Plato's "Republic" is possibly a reference to that). In order to approximate this interval, the musical antanairesis has been forced to stop, an operation which corresponds to finding a convergent of a continued fraction. Those, as the Phytagoreans, who did not accept this kind of intervals, solved the problem by using two incommensurable measures. This suggests an alternative usage of the cut of an antanairesis, that we analyze in the second part of the paper.

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Authors & Contributors
Bellissima, Fabio
Harari, Orna
Barker, Andrew D.
Krol, Z.
Yusta, Piedad
Petraki, Zacharoula A.
Archive for History of Exact Sciences
Bollettino di Storia delle Scienze Matematiche
Apeiron: Journal for Ancient Philosophy and Science
Kwartalnik Historii Nauki i Techniki
Historia Scientiarum: International Journal of the History of Science Society of Japan
Studies in History and Philosophy of Science
Cambridge University Press
Oxford University Press
Edizioni di Storia e Letteratura
Princeton University Press
Pavia University Press
Harmony (music theory)
Ptolemy, Claudius
Apollonius, of Perga
Time Periods
Early modern
18th century
16th century
Middle and Near East
Babylon (extinct city)

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