**Marina Baldissera Pacchetti**

**Abstract: **Aristotle bases the principle of harmony on the diesis being the indivisible unit that can be represented by number. This, in a passages of *De Anima* and *De Sensu, *sets aesthetic standards for ratios of numbers representing melodious concords (`symphonia’), which are demonstrated by arithmetic. This allows him to define harmonics to be one of the `more physical of the mathematical sciences’ because it shares the principle of indivisible unit number with arithmetic, and the demonstrations that provide knowledge about concords are arithmetical.

Aristoxenus says that this is not an appropriate treatment of harmonics and a proper explanation of what justifies aesthetic perception of consonant sounds. The principles of this science cannot rely on mathematics, but on movement with respect to pitch space. Harmonious sound arises from intervallic movement (to be defined), which is recognized only in terms of its phenomenology. He does not discard the use of arithmetic, but he rather sees it as a useful tool to calculate modes – as Barker says (note 50 GMW II p.135), Aristoxenus does not have a problem in including intervals smaller than a quarter tone (the diesis) in his `science of harmonics’. The inclusion of tones smaller than a quarter tone was problematic in explanations of concords for his predecessors (esp. Pythagoreans) and those who will come afterwards (Ptolemy 2C AD, Boethius 6C AD, Gafurio 15C AD, Zarlino 16C AD). Aristoxenus does not deny that the use of mathematics is not useful in harmonics, but as his `first principles’ are not mathematical, he seems to be able to accommodate standard calculations within his framework without incurring in inconsistencies in his philosophical framework.