Pythagorean Mathematics and Music

Verifying the frequency ratios in the musical scale of just intonation with “hear‐and‐see” learning tools

“…The musical scale of just intonation, especially used in the Western music until the 17th century, has an arithmetical basis already established by the ancient Pythagorean school in the 6th and 5th century B.C., conveniently extended by Ptolemy (130 C.E.) and theorists and historians of later ages . As is well known, the Pythagoreans saw the link between basic musical sounds and numbers.…”

“Mathematics” and “physics” in the science of harmonics

“…For in every subject it is inherent in observation and knowledge to demonstrate that the works of nature have been crafted with some reason and prearranged cause and completed not at all in random" [54, §I.2, § §I.5. [13][14][15][16][17][18][19][20][21]. 17 Like other scholars of his time, Ptolemy's "criterion of truth" is dictated by a strange kind of "concord" between theory and observation, where the model, although highly mathematized, has lost its meaning as a theoretical entity and taken over the former prescientific meaning of direct representation of the known reality.…”

“…Although one might regard this as a dialectical strategy to give maximum credibility to the position he wants to hold, in this way he ends up denying the method of his Hellenistic predecessors, deeming legitimate only one theory: that whose assumptions are entirely justified by the phenomena and at the same time reflect the "rationality" of nature's works. 18 Akin to Galen's craving to strike a balance between the "rationalist" and "empiricist" schools of medicine [30], 19 Ptolemy loudly distinguishes his approach to the study of consonances from that of the "excessively rationalist" Pythagoreans -accused of accepting rationally justifiable statements even when they are contradicted by the senses 20 -and that of the "overly empirical" Aristoxeneans, for whom audible harmonies are not subject to mathematical analysis at all. Since for him the objects of sense-perception and thought are (again) identical (cf.…”

“…18 One should perhaps also include evidence that these narrow epistemological bases will actively operate in the development of modern science. 19 A discussion on Ptolemy's epistemological affinity with his contemporary Galen can be found in [41]. 20 His concern focused in particular on the fact -already mentioned above -that the interval corresponding to the ratio 8 : observable heavenly bodies by reference to the formal features of the spheres or other bodies on which they are physically carried.…”

Understanding Musical Consonance and Dissonance: Epistemological Considerations from a Systemic Perspective

“…Pythagoras, in the VI century B.C., postulated that the simpler the frequency ratio, the more consonant the interval: frequency ratio of 2:1 gives the octave, which is the most consonant; frequency ratio of 3:2 give fifths and frequency ratio of 4:3 give fourths and so on decreasing in consonance. The Pythagorean "tetractys", composed by integers 1-2-3-4, gives reason for all the perfect consonances [14].…”