Ancient belief in a cosmos composed of spheres, producing music as angels guided them through the heavens, was still flourishing in Elizabethan times. ...There is a good deal more to Pythagorean musical theory than celestial harmony. Besides the music of the celestial spheres (musica mundana), two other varieties of music were distinguished: the sound of instruments...(musica instrumentalis), and the continuous unheard music that emanated from the human body (musica humana), which arises from a resonance between the body and the soul. ...In the medieval world, the status of music is revealed by its position within the Quadrivium—the fourfold curriculum—alongside arithmetic, geometry, and astronomy. Medieval students... believed all forms of harmony to derive from a common source. Before Boethius' studies in the ninth century, the idea of musical harmony was not considered independently of wider matters of celestial or ethical harmony.

Pythagoras... applied the proportion he had thus found by experiments, to the Heavens, and from thence learn'd the Harmony of the Spheres. And, by comparing these Weights with the Weights of the Planets, and the intervals of the Tones, produced by the Weights, with the interval of the Spheres; and lastly, the lengths of Strings with the Distances of the Planets from the Center of the Orbs; he understood, as it were by the Harmony of the Heavens, that the Gravity of the Planets towards the Sun (according to whose measures the Planets move) were reciprocally as the Squares of their Distances from the Sun.

For Pythagoras as he was passing by a Smith's Shop, took occasion to observe, that the Sounds the Hammers made, were more accute or grave in proportion to the weights of the Hammers; afterwards stretching Sheeps Guts, and fastning various Weights to them, he learn'd that here likewise the Sounds were proportional to the Weights. Having satisfy'd himself of this, he investigated the Numbers, according to which Consonant Sounds were generated. Whether the whole of this Story be true, or but a Fable, 'tis certain Pythagoras found out the true ratio between the sound of Strings and the Weights fasten'd to them.

Pythagorean thought was dominated by mathematics, but it was also profoundly mystical.

But, good God, what is this when compared with that deep and true music of the wise, whereby the proportions of natural things are investigated, the harmonical concord and the qualities of the whole world are revealed, by which also connected things are bound together, peace established between conflicting elements, and whereby each star is perpetually suspended in its appointed place by its weight and strength, and by the harmony of its lucent spirit

And since the proportions of the human voice and the gesticulations of the human body are regulated by the same modulation as that by which sound and the motion of other bodies are, musical thought is subalternated not only to the harmony of human voice and gesticulation, but also of instruments and of those whose delectation consist in motion or sound and with these the harmony of the celestial and non-celestial. And since the concordance of times and the composition and harmony of the lower world and of all things composed of four elements come from celestial motions, and, moreover, since it is necessary to find the harmony of causes in their effects, the study of music also extends to knowing the proportions of times and the constitution of the elements of the lower world, and even the composition of all the elements.

But it seems to have eluded all these philosophers in what way each of us is truly two fold and composite. For that other two fold nature of ours they have not discerned, but merely the more obvious one, the blend of soul and body. But that there is some element of composition, some two fold nature and dissimilarity of the very soul within itself, since the irrational, as though it were another substance, is mingled and joined with reason by some compulsion of Nature — E this, it is likely, was not unknown even to Pythagoras, if we may judge by the man’s enthusiasm for the study of music, which he introduced to enchant and assuage the soul, perceiving that the soul has not every part of itself in subjection to discipline and study, and that not every part can be changed from vice by reason, but that the several parts have need of some other kind of persuasion to co operate with them, to mould them, and to tame them, if they are not to be utterly intractable and obstinate to the teaching of philosophy.

I began to study trigonometry. There was solace in its strange formulas and equations. I was drawn to the Pythagorean theorem and its promise of a universal—the ability to predict the nature of any three points containing a right angle, anywhere, always. What I knew of physics I had learned in the junkyard, where the physical world often seemed unstable, capricious. But here was a principal through which the dimensions of life could be defined, captured. Perhaps reality was not wholly volatile. Perhaps it could be explained, predicted. Perhaps it could be made to make sense.

Happy are they that go to bed with grave music like Pythagoras.

There is a musicke where-ever there is a harmony, order or proportion; and thus farre we may maintain the musick of the spheares; for those well ordered motions, and regular paces, though they give no sound unto the eare, yet to the understanding they strike a note most full of harmony. Whatever is harmonically composed delights in harmony; which makes me much distrust the symmetry of those heads which declaime against all Church musicke. For my self, not only from my obedience but my particular genius, I doe embrace it; for even that vulgar and Taverne Musicke, which makes one man merry, another mad, strikes in mee a deepe fit of devotion, and a profound contemplation of the first composer; there is something in it of Divinity more than the eare discovers. It is an Hieroglyphicall and shadowed lesson of the whole world, and Creatures of God, such a melody to the eare, as the whole world well understood, would afford the understanding. In briefe it is a sensible fit of that Harmony, which intellectually sounds in the eares of God.

The association between religion and mathematically based science has its origins in the mists of history. ...the very dawn of Western culture in sixth-century B.C. Greece. ...[W]hen the Greeks were turning away from the mythological picture immortalized by Homer and Hesiod, the Ionian philosopher Pythagoras of Samos pioneered a worldview in which mathematics was seen as the key to reality. In place of the mythological gods, Pythagoras painted a picture in which the universe was conceived as a great musical instrument resonating with divine mathematical harmonies. ...[inspiring] mystics, theologians, and physicists ever since. ...But to Pythagoras and his followers, mathematics was the key not simply to the physical world, but more importantly to the spiritual world—for they believed that numbers were literally gods. By contemplating numbers and their relationships, the Pythagoreans sought union with the "divine." For them, mathematics was first and foremost a religious activity.

The musical language which made the classical style possible is that of tonality, which was not a massive, immobile system but a living, gradually changing language from its beginning. It had reached a new and important turning point just before the style of Haydn and Mozart took shape.

There is geometry in the humming of the strings. There is music in the spacings of the spheres.

And once more is this true in the case of music; not only because the absolute is prior to the relative, as 'great' to 'greater' and 'rich' to 'richer' and 'man' to 'father,' but also because the musical harmonies, diatessaron, diapente, and diapason, are named for numbers; similarly all of their harmonic ratios are arithmetical ones, for the diatessaron <nowiki>[</nowiki>] is the ratio of 4 : 3, the diapente <nowiki>[</nowiki>] that of 3 : 2, and the diapason [perfect ] the double ratio [2 : 1]; and the most perfect, the di-diapason <nowiki>[</nowiki>], is the quadruple ratio [4 : 1].