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For Pythagoras (c. 570 BCE – c. 495 BCE), the ancient Greek philosopher and mathematician, numbers were not merely a mathematical phenomena, but were representative all levels of existence, from the individual, to society, to the universe as a whole. In Metaphysica (Metaphysics), Aristotle wrote:
Pythagoreans applied themselves to mathematics, and were the first to develop this science; and through studying it they came to believe that its principles are the principles of everything.And since numbers are by nature first among these principles, and they fancied that they could detect in numbers, to a greater extent than in fire and earth and water, many analogues of what is and comes into being—such and such a property of number being justice, and such and such soul or mind…1328344
NUMBERS AND UNIVERSAL ORDER
The numbers 1 through 4 are central to Pythagorean theory:
| Number | Name | Magnitude | Purpose |
| 1 | Monad | Point. | Pythagorus explained: "The Monad is God and the good, which is the origin of the One and is itself Intelligence. The Monad is the beginning of everything. Unity is the principle of all things."2329345 |
| 2 | Dyad | Line | Diversity. Tension and desire to return to oneness. Matter. James Baldwin wrote: "Pythagoras took the next important step by subordinating the mere matter of nature to its essential form and order, identifying the latter with the reason or the soul."3330346 |
| 3 | Triad | Plane | The sum of monad and dyad (1 + 2 = 3). Unity and diversity are restored in harmony. "The triad signifies prudence, wisdom, piety, friendship, peace, and harmony."4331347 |
| 4 | Tetrad | Solid | Symbolizes justice, wholeness, completion. Four powers of the soul: mind, science, opinion, sense. Four ages of man: infancy, youth, adulthood, old age. Four lunar phases. Four elements of the earth. Four seasons. |
The sum of 1, 2, 3, and 4 is 10, which is described as "The fount of ever flowing nature."5332348 The universe was described in the terms of the number 10. There were 10 heavenly bodies in the cosmos.
RELATIONSHIP TO MUSIC
Pythagorean mathematical principles extended to music. Iamblichus 5333349, a fourth century (CE) scholar, recorded the anecdote that had been passed down for centuries before him. Pythagoras was pondering a mechanical aid for hearing, similar to compasses, rules, and optical instruments of sight. Pythagoras happened be passing by a blacksmith's workshop and listened to the resonances of the hammers pounding on the iron. What he heard was the first six notes shown below:6334350
What Pythagoras was hearing was the first few notes of the harmonic series. When a note is played on an instrument, the strongest sound you can hear is the lowest note, called the fundamental. Within this sound, however, you'll hear other resonances (called harmonics or overtones) which gives the note depth of tone and timbre. The more remote the overtones, the less audible to the ear.
What Pythagoras figured out was that each of these harmonics is in mathematical proportion. The fundamental is 1:1 (in unity with itself); the octave above is 2:1; the fifth above that, 3:2; fourth above that is 4:3. With this mathematical basis, Pythagoras was able to reconcile the principles of the sound with the cosmos: the harmony of the spheres (musica universalis). The orbit of each cosmic body (the earth, the moon, and the sun) emits musical tones: the faster the planet moves, the higher the pitch. Music became linked with astronomy. Plato wrote: "As the eyes, said I, seem formed for studying astronomy, so do the ears seem formed for harmonious motions: and these seem to be twin sciences to one another, as also the Pythagoreans say."8335351 With this comprehension of the universe, as Aristotle stated, there was greater potential to understand the soul, the mind, and society as a whole.9336352
If the monad is the central point, the central truth of the universe, then the fundamental tone is the central truth of sound, and must be in balance with the universe. The dyad is the line, the diversity creating a pull back to the unity. The more remote the musical interval, the more discordant and greater need to return to harmony. The triad, being the sum of monat and the dyad, is the restoration of harmony. If the tetrad represents completeness through the four powers of the soul, it can also represent music's ability to impact on the soul: to promote consciousness of being, stimulate the rational mind, empathize with others in society, and heal the soul. As Aristotle put it:
music ought to be used not as conferring one benefit only but many; for example, for education and cathartic purposes, as an intellectual pass time, as relaxation, as release after tension. While then we must make use of all the harmonies, we are not to use them in all the same manner, but for education use those which improve the character, for listening to others performing use both the activating and the emotion-striving or enthusiastic.9337353
What Aristotle is alluding to here is the affective power of music. The affects are created largely through the wide vocabulary of harmonies. The Pythagorean view of the universe is unity through diversity. There may be many cosmic bodies, but all are explained, all are proportioned. There may be a plethora of musical harmonies, but these are balanced by an underlying pull to unity and completeness. While the application of this theory (whether conscious or not) has manifested itself in several guises the centuries, the principles have largely remained untouched, and have served as the backbone of Western musical language to this day.
1.3 THE CENTRAL PULL: THE FUNDAMENTAL
(1) Aristotle, Metaphysia (360 BCE) book 1, section 185b. Translated by W. D. Ross. http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0052%3Abook%3D1%3Asection%3D985b338354, accessed 10 May 2016.
(2) Frank Higgins, The Beginings of Freemasonry (New York, 1916)p. 27. Reprinted at: http://www.rgle.org.uk/Higgins_Beginning_of_Masonry.pdf339355, accessed 10 May 2016.
(3) James Baldwin, History of Psychology: A Sketch and an Interpretation, volume II (London: Watts, 1913), p.153. Reprinted at: TARGET=_blank>http://psychclassics.yorku.ca/Baldwin/History/chap2-8.htm340356, accessed 10 May 2016.
(4) Manly Palmer Hall, Secret Teachings of All Ages, (San Francisco, Crocker Company, 1928), p.18. Reprinted at: http://www.sacred-texts.com/eso/sta/sta16.htm341357, accessed 10 May 2016.
(5) Sextus Empiricus, Against the Mathematicians (Adversus Mathematicos), vi, 94-5. Cited in: Andrew Barker, Greek Musical Writings: vol. 2, Harmonic and Accoustic Theory (Cambridge, Cambridge University Press, 2004), p.30.
(6) Iamblichus, The Pythagorean Life, trans. Thomas Taylor (London, J. M. Watkins, 1818), p.62-64. Reprinted at: https://classicalastrologer.files.wordpress.com/2012/12/iamblichus-the-pythagorean-life-1.pdf, accessed 10 May 2016.342358
(7) Iamblichus remarks that Pythagoras heard the harmonics of an octave above the note (diapason), a fifth above diapente), and a fourth above that (diatessaron). He also noticed that the note between the diatessaron and the diapente was dissonant, yet nevertheless gave the sound completeness.
(8) Henry Davis, The Republic The Statesman of Plato (London: M. W. Dunne 1901; Nabu Press reprint, 2010) p.252.
(9) Aristotle, op. cit.
(10) Aristotle, The Politics, trans. T. A. Sinclair, revised and re-presented by Trevor J.a Saunders. Cited in: Piero Weiss & Richard Taruskin, Music in the Western World: A History in Documents, Belmont, Thomson, 2008, p.12.