Pythagoras and Antiquity — Number vs. Ear

Music Theory Began with a “Discovery of Number”

The history of music theory begins where a single philosopher plucked a string. Pythagoras (c.570-495 BCE) — the mathematician-philosopher of Samos. What he discovered was that the pleasing combinations of sounds correspond to simple integer ratios.

This was an astonishing realization: that music can be made an object of science. The sensuous, elusive “beautiful sonority” can be written in the universal language of number. The enterprise of music theory begins its 2500-year walk here.

String Ratios and Consonance

The correspondences Pythagoras (and his school) found:

Ratio of string lengths	Interval	Sonority
2:1	Octave	Most fused (nearly the same note)
3:2	Perfect fifth	Bright and stable
4:3	Perfect fourth	Stable
9:8	Whole tone	—

Halving the string gives a note an octave higher; taking 2/3 gives a perfect fifth higher. The principle: the simpler the ratio, the more consonant; the more complex, the more dissonant. It became the core of the Pythagorean worldview that “the cosmos is made of numerical harmony” (musica universalis, the music of the spheres).

Pythagorean Tuning and Its Fate

The method of building a scale by stacking perfect fifths (3:2) is called Pythagorean tuning. But there is a fateful problem here: stacking twelve fifths does not land exactly on seven octaves. A tiny gap remains (the Pythagorean comma).

This is the discovery that “a mathematically pure tuning does not, in fact, close,” and it became the origin of the temperament problem (one of this series’ threads). Later, equal temperament (Article 5) would solve it by spreading this gap thinly across the 12 notes.

Number vs. Ear — Aristoxenus’s Rebuttal

Against the Pythagoreans’ view that “music is determined by numerical ratio,” Aristoxenus (c.4th c. BCE) — a pupil of Aristotle — argued head-on:

The correctness of an interval should be decided not by numerical calculation but by the trained ear.

He tried to treat music as “what is heard” — empirically and phenomenologically. Number (Pythagoras) vs. ear (Aristoxenus) — this opposition becomes the oldest, and longest-lasting, thread of music theory’s history. The modern debate over “theoretically correct but doesn’t sound good / outside the rules but good” is a repetition of the quarrel of these two men.

The Modes of Ancient Greece

Ancient Greece had a theory of harmoniai (modes) and tetrachords (units of four notes). The names Dorian, Phrygian, Lydian would be handed down — with shifted meaning — to the later medieval church modes (Article 2). Plato, in the Republic, discussed “which mode has a good influence on character,” giving rise to a view that music is tied to ethics and education (the doctrine of ethos).

Not the West Alone — China and India

What matters is that music theory is not the West’s exclusive property. At almost the same time, refined theories arose independently:

• China — the sanfen-sunyi method (fifth generation). By operations of taking 2/3 and 4/3 of a string (essentially the same as Pythagoras’s fifth and fourth), it derived the twelve lü. It tied tuning to astronomy, the calendar, and politics
• India — shruti (dividing the octave into 22 microtones) and the germ of the later raga (melodic type)

So the idea of “building a tuning from string ratios” was a universal discovery humanity reached independently in many places, while how to systematize it (the West toward harmony; India toward microtones and melodic types) diverged greatly by culture (detailed in Article 10).

From This Article to the Next

We have traced antiquity — Pythagoras’s number, Aristoxenus’s ear, and the independent theories of each civilization. But ancient theory had a decisive limit: there was no way to write sound down accurately. The next article turns to the notation revolution invented by the medieval Guido d’Arezzo (the four-line staff and the origin of do-re-mi), and how it made the complex theory and polyphony that followed possible.