We are used to thinking of mathematics and music as two separate worlds, one made of cold calculation, the other of warm inspiration. But for the ancient Greeks, and for Pythagoras in particular, music was literally audible mathematics. Through his experiments, Pythagoras discovered that harmonious sounds correspond to precise numerical proportions: a ratio of 3:2 generates a perfect fifth, and 4:3 generates a fourth.
The instrument he used to demonstrate these proportions was the monochord, a simple resonating box with a single stretched string. Yet there is a fascinating historical detail. Ancient theorists also called the monochord “canon” (rule, measure), the same word we use today to describe the most complex form of musical polyphony. By measuring the string, Pythagoras was already sowing the seeds of polyphony.
As in the Horizontal, So in the Vertical (and in the Diagonal)
If numbers govern sound, then the hermetic principle “as above, so below” applies perfectly to musical space. In our system, the numerical sequence 1-3-5 is not treated as a mere chord, but as an abstract seed. When we play 1-3-5 one after the other in time, we obtain a melody (horizontal dimension). When we play them together at the same instant, we obtain harmony (vertical dimension). But if we let them chase one another, staggered in time, the diagonal appears: the Canon.
Polyphony is not an artificial invention superimposed on music centuries later, but an intrinsic property of the numerical system itself. One simply needs to know how to activate it.
The Traditional Pythagorean Table
We all know the classic Pythagorean table learned in elementary school — the multiplication table. If we look at it as a matrix, it appears as follows:
1
2
3
4
5
2
4
6
8
10
3
6
9
12
15
4
8
12
16
20
5
10
15
20
25
These numbers grow rapidly toward infinity. But tonal music, as we know, is based on a cycle of seven notes that constantly repeats (Do, Re, Mi, Fa, Sol, La, Si), and at the eighth note (the higher Do), the cycle begins again.
The Magic of the Modular Table
To adapt the infinite numbers of mathematics to the musical staff, we must use modular arithmetic in base 7. In practice, every time we reach 7 or one of its multiples (14, 21...), we reduce it to 0. The octave, in fact, acts like the unison, bringing us back to identity (Do). By applying this simple reduction, the infinite Pythagorean table crystallizes into a magic square bordered with zeros:
1
2
3
4
5
6
0
2
4
6
1
3
5
0
3
6
2
5
1
4
0
4
1
5
2
6
3
0
5
3
1
6
4
2
0
6
5
4
3
2
1
0
0
0
0
0
0
0
0
Observe how magical and symmetrical it is. Excluding the zeros, the first row read from left to right equals the first column read from top to bottom, and the same regularities appear along the diagonals. This is not merely mathematics: it is a database of musical transformers, our “infinite keys,” which we shall henceforth call the matrix of keys.
The Miracle of Frère Jacques
The Pythagorean tables literally sound. Suppose you have written a simple canon at the unison (imagine Frère Jacques). Now you wish to transform it into a four-voice canon at the lower seventh, sixth, and fifth — an operation that usually requires hours of calculations, trial and error (and that drove nineteenth-century theorists mad). With our modular table, the problem disappears. To obtain a canon in which the voices imitate one another in this way, one adds to the Sound Matrix the Pythagorean progression derived from the matrix of keys — that is, from the row of 1 in the modular Pythagorean table (0, 1, 2, 3, 4...). Add 0 to the numbers of the notes in measures 1 and 2, 1 to those in measures 3 and 4, 2 to those in measures 5 and 6, and as if by magic, your canon will shift by the required intervals, without automatically generating the typical inconsistencies that arise in empirical procedures.
score of Frère Jacques, four-voice canon at the lower seventh, sixth, and fifth
Toward Automation
This mechanism demonstrates that complexity does not reside in the number of parts, but in the method used to generate them. Working with eight voices or more is not a nightmare; it becomes almost automatic if one respects the geometry of saturation and abandons the old rules of “harmonic linkage” that forced even academics of the stature of Luigi Cherubini into error. The system of the modular Pythagorean table and abstract seeds overturns the historical perspective. It transforms the art of counterpoint into a matrix algebra, opening the door to that automation which today allows us to generate these perfect architectures in programming languages such as C++. The numbers of Pythagoras, three thousand years later, have not ceased to create universes through their harmonies.
Partimento (1740 ca.), manoscritto musicale di Carlo Cotumacci, didatta al Conservatorio di Sant'Onofrio. Biblioteca del Conservatorio di San Pietro a Majella, Napoli (Pubblico dominio).
The magic of Pythagorean numbers is not merely theory, but an exceptional practical tool. Would you like to explore the system of abstract seeds and discover how to write a perfect canon in no time, just as Palestrina might have done?
