So let’s return two centuries back, to meet the great Pythagoras (a. 572-493. BC). He had an ear so fine that, legend has it, he heard singing the heavens. The construction of its range is entirely based on a progression of fifths: the octave is only an accessory and reduction. To simplify, we start from is:

fa – do – sol – re – la – mi – si – fa sharp. etc.

do re mi fa sol la si do

9/8 9/8 256/243 9/8 9/8 9/8 256/243

There are two fundamental intervals:

9/8 = 1.125

256/243 = 1.0535

for intervals with the tonic (first note) we have:

do re mi fa sol la si do

1/1 9/8 81/64 4/3 3/2 27/16 243/128 2/1

1 1.1250 1.2656 1.3333 1.500 1.6875 1.8984 2

For the way in which they are designed, the sounds in this range belong all to the same harmonic series based on fa:

do re mi fa sol la si do

384/256 486/256 586/256 729/256

432/256 512/256 648/256 768/256

Please note, in order to clarify the theory of harmonics, that if one starts from a sound “1” frequency “F”, all the integers that follow can be considered as harmonic, that is, multiple of the frequency 1F, so: 2F, 3F, 4F , etc. Joseph Fourier (1768-1839) did us this lovely service. In this series, some intervals are fatally musical and allow scholars to scientifically investigate the sound ranges. With a base fa, the other fa of the octave will then be 1F, 2F, 4F, 8F, 16F, 32F, 64F, 128F, 256F, 512F, etc.

It is thus understandable why, according to this reasoning, the Pythagoras tonic do corresponds to the harmonics n. 384 of the base fa (384 = 256 x 1.5).

This relative fa, of course, brings the number 512: it is sufficient to divide by the value of the tonic to restore the initial values.

However, apart from some details, this range, said cyclic, is not considered a natural range! That would not like to the philosopher, as based on the famous Tetraktys which sum is 10, the perfect number. In fact, the intervals of fourth (4/2), of fifth (3/2) and octave (2/1), give a preferred series, allowing Pythagoreans to define all other intervals. Continue at Atorène, Music Theory Course for Alchemists. Part 2.

A quite exhaustive essay on Pythagorean mathematics applied to music has been written by Arturo Reghini in “Numeri Sacri nella Tradizione Pitagorica Massonica”, or sacred numbers in the pythagorean masonic tradition, and published in this site as *The Pythagorean Acoustics Based on Tetraktys* and *The Pythagorean Acoustic: Geometry & Music of Sirens*.