Pythagoras' Harmonic Science and Harmonic Ratios

Pythagoras invented the harmonic science and harmonic ratios.

He wanted to invent a tool to aid the hearing which would be as firm and unerring as:

• a ruler or compass is unerring to the eyes.
• a dioptric instrument by Jupiter
• a balance scale

As he was walking near a brazier’s shop, he heard from a certain divine casualty the hammers beating out a piece of iron on an anvil.

These produced sounds that accorded with each other, with the exception of one combination.

• He recognized in those sounds, the diapason, the diapente, and the diatessaron, harmony.
• He saw, however, that the sound which was between the diatessaron and the diapente was dissonant by itself. Yet, nevertheless, it completed the greater sound among them.

He went into the brazier’s shop.

By various experiments, he found that the difference of sound arose from the magnitude of the hammers.* It was not from:

• the force of the strokes
• the shape of the hammers, nor
• the transposition of the iron which was beaten.

He accurately examined the weights and the equal counterpoise of the hammers.

He returned home and fixed one stake diagonally to the walls.

On it, he suspended 4 chords consisting of the same materials, same magnitude, and thickness, and likewise equally twisted.

He tied a weight to the end of each chord so that the chords were perfectly equal to each other in length.

He afterwards alternately struck 2 chords at once. He found the before-mentioned symphonies, viz. a different symphony in a different combination.

He discovered that the chord which was stretched by the greatest weight produced the symphony diapason, relative to the one stretched by the smallest.

The heaviest weight was 12 pounds.

• The lightest weight was 6 pounds.

Therefore, being double, it exhibited the consonance diapason which the weights themselves rendered apparent.

He found that the symphony diapente was produced from that heaviest weight compared with the weight next to the smallest, This next-to-the-smallest weight was 8 pounds.

Hence, he discovered that this symphony is in a sesquialter ratio, in which ratio also the weights were to each other.

He found that the chord which was stretched by the greatest weight, produced, when compared with that which was next to it in weight, and was 9 pounds, the symphony diatessaron, analogously to the weights.

This ratio, he discovered to be sesquitertian.

But that of the chord from which a weight of 9 pounds was suspended, to the chord which had the smallest weight [or six pounds,] to be sesquialter.

For 9 is to 6 in a sesquialter ratio.

Similarly, the chord next to that from which the smallest weight depended, was to that which had the smallest weight, in a sesquitertian ratio, [for it was the ratio of 8 to 6,] but to the chord which had the greatest weight, in a sesquialter ratio [for such is the ratio of 12 to 8.]

Hence, that which is between the diapente and the diatessaron, and by which the diapente exceeds the diatessaron, is proved to be in an epogdoan ratio, or that of 9 to 8.

But either way it may be proved that the diapason is a system consisting of the diapente in conjunction with the diatessaron, just as the duple ratio consists of the sesquialter and sesquitertian, as for instance, 12, 8, and 6; or conversely, of the diatessaron and the diapente, as in the duple ratio of the sesquitertian and sesquialter ratios, as for instance 12, 9, and 6.

In this manner and order, his hand and his hearing both conformed to the suspended weights. He established according to them the ratio of the habitudes. This let him easily transfer the common suspension of the chords from the diagonal stake to the limen of the instrument, which he called chordotonon.

But he produced by the aid of pegs a tension of the chords analogous to that effected by the weights.

Employing this method as a basis, and as it were an infallible rule, he afterwards extended the experiment to various instruments:

• to the pulsation of patellæ or pans
• to pipes and reeds
• to monochords, triangles, and the like.

In all these, he found and named the immutable concord with the ratio of numbers.

• Hypate was the sound which participates of the number 6.
• Mese was the sound which participates of the number 8. This is sesquitertian.
• Paramese and epogdous are those that participates of the number 9. This is more acute by a tone than mese.
• Nete was that which participates of the dodecad.

He filled up the middle spaces with analogous sounds according to the diatonic genus. This led to an octochord from symphonious numbers, viz. from the double, the sesquialter, the sesquitertian, and from the difference of these, the epogdous.

Thus, he discovered the [harmonic] progression, which tends by a certain physical necessity from the most grave [i. e. flat] to the most acute sound, according to this diatonic genus.

For from the diatonic, he rendered the chromatic and enharmonic genus perspicuous.

This diatonic genus, however, has physical gradations and progressions:

• a semitone
• a tone
• a tone

This is the diatessaron. It is a system consisting of 2 tones, and 1 semitone.

Afterwards, another tone being assumed, viz. the one which is intermediate, the diapente is produced, which is a system consisting of three tones and a semitone. In the next place to this is the system of a semitone, a tone, and a tone, forming another diatessaron, i. e. another sesquitertian ratio.

So that in the more ancient heptachord, all the sounds, from the most grave, which are with respect to each other fourths, produce every where with each other the symphony diatessaron; the semitone receiving by transition, the first, middle, and third place, according to the tetrachord.

In the Pythagoric octachord, however, which by conjunction is a system of the tetrachord and pentachord, but if disjoined is a system of two tetrachords separated from each other, the progression is from the most grave sound.

Hence all the sounds that are by their distance from each other fifths, produce with each other the symphony diapente. The semitone successively proceeding into 4 places:

• the 1st
• the 2nd
• the 3rd
• the 4th.

In this way, music was discovered by Pythagoras.