Chapter 26

# Pythagoras' Harmonic Science and Harmonic Ratios

September 14, 2015

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 itself by itself dissonant. Yet, nevertheless, it completed the greater sound among them.

Being delighted, therefore, to find that the thing which he was anxious to discover had succeeded to his wishes by divine assistance,

He went into the brazier’s shop and was delighted to find by various experiments 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, lest if there were many, a certain difference should arise from this circumstance, or in short, lest the peculiar nature of each of the stakes should cause a suspicion of mutation.

From this stake, he suspended four chords consisting of the same materials, same magnitude, and thickness, and likewise equally twisted.

To the extremity of each chord also he tied a weight.

When he had so contrived, that the chords were perfectly equal to each other in length, he afterwards alternately struck two chords at once, and 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, when compared with that which was stretched by the smallest, the symphony diapason.

But the former of these weights was 12 pounds, and the latter 6. Therefore, being double, it exhibited the consonance diapason which the weights themselves rendered apparent.

He found that the chord from which the greatest weight was suspended compared with that from which the weight next to the smallest depended, and which weight was eight pounds, produced the symphony diapente.

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, having conformed both his hand and his hearing to the suspended weights, and having established according to them the ratio of the habitudes, he transferred by an easy artifice 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 an immutable concord with the ratio of numbers.

But he called the sound which participates of the number 6 hypate= that which participates of the number 8 and is sesquitertian, mese; that which participates of the number 9, but is more acute by a tone than mese, he called paramese, and epogdous; but that which participates of the dodecad, nete.

Having also filled up the middle spaces with analogous sounds according to the diatonic genus, he formed 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, as we shall some time or other show when we treat of music. This diatonic genus, however, appears to have such physical gradations and progressions as the following; viz. a semitone, a tone, and then a tone; and this is the diatessaron, being a system consisting of two tones, and of what is called a 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 four places, viz. the first, second, third, and fourth. In this way, music was discovered by Pythagoras.