Music According to Pythagoras

Music was discovered by Pythagoras. Its principles are below.

Take two brazen chords, such as are used in harps. The chords made from sheep intestines are false or obnoxious to the change of the air.

A———————B C———————D | E

These chords should be perfectly equal and equally stretched, so as to be in unison. There may be only one sound, though there are two strings.

They should be placed on some oblong and polished rule. The ancients called a harmonic rule, or a monochord, by which instrument all consonances and dissonances, and likewise musical intervals, were tried.

Bisect one of these chords in E. The point under E is vulgarly called the tactus. But it was called by the ancients as a hemisphere from its shape.

The tactus presses the chord, making only half of it, as ED, stikable and produce sound. Striking AB and ED at the same time will create the sweetest of all consonances. It is made up of:

the sound of the whole chord AB, and
the sound of the half ED.

This consonance the ancients called diapason or “through all [the chords]”, because in their musical instruments, the two extreme chords, i. e. the most grave, and the most acute of all the chords, contained this consonance so that, from the gravest chord having made a transition through all the chords to the supreme and most acute of all, they would hear this sweetest consonance.

It was in a duple ratio of the proportion of one sound to the other. For the sound of the chord AB is doubly greater or more grave than the sound of the half ED. For as sounding bodies are to each other, so are their sounds. But the chord AB is the double of ED.

Chord AB is now commonly called the octave. This is because from the first sound, and that the gravest, which is called ut, as far as to that sound which corresponds to it in the consonance diapason, there are these eight sounds, ut, re, mi, fa, sol, re, mi, fa.

Of these the first ut, and the last fa, which is the eighth, produce the consonance diapason, or the double, or the octave*.

*Superphysics note: 7 tacti lead to 8 tones. In Chinese philosophy, four tacti lead to five tones.

“Again, let the same chord CD be divided into three equal parts in the points F, G.

A———————————B C———————————D | | F G

FD will be 2/3 of CD and AB.

Place the tactus in F, and strike AB and FD be at the same time. This will produce a very sweet consonance that is not as sweet as the diapason.

The ancients called diapente (i. e. through five chords), because the 1st and the 5th chord produce this consonance.

But according to proportion, it is called sesquialter because the chord AB is sesquialter to FD. Consequently, the sounds of these chords also are in the same ratio. But sesquialter ratio is when the greater quantity AB contains the less FD once, and the half of it besides. It is commonly called the fifth, because it is composed from the first sound ut, and the fifth, sol.

Again, let the same chord be cut into four equal parts in the points H, E, I, so that the chord HD, may be three-fourths of the whole CD.

The tactus, therefore, being placed in H, let AB and HD be struck at one and the same time, and a consonance will be heard, yet more imperfect than the preceding two.

This was called by the ancients diatessaron or “through 4 sounds”.

With reference, however, to the ratio of the chords and sounds, it is called sesquitertian, because the greater AB contains the less once, and a third part of it besides. But it is now commonly called a fourth, because it is found between the first sound ut, and the fourth fa.

If now the point F be added in the preceding figure, and at one and the same time two chords HD and FD are compared in arithmetical ratios, we shall find that the greater HD will have to the less FD a sesquioctave[105] ratio, and the sound of the greater HD to the less FD will have the same ratio, i. e. in modern terms, that between fa and sol there is a sesquioctave ratio.

But if these two sounds are heard together, they will be discordant to the ear. Again, the distance between these sounds fa, sol, or between the chords HD and FD, or between the two harmonic intervals HD and FD, the ratio of which was sesquioctave, was called by the ancients a tone.

Afterwards they divided the whole of CD into 9 equal parts. The first of which is divided in K, so that the whole CD may have to the remainder KD, which contains 8 of those parts, a sesquioctave ratio.

This, in like manner, will be the interval of a tone, the first sound of which, i. e, of the whole CD, is now called ut, but the second sound of the rest of the chord KD is called re. Afterwards, they in a similar manner divided the remainder KD into nine parts, the first part of which is marked in the point L.

For the same reason between the chord KD and the chord KD, and their sounds, there will be a sesquioctave ratio.

The sound of the chord LD is now called mi; but the interval which remains between the chord LD and the chord HD has not a sesquioctave ratio, but less than it almost by half, and therefore an interval of this kind was called a semitone, and also diesis or a division. But that interval which remains between the points F and E they divided after the same manner, as the space between C and H was divided, and they again found the same sounds.

Let those divisions be marked by the points M and N; and here, also, between N and E, or between mi and fa, there is in like manner another semitone.

These eight sounds of the whole diapason, therefore, are ut, re, mi, fa, sol, re, mi, fa.

Between ut and the last fa is the consonance diapason, or between the chord CD or AB, and the chord ED.

But from the intervals which are between the sounds there are two semitones, viz. one between mi and fa, denoted by the letters L, N, and the other between the last mi and fa, denoted by the letters N, E. The remaining five intervals are entire tones.

From ut to the first sol is the consonance diapente, which contains 3 tonic intervals, and 1 semitone. Nevertheless, in all there are 5 sounds:

From sol to the last fa, there are 4 sounds:

These are perfectly similar to the first 4:

Nevertheless these are more grave, but those are more acute.

The first diatessaron is from ut to the first fa.

The second diatessaron is from sol to the last fa.

The diapason is divided into one diatessaron, and one diapente.

For from ut to sol is the diapente.

But from sol to the last fa is the diatessaron. This will also be the case if we should say that from ut to the first fa is the diatessaron, as is evident from the division of the chord.

But from the first fa to the last fa is the diapente, as is evident from the four intervals of the chord, three of which are tones, and the remaining interval is a semitone, which also in the other diapente were contained between ut and sol.

Let the tactus be placed in I; but I is the fourth part of the whole CD.

Let, also, AB and ID be struck at one and the same time, and the sweetest consonance, called bisdiapason, will be produced; which is so denominated, because it is composed from two diapasons, of which the first is between AB or CD, and ED, but the second is between ED and ID; for the ratio of these is double as well as of those.

The ratio of the bisdiapason is quadruple, as is evident from 327 the division; and is commonly called a fifteenth, because from the first ut to this sound, which is also denominated fa, there would be fifteen sounds, if the interval EI were divided after the same manner as the first CE is divided.

Let GD be a third part of the whole CD.

Let the tactus be placed in G.

Then at one and the same time let AB and GD be struck, and a sweet consonance will be heard, which is called diapasondiapente.

Bbecause it is composed from one diapason contained by the interval CE, or the two chords CD, ED, and one diapente, contained by the interval EG, or the chords ED, GD.

The chord ED is sesquialter to the chord GD; which ratio constitutes the nature of the diapente.

The proportion, also, of this consonance is triple.

The chord AB or CD is triple of GD.

It is commonly called the 12th, because between ut and sol, denoted by the letter G. There would be 12 sounds if the interval EG received its divisions.

From all of this, there are altogether 5 consonances, 3 simple: