Music According to Pythagoras

Music was discovered by Pythagoras. Its principles are below.

Take two brazen chords, such as those 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 used a ‘harmonic rule’, or a monochord, to test all consonances and dissonances of instruments and musical intervals.

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, strikable 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.

The ancients called this consonance diapason or “through all the chords”. This is because in their musical instruments, the two extreme chords, i.e. the most grave, and the most acute, contained this consonance.

They would hear this sweetest consonance from the gravest chord having transitioned through all the chords to the supreme and most acute of all,

It was in a duple ratio of the proportion of one sound to the other.

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 the gravest sound, both called ut, there are 8 sounds, ut, re, mi, fa, sol, re, mi, fa.

These 8 correspond to ut in the consonance diapason.

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

Let the same chord CD be divided into 3 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 at the same time. This will produce a very sweet consonance that is not as sweet as the diapason.

The ancients called it 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 3/4 of the whole CD.

A——————————————————————————B
C——————————————————————————D
 |   |  | | | | |   | |
 K   L  H F M N E   G I

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”.

The ratio of the chords and sounds is called sesquitertian because the greater AB contains the less once.

• 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 is added in the preceding figure, and at one and the same time two chords HD and FD are compared in arithmetical ratios, then:

• the greater HD will have to the less FD a sesquioctave ratio, and
• the sound of the greater HD will have the same ratio to the less FD, i.e. in modern terms, there is a sesquioctave ratio between fa and sol.

But if these 2 sounds are heard together, they will be discordant to the ear.

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. Between N and E, or between mi and fa, there is in like manner another semitone.

These 8 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 5 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:

1. ut
2. re
3. mi
4. fa
5. sol

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

1. sol
2. re
3. mi
4. fa

These are perfectly similar to the first 4:

1. ut
2. re
3. mi
4. fa

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:

• the diapason
• the diapente
• the diatessaron.

But there are 2 composite:

• the bisdiapason
• the diapasondiapente

Finally, those ancient Greeks differently denominated these sounds, ut, re, etc.

For the first, i. e. the gravest sound or chord, which is now called ut, they, denominated hypate, and the others in the following order: