The Harmonic Division of the String

The origin of the harmonic proportions are twofold:

1. Immediately from the constructible figures, the same being also congruent

2. Through the mediation of double proportion, on which the identity of consonances depends.

Harmonic proportions are infinite. These, as far as our knowledge goes, are still rough, unpolished, unnoticed, and unnamed, and scattered like rough stones or timber.

We should polish them, attach names to them, and finally to construct from them the splendid edifice of the harmonic system, or musical scale.

Its construction is not arbitrary. It is not a human invention which may be changed. Instead, it is entirely rational and natural.

God Himself, the Creator, has given expression to it in adjusting the heavenly motions to each other.

The harmonic proportions are fitted together with them into a single system by the harmonic divisions of the string.

How many strings are there?

If the whole string is divided into parts such that they are individually iu cousonauce both with each other and with the whole, we shall call the division harmonic.

The middle term of this division, in musical (that is, consonant) proportions, is one of two equal parts, or if they are unequal, the greater of them: the outer terms of a consonant proportion are the other, or smaller part, and the whole string.

Let the geometer note the analogy of the divine proportion, that is the proportion of extreme and mean, in which the whole bears the same proportion to the greater part as the greater bears to the smaller.

For what in this geometrical division is the same proportion, in our musical division is the same quality, which is called concord, consonance, congruence, or harmony.

Beware however of assuming a consonance of the same kind, just as in the geometrical case the proportion is unique. The ancients did not mention this division in this sense, as they did not know the true cause of consonances.

But we shall deal below with their division of the string.

Proposition 9: The division of a string into 2 equal parts is harmonic

This is because:

• equal parts give out the same sound at any given tension, by Axiom 2
• the whole is twice the individual parts

Therefore, it is in identical consonance with each one of them, by Proposition 1.

Therefore there are 3 consonances*. Hence, by definition the string is divided harmonically.

Proposition 10: The division of a string into 2 parts which are in double proportion is harmonic

This is because:

• the parts in this proportion are in identical consonance, by Proposition 1.
• the greater part is double the smaller, therefore the whole is 3 times the smaller.

Therefore it is to the smaller as a circle is to the part cut off by a side of an equilateral triangle, which is consonant, by the final Corollary of the previous Chapter.

Hence the whole is itself in consonance with the smaller part, by Axiom 5.

Therefore, it is in consonance with the one which is double it, that is the remainder, by Proposition 4.

Therefore, 3 consonances are established by this division and the proposition follows.

Proposition 11: The division of a string into 2 parts which are in triple proportion to each other is harmonic

This is because the parts 1 and 3 are to each other as a consonant part of a circle is to the whole, they themselves are also in consonance with each other, by Axiom 5.

Since 1 + 3 = 4, the part 1 will also be in consonance with the whole by Axiom 1 and by Proposition 3.

The remainder 3 is in consonance with the part 1. So it will also be in consonance with 4 times the part, 4, that is with the whole string.

Hence in this case also there are 3 consonances.

Proposition 12: The division of a string into 2 parts which are in quadruple proportion to each other is harmonic

This is because:

• the parts are in quadruple proportion. They are therefore in identical consonance with each other, by Proposition 3.
• 1 + 4 = 5. Therefore, the part 1 is in consonance with the whole 5, by Axiom 1 and the Corollary mentioned.

Hence, the whole 5 is also in consonance with 4, the quadruple of the part 1, by Proposition 4. Therefore, 3 consonances occur.

Proposition 13: The division of a string into 2 parts which are in quintuple proportion to each other is harmonic

This is because:

• the part is 1, the remainder 5. They are therefore in the proportion to each other in which the whole circle is to a consonant part, by Axiom 1 and the Corollary mentioned. Hence they are also themselves in consonance with each other, by Axiom 5
• the part 1 together with the remainder 5 makes up the whole, 6, therefore (by Axiom 1 and its Corollary) the part 1 is in consonance with the whole 6.
• the remainder 5 is to the quarter of the whole circle 6 (that is to say, to in this division) as the whole circle 10 is to its part 3, which is consonant by the Corollary, hence the remainder 5 will also be in consonance with the whole 6, by Proposition 7.

Or, which comes to the same thing, because the remainder 3 is to twice the whole circle 6, that is 12, as a consonant part is to the whole, by the Corollary, hence this remainder, 5, will also be in consonance with 12, twice the whole, by Axiom 5.

Therefore it will also be in consonance with the simple circle, that is to say the whole circle 6 itself, by Proposition 4.

Thus 3 consonances occur.

Proposition 14: The division of a string into two parts, in sesquialterate proportion to each other, is harmonic

This is because:

• the part 2 makes the sesquialterate proportion with its remainder, 3. Terefore, the part is to the remainder as a consonant remainder 2 is to its circle, 3, by the Corollary. Hence this part 2 will also be in consonance with its remainder 3, by Axiom 5
• the part 2 together with its remainder 3 makes a whole 5, but a part 1 and its remainder 4 are in consonance with their whole 5 by the Corollary. Therefore, the whole 5 will also be in consonance with 2, which is twice its consonant part 1, which is our part at this point, or with 2 as half its remainder, 4, by Proposition 4.

The same also follows directly from the axiomatic first part of Proposition 5 because the chord subtended by two fifths is constructible, hence it is also consonant.

Lastly because the remainder 3 of a part 2 is to a quarter of the whole, 5, as a whole circle 12 is to its consonant part 5, by the Corollary, therefore our remainder 3 will be in consonance with the whole 5 by Proposition VII. Therefore three consonances exist.

Hence by Axiom V our part 3 will also be in consonance with our remainder, 3. And because the part 3 together with the remainder 3 makes a whole of 8, hence by the Corollary the part 3 will be in consonance with the whole, 8. Lastly because the remainder, 3, is to 4, the half of the whole, 8, as the whole circle, 3, is to a remainder, 4, which is consonant; or to the fourth part, 2, of the whole 8 as the whole circle 3 is to its part, 2, which is consonant by the Corollary, therefore our remainder will also be in consonance with its whole, 8, by Proposition 7.

Therefore in this case also three consonances occur.

Proposition 16: If a string is divided into two expressible parts, and between them and the whole, that is between the three terms, there is one dissonance, there must also be another dissonance between them.

For the cause of the dissonance will be that either the whole or the part has from that division a number of portions which belongs to an inconstructible figure.

But such a number is allied by consonance neither with any greater number, which belongs to a constructible figure, nor to any smaller than itself, by Axiom 3 and 5 and Proposition 5 and 7.

Therefore the term which is made up of such a number of portions is in dissonance with the two remaining terms in that division;

Thus there are two dissonances at the same time.

To this proposition the following proposition in geometry is similar, that if a straight line is divided into expressible parts, and one of them is incommensurable with a third (not with the whole made up of both of them as in this case) the other must also be incommensurable with the same third part.

Or, if a straight line is divided into parts which are incommensur­ able with each other, each will be incommensurable with the whole.

Proposition 17: If a string is divided into 2 parts expressible in length, and there are 2 consonances between them and the whole, that is, between the three terms, there must also be a third consonance.

For if there are two consonances, since there are not more than three proportions, therefore there cannot be two dissonances. If there are not two dissonances, therefore there is not one either, by the converse of XVI. Therefore all three proportions will be consonances.

Proposition 18: The division of a string into two parts in the proportion of one and 2/3 to 1, or 5 to 3, is harmonic.

For because the proportion of the part 3 to the remainder 5 is the same as that of any remainder 3, which is consonant by the Corollary, to the whole In the same way in geometry, if a straight line is divided into parts which are commensurable with each other, the whole will be commensurable with both the parts.

Proposition 18: The division of a string into two parts which are expressible in length, in which either the whole or one of the parts acquires the number of portions which belongs to an inconstructible figure (where in fact the numbers both of the whole and of the parts have no common factors), it is not harmonic.

It is proved like XVI. For at least two dissonances occur between the three proportions of the three terms, which is contrary to the foregoing Definition. In this case there are three examples. In the first the greater part is seven eighths; in the last the smaller is one ninth; in the middle one, the whole contains seven parts. All are dissonant.

Proposition 19

After the octagonal, no harmonic division of a string is produced.

This is because the subsequent divisions either occur:

• through inconstructible shapes and their stars
  • The parts might be consonant with each other then, yet they are dissonant with the whole, by Axiom 3 or
• through shapes which are constructible by an inappropriate construction, such as the pentekaedecagonl [fifteen-sided figure], and parts which belong to this division are in dissonance with the whole, by the Corollary to Axiom 3; or
• through shapes which are constructible by an appropriate construction, which after the Pentagon are all figures of an even number of sides: see Book 1.

Then the parts which belong to such divisions must consist of an uneven number of portions by the division. If they were represented by an even number, the part would belong not to this division, but to a previous one.

Thus, if a string is divided into 10 and you take 4 or 6 portions, it is just as if you were to divide the string into 3 and take 2 or 3 portions.

Since the part is of an uneven number, the whole is of an even number, and the part can indeed be in consonance with the whole if it is not greater than fivefold (by Proposition V.) But one consonance is not sufficient for a harmonic division, as is evident from the definition.

In that case then the remainder will be in dissonance; for the whole is assumed to have more than 8 portions and the definition of the remainder is that it is greater than half, that is to say greater than 4. Then the smallest remainder in an eightfold division is 5; in those of greater number it is greater than 5.

Then in all the divisions of the string subsequent to the eightfold, the remainders are of uneven number, greater than 3. But uneven numbers greater than 3 belong to inconstructiblefigures, by XLV and XLVII of thefirst Book. Then by Proposition 18 of this Book, these remainders bring about divisions which are not harmonic.