The Harmonic Means, and the Trinity of Consonant Sounds

I superfluously define harmonic proportion as a proportion wherein 3 numbers placed in their natural order, the amounts by which one of a pair of neighbors exceeds the other are in the same proportion as the outer numbers.

For example, in 3, 4, 6

• 6 is twice the smallest 3
• The difference between the greater neighbors 6 and 4 is 2
• The difference between the lesser neighbors 4 and 3 is 1

Thus, the differemce between the greater is twice that of the lesser.

Harmonious proportions are called ‘musical’ when they are applied to ethics and politics.

How can we find musical proportions numerically?

Given two numbers having no common factors, which contain the proportion both of the outer numbers (of 3 which are to be musically combined according to the scheme of the ancients) and of the differences of each from the mean. Multiply each by itself and both by each other.

Of the 3 results add together the two smaller for the smallest of the numbers which are to be found; add together the two greater to find the greatest; and double the mean to find the musical mean of the ancients.

Let there be 3 numbers in the musical proportion of the ancients. The outer ones are in the proportion of 3 to 5.

• 3 * 3 = 9
• 3 * 5 = 15
• 5 * 5 = 25

Therefore, the results are 9, 15, and 25.

• 9 + 15 = 24
• 15 + 25 = 40
• 15 * 2 = 30

Therefore the three required numbers are 24, 30, and 40.

Their differences (of the outer numbers from the mean) are 6 and 10.

As 3 is to 3, so 24 is to 40, and 6 is to 10.

In the lowest terms which have no common factors, 12, 13, 20.

I think that this is truly a harmonic proportion because not only is the proposed proportion between 3 and 5 harmonic, by the Corollary of Proposition 8, but also the mean number found, 15, makes consonant proportions with the outer numbers 12 and 20 by the same Corollary.

But this does not always occur.

Every time that the arithmetic mean, between two numbers proposed on this basis, marks out proportions with the outer numbers which are dissonant, there also emerge from this operation three numbers in a proportion which is in truth not harmonic, though the two originally proposed taken on their own form a proportion which is harmonic.

That occurs in the case of 1 and 6, of 1 and 8, of 3 and 4, of 4 and 5, of 5 and 6, of 2 and 5, of 3 and 8, and of 5 and 8.

For instance, between 2 and 5, that is 4 and 10, the arithmetic mean is 7, which is not harmonic, because 7 is not consonant either with 4 or with 10, by Proposition V.

Then operate according to the rule.

The resulting numbers will be 14, 20, and 35, with the excesses 6 and 15. Thus 20 ought according to the ancients to be declared the harmonic mean, because as 14 is to 25 (that is 2 to 5), so 6 is to 15.

But the ears completely repudiate 20^35 (in other words 4:7) and 14:20 (in other words, 7:10).

Therefore in the harmonic divisions of Chapter 2, the number of means emerging is the same as the number of divisions, minus one.

Also “mean” in those sections is in fact taken in its stricter sense, that in a string harmonically divided into unequal parts, it is the greater part, or the number expressing it. Thus 2 is the harmonic mean between 1 and 3; 3 is that between 1 and 4 and 2 and 5; 4 between 1 and 5; 5 between 1 and 6 and 3 and 8.

Apart from these there are also some other means, which are not subject to this law of the division of the whole string into two parts, but are included in our general definition, and divide not a single string, as in the previous Chapter, but the proportion of strings, into lesser consonant proportions.

First, all proportions greater than double are resolved into their components, by the extraction of the double proportion. Thus 1:24 is made up of four doublings (that is, from multiplying by sixteen) and multiplying by one and a half.

Hence as harmonic means under this heading 2, 4, 8, and 16 are interposed in this way between 1 and 24, taking the multiplication by 16 first, or 12, 6, 3, and 2 in this way, taking one doubling first and three later; for it can be done in various ways.

Secondly, a double proportion is resolved into the following consonances: 3:4 and 2:3, or 3:4 and 4:5 and 5:6, or 4:5 and 5:8, or 5:6 and 3:5. Lastly the sesquialterate proportion, 2:3, is resolved into 4:5 and 5:6. Similarly 5:8 is resolved into 5:6 and 3:4, and 3:5 into 3:4 and 4:5.

Therefore the 3 proportions 3:4 and 4:5 and 5:6 are the smallest of the consonances, that is, they are immediate, or without the harmonic mean, that is to say, they are consonant elements of the other proportions.

It follows that in one double proportion there can be 2 means, which are also consonant with each other, and in six ways. For because the double proportion has three smallest consonant elements, their order can be varied in six ways. For 3:4 is either in the first position on the smaller string, or in the middle place, or in the last; and in any given case, of the remaining elements either the greater with respect to the smaller string is 4:5, or the smaller is 5 ‘6.

The individual cases have to be expressed in individual sets of 4 numbers, as shown in the following table.

Then since strings in double proportion are in identical consonance, it is impossible for there to be between them in any one case more than two means, which are consonant both with each other and with their doubles.

Hence arose that celebrated observation of the musicians, who wonder that all harmonies can be accomplished by three notes.

For however many notes are assembled together additionally, each of them comes to the same as one of the three by the identical consonance of double proportion. For although one consonance emerges from strings of all these sizes —3, 4, 5, 6, 8, 10, 12, 16, 20, and 24 —yet every thing after the strings of length 3, 4, and 5 comes to the same as one of the following by identity of consonance: as 6 comes to 3 and 8 to 4 and 10 to 5; similarly 12 comes to 6 and 3; 16 to 8 and 4; 20 to 10 and 5; 24 to 12, 6 and 3.

The cause of this fact different people seek vainly in different ways: some in the threefold dimensionality of the perfect quantity, or body, as it appears in length, breadth and depth; some in the perfection of the threefold number; others in the revered Trinity itself of the Divinity.

All, I say, vainly. For neither does three-dimensional quantity enter into this affair, since we have learnt that the origin of the harmonic proportions is in plane figures. Also three-dimensional quantity is greatly different, as far as knowledge of it is concerned, from two-dimensional, inasmuch as the former employs two mean proportionals, and in knowing them it is impossible for there to be any confusion; nor can there be any power in a number, insofar as it is considered as a counting number; nor, furthermore, is the origin of this trinity immediately from the Divine Being, causing it by imitation, as it has been made clear above that the cause of this matter is in the basic principles which were expounded, which in no way imply any particular number of notes on their own, but by fitting together individual notes to individual notes harmonically, and thus while doing something else accidentally produce something similar to the Divinities on account of the number’s being the same.

The same thing also happens in very many other matters.

In short, this threefold number is not the efficient cause of the harmonies.

Instead, it is an effect, a concomitant of the harmony effected.

It does not give form to the harmonies, but is a splendor of their form. It is not the matter of the harmonic notes, but is an offspring begotten by material necessity. It is not the end “for the sake of which,” but it is an eventual product of the work.

Lastly, nothing results from harmony itself, but it is a secondary entity of the reason, and a concept of the mind, by second intention.

For it is no more important to ask why only three notes are harmonically consonant, and a fourth and all others come back somehow or other to the same thing by the consonance of double proportion, than why there are only six pairs in any given octave, six forms of triple consonances.

For as this sixfold does not come from the six days of Creation, so neither does that threefold depend on the Trinity of persons in the Deity.

But since the threefold is common to divine and worldly things, whenever it occurs the human mind intervenes and knowing nothing of the causes marvels at this coincidence.