Of the Various Species of Numbers

The Ten Species of Number

It would be both lengthy and beside the purpose to want to recount one by one all the various sorts of numbers and to wish to say of each what it is. But because the Musician considers certain species among them, I shall speak only of those that are relevant to our purpose, leaving aside the others as useless to this science. We shall say therefore that the species of numbers which it is necessary to know for the understanding of this Treatise, and which pertain to the Musician, are ten — that is: Even, Odd, Evenly Even, Prime and Incomposite, Composite, Mutually Prime, Communicant, Square, Cubic, and Perfect.

Even numbers are those which can be divided into two equal parts — such as 2, 4, 6, 8, 10, and others like them.

Odd numbers are those which cannot be divided into two equal parts — rather, of necessity one part exceeds the other by Unity — and these are 3, 5, 7, 9, 11, and the others.

Evenly Even numbers are those whose parts can be divided into two equal parts, down to the point where one arrives at Unity — from which they begin to have their being, continuing in double proportion to infinity — such as 2, 4, 8, 16, 32, 64, and the others.

Prime and Incomposite numbers are those which cannot be measured or divided by any other number except Unity — such as 2, 3, 5, 7, 11, 13, 17, 19, and others like them.

Composite numbers are those which are measured and divided by other numbers — and these are 4, 6, 8, 9, 10, 12, and the others proceeding to infinity.

Mutually Prime numbers are those which cannot be measured or divided except by Unity, the common measure of every number — such as 9 and 10, which are composite numbers, but when compared together are called Mutually Prime because they have no other common measure between them that measures and divides them except Unity. And these are found of three sorts: either both are composite, as just shown; or both are prime, such as 13 and 17; or one is composite and the other prime, such as 4 and 9.

Communicant numbers are those which are measured and divided by a number other than Unity, and neither of them is prime with respect to the other — and these are found of three sorts: either both are even, such as 4 and 6; or both are odd, such as 9 and 15; or they are one even and one odd, such as 6 and 9.

Square numbers are those which arise from the multiplication of a smaller number by itself — such as 4, 9, and 16, which arise from 2, 3, and 4, which are the square roots of such numbers.

Cubic numbers are those which arise from the multiplication of any number by itself, and the product again multiplied by that same number — such as 8, 27, 64, and similar ones, which arise from the multiplication of 2, 3, and 4 in themselves, which are called the Cubic Roots of such numbers; and the products again multiplied by them — as multiplying 2 by itself produces 4, which multiplied again by 2 gives 8, called a Cubic Number, of which 2 is the root.

Perfect numbers are those which are constituted from their own parts, and are Even and Composite numbers, always ending in 6 or 8 — such as 6, 28, 496, and the others — since their parts taken together and summed render precisely their whole. As the parts of the Senary, which are 1, 2, and 3, which divide it wholly — Unity first into six parts, the Binary then into three, and the Ternary into two parts — which parts summed together render completely the Senary itself.

The Senario and the Consonances

These are therefore the species of numbers necessary to the Musician — for their knowledge serves in Music for the investigation of the properties of its proper subject, which is the Harmonic or Sonorous Number, contained in the first perfect number, which is the Senario, as we shall see. In which number are contained all the forms of the simple consonances possible to find, apt to produce harmonies and melodies.

For the Diapason, which arises from the Duple proportion — the true form of such a consonance — is contained between the terms 2 and 1. And such proportion the Musician takes as the whole divisible into many parts. Then the Diapente is contained between the terms 3 and 2 in the Sesquialtera proportion; the Diatessaron between 4 and 3 containing the Sesquitertia proportion. And these are the two greater parts which arise from the division of the Duple, or of the Diapason. The Ditone then is contained between 5 and 4 in the Sesquiquarta proportion; and the Semiditone in the Sesquiquinta between 6 and 5. And these two parts arise from the division of the Sesquialtera, or of the Diapente.

And because all these are parts of the Diapason, or of the Duple, and arise through harmonic division — I therefore call them simple and elementary: since every consonance, or interval however small that is less than the Diapason, arises not through the addition of many intervals placed together, but through the division of the Diapason itself. And those which are greater are composed of it and of one of the named parts, or of many Diapasons added together, or of two parts — as their names make manifest. For from the Diapason and the Diapente placed together is composed the Diapasondiapente, contained by the Triple proportion between 3 and 1. The Disdiapason, composed of two Diapasons, is contained by the Quadruple proportion between 4 and 1. The major Hexachord and likewise the minor arise from the joining of the Diatessaron with the Ditone and the Semiditone respectively. But leaving off for now saying more of these and of the others, at another time we shall reason of them more fully.

Moses and the Senario

From the things we have said, we can therefore understand for what reason the great Prophet Moses, in describing the great and marvelous fabric of the world, chose the Senary number — God having had no need of time in His operations. For, as one who was the perfect master of every science, knowing through the work of the divine Spirit the harmony that was enclosed in such a number, and knowing that from visible and apparent things we come to know the invisible things of God, His omnipotence and His divinity — he wished by means of such a number to express at once and together show the perfection of the work, and within it the enclosed harmony, which is the confirmation of its being: without which it would in no way endure, but would be entirely annulled, and all things returning to their first being (if one may so speak), there would once again be seen the confusion of the ancient Chaos. The Holy Prophet therefore wished to manifest the mystery and perfect work of the Lord accomplished without any time, by means of the Senario — from which number how many things both of nature and of art are comprehended, from what follows we shall be able to know.