The Great Art of Consonance and Dissonance

within three octaves—that is, 22 notes—they can be changed with their intervals 12,893,126,400 times.
The marvelous power of combination

In like manner, you see that an interval of 5, or 5 numbers, can be changed 4,683,336,365,740,320,000 times. And so it is for the rest. But since these things may seem incredible to one unskilled in Arithmetic (Arithmetices): The study of numbers and their properties, here applied to the "science of music.", and because this combination could not be exhausted by 10,000 writers even if they labored for 1,000 years, it seemed appropriate to provide examples here in smaller numbers, so that the truth of the matter may be recognized through expanded mutations of combinations; for the method of proceeding is the same in small numbers as in large ones.

Examples in smaller numbers.

Suppose, therefore, someone wishes to know how many times, out of five proposed notes, only two notes can be changed so that they are always different, and the melody always produces different intervals; proceed as follows: let the numbers be placed in their natural order, as is evident in the example, in which the first number shows five notes, or the whole set. The second column, using Roman numerals, signifies the parts of the whole—namely, how many times the parts of the five notes can be changed within the interval of a fifth. The third column signifies the mutations themselves. Thus you see that a single note can be changed only five times, two notes twenty times, three notes sixty times, and four notes one hundred and twenty times. Finally, five notes can be changed as many times as the ordinary combination of the first table prescribes. This table is made very easily in this way: multiply 4 by 5, and the product 20 will give the mutation of two notes. Again, multiply 3 by 20 and you will have 60 mutations of 3 notes. Again, 2 times 60 will produce 120 mutations of four notes, and 1 times 120 will give the ordinary combination of the matter as stated, but let us show the matter with examples.

Mutations of two notes out of five—that is, within the interval of a fifth—such that they are never found twice upon the same pitch.

Two rows of musical notation on five-line staves illustrate musical permutations of note pairs within the range of a fifth. The first row contains 14 measures, numbered 1 through 14. The second row contains 6 measures numbered 15 through 20, followed by 5 measures showing "unison" note pairs (two notes on the same pitch) numbered 1 through 5. This visualizes how two notes can move relative to one another within a fixed five-note scale.

In this example, you see that the positions of two notes can be changed twenty times, and no more, if the notes are taken so that the intervals are always different. If, however, we wish to know how many times these two notes can be changed if two are placed on the same degree That is, a "unison" where both notes share the same pitch., nothing else is to be done except to add 5; or, what is the same thing, the number five original: quinarius is to be multiplied by itself to make 25. For if, out of 5, you establish two notes twice on the same degree, as is evident in the last 5 mutations of this diagram, these added to the previous 20 will give exactly 25 mutations of 2 notes out of 5. The reasoning is clearly evident from the positioning of the notes.

Since, however, in this place two intervals are always taken—one ascending original: ἀνάβατον (anabaton) and the other descending original: κατάβατον (katabaton), that is, in one of which the notes go up and in the other they go down—these are nevertheless the same intervals, such as 4 and 5, 3 and 6, 2 and 7, 1 and 8, and so on for the rest. It is now asked how we can know how many times two notes precisely within the interval of a fifth original: diapente...