Book VIII. On the Miraculous Art of Music-Making.
Signature: A 2

Therefore, led by these considerations, I wished to make this miraculous art of music-making Original: "Musurgia Mirifica," Kircher’s term for his system of automated musical composition. available to the public, lest I should seem to have omitted anything that might contribute to the propagation of divine worship and honor in this endeavor. But truly, lest we keep the reader's eager mind in suspense any longer, let us approach the task with favorable omens. However, so that we do not proceed in this new and unheard-of art without method Original: ἀμεθόδως (amethódōs), we shall proceed with that order and method which this miraculous art of music-making rightly claims for itself.

OF THE MIRACULOUS ART OF MUSIC-MAKING

PART ONE.

COMBINATORIAL MUSIC-MAKING

MUSURGICAL LEMMA I.

A "Lemma" is a preliminary proposition or a "stepping stone" theorem used to prove a larger point later on.

Whenever a ratio—for example, A to B—is said to be "composite" (as if made from the ratios A, E, and I), I say that this ratio is composed from those same parts regardless of the order in which they are placed. That is, between two terms A and B, it is permissible to continue these ratios as many times as they can change places among themselves. For example: let as many numbers be taken in a natural series as there are proposed things. When these numbers are multiplied by each other, they produce the total sum of mutations In this context, "mutations" refers to what modern mathematics calls permutations—the different ways a set of items can be arranged.; thus, for two things, two mutations result; for three things, six; for four, 24; for five, 120.

A B 1 & B A 2

1. Let there be, for example, two proposed things, A and B. I say they can be changed twice and no more, for each will occupy the first place only once, as is clear in the margin; for these numbers multiplied together (1 x 2) produce two.

2. But three things can be varied in six ways. For these numbers—1, 2, and 3—multiplied in order, produce six. That is to say, 1 times 2 makes 2, and 2 times 3 makes 6, etc. The reason for this is that each thing will hold the first place once, and the remaining two can be changed twice among themselves, as is clear in the following example.

3. Suppose there are four things to be changed among themselves; therefore, multiply four by the number produced from the previous multiplication—namely, by 6—and 24 mutations will be produced. This is because these numbers 1, 2, 3, and 4, multiplied in order, produce 24. For each thing will hold the first place only once, and the remaining three can be varied six times among themselves. Indeed, inspect the present example.