Hoc in volumine haec opera continentur

ngun
militur
thon
dalis
alis
modi
He
ralis

benti
goge,
me
no to
a uxe

et ph
d iux

The original text contains marginalia that appear to be fragmented mnemonics or indices related to the musical terms. These are partially illegible in the manuscript.

It is necessary for a strike original: "plagam" and a motion to exist before a sound can be heard; for all phthongi musical tones are made by some sort of strike, and no strike can occur unless a prior motion has preceded it. Some motions are denser, while others are rarer. Indeed, the denser motions produce sharper tones, while it is necessary for the rarer motions to be flatter because they consist of denser and more numerous motions. The flatter tones, however, occur because they consist of rarer and fewer motions. So, it should be said that tones consist of portions, since they are understood through addition and subtraction: they are sharper by the subtraction of motion, and flatter by the addition of motion, as is required for the work. Since all things consisting of particles are said to be compared to one another by the ratio of numbers, it is necessary that they be spoken of in terms of numerical ratio to one another. Some numbers are spoken of in a multiple ratio, others in a superparticular ratio, and others in a superpartient ratio. Thus, it is necessary that tones also be spoken of in such ratios to one another. Indeed, some of these, both multiples and superparticulars, are spoken of by a single name in relation to one another.

We also recognize that some tones are consonant, while others are dissonant. Consonants produce a single mixture from both, while dissonants do not. Since these things are so, it is equitable that consonant tones (since they make a single mixture from both voices) be spoken of by the same name as the numbers that relate to them, whether they are multiples or superparticulars.

If the interval b c is a multiple, and b is a multiple of c, and we create a ratio such that c is to b as b is to d, I state that b is also a multiple of c the author refers to a geometric progression of ratios here. For since b is a multiple of c, then c measures b. And since it was established that as c is to b, so is b to d, then c also measures d. Therefore, d is a multiple of c.

If an interval composed twice makes a whole that is a multiple, the interval itself will also be a multiple. Let there be an interval b c, and let us create a ratio such that c is to b as b is to d, and let d be a multiple of c. I claim that b is also a multiple of c. For since d is a multiple of c, then c measures d. And we have learned that if there are any numbers that are proportional, and if the first measures the last, it will also measure those situated in the middle. Therefore, c measures b. Thus, b is a multiple of c.

No single proportional number, nor several, will fall within a superparticular interval. For if a superparticular interval b of c is created, and the smallest numbers in that same ratio are b, c, d, f, h, these are measured only by the unit as a common measure. Subtract a value equal to h, which is K. F. The remainder is a unit. Therefore, the superparticular is d, f, and the excess is d, K. The common measure of d, f, and h is the unit. Therefore, d, K is not a common measure. Thus, no middle number will fall between d, f, and h, for the falling value would be smaller than d, f and larger than h, and therefore the unit would be divided, which can happen in no way. Therefore, no number will fall into d, f, h. Indeed, in the smallest ratios, no means will fall. But none falls into d, f, h, nor will any fall into b, c.

If an interval that is not a multiple, when composed twice, makes a whole, it will be neither a multiple nor a superparticular. For let there be a non-multiple interval b, c, and let us create a ratio such that c is to b as b is to d. I state that d is neither a multiple of c nor a superparticular. For if we assume first that d is a multiple of c, we have already learned that if an interval composed twice makes a whole that is a multiple, it is itself a multiple.