Arithmetic, Geometry, and Music

arithmetic and harmonic mediation are alternated between them, and concerning their generations. Chapter 50.
Concerning three mediations which are contrary to the harmonic and geometric. Chapter 51.
Concerning four mediations which later authors added to complete the limit of the decade. Chapter 52.
The arrangement of ten mediations. Chapter 53.
Concerning the greatest and perfect symphony which is extended by three intervals. Chapter 54.

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The Second Book Begins.

How every inequality may be reduced to equality. Chapter the first.

It is digested in the reasoning of the previous book how the entire substance of inequality proceeds from equality, which is the prince of its genus. But just as there are elements of a voice, from which all things are primarily composed and into which they are again resolved when a resolution is made—just as letters are the elements of articulate speech, from which a combination of syllables proceeds and into which it again terminates at the extremes—so too does sound hold the same power in musical matters. Now, we do not ignore that four bodies make up the world; for, as it is said, souls are generated from water, earth, and fire, but in these four elements, the final resolution of them is made again. Therefore, just as we see that all species of inequality proceed from the margin of equality, let all inequality be resolved by us back to equality as if to a certain element of its own genus. This is gathered again by a triple investigation. The art of resolving them is given to any three terms that are indeed unequal but proportionally constituted—that is, so that the middle term holds the same power of proportion to the first as the extreme does to the middle in any ratio of inequality, whether in multiples, or in superparticulars, or in superpartients, or in those which are procreated from these, that is, in multiple superparticulars or in multiple superpartients—the same and unique ratio will stand undoubtingly. For, having proposed three terms, as has been said, arranged in equal proportions, we shall always subtract the last from the middle, and we shall place that very last term as the first; what remains from the middle, we shall make the second. From the third sum of the proposed terms, we shall take away one first and two seconds—those which remained from the mediation—and what is left from the third sum, we shall constitute as the third term. You will see, therefore, that upon doing this, the sums return to a lesser mode and the comparisons and proportions are reduced to a more primary relationship. For instance, if there is a quadruple proportion, it returns to a triple, then to a double, and then to equality. And if it is a sesquiquartus a ratio of 5:4 superparticular, it returns to a sesquitertius a ratio of 4:3, then to a sesquialter a ratio of 3:2, and finally back to three equal terms. We shall teach this, for the sake of example, only in the multiple proportion. The diligent person, experiencing this in other species of inequality as well, will be helped by the same ratio of precepts. For let three quadruple terms be constituted to one another.

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a Take away, therefore, the lesser from the middle, that is, 8 from 32; let 24 remain, and you will place 8 as the first term, and 24 as the second, which is what remained from the middle, so that these two terms are 8 and 24. From the third, however, that is, 128, take away one first, that is, 8, and two seconds, which are the remainder—that is, twice 24—and 72 remains. With these terms disposed, the triple proportion has been reduced from the quadruple proportions. For these are the terms.

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e And if you do the same from these very ones, the comparison will return again to the double. For place the first as equal to the lesser, that is, 8; take away the first from the second, 16, and 8 will remain. But from the third, that is, 72, take away the first, that is, 8, and two seconds, that is, twice 16, and the remaining part will be 32. With these placed,

the relationship is reduced to double proportions.

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i Truly, if the same is done from these, we shall equate the whole matter to the sums of equality. For place the first equal to the lesser, that is, 8, and take the 8 from the 16; 8 remains. With these disposed, having taken the first from the third, that is, 32—that is, 8—and two seconds—that is, 8s—8 remain. With these disposed, the first equality falls to us, as the subordinate sums teach.

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b If anyone, therefore, directs their mind to other species of inequality, they will find the same correspondence without hesitation. Wherefore, it must be pronounced and doubted with no trepidation that, just as the unity of quantity established by itself is the beginning and element, so also the equality of quantity related to something is the mother. For we have demonstrated that from here both its first procreation would be, and into it is the final resolution again.

Concerning finding in each number how many numbers of the same proportion can proceed: their description, and the exposition of the description. Chapter 2.

e There is, however, in this matter a profound and wonderful speculation, and as Nicomachus says, an ennoeofaton thought-provoking theorem, extending to the Platonic generation of the soul in the Timaeus, and to the intervals of musical discipline. For there we are ordered to produce and extend three or four sesquialter 3:2 ratios, or any number of sesquitertius 4:3 proportions, and sesquiquarta 5:4 comparisons. We are ordered to extend them according to a proposed order, often continued. However, by this labor—although it may always be very great and most frequently happens in an interpolated manner—we must investigate by this method in how many numbers there can be superparticulars. For all multiples will be the princes of so many proportions similar to themselves, as far as they themselves stand distant from unity. But what I say "similar to themselves" is such that the multiplicity of the double, as has been shown above, creates sesquialters, and the triple is that of two sesquitertii, and the quadruple that of sesquiquarti. Therefore, the first double will have one single sesquialter; the second, two; the third, three; the fourth, four; and according to this order, the same progression is made to infinity. Nor can it ever happen that there is a diminution from the number of proportions, or an equal location from unity. Therefore, the first double is the binary number, which receives one single sesquialter, that is, the ternary. For the binary compared against the ternary effects a sesquialter proportion. The ternary, however, because it does not receive a half, is not another number to which it could be compared in a sesquialter ratio. The quaternary number, however, is the second double. Therefore, this one precedes two sesquialters. For the number six is compared to it, and to the six, because it has a half, the nine; and there are two sesquialters—namely 6 to 4, and 9 to 6. The nine, however, because it lacks a half, is excluded from this comparison. The third double, however, is 8. Therefore, this one precedes 3 sesquialters. For the number twelve is compared to it, 18 to the twelve, and to the 18, again, 27. But 27 lacks a half. It is necessary that the same also happen in the following ones, which we have placed below with their own arrangement. For always, by a certain divine and not human constitution, this occurs in speculations: that whenever the last number is found, which in the place of the double from unity becomes even, it is such that it cannot be divided or cut into halves.

i The same happens also in triples. For from them, sesquitertii are procreated. For since the first triple is the ternary number, it has one sesquitertius, that is, 4.