On the Institution of Music (De institutione musica)

Multiplicity, therefore, since it has no end to its growth, best preserves the nature of the infinite. Superparticularity original: "superparticularitas"; a ratio where the larger number contains the smaller plus one part of it, e.g., 3:2 or 4:3, however, since it diminishes into the infinitely smaller, preserves the property of continuous quantity. For the smaller always contains the smaller plus either its half part, its third, its fourth, or its fifth. For the part named by the larger number is itself smaller. For since a third is named from three, and a fifth from five, and since five exceeds three, the fifth is diminished more clearly than the third. Superpartience original: "superpariens"; a ratio where the larger number contains the smaller plus more than one part of it, e.g., 5:3, however, already departs in a certain way from simplicity. For it has two or three or four parts in addition, and departing from simplicity, it overflows into a certain plurality of parts. Again, all multiplicity contains itself in integrity. For the double contains the whole smaller part twice; the triple likewise contains the whole smaller part three times; and the others in the same manner. Superparticularity, however, preserves nothing of the whole, but exceeds it by either a half, a third, a fourth, or a fifth. But then it effects division into single and simple parts. Superpartient inequality neither preserves the whole nor takes away single parts. And therefore, according to the Pythagoreans Followers of Pythagoras who believed the universe was governed by numbers, it is least applied to musical consonances. Ptolemy Claudius Ptolemy, the Greco-Roman astronomer and music theorist, however, places even this proportion among the consonances, as I shall show later.

Which proportions are adapted to which musical consonances. Chapter VII.

This, however, ought to be known: that all musical consonances consist either in double, triple, quadruple, sesquialter 3:2 ratio, or sesquitertial 4:3 ratio proportions. And they are noted as follows: that which is in a sesquitertial ratio is a diatesseron the interval of a perfect fourth in sounds. That which is in a sesquialter ratio—

—is called a diapente the interval of a perfect fifth in sounds. That which is in a double proportion is a diapason the interval of an octave in consonances. The triple is a diapente and diapason combined an octave plus a fifth. The quadruple is a double diapason two octaves. And now indeed let this be said generally and without distinction; later, however, the whole logic of proportions will shine forth.

What sound is, what an interval is, and what consonance is. Chapter VIII.

Every falling of the voice is therefore emmeleis Greek: "in tune" or "musical", that is, fit for melody in a single tension. We do not wish to define "sound" in a general sense now, but that which in Greek is called phthongos a musical note of definite pitch, named by likeness to "logos," that is, properly of voices. An interval is the distance between a high and a low sound. Consonance is the mixture of a high and low sound reaching the ears sweetly and uniformly. Dissonance, however, is a harsh and unpleasant percussion of two sounds mixed together coming to the ear. For while they wish to be mixed together and each strives in a certain way to arrive whole, and one gets in the way of the other, it is transmitted unpleasantly to the sense. Not every judgment should be given to the senses, but more should be trusted to reason.

In which the fallacy of the senses is discussed. Chapter IX.

But concerning these things we propose this: that we do not give every judgment to the senses, although every principle of this art is taken from the sense of hearing. For if there were no hearing, no discussion of voices would exist at all. But hearing holds the principle, in a way, as a substitute for an admonition. The final perfection and the power of recognition, however, consists in reason. Which, holding itself to certain rules, never falls into any error. For what should we say about the error of the senses, when the power of perceiving is neither the same for all, nor always equal for the same man?