De institutione musica V.2: De vi armonicae et quae sint eius instrumenta iudicii et quonam usque sensibus oporteat credi.
But of these we shall speak a little later. Now we must say what the power of harmonics is, the subject on which we have completed four books in undertaking to treat it. Its nature and power, however, we have put off to be set forth in the course of this fifth volume. Harmonics is a faculty weighing the differences of high and low sounds by sense and by reason. For sense and reason are, as it were, the instruments of the harmonic faculty. Sense perceives something confused and only approximately such as the thing it perceives; reason, however, judges the whole and pursues the finest differences. So sense finds things confused and near to the truth, but receives their completeness from reason; reason itself finds the completeness, but receives from sense a confused and near likeness of the truth. For sense conceives nothing of completeness but comes only to what is near, while reason judges. It is as if someone should draw a circle by hand: perhaps the eye would judge it truly a circle, but reason in no way understands it to be what it pretends to be. And this is because sense is concerned with matter, and grasps in it forms that are as fluid, imperfect, undetermined and unpolished as matter itself. Hence confusion follows sense also; but mind and reason, since matter does not detain them, look upon the forms they perceive apart from their association with the subject, and so completeness and truth attend them, and afterwards reason either corrects or fills up whatever in sense is at fault or lacking. Perhaps, however, what sense recognizes not completely but confusedly and with less truth, like an unwary judge, contains little error in each single instance, but the errors, gathered together, are multiplied into a sum and so make a very great difference. For if sense should judge two little voices to be a tone apart when they are not, and again should think a third to be a tone distant from one of them, though the distance is not a true and complete tone, and likewise should judge a tone between the third and the fourth, and err in this also, the difference not being a tone, and should think the fifth to be a semitone from this fourth, and again not judge truly and completely; in each single case perhaps the error would seem small, but what sense left short in the first tone, and what was wrong in the second and third tones and in the fourth semitone, gathered into one and collected, will bring it about that the first voice to the fifth voice does not contain the diapente consonance, which would have had to be the case if sense had judged the three tones and the semitone completely. What was therefore seen too little in the single tones appeared plainly when gathered in the consonance. And so that it may be seen that sense gathers confused things and in no way rises to the completeness of reason, let us consider it thus. To find a line greater or smaller than a given line is no difficulty for sense. But when a measure is proposed, so that it must find a line so much greater or so much smaller, sense will not do this by its first conception, but the skilled invention of reason will. Or again, if it is proposed to double a given line or to cut it in half, this perhaps, though somewhat harder than finding confusedly a greater or smaller, can nevertheless be established by the invention of sense. But if it is commanded that a triple of the proposed line be set down, or that a third part be cut from it, or that a quadruple be established or a fourth part cut off, would it not be impossible for sense unless the completeness of reason comes to it? This is because in the course of the process room grows for reason, but fails sense. For if someone is ordered to take away the eighth part of the proposed line, or is compelled to give its octuple, he is forced to take half of the whole, and the half of the half, so that it is the fourth, and the half of the fourth, so that it is the eighth; and again the double of the whole, and the double of the double, so that it is the quadruple, and the double of the quadruple, so that it is the octuple. So in so great a number of operations sense accomplishes nothing, since all its judgment, sudden and lying on the surface, does not unfold completeness and perfection. Therefore the whole judgment is not to be given to the sense of the ears, but reason also must be brought in, to govern and temper the erring sense, so that sense, slipping and failing, may lean upon it as on a staff. For as each art has certain instruments, with some of which it sketches something confusedly, such as the chisel, and with others grasps what is exact, such as the compass, so also the harmonic faculty has two parts of judgment: one by which it grasps by sense the differences of the voices set before it, the other by which it considers the exact measure and standard of those differences themselves.