On the Institution of Music (De institutione musica)

And the whole sesquioctave sesquioctave: a ratio of 9:8, which represents a whole musical tone, when broken into proportions, cannot be divided into two equal parts; just as no portion of its kind—that is, the superparticulars original: superparticulares; a ratio where the numerator is the denominator plus one (e.g., 3/2, 4/3, 9/8). These cannot be divided into two equal rational ratios—can be. The Diatesseron Diatesseron: the interval of a fourth also consists of two tones and a semitone. Furthermore, there are two semitones, a minor and one even smaller. The Diapente Diapente: the interval of a fifth is filled by three tones and two minor semitones. It does not reach six whole tones in any way. All these things I will explain later through both the reasoning of proportions and the judgment of the ears. And so much for this.

What a Musician is. Chapter 34.

Now we must consider that every art and every discipline naturally possesses a more honorable character through reason than through the craftsmanship that is exercised by hand and manual labor. For it is much greater and nobler to know what one does than to merely perform what one knows. Indeed, physical craftsmanship serves like a slave, while reason rules like a mistress. And unless the hand immediately carries out what reason permits, the effort would be in vain. Therefore, the science of music is as much more excellent in the knowledge of reason than in the work of performing and acting, as the mind surpasses the body. For that which lacks reason lives in servitude, while reason commands and leads toward what is right. Unless its command is obeyed, the work—lacking reason—will stumble. Hence it follows that the contemplation of reason does not need the act of working, but the works of the hands are nothing unless guided by reason. Indeed, how great the glory and merit of reason is can be understood from this: that other physical craftsmen, so to speak, do not take their names from their discipline, but rather from their instruments. For the citharedus original: citharedus; a lyre-player is named from the cithara, the tibicen original: tibicen; a pipe or flute player from the tibia, and others are named from the titles of their instruments.

He, however, is a musician who assumes the science of singing through the power of reason, not by the servitude of work, but by the command of contemplation. We see this same principle in the works of buildings and wars, specifically in the naming of terms. For buildings are inscribed with the names, and triumphs are celebrated in the names, of those by whose command and reason they were established, not those by whose labor and completion they were finished. There are, therefore, three classes concerned with music. One class is that which performs with instruments. Another composes songs. The third judges the work of the instrument and the song. But that class which is occupied with instruments and spends all its effort there—such as the lyre-players and those who prove their skill on the organ and other musical instruments—is separated from the understanding of musical science, because they serve, as was said, and bring nothing of reason, being devoid of all contemplation. The second class of those dealing with music is that of the Poets, which is led to song not so much by contemplation and reason as by a certain natural instinct. And for this reason, this class is also to be separated from the true science of music. The third class is that which possesses the skill of judging, so that it can evaluate rhythms, melodies, and the whole song. Since this is entirely placed in reason and contemplation, this shall be properly assigned to Music. He is a musician who possesses the faculty of judging modes and rhythms, the kinds of melodies and their mixtures, and everything which must be explained later, as well as the songs of the poets, according to the contemplation or reason proposed and suitable for music...