On the Institution of Music (De institutione musica)

In what way hearing occurs. Chapter 14.

Now let us discuss the manner in which hearing takes place. A phenomenon occurs in the air similar to what happens in ponds when a stone, thrown from a distance, sinks into marshes or quiet waters. First, the stone gathers the water into a very small circle. Then, it spreads the waves into larger spheres, and so on, until the motion, exhausted by the surging waves, comes to rest. The later and larger wave is always spread by a weaker pulse. But if there should be something that can block the spreading waves, that motion immediately returns and is rounded back toward the center from which it started, following the same patterns. In this same way, then, when air is struck, it creates a sound: it pushes the next bit of air and excites a somewhat rounded wave of air. Thus, the sound is spread and strikes the hearing of all those standing around. The voice is more obscure to the person who stands further away, because the wave of air reaches them with a weaker pulse.

On the order of the theories, that is, the speculations. Chapter 15.

Diatonic original: "i dia" Common original: "commum" Chromatic original: "chromati" or Enharmonic original: "l' armoniam"

These things having been proposed, it seems necessary to say by what means and through which elements every song is woven. These are the matters considered by the discipline of discovering harmony original: "armoniae". These elements are the Diatonic, the Chromatic, and the Enharmonic These represent the three "genera" or types of musical scales used in ancient Greek music. We must explain these only after we have discussed the strings likely referring to the strings of the lyre or monochord and how the number of strings increased to the plurality that exists now. This will be done if we first enumerate the proportions by which musical consonances original: "symphoniae" are mixed.

On consonances, the tone, and the semitone. Chapter 16.

The octave original: "Diapason" is the consonance that is created in a double ratio 2:1, as is shown here. The fifth original: "Diapenthe", truly, is that which consists of these numbers 3:2. The fourth original: "Diatesseron", truly, is...

The tone, truly, revolves in this sesquioctave 9:8 ratio, but in this ratio there is not yet a consonance. The octave, however, is composed of the fifth and the fourth The text likely meant to say that adding a fifth (3:2) and a fourth (4:3) results in an octave (2:1). The transcription "tripla opacit" is likely a scribal error for the math of the fourth arranged in this way: the double octave original: "bis diapason" is completed by a quadruple comparison 4:1. The fourth and the fifth together complete one octave in this way:

Octave | Fifth | Fourth | Sesquialter | Sesquitertial

For if one voice is twice as high or low as another, an octave consonance is made. If a voice is higher or lower than another by a sesquialter proportion 3:2, or a sesquitertial 4:3, or a sesquioctave 9:8, it would return a fifth, a fourth, or a tone consonance. Likewise, if an octave (as 2 and 4) and a fifth (as 6 and 4) are joined: the triple ratio 3:1, which is a fifth and an octave together, creates a consonance. But if a double octave is made (as 2 to 4 and 4 to 8), a consonance is made which is the double octave. But if a sesquialter and a sesquitertial—that is, a fifth and a fourth—are joined (as 2 to 3 and 3 to 4): a double ratio, which is...