On the Institution of Music (De institutione musica) (c. 1490) - Page 15

Indeed, it is composed of five intervals, consisting of four of these tones and a minor semitone. original: "semitonij minoris"; in Pythagorean tuning, this is the limma, which is slightly smaller than a true half-tone. For let the number 192 be set down and its sesquialter sesquialtera: a 3:2 ratio, representing a musical fifth be taken, which creates a fifth original: "diapente" consonance compared to it. Thus, the number is 288. Therefore, let these and the numbers found above, 192, have these numbers placed in the middle: 216, 243, 256; 248-256 288 288. In the above description, 192 and 256 were shown to contain two tones and a semitone. It remains, therefore, to be seen that 256 compared to 288 is a sesquioctave sesquioctava: a 9:8 ratio, the mathematical definition of a whole tone, which is a tone. Their difference is 32, A small hand (manicule) pointing right toward the text. which is the eighth part of 256. Thus, it has been shown that the fifth original: "diapente" consonance consists of three tones and a semitone. But whereas previously the fourth original: "diatesseron" consonance had reached from the number 192 up to 256, now indeed the fifth is extended from the same number 192 up to 288. Therefore, the fourth consonance is exceeded by the fifth by the proportion that is contained between the numbers 256 and 288. And this is a tone. Therefore, the fourth symphony is surpassed by the fifth by a whole tone.

That the Octave is Joined by Five Tones and Two Semitones. 119.

The octave original: "diapason" consonance consists of five tones and two semitones, which, however, do not quite fill one whole tone. For since it has been shown that the octave consists of a fourth and a fifth, and the fourth has been proven to consist of two tones and a semitone, while the fifth consists of three tones and a semitone, joined together they produce five tones. But because those two semitones were not of a full half-measure, their joining does not reach a full whole [tone], but rather exceeds a half-measure while falling short of wholeness. And the octave is, according to this reasoning,

composed of five tones and two semitones, which, just as they do not reach a full tone, so they proceed beyond a single semitone. But because there is a reason or method by which the musical consonances themselves are perceived, it will be explained more clearly later. Meanwhile, for the present discussion, belief should be applied with a moderate understanding. One must proceed with caution regarding the highest faith in these matters until it has shone through its own proper and clear demonstration. With these things thus arranged, let us discuss for a little while the strings of the cithara A professional version of the lyre, often associated with Apollo and their names, in what way they were added, and the reason for their names. For once these first come to our knowledge, it will be easy to embrace the science that follows.

Concerning the Addition of Strings and Their Names. 120.

Nicomachus A Pythagorean mathematician and music theorist (c. 60–120 AD) reports that music was simple in the beginning, so much so that the whole of it consisted of only four strings. This lasted until the time of Orpheus, so that the first, fifth, and fourth strings sounded the octave consonance; the middle strings were tuned to each other and to the extremes as a tone, a fourth, and a fifth, so that nothing in them was out of consonance, namely in imitation of the World, which consists of four elements. Of this four-stringed instrument original: "quadrichordi" Mercury is said to be the inventor. Indeed, Chorebus, the son of Atys, added a fifth string; then Hyagnis added a sixth to these. But the seventh string was added by Terpander of Lesbos A famous Greek poet and musician of the 7th century BC according to the likeness of the seven planets. And among all these, the string which was the lowest in pitch was called hypate original: "hypate"; Greek for "highest." To the Greeks, this meant the "highest" in rank or the "highest" on the instrument's frame, though it produced the lowest sound., as if it were the greatest and most honorable. Whence they even call Jupiter Hypatos. They also name the Consul by the same name because of the excellence of his dignity. And the Latins...