On the Institution of Music (De institutione musica)

...they joined. The diapason original: "diapason"; the octave interval (2:1 ratio) sounded in a double ratio. Truly, a hammer of 12 weights compared to a hammer of 8, and a hammer of 8 weights compared to a hammer of 6, joined the diatessaron original: "diatessaron"; the interval of a fourth (4:3 ratio) consonance according to the epitrite original: "epitrita"; the ratio 4:3 proportion. But the 8-weight hammer compared to the 6, and 8 the 12 to the 9, mixed together the diapente original: "diapente"; the interval of a fifth (3:2 ratio) consonance. Finally, the 8-weight hammer compared to the 9 sounded a tone original: "tonum"; the fundamental interval of a major second in a sesquioctave original: "sesquioctava"; the ratio 9:8 proportion.

Chapter XI: In what ways the proportions of consonances were variously weighed by Pythagoras

Returning home from there, he weighed through various examinations whether the whole system of harmonies consisted entirely in these proportions. First, he fitted equal weights to strings, judging their consonances by ear. Then, in the length of pipes, he restored the doubling and halving of the length and fitted the other proportions, gaining the most complete trust through diverse experiments. Often, he would pour liquid into four jars of equal weight according to the measure of the proportions; often, he struck these same jars, formed of different weights, with a copper rod, and was delighted to find he had discovered nothing different. Driven by this, he also set out to examine the length and thickness of strings. Thus, he discovered the rule original: "regulam"; here referring to the monochord, a single-stringed instrument used for measuring musical intervals of which we shall speak later, which took its name from the thing itself. Not that it is merely a wooden rule by which we measure the sizes of bodies or sounds, but because such an inspection is a fixed and firm "rule," so that it might never deceive any inquirer with a doubtful judgment.

Chapter XII: On the division of voices and their explanation

But enough of these things for now. Now let us collect the

differences of the voice. Every voice is either SYNECHE original: "CYNEXHE"; continuous, which is sustained, or DIASTEMATIKE original: "DIACRHEMATIKE"; intervallic, which is suspended by intervals. Indeed, the continuous voice is that which we run through when speaking or reading prose. For then the voice hastens not to linger on high or low sounds, but runs through the words as quickly as possible; the impulse of the continuous voice works to facilitate the senses and express speech. On the other hand, the DIASTEMATIKE is that which we sustain in singing. In this, we serve not so much the words as the sounds. The voice itself is slower, and by modulating through varieties, it creates a certain interval—not a path of silence, but rather of a suspended and slow song. To these, as Albinus A 2nd-century philosopher who wrote on music and geometry asserts, a third difference is added which can include intermediate voices: namely, when we read heroic poems, neither in a continuous prose run, nor in the suspended and slower manner of singing.

Chapter XIII: That human nature limits the infinity of voices

But as for the continuous voice and that which we use for singing, they are by nature infinite. For if one considers it, there is no set limit for drawing out speech, nor for lifting the voice to high pitches or relaxing it in volume. But human nature has established a proper limit for both. For human breath has made a boundary for the continuous voice, beyond which it cannot pass by any means. For each person speaks continuously only as long as their natural breath allows. Likewise, the nature of men makes a limit for the DIASTEMATIKE voice, which determines their high and low sounds. For each person is only able to raise or lower the peak? original: "p'acimen"; likely acumen (sharpness/height) of their volume as much as the natural measure of their voice permits.