Dimostrationi Harmoniche

...and truly a thing to be refused: yet desiring to satisfy in some part your desire—since this is the first thing, and the first favor, that you have requested of me—I will not fail to tell you all that I feel concerning this request of yours. All the more so, because I see you all of one and the same will, and kindled by a good desire. Therefore, to satisfy you, I will set aside no time. Pray then to God, that He may illuminate my mind to tell you things that will bring you satisfaction.

ADRI. We shall do so; and we all pray you to begin.

GIOS. Take note then, that Pythagoras held the opinion—as you have often been able to hear—that all those intervals that are consonant had their forms contained solely by the ratios of the Multiple or Superparticular In Renaissance music theory, a "Multiple" ratio is one where the larger number is a multiple of the smaller (like 2:1, an octave). A "Superparticular" ratio is one where the larger number is exactly one unit greater than the smaller (like 3:2, a perfect fifth). These were considered the most "pure" mathematical relationships. genus. He held it as certain that all those which had their forms contained under other genera, other than one or the other of the two named, were entirely dissonant. Therefore, holding this opinion—and seeing already that the Tetrachords of the ditonic diatonic genus A specific ancient Greek tuning system that used two equal whole tones and a small remainder for a semitone., which more than any other was received by him and his followers, proceeded from low to high by two Tones of Sesquioctave ratio The ratio 9:8, representing a standard whole tone., and by a Semitone contained by the ratio of 256 to 243 original: "Super 13. partiente. 243"—this is the Pythagorean limma, a mathematically complex semitone.; and that the two Tones, which formed the Ditone The Pythagorean major third., were contained in their extremes by the ratio of 81 to 64 original: "Super. 17. partiente. 64"; and that a Tone with the named Semitone, from which one could form a Semiditone The Pythagorean minor third., were contained by the ratio of 32 to 27 original: "Super. 5. partiente. 32"—finding these two ratios among those of the Superpartient genus A mathematical genus where the larger number contains the smaller plus several parts (not just one), such as 81:64. Pythagoras considered these too complex to be consonant., he came to conclude (by the first reason I can tell you) that those intervals contained between these forms at their extremes were, as they truly are, dissonant.

From this Rule he did not exclude the two Sixths original: "Hexachordi", major and minor, since they have their forms in such a genus. And this is all too true: because such intervals, when put into practice, are known to be little pleasing to the ear. Thus, such an opinion is not to be judged false regarding this reasoning, and it should not seem a strange thing.

ADRI. What you say is very true: but this seems to me a great thing to say: since (as is clearly understood by anyone of judgment) all the beauty and the grace of Music—and I will even say all its diversity—is placed in the two Consonances smaller than the Fourth original: "Diateſſaron": that is, in the Ditone and the Semiditone; and also in the two Sixths, major and minor. It seems strange that the Ancients would never have heard these among the seven spaces contained in the Octave original: "Diapaſon", and did not recognize the named intervals to be consonant. It is indeed true, that their not holding them as consonant, I believe, was done not without some reason.

GIOS. Sir, to what you have said, I will respond with this other reason. You must consider that if the Ancients wanted to hear the intervals we have named, it was necessary that they hear them in two ways: first, under the forms contained among the seven named spaces or intervals of the Octave; afterwards, under other forms varied from those. As for having heard them in the first way, believe me, they heard them as dissonant: because the said forms are subject to the Superpartient genus. But as for hearing them under other forms—whether in voices or in sounds—this is indeed possible to have heard them as consonant.

Take note, however, that they could hear such intervals in the second manner in two ways: first in their proper, true, and natural places; and then outside of their aforementioned places. If they wanted to hear them in their proper and true places upon their instruments, this was impossible: because such instruments were not sufficient to make them hear such a thing. For (as I said in chapter 2 of the Second part of the Institutions Zarlino's own famous work, Le Istituzioni harmoniche (1558).) the Ancients never surpassed the fifteenth voice or string of their instruments; nor did they ever pass (according to the precept of Pythagoras) the Quadruple ratio The ratio 4:1, representing two octaves.. Hence they necessarily heard them outside their places, and in improper places. And if they heard them in improper places, they could not fully satisfy the sense. Thus, by necessity, they judged them dissonant rather than consonant. Therefore, I am of the opinion that they judged the intervals smaller than the Fourth to be dissonant for no other reason than because they had no knowledge—or to put it better, they did not understand—the true, legitimate, proper, and natural places of the consonances: that is, where each one should naturally be placed. For (as you all know) even if the Ditone is a consonance, nevertheless, placed outside its natural place and located in the place of another consonance, it renders a dissonance rather than a good...