Harmonic Demonstrations

First.

...and truly a thing to be refused. Nevertheless, desiring to satisfy your desire in some part, since this is the first thing and the first favor that you have asked of me, I shall not hesitate to tell you all that I feel regarding this request of yours. All the more so, since I see you all of one same will and ignited by a good desire. Wherefore, to satisfy you, I will not put any time in between. Pray, therefore, to God that He may illuminate my mind to tell you things that may be of satisfaction to you.

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

GIOS. Take note, then, that Pythagoras, having held the opinion—as you have been able to hear many times—that all those intervals which are consonant had their forms contained only by the ratios of the Multiple or Superparticular genus, held it for certain that all those which had their forms contained under other genera, other than one or the other of the two named, were altogether dissonant. Wherefore, having this opinion, and already seeing that the tetrachords of the diatonic a scale pattern based on whole and half steps genus—which was received by him and his followers more than any other—proceeded from the low to the high by two tones of sesquioctave ratio 9:8 proportion, and by one semitone contained by the ratio of 13 parts to 243, and that the two tones which formed the ditone major third were contained in their extremes by the ratio of 17 parts to 64, and that one tone with the named semitone, from which one could form a semiditone minor third, were contained by the ratio of 5 parts to 32, finding these two ratios among those of the superpartient a ratio of n+x:n where x > 1 genus, he concluded, by the first reason that I can tell you, that those intervals which were contained between these forms in their extremes were, as they truly are, dissonant. From this rule, he did not exclude the two greater and lesser hexachords six-note scale segments, 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. Wherefore such an opinion is not to be judged false regarding this reason, 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: given that (as is clearly understood by anyone of judgment) all the charm and grace of music, and I will also say all its diversity, is placed in the two lesser consonances of the diatessaron perfect fourth, that is, in the ditone and the semiditone, and also in the two greater and lesser hexachords, that the ancients should never have heard them among the seven spaces contained in the diapason octave, and did not know the named intervals to be consonant. It is very true, however, that not holding them as consonant, I believe, was done not without some reason.

GIOS. Sir, to this which you have said, I will answer with this other reason. You must consider that if the ancients wished to hear the intervals that we have named, it was necessary that they had heard them in two ways: first, under the forms contained between the seven named spaces or intervals of the diapason, and then under other forms varied from those. Regarding having heard them in the first way, believe me that they heard them as dissonant, because the said forms are subject to the superpartient genus. But as to the having heard them under other forms, be it in the voices or in the sounds, it 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 own, true, and natural places, and then outside their aforementioned places. If they wanted to hear them in the proper and true places on their instruments, this was impossible, because such instruments were not sufficient to make them hear such a thing, since (as I have said in Chapter 2 of the Second Part of the Institutions) the ancients never passed the fifteenth voice or string of their instruments, nor did they ever pass (according to the precept of Pythagoras) the quadruple ratio. Wherefore they necessarily heard them outside their places, and in non-proper places. And if they heard them in non-proper places, they could not fully satisfy the sense. Wherefore they forcedly judged them as dissonant rather than consonant. For this reason, I am of the opinion that they judged the intervals that are smaller than the diatessaron as dissonant for no other reason than because they did not have knowledge, or to speak better, did not understand the true, legitimate, proper, and natural places of the consonances, that is, where each one should naturally be placed. Since (as you all know), even if the ditone is a consonance, nevertheless, when placed outside its natural place and located in the place of another consonance, it renders a dissonance rather than a good...