Harmonic Demonstrations

Reasoning

...concert.

ADRI. This is all too true, and experience demonstrates it to us. Because when this interval is heard in the voices or in the sounds placed in the low range, then it seems to me I hear a certain sadness that is born in the composition from such an interval, which greatly offends my feeling.

GIOS. This, truly, no one of sound judgment could deny. But if this interval is carried toward the high range, the more it is transported, the more delight it gives to the sense, such that if that ditone which is placed in the low part of a composition offends the ear, the one which is placed between the low and the high not only does not offend, but also delights. When it is then placed in the more acute part of the composition, it gives even greater delight than that which the two named ones give, placed in the manner already said. And because this thing is known to you, therefore I will not extend myself to speak more words to you.

DESI. It seems to me that this thing is of no small importance. And this discourse of yours has pleased me much. But let us suppose that the ditone, which you have named, placed in a low place in such a manner makes a sad effect: will the semiditone perhaps make the same effect?

GIOS. It will not only make it sad, but most sad, in such a way that it can almost not be heard.

CLAV. This is true, Signor Desiderio, and believe it also from me, that I often and almost every day test it on the organs that I play: that when I come to touch the ditone in their low part, a most sad effect is heard. And if by chance I come to touch the semiditone, it makes a ruin so great that it can barely be heard. But when these intervals are touched in the middle of such instruments, they make a pleasing and soft sound heard. And if they are touched even more toward the high range, they make better effects. So that what M. Gioseffo and M. Adriano have said is very true.

DESI. Indeed, this thing pleases me, and I believe that few are those who consider such things.

CLAV. Few they are indeed, my Lord, and so few that I would not know how to find many for you. Nor have I ever heard this thing from anyone else, if not from M. Gioseffo.

GIOS. I want to say even more: that not only these intervals, when they are placed in the low, can offend the ear, but also when they are placed in the high. Because when the ditone takes the place of the semiditone, or to the contrary, they give little delight. And know that the greater part of all those musical compositions which little delight, among the other defects they have, this is one of them.

CLA. Truly it is so, because many times I have paid attention to some things that I play, and in all those that delight me, I find the ditone to be repeated between the parts above the bass part. To the contrary, in those that little please me, I have understood that the bass often has the semiditone above itself. I believe that you too, Sir, have often paid attention to this thing.

ADRI. In truth, it is so.

DESI. Pray, M. Gioseffo, give me to understand this thing better.

1. Part, ch. 13.

3. Part, ch. 60.

GIOS. You must know, Sir, as I have said and repeated many times in the Institutions, that the consonances in music have their degrees, and naturally occupy those places that their forms hold among the harmonic numbers. And when such consonances are placed to the contrary, if they do not make a sad effect, at least they make one less good than they would if they were in their proper places. Therefore, just as the dupla ratio 2:1, which is the true form of the diapason, located among the numbers between 2 and 1—to give you an example—holds the first place among them, and among the proportions is the first, seeing that before these two terms 2 and 1, there is no number to be found that is smaller; so among the consonances, there is no other to be found that by origin is before the diapason. Wherefore the diapason holds the first place in the low range, and before it is found no consonance that is greater or smaller than it. For which reason I have said many times that the diapason is the first consonance, from which all the others are born, whether they be greater or smaller than it. I add also anew that it is not only a principle but an element of all the others. Wherefore, just as its simple form, contained in its radical terms 2 and 1, does not receive another number or middle term that divides it into two parts, so it does not admit in the first, most low place of the order of consonances any middle string, nor even in the most low part of any instrument whatsoever that might divide it into two intervals, whereby one could hear any effect that is not less pleasing than that which is heard when it is played simply. In the second place is found the diapente perfect fifth, whose form is 3 and 2, which among the natural order of the numbers holds the second place, for which reason it must be placed without any middle after the diapason. And just as between 3 and 2 there cannot contain any middle number, so between the extreme strings of the dia-