Harmonic Demonstrations

Therefore, from not knowing the degrees, the order, and the proper places of the consonances, it happened that the Ancients denied that those intervals which are smaller than the Diatessaron were consonant.

DESI. Was it not enough for them to at least know that, placed in the high range and in their proper places, they were and are consonant?

GIOS. I have already said that they did not consider this; nay, they never knew it. Since, if they had known it, there is no doubt that they would have put it into use, or at least would have left some memory of it, as some of our moderns have done. Although these have not demonstrated that the Ditone, contained by the sesquiquarta 4:5 proportion, and the Semiditone, contained by the sesquiquinta 5:6 proportion, were consonant, they have at least asserted with great hesitation that the Ditone, composed of two tones major whole steps literally "sesquioctave tones", and the Semiditone, which contains a tone and a minor semitone, considered from the side of their proportions, were dissonant. They were supported by this opinion: that consonances could not have their forms from proportions other than the Multiple and the Superparticular, and that when considered in act in the sounds or in the voices, they were consonant. For when they wanted, they heard them as consonant. But they were mistaken, because when they heard them as consonant, they were contained by Superparticular proportions, and not by Superpartient, as they believed.

DESI. Whence could the error of these men arise, if I may ask?

GIOS. From not having had sufficient principles in their demonstrations, and having little experience in matters of music. For it was not enough to say only that the consonances were those that had their form from the Multiple or Superparticular genus, contained between the parts of the quaternary number, but it was necessary rather to say that the consonances were those that had their forms from those proportions that are found in act between the parts of the senary the number 6, which Zarlino considers the limit of perfect harmony.

FRAN. Therefore, they could not demonstrate the things of music perfectly, not having had sufficient principles, since from them one has the knowledge (as I have often heard said) of all things that are treated in any science whatsoever.

GIOS. You say too true. Wherefore one must know that to want to demonstrate the things of music perfectly, one must add to those principles, by the means of which others have demonstrated, those things that lead us to the end of the thing that we seek. And remember what I said in the Istitutioni Zarlino's treatise "Istitutioni harmoniche": that the consonances or musical intervals do not arise primarily, as some have held, by the addition of many smaller intervals placed together, but by the division of the Diapason, which I called the Mother and Source of every other consonance and interval.

ADRI. I believe that each one of us has this in memory, but it seems to me that it is not entirely poorly said that the Diapason is composed of three major tones, two minor, and two major semitones. Because one sees indeed that it truly contains in itself, and travels through, such intervals.

GIOS. You speak well, Master. In music you have to consider two things: first, the interval, which by ancient Greek musicians is called Diastema interval, and the orders or scales—so called by some moderns—or even Constitutions, if we wish to name them, which are called by the Greeks separately and each by itself Systema system. Therefore I say that if you speak of this latter, it is not inconvenient to say that a Diapente is composed secondarily of two major tones, one minor, and one major semitone. But speaking of the first, this is not verified, because they arise from the division of the Diapason. And it is not without purpose to say that, having made many parts of a Diapason, from those same parts one can reintegrate and compose an order that contains as many tones and semitones as can happen in that composition, according to the quality of the order that you want to compose, as it would be to say: to compose a Diapason into which enter the intervals named above and others similar. And in this way it is not an error to say that a consonance—that is, one of these orders—is composed. But it would be an error to say that an interval of the first ones named was composed.

ADRI. You speak well. But for heaven's sake, do me a favor, and all the others who are here gathered and delight in music: discuss these things a little with us, so that we also know how to discuss something when there is need. Because I desire greatly to see one day the things of music demonstrated as they ought to stand. And if you would now wish to take...

1. Par. cap. 13.

Inst. 3. par. cap. 3.