First.

that we seek. And remember what I have said in my Institutions referring to Zarlino's Istitutioni harmoniche; that Consonances or musical Intervals do not arise primarily, as some have maintained, from the addition of many smaller intervals placed together, but from the Division of the Diapason octave, which I called the Mother and Source of every other consonance and interval. I believe everyone among us remembers this, said Master Adriano; but it seems to me that it is not entirely wrong to say that the Diapason is composed of three major tones, two minor ones, and two major semitones; because one clearly sees that it truly contains them, and proceeds through such Intervals. You speak, Sir, I replied; in Music you have to consider two things; first the Interval, which by the Greeks is called Διάστημα diastēma / distance or interval; and the orders, or scales, as called by some moderns, or even Constitutions, if we wish to name them, called separately, and each one by itself, Σύστημα systēma / system; however I say, that if you speak of this latter; it is not inconvenient to say that a Diapente fifth is composed of two major tones, one minor, and one major semitone, as of two parts: But speaking of the first, this is not verified; because they are born 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 ones one can reintegrate and compose an order, which contains as many Tones and Semitones as might fall into that composition, according to the quality of the order that you wish to compose; as if to say; composing a Diapason in which enter the Intervals named above, and others similar. And in this way, it is no error to say that a Consonance; that is, one of these orders, is composed. But it would be wrong, when one said that an Interval of the first sort was composed. You speak well, added Master Adriano, but for goodness' sake do me a favor, and to all the others who are gathered here and delight in Music; reason with us a little about these things; so that we too may know how to discuss them, when it is needed, some things; because I desire greatly to see one day the things of Music demonstrated, as they ought to stand. And if you would take this task upon yourself, you would do us a pleasing thing; because I believe that there is no one among us who would not take pleasure in it, and would not have an obligation for it. So it is truly, replied Signor Desiderio; and I in particular desire it much; because from these I will learn the way that the Musician takes in demonstrating his conclusions, and I will see the difference that is between the demonstrations that the Musician makes, which I have never fully seen; and those which the Arithmetician and the Geometer make; having seen these last two many times put into action. Sir, I replied; there is no need to have haste in these things; but there is need to go slowly. And I promise you, that it would take more time than you imagine, if I wanted to undertake the task of reasoning, as one ought, about the things of Music; because it would first be necessary to declare some things to those who have not seen or read the things that concur in Demonstration, wishing to reason with you about this matter with Demonstration in hand; and it would be necessary besides this to declare what a Demonstration is, and of what it is composed, and many other things, which would bring with them much time. And it would be necessary that you imagine having more than one conversation; being in Music one thing chained with the other, in the manner as are the rings that are found joined together in a chain. Then Master Adriano replied, you increase my desire, saying these things; wherefore I pray you to take willingly this task; because it will be a great relief for my ailment. And I swear to you, that if I were younger than I am; I would wish to become a Disciple anew, and set to work in such a manner to understand the reason of the things of Music; that I would not wish that anyone asked me anything, which I could not satisfy. This would truly be to me also beyond measure pleasing, said Master Francesco; so that while I stay in Venice, I might acquire something to take with me to Ferrara; wherefore I exhort and pray Master Gioseffo to take this charge. But I tell you now, Sir, that you should content yourself with being the first of our times in the things of practice, which you also have not without some knowledge of Theory; having always delighted to practice with men learned in this profession. And if you are not entirely colored by Theory, at least you are well-dyed. And you should rejoice in this, because these are gifts, which are not

1. Part. ch. 13. and 3. part. ch. 5.

Harmonic Demonstration.