Reasoning

Page 6

natural places they are situated. Thus they judged that in every way they were dissonant. Therefore, from the failure to know the degrees, the order, and the proper locations of consonances, it followed that the Ancients denied that those intervals smaller than the Perfect Fourthoriginal: "Diateffaron" (Diatessaron). An interval spanning four scale degrees. were consonant.

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

GIOS. I have already said that they did not consider this; indeed, they never knew it. If they had known it, there is no doubt they would have put it into use, or at least left some memory of it, as some of our moderns have done. These moderns, although they have not demonstrated that the Major Thirdoriginal: "Ditono". Literally "two tones." contained by the SesquiquartaThe ratio of 5:4, which defines the "just" major third. proportion and the Minor Thirdoriginal: "Semiditono". Literally "half-two-tones." contained by the SesquiquintaThe ratio of 6:5, defining the "just" minor third. were consonant, they have at least asserted with great hesitation that the Major Third composed of two 9:8 tonesoriginal: "Tuoni sequiottavi". The Pythagorean major third (81:64), which is slightly wider and more dissonant than the just third (5:4). and the Minor Third containing one 9:8 tone and one minor semitone, when considered from the perspective of their proportions, were dissonant. They relied on the opinion that, except for MultipleRatios like 2:1 (octave) or 3:1 (twelfth). and SuperparticularRatios where the numerator is one greater than the denominator, like 3:2 or 4:3. proportions, consonances could not have their forms—yet when considered in practice in sounds or voices, they were consonant. This was because when they heard them as consonant, they were actually contained by Superparticular proportions, and not by SuperpartientRatios where the numerator is more than one greater than the denominator. The ancients believed these could not be consonant. ones as they believed.

DESI. From where could their error arise, if I may ask?

GIOS. From not having had sufficient principles in their demonstrations, and having little experience in musical matters. For it was not enough to say merely that consonances were those whose forms came from the Multiple or Superparticular genus contained within the parts of the number fouroriginal: "Quaternario". The "Tetraktys" of the Pythagoreans, who believed only the first four numbers could create harmony.; but one should rather say that consonances were those whose forms are found in practice among the parts of the number sixoriginal: "Senario". Zarlino's groundbreaking theory that expanded the "number of harmony" to 6, allowing for the inclusion of thirds and sixths as consonances..

FRAN. Therefore, they could not demonstrate musical matters perfectly, not having had sufficient principles; since from principles one gains knowledge (as I have often heard said) of all things treated in any science whatsoever.

GIOS. You speak the truth indeed. Thus, one must know that to perfectly demonstrate musical matters, one must add to those principles used by others those things that lead us to the goal of what we seek.

Part 1, Chapter 13

And remember what I said in the InstitutionsZarlino's Istitutioni harmoniche (1558), the most influential music theory treatise of the 16th century.: that consonances or musical intervals do not primarily arise, as some have held, by
Institutions, Part 3, Chapter 3

the addition of many smaller intervals placed together, but by the division of the Octaveoriginal: "Diapason"., which I called the Mother and Source of every other consonance and interval.

ADRI. I believe everyone here remembers this; but it seems to me it is not entirely wrong to say that the Octave is composed of three major tones, two minor tones, and two major semitones. For one clearly sees that it truly contains and proceeds through such intervals.

GIOS. Listen, sir: in music you have to consider two things. First, the interval, which the ancient Greek musicians called Diastemaoriginal: "Διάστημα". An interval considered as a distance between two pitches., and the orders or scales—as some moderns call them—or Constitutions, if we wish to name them so, which the Greeks called separately, each by itself, Systemaoriginal: "Σύστημα". A system or collection of intervals forming a scale.. Therefore, I say that if you speak of the latter, it is not incorrect to say that a Perfect Fifthoriginal: "Diapente". is composed secondarily of two major tones, one minor tone, and one major semitone. But speaking of the former [the interval as a single entity], this is not true; for they arise from the division of the Octave. It is not inappropriate to say that once many parts of an Octave are made, one can reintegrate and compose an "order" from those same parts, which contains as many tones and semitones as may occur in that composition, according to the quality of the scale you wish to compose—such as composing an Octave in which the intervals named above and others like them enter. 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 one of the first-named intervals was composed.

ADRI. You speak well. But please, do me a favor—and for all the others gathered here who delight in music—discourse a little on these matters, so that we too may know how to reason when necessary. For I greatly desire to see musical matters demonstrated as they ought to be one day. And if you would now like to take...