What Simple and Composite Consonance Is; and That in the Senario Are Found the Forms of All Simple Consonances; and Whence the Minor Hexachord Has Its Origin

The Definition of Simple and Composite Consonance

Although some are in doubt whether the Hexachord is to be placed among the number of consonances — its proportion being contained in the genus Superpartient, which (as they say) is not apt to produce them — nonetheless, since this interval has hitherto been approved and received as consonant by musicians, I have placed it also in the number of them. But since I have said that the Hexachord is a composite consonance, we shall therefore see at present what is to be understood by simple or composite interval.

I say then that by Composite Consonance or Interval I understand that of which the smallest terms of its proportion will be found in such a way distanced from each other that they can be mediated and divided by two or more mean terms — so that from one proportion we can derive two or more. So on the contrary, Simple Consonance or Interval I say to be that which, taking the smallest terms of its proportion, will be ordered in such a way that they cannot receive between them any mean term that would divide such proportion into more parts — since they will always be distanced from each other by Unity.

The Major Hexachord as Composite

Whence I have said that the major Hexachord is a composite consonance: because the smallest terms of its proportion, which are 5 and 3, are capable of a mean term, which is 4, as I showed above — and the Diapente I say to be a simple consonance, because the smallest terms of its proportion, which are 3 and 2, cannot receive any mean term between them that would divide it into more parts, since they are distanced from each other by Unity.

The Three Ways Consonances Are Composite

It is necessary however to observe that in three ways it can be said that consonances are composite — as was also said above. First, when they are composed of two parts of the Diapason, which joined together do not reconstitute the Diapason itself. Then, when they are composed of the Diapason and one of its parts. And thirdly, when they are composed of more than one Diapason.

In the first way is considered the Hexachord named above, which is composed of the Diatessaron and the Ditone — as is seen between the smallest terms of its proportion, which are 5 and 3, mediated by 4, as is seen here: 5 · 4 · 3. To which I shall add the minor Hexachord, which arises from the joining of the Diatessaron to the Semiditone, whose smallest terms contained in the genus Superpartient by the proportion Supertripartientquinta can be mediated by a mean term. For finding such proportion between 8 and 5, those terms are capable of a harmonic mean term, which is 6 — which divides it into two lesser proportions: that is, into a Sesquitertia and a Sesquiquinta, as is seen here: 8 · 6 · 5. So that for this reason we may call this consonance composite — which has hitherto been embraced and placed by musicians in the number of the others. And although it is not found in act among the parts of the Senario, it is found nonetheless in potency: since from the parts contained within it, it takes its form — that is, from the Diatessaron and the Semiditone, of which two consonances it is composed. Wherefore it comes to have its form in act between the first cubic number, which is 8, and the 5.

In the second way is considered the Diapasondiapente, which is composed of the Diapason joined with the Diapente — since the smallest terms of its proportion, which are 3 and 1, are divided naturally into a Duple and a Sesquialtera, which are the proportions containing this consonance, as is seen here: 3 · 2 · 1.

In the third way we can place the Disdiapason: since the smallest terms of its proportion, which are 4 and 1, are capable of a mean term which divides it into two Duple proportions in Geometric proportionality, as is seen here: 4 · 2 · 1. Moreover we can consider this consonance to be composed of the Diapason, the Diapente, and the Diatessaron — since those terms are capable of two mean terms that divide it into three parts containing the proportions of the named consonances, as is seen here: 4 · 3 · 2 · 1.

Proper and Improper Composite Consonances

Nonetheless we must observe that however much such consonances can be considered composite in so many ways, I properly and truly call composite only those which are composed of the Diapason and some one of its parts, according to the latter two modes shown above. But those which are considered composite in the first mode I call such only improperly, and composite in a certain sense — because being less than the Diapason, they appear to be almost simple and elementary: which does not occur in the others, for the reason I shall say elsewhere.

Conclusion: The Senario Contains All Simple Consonances

And since it is impossible to find new consonances which are simple, beyond the five shown — which are the Diapason, the Diapente, the Diatessaron, the Ditone, and the Semiditone, from which every other consonance is composed — I therefore say and conclude that in the Senario, that is among its parts, every simple musical consonance is found in act, and the compound consonances in potency as well — from which arises every good and perfect harmony: understanding however the forms or proportions, and not the sounds.

But so that we may more easily grasp what I have said, I shall reason first of the things that are necessary for the knowledge of proportions, and afterward we shall see how they are put into practice — for without knowledge of them it would be impossible to have any understanding of Music.