For What Reason the Ancients Did Not Use the Imperfect Consonances in Their Harmonies, and Pythagoras Forbade Passing Beyond the Quadruple

Nor ought we to marvel that the ancients did not receive such consonances: for they gave the greatest credence to the doctrine of Pythagoras, who, being a most diligent investigator of the profound secrets of Nature, did not choose to accept them among the consonances, being himself a lover of things simple and pure. And he took delight in all things, so long as their matter did not depart from simplicity; and in that matter he investigated the secret things, that is to say their causes, holding the opinion that where things were found to be simple there would be in them both firmness and stability, and where they were mixed and diverse, inconstancy and variety. And because he judged that of these latter no firm reason could be had, he therefore rejected them without proceeding any further.

Whence only those consonances pleased him which agreed together by reason of numbers that were simple and had their nature most pure, such as are those which arise from the Multiple and from the Superparticular genus, and are the five already shown, contained within the Quaternary number. And he rejected those which are comprehended by the numbers found beyond the Quaternary, and which enter into the other genera of proportion, from which arose their Ditone, the Trihemitone or Semiditone, and the other like intervals, as we shall see. Nor did he place among the consonances the Ditone and the Semiditone, contained in the Superparticular genus, which I have shown in the first part: for he knew their nature very well (as I believe), and saw that from the mixture of such imperfect consonances with the perfect ones the two Hexachords could arise, namely the greater and the lesser, which are contained in the Superpartient genus, as their forms make manifest to us.

He approved, then, only those consonances, as the more simple and the more noble, which have their forms among the parts of the Quaternary number: for from them no sound can arise which is not consonant. And perhaps the Pythagoreans held this number in the highest veneration for no other cause than that they saw such simplicity of concord arise from it; whence they came to the opinion that it pertained to the perfection of the Soul. And they held this for true to such a degree that, when they wished undoubted credence to be given to what they affirmed (as Macrobius says), they would say: I swear to you by him who gives to our soul the Quaternary number.

The Divine Philosopher, therefore, forbade the passing beyond the Quadruple: for beyond it (as Marsilio Ficino, the Platonic Philosopher, says in the Compendium on Plato’s Timaeus) he heard no harmony, seeing that by proceeding further there arises the Quintuple between 5 and 4, and the Superbipartient between 5 and 3, which generates dissonance. It is quite true that if Ficino’s words were taken as they sound, what is false would be understood: for the Quintuple is not found between 5 and 4, but rather between 5 and 1. Therefore I judge that this text is truly corrupt, and that in place of the 4 the Unity ought to be set; or else that such words are to be understood in this manner: that in proceeding beyond the Quadruple, the Quinary being added to the Quaternary number — that is, the Sesquiquarta being added to the Quadruple proportion in this form, 5. 4. 3. 2. 1. — there arises the Quintuple proportion between 5 and 1, and likewise the Superbipartient-third between 5 and 3, which departs from the simplicity of the numbers and is contained in the third genus of proportion, which is called Superpartient; which genus, Pythagoras said, was not fit for the generation of the musical consonances, as is seen in the example set below.

For this cause, then, and for no other, I judge that Pythagoras forbade the passing beyond the Quadruple. It is quite true that certain others say that the Philosopher meant that the Quadruple was not to be passed in melodies, that is, the number of the Fifteen strings contained within the Disdiapason: for he judged that every excellent voice, Nature having set a limit to all things, might without its own distress naturally ascend from the low to the high, or contrariwise descend, through Fifteen notes; and that whenever one passed further, whether in the low or in the high, such notes would no longer be natural but forced, and would bring annoyance to the listeners. But of these two reasons the first (according to my judgment) is the better, and is more to the purpose. It is not to be marvelled at, then, that the ancients did not receive such consonances, since by the Pythagorean laws they were forbidden to pass beyond the Quadruple.

[Editorial note: Here Zarlino’s original text contains a large framed woodcut, which the compositor has carried over to the head of the following page; the text refers to it above, at “as is seen in the example set below,” in the paragraph on the Superpartient genus. Along the top of the frame stand the numbers 5, 4, 3, 2, 1, set from left to right, and beneath them four descending tiers of arcs connect the terms in every combination. The uppermost tier joins adjacent terms: Sesquiquarta (5:4), Sesquiterza (4:3), Sesquialtera (3:2), and Dupla (2:1). The second tier skips one term: Superbipartiente terza (5:3), Dupla (4:2), and Tripla (3:1). The third joins Dupla sesquialtera (5:2) and Quadrupla (4:1). The lowest single arc spans the whole, Quintupla (5:1). The figure shows what the argument has just claimed: extending the Quaternary by the Quinary introduces the Quintupla and the Superbipartiente terza, the latter falling into the Superpartient genus that Pythagoras excluded from consonance.]