Giuseppe Tartini - Lettere e documenti / Pisma in dokumenti / Letters and Documents - Volume / Knjiga / Volume II

401 LETTERS subsequently as 1 : 4, 1 : 3, 2 : 3 etc., from which all the consonances result. If there are more sounds, I express their harmony with the lowest number divisible by the numbers which express the sounds, which is called by me exponent of the consonance, the nature of which can be judged by the ease with which one detects the consonance itself. So 6 is the exponent of the consonance composed by the sounds expressed by the numbers 3 and 2 which form a fifth; and reciprocally, when the exponent is given, to find the relative consonance it is sufficient to take all the divisors of this exponent, and the consonance will be formed by the sounds expressed by the divisors found. So the exponent 12 includes the consonance expressed by the sounds that are like the numbers 1, 2, 3, 4, 6, 12. When all these sounds are taken together, the consonance shall be complete, as one would not know how to add a new sound without it becoming more complicated. But these two sounds 3 and 4 already form a consonance which has as exponent 12, but that is not complete, because it does not become more complicated even if one adds to it the sounds expressed by the numbers 1, 2, 6, 12. But to judge the degree of harmony of a consonance it is not necessary to look to the quantity of its exponent, rather to the composition of its factors, as 12 is the product of 2, 2, 3: the simplicity of these factors is what makes the consonance pleasant; and in my essay I have ordered all the consonances possible according to the degrees of pleasantness, and I shall mark this as much, given that hitherto one has not received in Music other consonances, other than those whose exponents are formed by the factors 2, 3, 5; these are the sole prime numbers that will be able to enter in the composition of the exponents of the consonances. And it is clear that the factor 2 introduces the octave, the 3 the fifth, and the 5 the major third, and the exponent 15 encompasses the perfect chords of musicians. The complete consonance of this exponent includes the sounds expressed by the numbers 1 : 3 : 5 : 15, in which the two extremes that are distant almost 4 octaves from one another cannot be united if not rarely; therefore if the highest 15 is removed, the other three 1: 3 : 5 produce a chord called hard, and if the lowest is removed, the remaining three 3, 5, 15, produce a chord called soft. It would be too long to report here everything to which the consideration of these exponents has led me with regard to the succession of more consonances, of the genres of music, of the ways, of the systems: matters that I have abundantly set out in my essay, and that seem to me to conform very well to the rules of counterpoint, although it seems to me very far from being able to bring perfection to this practical science, and that the perfection of this part cannot be expected if not from a great musician who would like to provide these principles with a peculiar attention. The excellent observation by Signor Tartini that two sounds which are together produced and kept vigorous, produce a lower third sound, which is as perceptible as if it were really played, follows necessarily from the established principles. He speaks quite at length of this harmonic phenomenon on page 13 and following in his work. And the reason is that when two sounds at once strike the ear,