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

408 else in substance and in precision if not the series and the physical nature of the third sound, produced by the two given sounds, and joined in harmony with the same. Here you will detect better than me that for example in the integral sesquitertian harmony 12, 6, 4, 3, intended in my sense, one finds the divisors 1, 2, 3, 4, understood in your sense: that in the integral sesquifourth harmony 20, 10, 7:1/2, 5, 4, intended in my sense, one finds the divisors 1, 2, 4, 5, intended in your sense, etc. But let us observe the thing more closely. In my first position sesquitertian 12, 6, 4, 3, compared to your sense, the term 6 as divisor of 12 is missing, which is your exponent, and is my third sound. In my second position sesquifourth 20, 10, 7:1/2, 5, 4, compared to my sense the term 7:1/2 is missing, which compared to your sense is not, nor can be divisor of 20, your exponent, and my third sound. Let us examine what derives therefrom in both positions. Having turned around your first position of divisors of 12, 1, 2, 3, 4, 6, in my sense in 12, 6, 4, 3, 1:1/2, I find that the term 1:1/2 added to my position does nothing but destroy the continuous harmonic proportion, in which per se one finds only the four terms 12, 6, 4, 3. I find that it comes out badly indeed in our practice, because having supposed that to the four aforementioned terms one must add the fifth term, it is beyond any doubt that if the proposed musical composition is in the tone (as it is called by us) of a major third, the fifth term must not be 1:1/2 but 2:2/5; if it is of a minor third, it must be 2:1/2. Having turned around your second position of the divisors of 20, 1, 2, 4, 5, 10, in my sense in 20, 10, 5, 4, 2, besides finding the continuous harmonic proportion of my position 20, 10, 7:1/2, 5, 4 destroyed; besides the incongruous disposition of the parts either sung, or played, if they were to be set out according to the dividing results, there is a substantial lack of the fifth in the harmony, of which it is an integral part, and that with regard to 20 as exponent, cannot be assigned in any way, because 7:1/2 forming with 10 the fifth necessary to the integral harmony; it is clear that 7:1/2 cannot be divisor of 20. This, and others similar are the difficulties which I find in your rule, which being generally true, does not then hold up individually in practice to all our musical needs, to which both generally and particularly the rule of the third sound holds up. Therefore I reply and confirm that if you had had by your side a composer by whom our musical needs had been indicated with precision, since you have already grasped the substantial point, you would have adapted it both to the universal and to our particular cases and needs, and from then on would have been convinced of the truth. The difference then that there is between the two of us, is not of substance, it is of order alone, and of greater or smaller scope, and I shall also say of easier or less easy understanding of the rule. In such case the learned world should, as I do, do you that justice, which distinctly you deserve from everybody, for having discovered the substance of the thing, so it is that you should (if I am allowed to say so) have towards me the goodness of believing me, that of our two rules in substance equal and true, mine is more suitable to practice both for its greater expansion to particular cases, and for its easier comprehension by musicians, who certainly are not the most learned men. But I