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

407 LETTERS given sounds, and the proportion in which the third sound resulted will have to find itself with the two given sounds. But given three sounds in harmonic proportion such as 6, 3, 2, the equitemporaneous vibrations of the relative strings are like 1, 2, 3, therefore equal to divisors 1, 2, 3, of 6. So one should say that given two sounds in sesquitertian such as 4 to 3, from the multiplication of which one obtains 12, that shall be the third sound: of sesquioctavian like 9 to 8, from the multiplication of which one obtains 72, and that shall be the third sound etc., etc. But this rule proceeds infinitely, it is always true, and constantly determines the third sound and the equitemporaneous vibrations of the relative strings. Therefore, in substance it is the same as for the exponent and its relative divisors. Descending to greater precision, just as you infer the relative consonance from the rule of the exponent, so do I from the third sound as harmonic bass of the two given sounds. Because given the two sounds in sesquitertian, that is to say as 4 to 3, given the product of 4 by 3, which is 12, and that is the third sound, the three terms set in harmonic series 12, 4, 3, I add and suppose nothing more thereto if not the term 6, harmonic mean between 12, 4: so the continuous harmonic proportion 12, 6, 4, 3 is the consonance, that is relative integral harmony. Given two sesquifourth sounds, that is as 5 to 4, given the product of 5 by 4, which is 20, and is the third sound, set in harmonic series the three terms 20, 5, 4, I add thereto and suppose the two terms 10, 7:1/2, as two harmonic means between 20, 5; and the continuous harmonic proportion 20, 10, 7:1/2, 5, 4, shall be the consonance, that is relative integral harmony. That I suppose and add thereto the above-mentioned harmonic means it is not due to my whim, but to demonstrative necessity. Because, we being in agreement upon the fact that from the duple ratio 2,1, neither can you have another exponent, nor I another third sound other than 2, it is certain between us that our position, i.e. result, consists in two terms alone: being for you the only divisor of 2 the unit, for me the third sound 2, which (supposed two duple sounds like 2 to 1) being unison to the given 2, remains equal to 2, nor does it form a proportion. Therefore, we likewise agreeing upon the fact that, given the sesquialter sounds as 3 to 2, your exponent being 6, my third sound being 6, it is demonstratively certain that the principle of proportion is in the sesquialter, from which one solely has as first principle the third term, different from two given terms. It is demonstratively certain that the proportion is harmonic, because in my sense of the strings it is 6, 3, 2: it is 1, 2, 3 in your sense of the equitemporaneous vibrations, but as consecutive and inseparable from the supposed strings 6, 3, 2 and, I add, necessarily supposed in harmonic proportion: conditio sine qua non. And by corollary, since one must suppose the continuation of the series in the same nature as its principle, which is in harmonic proportion, I will necessarily have to add and suppose the harmonic mean 6 between 12, 4, given the three terms 12, 4, 3: the two harmonic means 10, 7:1/2, between 20, 5, given the three terms 20, 5, 4: the three harmonic means 15, 10, 7:1/2, between 30, 6, given the three terms 30, 6, 5, etc. Nor can there be here objection or reply, because what I have set out here in the demonstrative form of numbers is nothing