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

351 LETTERS adapted to an extendible sounding string; and it has been demonstrated in what precise mathematical ratio the sounds physically produced are amongst them. Omitting the second and third objection (and the reason shall be given at the end), it is said in the letter received: The second thing which is inferred remains quite confused; and it is: the mean 2 not being able to have progress with the indefinite term, in consequence the subduple 1,2 is converted into duple 1, 1/2 etc.: how this is inferred from the antecedents is not sufficiently clear etc. If the 2 inferred is the harmonic mean between the given unit and the indefinite term, be it x, it shall be the natural constitution of the three terms determined by the mean in the harmonic progression, x, 2, 1; nor shall it ever be 1,2, x. X not being realisable, and hence subtracted, the two concrete terms 2, 1 shall remain; never 1, 2; in the same way, sense and concept in which, given three terms of an arithmetical series 1, 2, 3, if one subtracts either the first or the third or even the second, always leaving two of them, the remaining two will always proceed from the lesser to the greater; 1, 2, subtracted 3. 2, 3, subtracted 1. 1, 3, subtracted 2. Nor will it ever be said 2, 1, subtracted 3, 2, subtracted 1. 3, 1, subtracted 2; and this, according to the nature of the series. This deduction is then a natural consequence of the proportion. This will be better understood supposing, by hypothesis, that the term 2 had been deduced as an arithmetical mean. Then, the natural state of the three terms would be 1, 2, x; subtracted x, 1, 2 subduple would remain. If the hypothesis is converted into the thesis, the deduction will clearly be seen. It is said in the letter: The third thing demonstratively inferred is that, if the subduple ratio 1, 2 is permuted in the duple 1, 1/2, the third term shall be 1/3 whose third term does not stand up either in harmonic progression, or geometrical, or arithmetical. I confess that here I do not understand the fault or the correction. If rigorously speaking, mathematicians wish to say that these three terms, 1, 1/2, 1/3, are in proportion, and not in harmonic progression, although this rigour appears childish to me, I shall grant it to them. But it must then necessarily be granted to me that the three terms, 1, 1/2, 1/3, are the first three terms of the infinite harmonic progression, which is enough for my intention, and actually is precisely my intention. So as not to be mistaken I decided to ask Signor Abate Succi (public professor and eminent mathematician). Not only did he not raise any objections to me, but neither is he able to understand how there could be any objection over said proposition. So you should explain yourselves better, because neither Succi nor I understand the objection. Coming to practical music, one observes in the third sound a certain inconstancy, that is to say, conduct which cannot be ordered. That being given, it can be observed from Figure III, that the minor 3rd C# and E, produces as a third sound below an A, which is a 17th below. The third sound A is not a 17th; it is a 12th