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

260 strength which is necessary to proceed from the higher with an always smaller degree. How then can the Greek system ever fit nature, where after a semitone of the proportion 15/16 it ascends for an entire tone of proportion 8/9, and so on to others proceeding upwards? You shall indeed not find this disorder in my system, rather, you shall see, on the contrary, that as one ascends upwards, where degree by degree there is a need for greater strength, the degree becomes smaller and smaller in proportion, and this with such an order that it is unlikely to find another one better. The instruments mentioned, then, are clear enough per se , nor do they need examination in this respect. The other objection which you propose to me with regard to the consideration of the number as geometrical is solved by itself, if you consider the order of the number in its prime root, which I say is the progression of the duple 1/2, 2/4, 4/8, 8/16 etc., the said progression being certainly geometrical. The division of the string on the monochord, done though not as it is taught to us by the tutors either ancient or modern, but as I intend that it should be done, namely, the whole string, 1, half the string, a second, a third of the string, a third, etc., a fourth of the string, a fourth, etc., this (I say) explains the truth enough, as in it one sees most clearly that every division ends in a duple, as a most perfect term, from which it starts and in which it always ends. The first duple 1/2 is indivisible. The second duple 2/4 is divisible by 3, but is perfected in the 4. The third 4/8 is divisible by 5, 6, 7, but ends and is perfected in the 8, and so on for all the others. Therefore, taking just the terms of completion and perfection, we find the only geometric progression in many duples, and the only harmonic division of the same in the terms that these duples divide. With regard to what these Signori Maestri say, that if the system is useful for practical purposes, they shall embrace it, and if solely theoretical, it shall not concern them, I tell you, that if in this system there were no other utility than to scire rem per causam , and know it a priori , its utility would be excessive, and this is certainly the case. There is the other way of operating with such certainty that it is impossible to be mistaken. For the practical aspect there is the correct understanding of the placing of the notes when we want them to create the best harmony between them, whether they be consonances or dissonances. There is the acquisition of a quantity of parts really distinct from one another, and not either in unisons, or in octaves, or in octaves of the fifth and the third. There is a better understanding about modulations, and many other things which we shall see. Meanwhile, be patient if I am a little late in sending you the other two papers, one on the nature of consonances and dissonances, the other on the nature of the practical harmonic number. I have on my hands a serious complication, which has hitherto stopped me from being able to serve you, but it shall soon cease, and I shall fulfil my debt. Meanwhile, never hesitate to write to me clearly and distinctly all the doubts that come to you or the objections that are raised. The greatest pleasure which I have is this,