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

336 prove it to be impossible, hence it is greatly doubted that the physical-harmonic art may achieve it, unless we should wish to achieve it by approximation, as indeed the mathematicians can do. The tuning of an instrument according to a certain rational interval will have a relation to numbers, which express the proportion of the sounds between which the interval stands, and if the numbers are rational, one may determine the interval, but if they are irrational, one cannot well understand how to determine it, given that irrational numbers are the same as those they call ‘deaf numbers’, and they cannot be expressed by integers or fractions. One therefore needs to know clearly what is intended by the term rational or rational interval. If the two bodies of air generated by the vibrations of the two strings of the violin, viola or cello generate a third sound on striking one another, the same should happen on other instruments, especially wind instruments such as the organ, oboe, flutes, etc. as also on instruments with metal strings. Attempts are still being made to see if the same phenomenon is found on these. If the two volumes of air generated by the sounding strings succeed in generating the third sound in the impact, it is not clear why the interval of the octave should not generate it, when two strings tuned a fifth apart succeed in creating the unison of the fundamental, given that there is almost the same diversity between the vibrations of two strings at the octave and of two strings tuned as a fifth, fourth, etc. It is said that irrational numbers also produce their third sound; it would be much desired to understand with some example what third sound they produce. If the terms are taken at their own value, the definition of a mathematical point is physically impossible, while the mathematical point is so precise that an infinitely small difference excludes it and yet this very difference cannot remove a physically indivisible point, which is the point that our efforts can reach. By the term harmonic proportion, in which one assumes the numbers in the exposition are arranged, it is to be desired to know if one means the same proportion that the geometers call harmonic, in which the two extremes stand in the ratio of the differences of each of these extremes from the third, or if the term harmonic proportion should understood as some other musical proportion.