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

348 measure or not. If they have a common measure, they are rational, if they do not have a common measure, they are irrational; like the side of the square with regard to the diagonal. Therefore, due to the lack of a common measure, no proportion can ever be determined between them. From here it follows that if ever two sounds are between them as two lines are between each other, which have no common measure, as between these lines, so between those sounds no real proportion can be determined in any way, and therefore it will not be necessary to resort to the monochord, which always has sounds expressed by means of determined lines, and that between them serve a determined proportion. Let us come to our case in which two simultaneous sounds produce a very defined third sound, for example when D and F# produce the octave. This octave can certainly be expressed by means of lines, which have between them a determined ratio, for if they did not have it, the sound produced would not be an octave. This stated, one does not well understand how the numerical, or irrational linear, quantities are introduced, while these cannot serve to express the sounds produced, which are all determined, so one wishes to know what use can come, in this treatise on sounds, from irrationality, which in itself removes any proportion. Page 4. But how is it that &. It is said that two can be the term of the arithmetical progression 1, 2, 3, which requires the constant difference, which in this series is the unit. This is as it should be. One moves on to consider that 1 and 2 can be the first terms of a geometric progression; which is quite true. The reason, though, for how 1 and 2 can be first terms of this progression may seem obscure, it being far from the ordinary way of expressing geometry; whereas in the geometric progression the proportion is considered, and not the difference of the terms. The difference between 1 and 2 is always a unit, nor can it form any subduple proportion. The sole ratio of the unit twice contained in 2 constitutes it. In the position of these three terms, three things are said to be demonstratively true. The first, that there can be no progression between two terms when the third is undefined. This is very true, because undefined is the same as indeterminate. The progression consists in determined terms, so if the third is indefinite, one will not have a progression, at least a determined one. It is indeed true though that this impotence of progression is due to the third term being indefinite, but not due to there being no proportion between the finite and the infinite. Since the indefinite is as different from the finite as the finite from the infinite, given that the indefinite is indeed unknown, but finite by nature. Page 4. 5. The second thing demonstratively inferred remains quite confused. It is very true that the middle in progressions determines the extremes. It is also very true that the indefinite term cannot be fixed. Therefore, the 2 cannot be the middle of this progression and, if one wants, it cannot determine it. But that for this reason the subduple proportion of 1 to 2 is changed to a duple proportion of 1 to 1/2 by force of natural laws, which, as is inferred from the antecedents, is not sufficiently seen. The