Acoustics

note to §. 103.) one cannot produce every tone at will; rather, every figure—that is, every possible way of dividing the disc into simultaneously vibrating parts—stands in certain tonal ratios to the other figures. These ratios (which mostly correspond with the squares of certain numbers, or are sometimes irrational In this context, "irrational" refers to mathematical ratios that cannot be expressed as simple fractions, common in complex acoustic vibrations.) always remain the same relative to one another, regardless of what the specific tone of a figure may be, which itself depends on the thickness and size of the disc. Furthermore, the same tone can often occur with entirely different figures or modes of vibration, of which several examples have been given, particularly regarding rectangular and elliptical discs.

Since some contradictions to my acoustic observations mentioned in the notes to §. 109. and 206. were caused merely by misunderstandings—where the other party was not speaking of the exact same matter, and thus each of us was right in our own way—I request everyone who believes they have found something different than I have to first investigate precisely whether it concerns the exact same matter of which I spoke. For example, when investigating certain modes of vibration and their corresponding tonal ratios, one should check whether they are the same modes of vibration that I intended, or whether they are others; and when investigating certain properties of a sounding or sound-conducting body, whether what was found relates to the same properties or to others, and whether the same circumstances I presupposed are present. I mention this not because I claim never to have erred (which has happened several times, and which I have afterwards admitted and corrected), but rather because the time and effort spent by both sides on merely apparent contradictions and their rectification could be far more advantageously applied to actual progress in science.