C 264 33 D 297 33 E 330 22 F 352 44 G 396 44 A 440 55 B 495 33 C^1 528 It will thus be seen that the differences between the frequencies of adjacent notes are such as to make beats between them which have a number per second so near to the limits of 30 to 40 that adjacent notes sounded together are discords. Suppose, however, we sound the _seventh_, viz. _C_ and _B_, together. The frequencies are 264 and 495, and the difference is 231. Since, then, the difference between the frequencies lies far beyond the limit of 30 to 40 per second, how comes it that in this case we have a discord? To answer this question we must consider the harmonics present with the fundamentals. Write down each frequency multiplied respectively by the numbers 1, 2, 3, 4, etc.— C. B. Fundamental 264 495 First harmonic 528 990 Second ” 792 1475 Third ” 1056 1980 Fourth ” 1320 2475 Fifth ” 1584 2970 On looking at these numbers we see that although the difference between the frequencies of the two fundamentals is too great to produce the disagreeable number of beats, yet the difference between the frequencies of the fundamental of note B (495) and the first harmonic of note C (528) is exactly 33, which is, therefore, the required number. Accordingly, the discordant character of the seventh interval played on a piano is not due to the beats between the primary tones, but to beats arising between the first harmonic of one and the fundamental of the other. It will be a useful exercise to the reader to select any other interval, and write down the primary frequencies and the overtone frequencies, or harmonics, and then determine whether between any pairs disagreeable beats can occur. The presence of harmonics or overtones is, therefore, a source of discord in some cases, but nevertheless these overtones communicate a certain character to the sound. Helmholtz’s chief conclusions as regards the cause of concord and discord in musical tones were as follows:— (1) Musical sounds which are pure, that is, have no harmonics mixed up with them, are soft and agreeable, but without brilliancy. Of this kind are the tones emitted by tuning-forks gently struck or open organ-pipes not blown violently.