The theory was first put forward by the famous physicist, Von Helmholtz, that the reason certain musical intervals are not agreeable to the trained ear is because the difference between the frequencies of the constituent fundamental tones or the harmonics present in them give rise to beats, approximately of 30 to 40 per second.
In order to simplify our explanations we will deal with two cases only, viz. that of the octave interval and that of the seventh. The first is a perfect concord, and the second, at least on stringed instruments, is a discord. It has already been explained that when a string vibrates it does so not only as a whole, but also in sections, giving out a fundamental note with superposed harmonics. Suppose we consider the octave of notes lying between the frequencies 264 and 528, which correspond to the notes C and C1 forming the middle octave on a piano. The frequencies and differences of the eight tones in this octave are as follows:—
| Frequencies of the Notes of the Middle Octave of a Piano. | ||||
| Notes. | Frequency. | Difference. | ||
| C | 264 | |||
| 33 | ||||
| D | 297 | |||
| 33 | ||||
| E | 330 | |||
| 22 | ||||
| F | 352 | |||
| 44 | ||||
| G | 396 | |||
| 44 | ||||
| A | 440 | |||
| 55 | ||||
| B | 495 | |||
| 33 | ||||
| C¹ | 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.