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Scan 29
Harvard Dictionary Of Music, 1944
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1ACOUSTICS
3tones, all the others of the diatonic scale can be derived [see *Intervals, Calculation of, II]. The result is as follows:
5<table>
6 <tr>
7 <th></th>
8 <th>c</th>
9 <th>d</th>
10 <th>e</th>
11 <th>f</th>
12 <th>g</th>
13 <th>a</th>
14 <th>b</th>
15 <th>c'</th>
16 </tr>
17 <tr>
18 <th>Frequency (n = 1):</th>
19 <td>1</td>
20 <td>9/8</td>
21 <td>5/4</td>
22 <td>4/3</td>
23 <td>3/2</td>
24 <td>5/3</td>
25 <td>15/8</td>
26 <td>2</td>
27 </tr>
28 <tr>
29 <th>Frequency (n = 24):</th>
30 <td>24</td>
31 <td>27</td>
32 <td>30</td>
33 <td>32</td>
34 <td>36</td>
35 <td>40</td>
36 <td>45</td>
37 <td>48</td>
38 </tr>
39 <tr>
40 <th>Vibrating length:</th>
41 <td>1</td>
42 <td>9/8</td>
43 <td>4/3</td>
44 <td>3/2</td>
45 <td>5/3</td>
46 <td>8/5</td>
47 <td>15/8</td>
48 <td>1/2</td>
49 </tr>
50</table>
52The illustration [Ex. 7] shows a number of frequencies calculated for the tone f' = 360 (the correct frequency for f' is 352). It must be noted that these frequencies give the tones of *just intonation, not of equal temperament [see *Temperament].
54IV. Harmonics. The acoustic effect produced by a single vibration of the type described above is called a pure sound; but practically no vibrating body produces a pure sound. All the musical instruments produce composite sounds,
56
58Frequency; Harmonics
60consisting of the main sound, or fundamental, plus a number of additional pure sounds, the so-called overtones, which, however, are not heard distinctly because their intensity (amplitude) is much less than that of the main sound. The frequencies of the overtones are exact multiples of the frequency of the fundamental. In other words, an instrument which produces the tone of the frequency n actually produces vibrations (pure sounds) of the frequencies n, 2n, 3n, 4n, … (up to 20n and more). The illustration [Ex. 8] shows the first 15 overtones of the tone C. A more common designation for these tones is partials or harmonics. It should be noted, however, that these terms (if properly used) include the fundamental, while the term overtone (if properly used) excludes it. Thus, the first overtone is the second harmonic, etc. Although the terms harmonics and partials are frequently used as interchangeable, the latter has, in scientific studies, a wider significance, since it includes also non-harmonic overtones, such as occur in noises, also in bells. With the exception of the octaves (2,4,8) none of the harmonics are tones of equal temperament. Those which result from the factors 3 and 5 (3,5,6,9,10,12, etc.) are tones of *just intonation (see the above table of frequencies) whereas the harmonics 7, 11 and 13 (indicated by black notes) can only approximately be identified with tones available in our system of tuning and notation. As can easily be seen, the 7th harmonic, which is 7 = 6 3/9, is lower than the B-flat of just intonation which is 16/9 × 4 = 6 4/9; this, in turn, is slightly lower than the B-flat of equal temperament (in *cents, the three tones are: 972, 996, 1000, respectively). Similarly, the 11th harmonic, which is 11 = 4 4/4, is lower than the F-sharp of just intonation (15/2 × 3/2 = 4 5/4) and, in fact, nearer to the F than to the F-sharp of equal temperament. Finally, the 13th harmonic is 13 = 3 9/3, whereas the A of just intonation is 5/3 × 8 = 4 0/3.
62The physical cause of the harmonics is to be found in the fact that a vibrating body, such as a string, vibrates simultaneously as a whole and in sections of one-half, one-third, one-fourth, etc., of the entire length. The secondary vibrations, however, have a much smaller amplitude, approximately between one-fifth and one-fiftieth of that of the fundamental [Ex. 9].
64The existence of these additional tones in what the ear believes to be a single sound was shown first by Helmholtz (1821-94), by means of *resonators of various sizes which reinforce one frequency and eliminate all the others. The harmonics can easily be demonstrated by the following simple experiment on the pianoforte: Depress the key of C with-
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