Another term sometimes applied to these standing waves is overtones. The second harmonic is the first overtone, the third harmonic is the second overtone, and so forth. “Overtone” is a term generally applied to any higher-frequency standing wave, whereas the term harmonic is reserved for those cases in which the frequencies of the overtones are integral multiples of the frequency of the fundamental. Overtones or harmonics are also called resonances. In the phenomenon of resonance, a system that vibrates at some natural frequency is subjected to external vibrations of the same frequency; as a result, the system resonates, or vibrates at a large amplitude.

The sequence of frequencies defined by equation (25), known as the overtone series, plays an important part in the analysis of musical instruments and musical tone quality. If the fundamental frequency is the note G₂ at the bottom of the bass clef, the first 10 frequencies in the series will correspond closely to the notes shown in Figure 5. Here the frequencies of the octaves (harmonics 1, 2, 4, and 8) are exactly those of the notes shown, but the other frequencies of the overtone series differ by a small amount from the frequencies of the notes on the equal-tempered scale. The seventh harmonic is quite out of tune when compared with the actual note, so it is enclosed in parentheses.

During the Middle Ages in Europe, keyboard instruments were sometimes tuned to a scale in which the primary chords were true frequencies of the overtone series. This tuning method, called just intonation, provided beatless chords, because the notes in the chord were members of a single overtone series.

Mersenne’s laws

From equation (22) can be derived three “laws” detailing how the fundamental frequency of a stretched string depends on the length, tension, and mass per unit length of the string. Known as Mersenne’s laws, these can be written as follows:

1.The fundamental frequency of a stretched string is inversely proportional to the length of the string, keeping the tension and the mass per unit length of the string constant:

2.The fundamental frequency of a stretched string is directly proportional to the square root of the tension in the string, keeping the length and the mass per unit length of the string constant:

3.The fundamental frequency of a stretched string is inversely proportional to the square root of the mass per unit length of the string, keeping the length and the tension in the string constant:

Mersenne’s laws help explain the construction and operation of string instruments. The lower strings of a guitar or violin are made with a greater mass per unit length, and the higher strings made thinner and lighter. This means that the tension in all the strings can be made more nearly the same, resulting in a more uniform sound. In a grand piano, the tension in each string is over 100 pounds, creating a total force on the frame of between 40,000 and 60,000 pounds. A large variation in tension between the lower and the higher strings could lead to warping of the piano frame, so that, in order to apply even tension throughout, the higher strings are shorter and smaller in diameter while the bass strings are constructed of heavy wire wound with additional thin wire. This construction makes the wires stiff, causing the overtones to be higher in frequency than the ideal harmonics and leading to the slight inharmonicity that plays an important part in the characteristic piano tone.