Final answer:
The frequencies of harmonics for a real string are slightly higher than the pure integer multiples due to its diameter affecting the tension and wavelength, causing inharmonicity and impacting the instrument's sound quality.
Step-by-step explanation:
The formula for calculating the frequencies of the harmonics of a real string taking into account its diameter is not provided explicitly, but it is mentioned that the frequencies of real string harmonics will be slightly higher than ideal integer multiples of the fundamental frequency. This discrepancy arises because the additional mass of real-world strings slightly alters the tension and therefore the wavelength for each harmonic. This effect is known as inharmonicity, which causes the harmonics to be sharper than the exact integer multiples.
The quality of the sound produced by an instrument can be affected because inharmonicity can lead to a less pure and potentially more complex sound. In some cases, this can add to the musicality by providing a richer, more distinctive timbre, but excessive inharmonicity might make the instrument sound out of tune or discordant.
The fundamental frequency of a string is f1 = Vw/2L, where Vw is the wave speed on the string, and L is the string's length. The frequencies of the first and second harmonics are f2 = 2f1 and f3 = 3f1, respectively, for an ideal string with no thickness. However, for real strings, these frequencies are slightly modified by the string's physical properties.