6 The harmonic series in theory and practice.

Many musicians are unaware of the historical process described by Slaymaker, and assume that the higher modes of vibration of wind instruments and stretched strings are automatically precisely harmonic. Actually the way in which brass instruments are made to produce a set of frequencies which approximate to the harmonic series is a triumph of experimental ingenuity, aided in the recent past by computer aided design. Moreover, the match is only approximate, the natural resonances differing from the harmonic in predictable ways. For instance, in this writer's experience, the shorter configurations of the orchestral horn, from F# up to B flat, have resonances which become progressively closer, while on the longer configurations, from E down to B basso, the resonances become more widely spread. The clarinet exemplifies this process in much greater degree: its characteristics are only slightly modified from those of a parallel tube closed at one end, and the first two resonances have a frequency ratio which is just under 3. This effect is discussed by Benade.[20] It should be noted that the second resonance has not been eliminated; the process by which its frequency moves from twice to three times the fundamental frequency is continuous and can be followed progressively with tapered tubes in which the closed end expands from zero diameter (a complete cone) up to equality with the open end (a parallel pipe closed at one end).

That the higher modes of vibration of bowed strings (string "harmonics") are so close to the harmonic frequencies is also a triumph, but in this case of high quality workmanship, or of fine quality control of machine processes, which make the commercially available strings of the violin family remarkably uniform in cross-section and linear density. That this was not always the case is recorded by Spohr, in 1832. He warns his readers that nearly all gut strings are thinner at one end than the other, and recommends than all the thin ends of the strings be at the same end of the violin, so that moving a finger straight across from one string to the next will produce a fifth, despite the defects.[21] Such strings would have had a compressed set of resonances, so that the result of damping near the middle (a so-called harmonic) would have been flat to the octave, but Spohr does not record this, possibly because he was rather disapproving of the technique, warning young violinists not to emulate Paganini.[22]

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