![]() ![]() (For more on the history of valved brass, see History of the French Horn. So a valved brass instrument can find, in the comfortable middle of its range (its middle register), a valve combination that will give a reasonably in-tune version for every note of the chromatic scale. ![]() The valves can be used in combination, too, making even more harmonic series available. Usually one valve gives the harmonic series one half step lower than the valveless intrument another, one whole step lower and the third, one and a half steps lower. Each valve opens an extra length of tube, making the instrument a little longer, and making available a whole new harmonic series. The solution to these problems, once brass valves were perfected, was to add a few valves to the instrument three is usually enough. (An important exception was the trombone and its relatives, which can easily change their length and harmonic series using a slide.) The upper octaves of the series, where the notes are close enough together to play an interesting melody, were often difficult to play, and some of the harmonics sound quite out of tune to ears that expect equal temperament. As a known series, only a handful are used as often in. As a counterexam-ple, few series more clearly illustrate that the convergence of terms to zero is not sucient to guarantee the convergence of a series. ![]() A brass instrument could play only the notes of one harmonic series. The harmonic series, X n1 1 n 1+ 1 2 + 1 3 + 1 4 + 1 5 +···, is one of the most celebrated innite series of mathematics. For example, an a capella choral group, or a brass ensemble, may find themselves singing or playing perfect fourths and fifths, "contracted" major thirds and "expanded" minor thirds, and half and whole steps of slightly varying sizes.Īlthough limited by the fact that it can only play one harmonic series, the bugle can still play many well-known tunes.įor centuries, all brass instruments were valveless. Interestingly, musicians have a tendency to revert to true harmonics when they can (in other words, when it is easy to fine-tune each note). For more about equal temperament, see Tuning Systems. From the Fourier series we know that any linear combination of sine and cosine functions, whose frequencies form a harmonic sequence, results in a periodic. The negative aspect is that it means that all intervals except for octaves are slightly out of tune with regard to the actual harmonic series. (They do have the same frequency ratios, unlike the half steps in the harmonic series.) The positive aspect of equal temperament (and the reason it is used) is that an instrument will be equally in tune in all keys. In fact, modern Western music uses the equal temperament tuning system, which divides the octave into twelve notes that are equally far apart. But 7:8, 8:9, 9:10, and 10:11, although they are pretty close, are not exactly the same. For example, the interval between the seventh and eighth harmonics is a major second, but so are the intervals between 8 and 9, between 9 and 10, and between 10 and 11. If you have been looking at the harmonic series above closely, you may have noticed that some notes that are written to give the same interval have different frequency ratios. ![]()
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