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|These graphics are reprinted with permission from the Fidelio Magazine, Volume I, No. 4, 1992. To read the complete article, click here.
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The six species of the human singing voice, and their registers
The violin family of instruments was developed in order to imitate, and then extend, the principles of the bel canto singing voice. Each member of the violin family has four strings, with each string turned at the musical interval of a fifth above or below the adjacent string or strings. (a) In the simplest case, each open (unfingered) string of the violin can be used as the lowest tone of a new surrogate vocal register. The succeeding higher tones fingered on that string remain in the same register, until the player changes to the next-higher string. For example, a register shift is simulated by moving from the C♯ played on the G string, up to the open D stringsimulating, for instance, the contraltos shift from first to second register. (b) The same principle applies to the open strings of the violaC, G, D, and A. Because the violas range straddles the usual treble and bass clefs, for clarity the same four strings are shown here using four different clefs; the treble clef, the modern tenor clef (sounds one octave lower than the treble clef), the alto clef (in which most viola music is written), and the bass clef. (c)The violoncellos open strings.
The violin plays the passage sung by the soprano, but as a displacement one octave lower. The octave displacement enables the violinist to imitate the II-III register shift by changing form the G string to the D string, and then the III-IV shift by switching to the A string.
By the nature of their construction, the woodwing instruments have registrations which are essentially fixed, even thought they can be modified to some degree by choosing altgernate fingerings for the same note. The wind instruments therefore tended to be designed and produced in sets or chest, whose member mostly corresponded to a particular species of singing voice. Above: a woodcut diagram of various wind instruments in use in the early seventeenth century.
Squaring the circle: Estimating the area of a square approximately equal to that of a given circle, as the average area of two regular polygons.
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An incribed polygon of 216 (65,536) sides may seem to closely approach a circl. but the perimeter of the polygon nca never become congruent with the circles perimter.
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In his 1673 On the Pendulum clock, Huygens demonstrated that a pendulum made to follow the path of a cycloid (curve MPI) will have the same period, no matter what the amplitude of the swingthat is, the cycloid is tautochronic.
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In 1697, Johann Bernoulli demonstrated that cycloid AMK had both the tautochronic property shown by Huygens, and the property of a brachistochronethat is, it is the least-action, least-time pathway of descent. Figure 10
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