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Comparing Sounds: The Distinctive Acoustics of Piano and Guitar Strings

This chapter explores the fundamental differences in sound production between a piano and a drum. The piano, with its struck strings, produces clear and sustained pitches that contribute to melodic and harmonic completeness in music. It highlights the unique acoustical properties of long, thin strings that enable their natural modes of vibration to form a harmonic series, providing a more musical tone compared to percussion instruments. Additionally, the chapter delves into natural modes, standing waves, and the specific characteristics that establish musical pitch.

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Comparing Sounds: The Distinctive Acoustics of Piano and Guitar Strings

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  1. Chapter 10 (Hall) Piano and Guitar Strings PHY 1071

  2. Comparing sounds from a piano and a drum • The piano, with its struck strings, sounds very different from drums. • Piano sounds have clear, sustained pitch and provide all the necessary melodic, harmonic, and rhythmic elements to make a complete piece of music. • What underlying acoustical characteristics of the piano set it apart in this way? PHY 1071

  3. Answer: Special property of long, thin strings • Long, thin strings have a special property: Their natural mode frequencies form a harmonic series. • This is what makes their tone more musical than that of other percussion instruments. PHY 1071

  4. Outline • Natural modes of a thin string • Vibration recipes for plucked strings PHY 1071

  5. Natural modes of a thin string • Natural modes: Each of the special patterns of vibration that gives simple harmonic motion is called a natural mode. • These natural mode vibrations are sometimes called standing waves. • Natural mode frequency: Each natural mode has its own characteristic frequency, called the natural mode frequency. • Nodes N and antinodes A PHY 1071

  6. Natural mode frequencies • About each natural mode: the nth natural mode fits n “loops” (vibrating sections between adjacent nodes) into the available string length L. • About nodes: The distance between two adjacent nodes (or each loop) is half a wavelength. • So, we have n(1/2)n = L and n =2L/n. • Natural mode frequency fn = n(vt/2L), n = 1,2,3…, where • vt = (T/)1/2 is the velocity of transverse waves on the string. • T: the tension in the string; : the string's mass per unit length. • Fundamental frequency of the string: f1 = (1/2L)(T/)1/2. PHY 1071

  7. Most remarkable thing about the natural modes of a thin string • The natural mode frequencies of a thin string form a harmonic series. • All these modes have intimate musical relationships and cooperate in establishing a sense of pitch – the presence of definite pitch for a piano. • The pitch corresponds to the fundamental frequency f1. PHY 1071

  8. Examples • Suppose you have a string of length L = 0.5 m on which waves travel at speed vt = 150 m/s. What is the wavelength 6 and frequency f6 of the 6th mode of this string? • Suppose that this string vibrates in mode 4. Sketch its appearance at several successive moments during a cycle, also indicating direction of motion. PHY 1071

  9. Vibration recipes for plucked strings • The recipe of vibrations (Fourier spectrum) is determined by the place and manner in which we first excite (strike or pluck) the string. • The excitation of any mode is proportional to how much motion that mode has at the plucking point, meaning in particular that any mode with a node at the plucking point is omitted from the recipe. PHY 1071

  10. Example: plucking at a point located at L/5. Modes 5, 10, 15,… are absent because they have nodes at the plucking point. PHY 1071

  11. Homework • Ch. 10 (Hall), P. 195, Exercises: #1, 2, 3, 4, 7, 20. PHY 1071

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