Review

Unit 3 Pipes, Voice and Percussion

Essential Facts and Concepts

Pipes

  • Open (cylindrical) pipes: fundamental frequency f1 = v/2L’ ,
  • v= 343+0.6 (T-20C) the speed of sound
  • L’ = length of pipe (with end correction δL = 0.3 Diameter
  • Harmonics: all present fn =n f1
  • Stopped (cylindrical) pipes: fundamental frequency f1 = v/4L’,
  • v= 343+0.6 (T-20C) the speed of sound
  • L’ = length of pipe = end correction
  • Harmonics: only odd present fn =(2n-1) f1
  • Conical pipes: fundamental frequency f1 = v/2(L’+c)
  • Cones are stopped pipe but act like open pipes.
  • v as above
  • L’+c = length of cone plus end correction.
  • Standing Waves
  • Standing waves result from the combination of the wave traveling down the pipe and the wave reflected from the end and holes.
  • Woodwinds and brass change pitch by changing the length of the pipe and by exciting different harmonics.
  • Impedance
  • The impedance is the ratio of the pressure fluctuation to the flow fluctuation: Z =p/U
  • When the impedance Z changes abruptly there are reflections of sound.
  • Therefore there are reflections at the ends (open and stopped) and from holes along the pipe.
  • The holes produce (by wave interference) standing waves for a pipe of an effectively shorter pipe.
  • Sounding the pipe
  • Flue and Fipple pipes:
  • The transverse flute (and other fipple woodwinds) and flue pipe are sounded by the edge tone of the fipple or embouchure.
  • Edge tone occurs because Bernoulli principle causes turbulent oscillation.
  • Bernoulli principle: higher velocity air is a lower pressure.
  • Pressure node near the open fipple.
  • The frequency of the edge tone is fedge = 0.2 vjet /b, where vjet is the velocity of the jet of air and b is the width of the opening in the fipple.
  • The transverse flute is an open cylindrical pipe in which the standing waves in the pipe produced by reflections from the ends and holes feedback to control the precise frequency of the edge tone.
  • Reed Pipes
  • Reeds are soft reeds if their they are responsive to feedback, i.e. they have a broad resonance; the reeds are hard reeds if they are unresponsive to the feedback because they have a sharp resonance.
  • Reeds work by Bernoulli principle: higher velocity air is a lower pressure.
  • Pressure anti-node at the reed.
  • The clarinet reed is a soft reed.
  • The clarinet is a stopped cylindrical pipe.
  • Fundamental frequency = v/4lL’
  • Only odd harmonics for first few overtones
  • Other woodwinds: sax, oboe, bassoon, etc have conical bore.
  • Fundamental frequency = v/2(L’+c)
  • All harmonics present.
  • All orchestral woodwind instruments have soft reeds.
  • Brass
  • Brass instruments are stopped pipes with combinations of cylindrical and conical pipes with a pronounced bell and a mouthpiece.
  • The pipe is sounded by buzzing lips.
  • The lips are a soft reed/valve.
  • Pressure anti-node at the players lips.
  • The pitch is changed primarily by changing the length of the pipe with slides and valves and by exciting different harmonics.
  • The combination produces a nearly complete harmonic series expect that the lowest harmonic is not the fundamental of the series.
  • The lowest frequency of the pitch is inharmonic and is called the “pedal tone.”
  • The trumpet, cornet and Flugel horn are examples of similar instrument in which the percentage of conical tubing varies. (More cone for Flugel, less of cornet, least for trumpet).
  • The mouthpiece cup volume and diameter of the tapered back-bore determine the resonance frequencies of the mouth piece, similar to a Helmholtz resonator.
  • The “flare constant” m in “exponential horns” determines how fast the bell opens up; a large m means an abrupt widening; a small m a gentle taper.
  • A Bessel horn is a slightly different kind of horn, similar to French Horns in shape.
  • Timbre of Pipes
  • The shape and diameter of the pipe determine the cutoff frequency for 3-D modes of oscillations of the air.
  • Only frequencies higher than the cutoff frequencies can participate in the 3-D modes.
  • The shape and size of the pipe and any bell affect the harmonic recipe and therefore the timbre of the pipe.
  • Mutes filter the sound of the pipe, changing its timbre.
  • The French Horn player’s hand changes the timbre of the pipe as well as changing its effective length.
  • The finish of the inside of the bore and the edges of holes and joints affect the harmonic recipe in fine instruments.

Voice

  • The vocal tract is the instrument of the human voice.
  • Sound originates in from the modulation of air by the vocal folds located in the larynx.
  • The vocal folds are hard reeds, not responsive to the standing waves in the vocal tract.
  • The pitch of the voice is determined by the density, length and tension (external and internal) of the vocal folds.
  • The vocal tract produces broad resonances called “formants” at about 500 Hz, 1500 Hz and 2500 Hz.
  • The intelligibility of speech is due to the relative frequency of the first two formats.
  • In singing the formants are modified from speech to produce matched resonance and harmonics of the vocal fold frequencies.
  • The singer’s formant is the third formant (~2500 Hz) that permits a singer to produce a significant amount of acoustic energy at a very high frequency that has little competition from other instruments by modification of the vocal tract.

Percussion

  • Percussion instruments are instruments that are struck.
  • Percussion instruments produce sound from the vibration of their natural modes of oscillation.
  • Most percussion instruments do not have pitch.
  • Pitch results from a harmonic series.
  • The fundamental frequency of a circular membrane scales as the inverse of the diameter and the thickness and as the square root of the ratio of the surface tension and the areal density, that is, the bigger the drum head or the thicker or the more dense the material the lower is the fundamental frequency. The tighter the stretch, on the other hand, the higher the range.
  • Bars and flexing sheets are dispersive (velocity of the wave depends on frequency) in their behavior.
  • For circular disks, the fundamental frequency depends on the inverse square of the diameter and is also proportional to the thickness.
  • The stiffer the bar or disk, the higher the range; the longer bar or larger diameter the lower the range.
  • The natural modes of membranes, circular sheets of metal and rectangular bars are not harmonic (whole number rations of the fundamental).
  • Air lowers the range of real drum heads.
  • Timpani achieve approximate pitch by (1) air loading; (2) resonances of the air in the kettle; (3) excitation of only certain modes that are nearly harmonic.
  • The fundamental of timpani does not “speak.”
  • Toms achieve near pitch by accentuating the fundamental with a resonance of the cavity beneath the membrane.
  • Tabla have approximate pitch by use of a Helmholtz resonator.
  • Xylophone, Marimba and vibraphone use modified bars and resonant cavities to achieve pitch.
  • The differences between a xylophone and marimba are (1) marimba is under cut more than xylophone; (marimba has curved undercut; xylophone, abrupt); (2) first overtone of marimba is 4x times fundamental; xylophone is 3x.
  • Marimba sound more “mellow;” xylophone sounds “brighter.”
  • Vibraphone is like xylophone but with metal bars and tremolo mechanism.
  • Resonate pipes absorb energy of sound from one side of bar, eliminating destructive interference.