Notes

Outline

Alan Hovaness (1911 –   ) Alleluia and Fugue, Op. 40b and Prayer of Saint Gregory

Chapter 6 - Outline Standing waves Basics Frequencies and wavelengths Longitudinal waves Complex Waves Timbre

Standing Wave Produced when incident and reflected waves interfere. Principle of superposition.

Standing Wave There is no apparent motion along the direction in which the two individual waves move.

Standing Waves

Wavelengths of Standing Waves in a Rope n (ln/2) = L ln = 2L/n

Frequencies of Standing Waves in a Rope Since      f = v/l then fn = n(v/2L)

Video I-4 Transverse standing waves

Harmonic Series A series of frequencies in which all members are an integral multiple of the lowest frequency

Harmonic Series The lowest frequency is called the  fundamental frequency  or  first harmonic.

Harmonic Series The higher frequencies are called the second harmonic, third harmonic, fourth harmonic, etc.

Harmonic Series f2 = 2f1  f3 = 3f1 f4 = 4f1 etc.

Harmonic Series Harmonics above the first are also called overtones.

Standing Sound Waves Tube Open at Both Ends

Standing Sound Waves Tube Open at Both Ends: ln = 2L/n       fn = nf1

Standing Sound Waves

Standing Sound Waves Tube Closed at Both Ends: (same as rope - fixed ends) ln = 2L/n        fn = nf1

Standing Sound Waves

Standing Sound Waves Tube Closed at One End: ln = 4L/(2n-1) fn = (2n-1)(v/4L)

Standing Sound Waves Tube Closed at One End: f2 = 3 f1 f3 = 5 f1 f4 = 7 f1

Video III-1 and 2 Resonance in tubes

Complex Waves Created when frequencies which are members of a harmonic series are added.

Complex Waves

Complex Waves Same Amplitude

Complex Waves Same Amplitude

Complex Waves Same Amplitude

Complex Waves Same Amplitude

Complex Waves When waves whose frequencies are members of a harmonic series are added, the frequency of the resultant wave is always the same as that of the fundamental.

Auditory Demo  Missing Fundamental (virtual pitch) Track 37. Virtual Pitch with Random Harmonics Track 43-45

Fourier Synthesis Any periodic wave of frequency f1 can be produced by adding together sine waves of frequency f1, 2f1, 3f1, 4f1, 5f1, etc.

Fourier Synthesis Different Amplitudes

Fourier Synthesis Square Wave

Fourier Synthesis Square Wave

Fourier Synthesis Square Wave

Fourier Synthesis Square Wave

Fourier Synthesis Square Wave

Fourier Synthesis Square Wave

Fourier Synthesis

Fourier Analysis Any periodic wave of frequency f1, no matter how complex, can be broken down into sine waves of frequency f1, 2f1, 3f1, 4f1, 5f1, etc.

Fourier Analysis The set of sine waves that make up a complex wave are called the complex wave’s  Fourier Components.

Fourier Spectrum or Harmonic Spectrum A listing of the amplitudes of each component in either tabular or graphical form

Fourier Spectrum or Harmonic Spectrum

Fourier Spectrum or Harmonic Spectrum

Timbre The different combinations of harmonics gives different qualities or timbers to sounds.

Timbre

Timbre

Timbre

Auditory Demo The Effect of spectrum on timbre Track 53

Videos III-6 Vibrations on a Guitar String III-7 Fourier Analysis and Synthesis

Summary Any periodic wave of frequency f1 can be produced by adding together sine waves of frequency f1, 2f1, 3f1, 4f1, 5f1, etc.

Summary Any periodic wave of frequency f1, no matter how complex, can be broken down into sine waves of frequency f1, 2f1, 3f1, 4f1, 5f1, etc.

Summary The pitch we hear always corresponds to that of the fundamental frequency.