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 (l_n/2) = L

l_n = 2L/n

Frequencies of Standing Waves in a Rope

Since

f = v/l then

f_n = 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:

l_n = 2L/n

f_n = nf₁

Standing Sound Waves

Standing Sound Waves

Tube Closed at Both Ends:

(same as rope - fixed ends)

l_n = 2L/n

f_n = nf₁

Standing Sound Waves

Standing Sound Waves

Tube Closed at One End:

l_n = 4L/(2n-1)

f_n = (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

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

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

Timbre

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

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.


	Standing waves
		Basics
		Frequencies and wavelengths
		Longitudinal waves
	Complex Waves
	Timbre


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


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


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


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


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


	Tube Closed at Both Ends:
	(same as rope - fixed ends)
	l_n = 2L/n
	f_n = nf₁


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


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


	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.


	Missing Fundamental
	(virtual pitch)
	Track 37.

	Virtual Pitch with Random Harmonics
	Track 43-45


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


	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.


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


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


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


	III-6 Vibrations on a Guitar String

	III-7 Fourier Analysis and Synthesis


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