Square waveSquare wave

Fourier AnalysisFourier Analysis

+

+

=

sin(2 )f t

1sin(2 3 )

3f t

1sin(2 5 )

5f t

1sin(2 ) sin(2 2 )

31

sin(2 5 )5

f t f t

f t

Adding sines with Adding sines with multiple frequencies we multiple frequencies we

can reproduce ANY can reproduce ANY shapeshape

Joseph Fourier (1768-1830)Joseph Fourier (1768-1830)

1 1 2 2 3 3( ) sin(2 ) sin(2 2 ) sin(2 3 ) ...P t A f t A f t A f t

ANYANYperiodic periodic functionfunction

fundamentalfundamental 11stst harmonic harmonic 22ndnd harmonic harmonic

integer multiples of integer multiples of fundamental frequencyfundamental frequency

AAii and and i i wave shapewave shape

timbretimbre

Ohm’s lawOhm’s law

We (We (pretty muchpretty much) can’t hear the phases) can’t hear the phases

sin(2 ) 0.5sin(2 2 )

0.2sin(2 3 ) sin(2 4 )

f t f t

f t f t

sin(2 ) 0.5sin(2 2 )

0.2sin(2 3 ) sin(2 4 )4

f t f t

f t f t

AA i i timbretimbrebut not but not ii

Fourier spectrumFourier spectrum

same same information information (except the (except the

phases)phases)

ff

AA

Examples of Fourier spectraExamples of Fourier spectra

Clarinet nowClarinet now

It is now a great time to read It is now a great time to read chapter 4 of Berg & Storkchapter 4 of Berg & Stork

Roughly …Roughly …

Amplitude LoudnessAmplitude Loudness

Frequency PitchFrequency Pitch

Wave shape TimbreWave shape Timbre

But …But …

Tone qualityTone quality

other things contributing to other things contributing to timbre besides the waveform of timbre besides the waveform of

the steady tonethe steady tone

Roughly …Roughly …

Amplitude LoudnessAmplitude Loudness

Frequency PitchFrequency Pitch

Wave shape TimbreWave shape Timbre

It is the wave shape of the It is the wave shape of the whole sound that matters, whole sound that matters,

not only of the “steady not only of the “steady state”state”

Regarding timbre …Regarding timbre …

Attack and decay transientsAttack and decay transients

Spectrum decaySpectrum decay

InharmonicitiesInharmonicities

sin(2 ) sin(2 2.01 ) sin(2 3.1 )f t f t f t

sin(2 ) sin(2 2 ) sin(2 3 )f t f t f t

higher harmonics higher harmonics slightly off the slightly off the

integer x f valueinteger x f value

VibratoVibrato

TremoloTremolo

oscillation in oscillation in frequencyfrequency

oscillation in oscillation in amplitudeamplitude

FormantsFormants

Chorus effectChorus effect

[let us all sing together][let us all sing together]

Some real examplesSome real examples

but first, but first, spectrographsspectrographs