Adding Sinusoids at Different
Frequencies
Music 175: Spectrum
Tamara Smyth, trsmyth@ucsd.edu
Department of Music,
University of California, San Diego (UCSD)
7
6
April 5, 2016
amplitude (offset)
5
4
3
2
1
0
−1
−2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time (s)
Figure 1: Adding 2 sinusoids at different frequencies.
• When adding sinusoids at different frequencies, the
resulting signal is no longer sinusoidal.
• But is it periodic?
1
Music 175: Spectrum
A Non-Sinusoidal Periodic Signal
2
Spectrum: Viewing in the Frequency
Domain
• If the frequencies of the added sinusoids are integer
multiples of the fundamental, the resulting signal will
be periodic.
• When viewing in the frequency domain, a “spike” at a
frequency indicates there is a sinusoid in the signal at
that frequency.
amplitude (offset)
12
10
6
8
6
4
2
0
−2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time (s)
4
1
2
0.8
magnitude
amplitude (offset)
8
10
0
0.6
0.4
0.2
−2
0
−4
0
5
10
15
20
25
30
frequency (Hz)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time (s)
Figure 3: Adding sinusoids at 5, 10, 15 Hz in both time and frequency domain.
Figure 2: Adding sinusoids at 3, 6, 9 Hz produces a periodic signal at 3 Hz.
• The height of the spike indicates the amplitude—all
sinusoids here have a peak amplitude of 1.
• The harmonic relationship between the partials can be
seen by the even spacing between the “spikes”.
• Sinusoidal components, or partials that are integer
multiples of a fundamental are calle harmonics.
Music 175: Spectrum
3
Music 175: Spectrum
4
Standard Periodic Waveforms
Spectra of Standard Waveforms
• square, triangle and sawtooth by adding sinusoids.
Square Wave Spectrum
Magnitude
1
Amplitude
1
0.5
0
−0.5
−1
0.8
0.6
0.4
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
Time (s)
0
5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
Amplitude
2
1
0.8
0.6
0.4
0.2
0
0
−1
−2
15
1
0
−1
Magnitude
Amplitude
1
−2
10
Frequency (Hz)
Triangle Wave Spectrum
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
5
10
15
Frequency (Hz)
Sawtooth Wave Spectrum
1
Time (s)
Magnitude
1
Table 1: Standard Waveforms Synthesized by Adding Sinusoids
Type
Harmonics
Amplitude
Phase (cos)
Phase (sin)
square
n = [1, 3, 5, ..., N]
1/n
−π/2
0
triangle
n = [1, 3, 5, ..., N]
1/n2
0
π/2
sawtooth
n = [1, 2, 3, ..., N]
1/n
−π/2
0
0.8
0.6
0.4
0.2
0
0
5
10
15
Frequency (Hz)
Music 175: Spectrum
Figure 4: Spectra of complex waveforms
5
Music 175: Spectrum
Harmonics and Pitch
6
Missing Fundamental
• Notice that even though these new waveforms contain
more than one frequency component, they are still
periodic.
Square wave (up to 7th harmonic)
1
0.8
0.6
0.4
• Because each of these frequency components are
integer multiples of some fundamental frequency,
they are called harmonics.
0.2
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Square wave (up to 7th harmonic) with MISSING FUNDAMENTAL
• Signals with harmonic spectra have a fundamental
frequency and therefore have a periodic waveform
(the reverse is, of course, also true).
1
0.8
0.6
• Pitch is our subjective response to the fundamental
frequency.
0.4
0.2
0
• The relative amplitudes of the harmonics
contribute to the timbre of a sound, but do not
necessarily alter the pitch.
200
400
600
800
1000
1200
1400
1600
1800
2000
Frequency (Hz)
• Listen to:
– square wave: squaref0.wav:
– square wave, NO fundamental: squareNOf0.wav:
• harmocity.pd
• Does the sense of pitch change? How about timbre?
Music 175: Spectrum
7
Music 175: Spectrum
8
Inharmonicity
Clarinet Analysis
• The (steady-state) tone of clarinet, mostly
closed-open, is sone in time and frequency domain.
• Generally, inharmonic overtones lack a clear sense of
pitch (difficult to hum).
• The perception of pitch
0.15
0.1
amplitude
– may vary with individuals;
– tends to be clearer when notes are played in
succession (particularly with inharmonic tones).
0.05
0
−0.05
−0.1
1.022
• Listen to:
1.024
1.026
1.028
1.03
1.032
1.034
1.036
1.038
1.04
1.042
time (s)
145 Hz
1
– bellsclip.wav: bell in isolation—pitch?
– bells.wav: bells in melodic context—pitch?
• The context allows us to focus on the change in notes
rather than on any one note itself.
magnitude
0.8
732 Hz
435 Hz
0.6
1171 Hz
0.4
1025 Hz
0.2
0
0
200
400
600
800
1000
1200
1400
1600
1800
2000
frequency (Hz)
Figure 5: Frequency analysis of a clarinet note.
• Summing sinusoids at 145, 433, 732, ..., can
approximate a steady-state synthesis of the clarinet:
– BbClar.ff.D3.wav
– clarsynth.wav
– clarSteadyState.wav
Music 175: Spectrum
9
Music 175: Spectrum
0.15
1
0.1
0.8
magnitude
tenor sax:
amplitude
Harmonicity and Pitch
0.05
0
−0.05
−0.1
9.08
9.085
0.4
0
9.09
magnitude
amplitude
0.4
0.2
0
−0.2
0.23
0.235
0.24
• There is a nonlinear relationship between pitch
perception and frequency in Hz:
0
500
1000
1500
2000
frequency (Hz)
– an octave above 220 Hz is an increase of 220 Hz;
– an octave above 440 Hz is an increase of 440 Hz.
1
• In equal-tempered tuning, there are 12 evenly spaced
tones in an octave, called semi-tones:
0.8
0.5
magnitude
amplitude
• An octave (P8) corresponds to a doubling
of frequency.
0.4
0
0.245
1
0
−0.5
0.6
0.4
– The frequency n semitones above A440 is
0.2
0
0.005
0.01
0.015
0
0.02
0
time (s)
500
1000
1500
2000
frequency (Hz)
0.3
440 × 2n/12 Hz.
1
0.2
0.8
magnitude
amplitude
2000
0.6
time (s)
0.1
0
−0.1
−0.2
– The frequency n semitones below A440 is
0.6
0.4
440 × 2−n/12 Hz.
0.2
−0.3
0.02
0.025
0.03
time (s)
Music 175: Spectrum
1500
0.2
−0.6
−0.4
1000
0.8
−0.4
Thunder:
500
1
0.6
−1
0
frequency (Hz)
0.8
Snare Drum:
• Listeners usually compare tones on the basis of the
musical interval separating them: m3, P5, P8 etc.
0.6
time (s)
Japanese Bell:
Pitch and Frequency
0.2
9.075
10
0.035
0
0
500
1000
1500
2000
• pitchFreq.pd
frequency (Hz)
11
Music 175: Spectrum
12
Beat Notes
• What happens when we add two frequencies that are
close in frequency?
2
1.5
1
0.5
0
210
212
214
216
218
220
222
224
226
228
230
0.45
0.5
Frequency (Hz)
Beat Note (f0 = 220 Hz, 2Hz envelope)
2
1
0
−1
−2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Time (s)
• The waveform shows a periodic, low frequency
amplitude envelope superimposed on a higher
frequency sinusoid creating a beat note.
• This can be explained by the Cosine Product formula:
cos(a + b) + cos(a − b)
cos(a) cos(b) =
2
Music 175: Spectrum
13