Announcements 10/24/11
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Prayer
Term project proposals under review, I’ve responded to about 2/3
of them. I’ll send an email when I’m done, and when scores
should show up in computer.
You can change your project idea, but if so you’ll need to send me
a new proposal
Due today: HW 22 and HW 23
Exam 2 review session: Tuesday 5-6 pm. Room: C255 (not C460
where we were last time)
Exam 2 starts on Thursday morning, goes until next Tuesday
evening
Frank &
Ernest
Tone “quality”
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Why does a trumpet playing 440 Hz sound
different than when I whistle or sing the same
frequency?
The wave: Spectrum Lab as oscilloscope
The sounds have different ____________
… but both sounds have the same ____________
What does that imply about their Fourier
frequency components?
Tone quality, cont.
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Spectrum Lab as frequency analyzer
From unknown website
Tone quality, cont.
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Odd-sounding instruments (“tonal percussion”:
bells, xylophone, tympani, etc.)
From http://web.telia.com/~u57011259/Bellspectra.htm
Piano keyboard layout
Image: http://www.music-for-music-teachers.com/piano-keyboard.html
C-sharp/D-flat
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C
E
D
G
F
B
A
C
Half step: C to C-sharp (or, e.g. E to F)
2 half steps
Whole step (C to D): ___
12 half steps
Octave (C to C): ___
7 half steps
Fifth (C to G): ___
5 half steps
Fourth (C to F): ___
4 half steps
Major Third (C to E): ___
3 half steps
Minor Third (C to E-flat): ___
Chords
Image: http://www.music-for-music-teachers.com/piano-keyboard.html
C
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E
G
Why does this sound “good”?
Because they are all harmonics of the same note!
f  a1 cos(0t )  a2 cos(20t )  a3 cos(30t )  a4 cos(40t )  ...
What is the note?
– It’s actually a C, two octaves below the C that’s being
played!
– The frequencies of the three notes are 4:5:6
f  cos(40t )  cos(50t )  cos(60t )
(plus higher harmonics of each term)
C, E, G
G combined with G#
Chords, cont.
Chord
Freq. Ratios
Octave (C-C)
2:1
Major triad (C-E-G)
4:5:6
Minor triad (C-Eflat-G)
10:12:15
Major 7th (C-E-G-B)
8:10:12:15
Major-minor, “dominant” 7th
(C-E-G-Bflat)
Minor-minor, “minor” 7th
(C-Eflat-G-Bflat)
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4:5:6:7
10:12:15:18
“nice” chords: simple frequency ratios (small
integers), many harmonics of each note overlap
“ugly” chords: not many harmonics match
Trumpets
(Lets suppose a “C trumpet” instead of a regular “B-flat” trumpet, so we don’t have
to worry about the usual whole-step shift between piano and trumpet scales.)
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The notes you can play with no valves pushed in:
Note
Frequency
Ratio to Fundamental
1st harmonic: Low C
(with difficulty)
130.8 Hz
(fundamental)
1:1
2nd harm: Middle C
261.6
2:1
3rd harm: G
392.4
3:1
4th harm: C above
middle C
523.3
4:1
5th harm: E
654.1
5:1
6th harm: G
784.9
6:1
7th harm: B-flat??
915.7
7:1
8th harm: High C
1046.5 Hz
B-flat on piano = 932.3 Hz
8:1
Back to Pianos
A = 440 Hz
(defined as
reference)
(middle C)
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high B-flat
Why is a high B-flat on a piano 932.3 Hz?
How many half steps is it?
How many half steps in an octave?
How much frequency change in an octave?
12
12
2
Each half step = increase freq by a factor of ______
440 
 2
12
13
2
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So, why are there 12 half-steps in an octave?
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Smallest number of tones that can give you close to the right ratios
needed for harmonics and chords
 Fewer equally-spaced tones in a scale wouldn’t get close enough
 More equally-spaced tones in a scale adds unnecessary complexity
Note on piano
Frequency
How calculated
Ratio to
Fundamental
Low C
130.8 Hz
f1 = 21 half steps
below A (440 Hz)
1:1
Middle C
261.6
f1  212/12
2:1
G
392.0
f1  219/12
2.997:1
C above middle C
523.3
f1  224/12
4:1
E
659.3
f1  228/12
5.040:1
G
783.9
f1  231/12
5.993:1
B-flat
932.3
f1  234/12
7.127:1
High C
1046.5
f1  236/12
8:1
Which is better? The debate
“Equal-tempered”
“Just-intonation”
Advocated by Galileo’s father,
1581; Extremely influential work
by J.S. Bach, 1782: “The WellTempered Clavier”
Still used in many instruments,
without even thinking about it
(just not piano)
Same ratio between successive
notes: all halfsteps are the
same. C to Dflat = same as Bflat
to B
All halfsteps are not equal. In
fact, what’s a halfstep?
Makes key changes possible
without retuning instrument
Key changes sound very bad
unless you re-tune
Chords are a little off (not exact
Chords are precise (integer
integer ratios), e.g. C-E-G =
ratios exact), e.g. C-E-G = 4:5:6
4.000 : 5.040 : 5.993
No beats
Creates beats (see PpP Fig 7.1)
Disclaimer: In actuality, piano tuners don’t use a strict equal-tempered scale
The Exam
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“What’s on the exam?” (you ask)
Light
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Textbook: “Sometimes light acts like a wave, and other times
it acts like a particle.”
Colton: Light is made up of quantum-mechanical particles,
called “photons”. Electrons, protons, etc., are also quantum
mechanical particles. Quantum-mechanical particles are
neither waves nor particles in the macroscopic sense, but
rather we should think of the converse: “waves” and
“particles” as we typically use the words are based on our
observations of large-scale effects of these quantummechanical particles.
Colloquium speaker a few weeks ago: “Photons don’t exist.
They are only quantized oscillations of electro-magnetic
fields.”
Advertisement for grad school. 
The wave nature of light
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What is “waving”?
http://stokes.byu.edu/emwave_flash.html
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Medium?
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Polarization: quick definition