Announcements 10/25/10
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Prayer
Change to TA’s office hours: Monday will now be 5-6 pm (to match Wed
and Fri schedule).
Project proposals: in process of reading through & giving feedback
Exam 2: starts Thursday, goes through next Thurs
a. Covers through today’s lecture, Wed’s HW
– Is cumulative!
– Specifically, there are 3 multiple choice questions very similar
to ones from Exam 1.
– There is 1 worked problem very similar to one from Exam 1.
No partial credit on this one!
b. Exam review session: Wed Oct 27, 5:30 – 7 pm. Room: C295 (next
door)
c. Can bring handwritten 3x5 card (both sides)
– But just one!
d. Time: took me 35 minutes (just like last exam)
Tone “quality”
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Why does a trumpet playing 440 Hz sound
different than when I whistle 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 (aka overtones) of the same note!
 Their fundamental frequencies are all integer multiples of the
same number
 What happens when you add frequencies that are multiples of
the same number?
What is the note?
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It’s actually a C, two octaves below the C that’s being played!
The frequencies of the three notes are 4:5:6
Chords, cont.
Chord
Octave (C-C)
Major triad (C-E-G)
Minor triad (C-Eflat-G)
Major 7th (C-E-G-B)
Dominant 7th
(C-E-G-Bflat)
Minor 7th
(C-Eflat-G-Bflat)
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Freq. Ratios
2:1
4:5:6
10:12:15
8:10:12:15
4:5:6:7
10:12:15:18
Consonant chords: simple frequency ratios (small
integers), many harmonics of each note overlap
Dissonant 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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Handout: “What’s on the exam?”
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. (Same with electrons, protons, etc.)
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 quantum-mechanical particles.
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