Announcements 10/22/12
Prayer
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Term project proposals: I responded to all I received, and
everyone should have a score.
a. You can change your project idea, but if so you’ll need to
send me a new proposal
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Careful: Lab 6 due Wed night
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Exam 2 starts Thurs morning, goes until next Tues evening
a. Review session: this Tues, 5:30 – 7 pm, C255
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Frank &
Ernest
Demos
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f(x,t) tests with Slinky (results from last lecture)
From warmup
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Extra time on?
a. (nothing in particular)
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Other comments?
a. Liked it!
b. If i did the lab on time but forgot to turn it
in...is it still late?
From warmup
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Why does a trumpet playing a 440 Hz note sound
qualitatively different than when a violin plays the same
note? Both are producing 440 Hz waves, aren't they?
a. Yes they are, but they have different timbresmeaning that they are both playing the fundamental
and then certain levels of the harmonics above the
fundamental. Due to the different levels of
harmonics played there is a different feel to the final
tone.
wave shape
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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Frequency analysis
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
From warmup
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What is a chapter on musical scales doing in a physics
textbook? What's the connection to physics?
a. It is acoustics and AWESOME.
b. Relation between notes of frequencies. Types of scales
used were mathematical relations between frequencies.
c. Everything's connected to physics. Physics defines how
things happen. Music created by sound waves of certain
frequencies, and as we've been learning in class, waves,
frequencies, and harmonics have a lot of physics
applications.
d. Just cause we are physicists doesnt mean we are
incapable of doing anything else...like art or music. This
statement is not reflexive however, If you are a
musician....you probably cant do physics.
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, aka “dominant 7th”
(C-E-G-Bflat)
Minor-minor, aka “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
?
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
Start with 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)
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