Download Unit 10.2 The Physics of Music Objectives Notes on a Piano

Unit 10.2 The Physics of
Music
Teacher: Dr. Van Der Sluys
Objectives
• The Physics of Music
– Strings
– Brass and Woodwinds
• Tuning - Beats
Notes on a Piano
Key
Note
Frquency (Hz)
Wavelength (m)
52
C
524
0.637
51
B
494
0.676
50
A# or Bb
466
0.717
49
A
440
0.759
48
G# or Ab
415
0.805
G
392
0.852
370
0.903
47
46
F#
or
Gb
45
F
349
0.957
44
E
330
1.01
43
D# or E b
311
1.07
42
D
294
1.14
41
C# or Db
277
1.21
40
C (middle)
262
1.27
http://en.wikipedia.org/wiki/Piano_key_frequencies
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Vibrating Strings Fundamental and Overtones
A vibration in a string can
produce a standing wave.
Usually a vibrating string
produces a sound whose
frequency in most cases is
constant. Therefore, since
frequency characterizes the
pitch, the sound produced
is a constant note. Vibrating
strings are the basis of any
string instrument like guitar,
cello, or piano. For the
fundamental, λ = 2 L where
L is the length of the string.
L = 1/2 λ 1
L = λ2
L = 3/2 λ 3
L = 2 λ4
L = 5/2 λ 5
L = 3 λ6
L = 7/2 λ 7
Vibration, standing waves in a string,
The fundamental and the first 6
overtones which form a harmonic series
http://en.wikipedia.org/wiki/Vibrating_string
Length of Piano Strings
The highest key on a piano corresponds to a
frequency about 150 times that of the lowest
key. If the string for the highest note is 5.0 cm
long, how long would the string for the lowest
note have to be if it had the same mass per
unit length and the same tension?
If v = fλ, how are the frequencies and length of
strings related?
Other String Instruments
• All string instruments produce sound from one or more
vibrating strings, transferred to the air by the body of the
instrument (or by a pickup in the case of electronicallyamplified instruments). They are usually categorized by
the technique used to make the strings vibrate. The three
most common techniques are plucking, bowing and
striking.
• A vibrating string on its own makes only a very quiet
sound, so string instruments are usually constructed in
such a way that this sound is coupled to a hollow
resonating chamber, a sounding board, or both. On the
violin, for example, the taut strings pass over a bridge
resting on a hollow box. The strings' vibrations are
distributed via the bridge and soundpost to all surfaces of
the instrument, and are thus made louder.
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Production of Multiple Notes
A string at a certain tension will only produce one note, so to
obtain multiple notes string instruments employ one of two
methods. One is to add enough strings to cover the range of notes
desired; the other is to allow the strings to be stopped. The piano
is an example of the former method, where each note on the
instrument has its own set of strings. On instruments with
stoppable strings, such as the violin or guitar, the player can
shorten the vibrating length of the string, using their fingers directly
(or more rarely through some mechanical device, as in the hurdy
gurdy). Such instruments usually have a fingerboard attached to
the neck of the instrument, providing a hard flat surface against
which the player can stop the strings. On some string instruments,
the fingerboard has frets, raised ridges perpendicular to the strings
that stop the string at precise intervals, in which case the
fingerboard is called a fretboard.
An Air Displacement Wave is
also an Air Pressure Wave
The nodes of the displacement wave, where the air is not rushing
back-and-forth but is doing the most piling-up-and-spreading-out,
are the antinodes of the pressure wave. The antinodes of the
displacement wave, where the air is rushing back-and-forth the
most, but is not piling up or spreading out at all, are the nodes of
the pressure wave. Both waves must have exactly the same
frequency, of course; they are actually just two aspects of the
same sound wave.
http://cnx.org/content/m12589/latest/
Vibrations and Open Tubes
These are the first four harmonics allowed in an open
tube. Any standing wave with a displacement antinode
at both ends is allowed, but the lower harmonics are
usually the easiest to play and the strongest harmonics
in the timbre. The longitudinal waves are represented
as pressure waves
3
Vibrations in Closed Tubes
Again, these are the lowest (lowest pitch and lowest
frequency) four harmonics allowed. Any wave with a
displacement node at the closed end and antinode at
the open end is allowed. Note that this means only the
odd-numbered harmonics "fit".
Woodwind and Brass
Instruments
The two shapes that are
useful for real wind
instruments are the cylinder
and the cone. Most real
wind instruments are a
combination of cylindrical
and conical sections, but
most act as (and can be
classified as) either
cylindrical bore or conical
bore instruments.
Higher Notes in Woodwind
and Brass Instruments
Just as on a string, the actual wave inside the instrument is a
complex wave that includes all of those possible harmonics. A
cylinder makes a good musical instrument because all the waves
in the tube happen to have simple, harmonic-series-type
relationships. This becomes very useful when the player
overblows in order to get more notes. As mentioned above,
woodwind players get different notes out of their instruments by
opening and closing finger holes, making the standing wave tube
longer or shorter. Once the player has used all the holes, higher
notes are played by overblowing, which causes the next higher
harmonic of the tube to sound. In other words, the fundamental of
the tube is not heard when the player "overblows"; the note heard
is the pitch of the next available harmonic (either harmonic two or
three). Brass players can get many different harmonics from their
instruments, and so do not need as many fingerings.
4
Tuning Instruments - Beats
A 110 Hz sine
wave (cyan), a
104 Hz G# sine
wave (magenta),
their sum (blue)
and the
corresponding
beat frequency,
BF (red)
Bf = f1 - f2
A Beat Problem
A tuning fork produces a steady 400 Hz
tone. When this tuning fork is struck and
held near a vibrating guitar string, twenty
beats are counted in five seconds. What
are the possible frequencies produced by
the guitar string?
Musical Scales
http://en.wikipedia.org/wiki/Chord_(music)
http://en.wikipedia.org/wiki/Musical_keyboard
•
•
•
•
Each octave represent a doubling of frequency.
In western music there are twelve notes per octave. Each note
increases in frequency by a factor of 21/12 or a half step.
On the piano, the white keys represent a C scale, or the notes
C, D, E, F, G, A, B, C. The difference in frequencies between
C, D and E as well as F, G, A and B are full steps or a
difference of 22/12, where as the difference between E, F and B,
C is only a half step or 21/12
If two note are played simultaneously and only separated by a
half step, the sound will be discordant.
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