Transcript sound - The Bennett Conservatory of Music

The Physics of Music
Charles H. Bennett
Bennett Conservatory of Music
April 12, 2015
What is sound?
Aristotle understood the gist of it over 2000 years ago.
“Sound takes place when bodies strike the air, . . . by its being
moved in a corresponding manner; the air being contracted and
expanded and overtaken, and again struck by the impulses of the
breath and the strings, for when air falls upon and strikes the air
which is next to it, the air is carried forward with an impetus, and
that which is contiguous to the first is carried onward; so that the
same voice spreads every way as far as the motion of the air takes
place.” —Aristotle (384–322 BCE), Treatise on Sound and Hearing
But light was poorly understood until much more recently.
(see me after class)
Click to show Aristotle’s correct idea of sound propagation
( You may have to click it a few times)
Physical versus Perceptual properties of Sound
Loudness ---------- Amplitude
Pitch
---------- Frequency
Timbre ---------- Wave shape
If the vibrations repeat, exactly or approximately, between
about 40 to 15,000 times per second,
the sound is perceived as a musical tone with a definite pitch.
Otherwise it is perceived as a non-musical sound.
We can see this on an Oscilloscope display, showing air
pressure versus time in milliseconds, or a Sound Spectrograph
display, showing frequency and intensity versus time in
seconds.
Most musical instruments, except tuning forks, produce a mix of frequencies.
For string and wind instruments these are whole number multiples of some
lowest, or “fundamental” frequency.
Mathematically, the repeat frequency, which corresponds to the pitch, is the
Greatest Common Divisor of all the frequencies the mixture.
For example the B above middle C corresponds to 500 vibrations per second.
A tuning for tuned to that note would produce only that frequency. But a
piano playing that note also produces 1000, 1500, 2000 frequencies. If the
500 frequency were removed, the note would still sound like the B above
middle C, but with a different timbre or tone quality, because 500 is the
greatest common divisor of 1000, 1500, and 2000.
If the 1500 frequency were removed, what pitch would the remaining mix of
1000 and 2000 vibrations have?
n=1
n=2
n=3
n=4
n=5
n=6
First 6 Normal Modes
of vibration of a string.
n’th mode frequency is
n times the fundamental.
Red dots denote nodes,
i.e. places on the string
that don’t move.
Harmonics on a string instrument:
n=1
n=3
n=5
n=2
n=4
n=6
Placing a finger gently against the
string damps out all modes except
those with a node at that position.
So, placing a finger half way
along the string leaves only
the even modes sounding.
The pitch is an octave above
the fundamental.
Placing a finger 1/3 of the
way along the string leaves
only the modes divisible by 3
sounding (e.g. n=3 and 6).
The pitch is an octave and a
fifth above the fundamental.
Harmonious
musical intervals
correspond to small
whole number
frequency ratios.
Using the “circle of
fifths” one can tune a
piano by alternately
going up a fifth (3/2)
and down a fourth
(3/4) until you have
gone up a whole
octave.
Octave
Fifth
Fourth
Major third
Minor third
2:1
3:2
4:3
5:4
6:5
But the resulting octave will be a
little sharp, because (9/8)6 =
Perfect
Tempered
2.0273 is a little more than 2. 2:1 = 2.000 2.0000
Click speaker to hear
3:2 = 1.500 1.4983
the 1% difference.
4:3 = 1.333 1.3348
To mitigate this problem a
5:4 = 1.250 1.2599
slightly flattened fifth is used
6:5 = 1.200 1.1892
instead, causing the octave to
be in perfectly in tune but the throwing the other intervals
slightly out of tune. This is called “Equal Temperament.”
Unlike a string or
wind instrument,
a drumhead has
complicated
vibration modes,
whose frequencies
are not integer
multiples of the
fundamental.
Therefore most
drums
are not
perceived as having
a definite pitch.
Click picture for a
movie clip of drum
vibration modes
(ignore movie audio)
Indian Tabla
A pair of drums.
Each drumhead
is thickened with
a dark spot of dried
paste to shift the
normal mode frequencies.
In the smaller drum, the centered spot causes most of
the frequencies to lie at near-whole number ratios, giving
the smaller tabla a bell-like sound with a definite pitch.
Leftovers and references
Ear largely insensitive to phase, <detuned waves on scope>
Sound spectrogram looking again at whistle, sing, pot lid, and various instruments:
Violin and harmonics French horn, clarinet
Sound propagation in 1d, 2d, and 3d. Trouble for whale communication
Mosquito ring tones
UNSW animations and movies of wave refl and superpos
http://www.animations.physics.unsw.edu.au/jw/waves_superposition_reflection.htm#pulses
wave or tone files in MyDocuments.
Dancing hair cell, http://auditoryneuroscience.com/acoustics/sound_propagation
Hearing missing fundamental
Mosquito ringtones http://www.noiseaddicts.com/2011/06/mosquito-ringtones/
Tabla 145 02 M idea on http://www.soundsnap.com/tags/tabla
Tympani musical element tympani roll and hit
httptp://www.soundsnap.com/search/audio/tympani/score