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Musical Sounds
When we examine the waveforms (amplitude versus time) of noise and music, a clear
distinction can be made. Music has regular, repeating patterns, whereas noise is random
and chaotic.
Musical sounds have three distinguishing characteristics: pitch, loudness, and
quality (timbre)
Pitch is very closely tied to the physical wave property of frequency. The higher the
frequency is the higher the pitch will be. The octave is a very important pitch concept.
Two tone separated by an integer frequency ratio will sound similar in tone to the human
ear. As an example middle C on the piano is 262 Hz. High C, one octave above middle
C has a frequency of 524 Hz.
Loudness is defined in terms of the intensity, or the square of the amplitude. If we
call b the loudness, I the intensity, and Io the intensity of the threshold of hearing, then
  10 log  I I 
 o
and loudness is measured in decibels. If the intensity increases ten fold, the loudness
increases by 10 decibels. Zero dB corresponds to the threshold of hearing and 120 dB is
the threshold of pain. The perceived loudness of a sound depends on the frequency.
Humans do not hear well at either end of the frequency range (20 – 20,000 Hz). Our best
hearing is around 3500 Hz.
Quality is also called timbre. It is the number and relative intensity of the
harmonics produced by a musical instrument. Quality makes it possible to distinguish
between musical instruments, even when they play the same note.
When an instrument plays a note, an entire harmonic series is generated. The lowest
frequency (and usually the loudest and the one we use to assign pitch) is called the
fundamental. Harmonics are integer multiples of the fundamental. Assignment of
harmonic and overtone numbers works like this…
Frequency
f
2f
3f
4f
Harmonic
1
2
3
4
Overtone
0
1
2
3 etc.
The amplitudes of the higher harmonics are generally much less than the first few.
The appearance of the waveform of the various instruments is dominated by the first few
harmonics. Nevertheless, the presence of all the harmonics gives the instrument its
characteristic quality. It is for this reason that electronic instruments generally fail to
faithfully reproduce the sound of a real instrument. The high harmonics have frequencies
that the electronic instrument is not designed to reproduce.
Because the ear/brain is accustomed to hearing harmonic series from instruments, it
can fill in the necessary information when one of the harmonics is missing. Thus you
hear bass notes from a tiny headphone speaker that cannot possibly reproduce tones of
low frequency. The rest of the harmonic sequence that is present provides the clues to
help fill in the missing members.
Instruments come in three basic types – stringed, air column, and percussion. The
later two will generally move more air and thus be louder. This is why an orchestra
requires many more stringed instruments.
The Principle of Superposition can be used to make complex waveforms out of
simple sine waves. This is the direct problem and is quite easily understood. The inverse
problem was demonstrated by Baron Jean Baptiste Joseph Fourier that complex
waveforms can be built up by a series of simple sine waves. The ear/brain can respond to
the complex waveforms that vibrate the eardrum, yet you can hear a single instrument or
voice in the piece.
Musical scales are numerous but each is built on a certain interval between notes.
The diatonic scale is the familiar “do, re, mi” scale and uses whole number ratios of the
frequencies as the interval between notes. The piano is tuned to the equitempered scale,
which uses twelve notes to the octave, all evenly spaced. Recalling that one octave is a
factor of two in frequency, we see that the interval in the equitempered scale is 12 2 or
about 1.059. Other scales include the Chromatic scale, Harmonic Minor scale, Pentatonic
scale, and Pythagorean scale.