Basic Concepts :

The human ear perceives a sound when it is subjected to a series of
alternate compression and rarefactions of the air, above and below
the ambient pressure at rates of ca. 15 to 17.000 times per second,
or 15Hz to 17.000 Hz. (In acoustics, we measure the frequency
of a sound in *cycles per second, or Hertz (Hz).*

One cycle of vibration (1Hz) is one complete excursion from the point of
equilibrium (0°) through the extremes of displacement in the positive
and negative domain (360°). The motion of this system is called :
*simple harmonic motion *: it is *periodic* and repeats as
long as there is no energy loss.

The time in seconds required to complete one cycle of vibration is called
the *period *of vibration. Another term, *frequency*, indicates
the number of cycles completed per unit of time. Thus *frequency, f= 1/period
* (frequency is the inverse of period). Remember, that frequency,
is measured in cycles per second or Hz.

Now look at the grandfather clock on the wall. This old clock mechanism
is powered by a pendulum.(the pendulum swings to the extreme right
(+) then swings back to the left to the perpendicular point (0) then,
to the extreme left (-) in an endless process . Now take the clock off
the wall, and let it drop on the floor: an outsider watching the scene
would clearly see a sine wave oscillation before the clock hits
the ground. Why? Because, in addition to the *Amplitude (A)*
of the oscillation, we have incorporated a *time (t)
*continuum.

What is needed for a system to oscillate? Let's find out..As you can see in
Figure 1 a forced manual movement of the pendulum's
arm to the extreme right position (90° angular phase/Positive polarity)
will start the whole oscillation process: when leaving the extreme
right position,the pendulum will pass through the inertia point (180°
angular phase/zero polarity), then to the extreme left position (270°
angular phase/Negative polarity), then back to the inertia point (360°
angular phase/zero polarity) before reaching its final destination to the
extreme right (90° angular phase /Positive polarity). Notice,that this
whole process has taken only one *cycle *in time.

Look now to Figure.2a : the sine oscillation
has a lesser Amplitude (A) than the one in Figure
2b. So, we can safely say, that the sine oscillation of figure 2b
has more *amplitude of vibration *and will cause *greater pressure
changes *in the surrounding air molecules: this will result in *louder
sound* than the one in figure 2a...

In general, the sound pressure level is measured in '*dynes*'
per square centimeters (dynes/cm2) or *microbars (µB)
*:* one microbar equals one dyne per square centimeter;
*or 1.4504 x 10^{ -5} pounds per square inch. The sound pressure
level which corresponds to the threshold of human hearing is *0.0002 µB
(micro Bar)* at a frequency of 1000Hz (1.000 cycles per second).

Pure tones made out of sine waves occur infrequently in emusic... Most sound
sources produce complex tones which are *collections of
single-frequency*, or pure tones, of different *amplitudes *and
*frequencies*. The number of these pure tones , along with their
individual frequencies and amplitudes, determine the tone quality, or
*Timbre,* of a complex sound. The pure-tone, single frequency
components of a complex sound are called *"partials" or
"harmonics"(H)*, depending on the *ratio *of their frequencies
to the* lowest frequency component or
'Fundamental"(F)*.(see Figure.3)

Now, let's consider two complex tones, both of which have 100Hz as their
lowest frequency component, or *fundamental *.(see
Figure 4a and 4b). In the first example, the first
tone contains the following frequency components: 100Hz, 200Hz, 300Hz, 400Hz,
500Hz, 600Hz, 700Hz, 800Hz and 900 Hz. The second tone contains the following
frequency components : 100Hz, 190Hz, 272Hz, 313Hz, 451Hz, 559Hz, 707.2Hz,
and 845 Hz. As you can see in both examples, each sine frequency component
has an amplitude value(A) associated with it. This kind of graph showing
a frequency versus an Amplitude distribution is called a *"line
spectrum".* For example, the frequencies of the partials
of the complex sound pictured in figure 4a,
are related one to another by *"integers"* i.e
1,2,3,4,5,6,7,8,9. This is not the case in the second example
(Figure 4b) ,where the frequencies of the partials
are related one to another by *'non-integers':* 1, 1.9, 2.72,3.13,
4.51, 5.59, 7.072,and 8.45..

In Mathematics, the series 1/2, 1/3, 1/4...1/n is called the
*'harmonic series' *(Frequencies components related to a fundamental
by integer ratios are called *"harmonics"*). Generally, one considers
that the *Fundamental(F) is the first harmonic (or first partial*).
In figure 4a, the frequency components are
*"harmonics"*, while the frequency components of
Figure 4b are called *"inharmonic partials",
but not harmonics.*( note that the frequencies components of many
emusic sounds are often non-integrally related ).

Any *periodic waveform *may be broken down into a number
of single frequency sine components. Indeed, simple addition of multiple
sine waves of different frequencies and amplitudes produce very complex waveforms
such as in *additive synthesis*.

An *oscillator, *is a device which produces a *periodic
voltage *or current *versus time function*. Most oscillators designed
for emusic are capable of producing several different waveforms simultaneously:
such an oscillator is called a *'function generator'*: these waveforms
differ from one another in the number and amplitudes of their harmonic
components.

The *Sine wave,* contains a *fundamental frequency or
first partial and no harmonics*: it is the purest of all waves
and is considered '*the Mother of all Waves'*.

The *Sawtooth wave *(see Figure 5) is commonly
available from most function generators: it is the *richest wave available
for synthesis*(Brass and Violin sounds)*.* Indeed, the sawtooth
contains a *fundamental frequency (F) and all integers harmonics *:
H2,H3,H4,H5,H6,H7,H8 and so on...

The *Square Wave *(Figure 6) is another waveform
commonly available from most function generators (the "timbre" can be described
as *'hollow and clarinet- like).* The Square wave is a particular
case of a *rectangular pulse wave* for which the *pulse width
*(PW) *is one half the period*: the frequency components
of a square wave are *ODD number* multiples of a fundamental
frequency: i.e the *Fundamental (F) or first partial and Harmonics
H3,H5,H 7,H 9, H11,H13,H15..*.

.It is to be noted, that the *Amplitude (A)* of any given component
is the* amplitude of the fundamental divided by the harmonic number.*
Many synthesizers used in emusic provide a Pulse Wide Modulation (PWM )i.e
a *variable width pulse waveform. * The "spectrum "of the pulse
wave depends on the *pulse width*. The* ratio* of the pulse
width to the total period is called the *'duty cycle'*, which
is usually expressed as a* ratio *e.g 1:2 or a
*percentage* 50%.(see Figure 7a-b-c-d-e-f).By
varying the *pulse duration *one can obtain a large number
of complex sounds with integrally related frequency components (Oboe,
Hindu Shanai sounds).

The *Triangle wave *(see Figure 8)
is another waveform commonly available from analog synthesizers: the
timbre of the *Triangle wave* is much like that of the sine wave,
but with a bit of *"edge".*Like the Square wave, the *Triangle
wave *contains a* Fundamental(F)or first partial and
ODD Harmonics H3,H5,H 7,H 9, H11,H13,H15..*..It is to be noted,
that the Amplitude (A) of any given component equals 1/(harmonic number)^{
2 }.

In general, there are *four operations *which are employed for generating
and/or modifying signals for emusic production : *additive synthesis*
(add sine waves to create complex waveforms), *subtractive synthesis
*(subtract harmonics by filtering), *modulation synthesis *( analog
FM,AM synthesis) and *waveform synthesis *(digital FM of waveforms
using operators, digital PCM and Waveform Resynthesis of Sampled sounds).

**Questions :**

1.If the period of oscillation of a given waveform is 0,150 ms, what is the frequency of oscillation in Hertz (Hz)?

2.What is the Amplitude of a signal?

3.What is an harmonic serie?

4. What is the duty cycle of a Square Wave?

5. Compare the frequency components of a 100Hz Sawtooth Wave with those of a Square Wave? ( make a graph showing the Fundamental and harmonic distribution of both waves).

6.Why is a Triangle wave not as rich as a Sawtooth Wave?

7. What type of synthesis is used in Analog Modular Synthesis?

End of Chapter 1 (Basic Electro-acoustic Concepts)

To Chapter 2