 ## CHAPTER 1

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