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asifadio · Posted: Sun Apr 01, 2012 3:47 am Post subject:

ni adalah thesis sy.. sy memerlukan bantuan otai2 untuk mengomen draf ni.. draf ni sudah di terima oleh fakulti sy.. ni sy pastekan

OBJECTIVE:
To familiarize with complex mathematical analysis such as fourier analysis, fourier transform and laplace
To develop the mathematical model for acoustical character for each timbre of music
Concept involved:
-Classical mechanic
-Calculus
-Fourier analysis
-Acoustical physic

INTRODUCTION:
There is a problem arises with military technology which is how do we distinguish civilian radar and military radar since both of it use a X-band frequency range (8–12 GHz). The fourier serier may done the job to seperating frequency but it is useless if 2 source of wave frequencies spectrum overlap each other. This complex and big model of situation is anologous to acaustical wave. The different instrument produce the same frequencies range but listener still can easily recognize what instrument did it came from for example, in the actual system like mixed musical signal such as sound produced from guitar and piano at same note, both instrument produce the same frequency(fundamental natural frequency) at that note. Even with the same loudness and pitch, the listener can easily distinguish both of instrument because of each instrument happen to have differrent overtones composition. This commonly known as musical timbre or tone color. This project is to modelling mathematical representation for these timbre.

Musical note:
Note in music exist because of the different fundamental natural frequency for example in note C , the fundamental natural frequency is 262Hz for both first C octave at guitar and middle C at piano. As we let the instruments to vibrating for a while, the frequency will decreasing due to the damping and the frequency shift from overtone to next lower overtones or harmonics. This is what commonly known as chromatic for example from C, it will descending in note to B, A#, A, G#, G and so on. Interestingly each instrument possessed its own overtones composition or timbre. In 1975, Robert Erickson in his book, Sound Structure in Music state a several parameter for a timbre which is the range of tonal character, spectral envelope, and time envelope (ADSR envelope).

ADSR envelope:
Is time-domain acoustical signal produce. ADSR stand for Attack Decay Sustain Release which is what make timbre differ from the other. There are device called synthesizer which can manipulate ADSR character thus making it hard to determine what instrument the signal come from.

The figure above shows the example of ADSR envelope where the red line is the envelope and the black is the superposition of natural frequency.

Spectral Envelope:
Is the frequency domain of acoustical signal that have been transfromed from ADSR envelope.

Figure shows the spectral envelope for clarinet, piano and violin timbre.

Procedure:
Obtain the acoustic signal:
The timbre can be obtain by using definite-pitched musical instrument play in the particular note and recorded to music production software which is graphing the signal in ADSR envelope.
Modelling a ADSR envelope:
Since ADSR consist of attack, decay, sustain and release, the mathematical representation for ADSR envelope generally is a piecewise function as shown below

1. Attack- The volume bursts to the maximum level in a short period of time
2. Decay- The volume drops a little bit
3. Sustain- The volume holds constant for a period
4. Release- The sound starts to fade out.

Separating Pure Tone by fourier series:
Collection of natural frequency which is also called pure tone can be seperate by fourier series where the function of ADSR envelope as a input signal.
Changing to frequency domain using mathematical transformation:
To obtain the characteristic of the timbre clearer, time-domain fourier series are then transform to a frequency-domain using fourier transform or laplace transform. This will lead to spectral envelope.
Comparing spectral envelope:
Spectral envelope obtained from 2 instrument produce the same note then will be compare. The previous procedure will be repeat for a different note and check whether the pattern of spectral envelope associate with ADSR it is the same.
Project outcome and continuity:
Modelling mathematical representation for acoustical signal may give a glimpse to a different perspective on looking toward analogue system which is 99% of our universe. The project can be expand to a indefinite-pitched instrument such as percussion since it is almost similar to the natural mechanical system. Before this, seperating the timbre from a single mixed signal are not possible because there is range of frequencies shares by different timbres. This mathematical model could be a first step to solve this problem since characteristic of timbre are known.
_________________
'we are not musician, we're physicist'-Dr. Nazari
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