modeling music signal using harmonics filter bank

[asifadio]

hello there, im a undergraduate physics student currently working for my final year project and need some help from engineering expert.

i tried to model/analysis a music signal using harmonic filter bank. but there is a problem. i cannot extract the parameter that needed in heterodyne filter function( which is a time, t).. here i show you what exactly my problem is

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[FvM]

You are presenting a lot of mathematic expressions that you apparently don't fully understand. But you don't tell what you exactly want to achieve.

E.g. what's the expected output of the signal analysis? Is it a "static" spectrum, or spectral distribution versus time (a spectrogram) or something else? In which regard do you want to model musical instruments?

You should be able to describe the problem in common language and then see which mathematic or signal processing method is suitable to handle it.

 

[asifadio]

You are presenting a lot of mathematic expressions that you apparently don't fully understand. But you don't tell what you exactly want to achieve.

E.g. what's the expected output of the signal analysis? Is it a "static" spectrum, or spectral distribution versus time (a spectrogram) or something else? In which regard do you want to model musical instruments?

You should be able to describe the problem in common language and then see which mathematic or signal processing method is suitable to handle it.

Thank you so much for reply on this thread,
Actually i just want to get a mathematical expression of music signal... since i'm not very familiar with signal processing, i believed i can get a mathematical expression of signal by analyze the spectral representation of the signal.. my music signal is in note C which i know that it also composed with other harmonics within it.. thats why i choose harmonic filter bank method...

I hope you understand what is my purpose are.. if you don't please let me know... and thanks again for reply

 

[BradtheRad]

It sounds as though you are looking for software that shows a sound spectrogram over time.

Here is a screenshot I made of a program called Audacity, a free sound processing program.

The sound is a guitar chord. It lasts about 3 seconds.

The pitches of the six strings can be recognized. Overtones can be seen although I believe harmonics of a low note might coincide with the fundamental of a higher note.

Changes in volume at various frequencies can be seen over time.



There is the time variable that applies to the period of a sinewave frequency.
Then there is the time variable that applies to attack and decay of a sound, or volume changes, and the like.

One is slow enough that we can measure with our ears and a stopwatch.
The other we cannot, even though we hear it present in the pitch, or in phasing, or waveform, etc.

 

[asifadio]

It sounds as though you are looking for software that shows a sound spectrogram over time.

Here is a screenshot I made of a program called Audacity, a free sound processing program.

The sound is a guitar chord. It lasts about 3 seconds.

The pitches of the six strings can be recognized. Overtones can be seen although I believe harmonics of a low note might coincide with the fundamental of a higher note.

Changes in volume at various frequencies can be seen over time.



There is the time variable that applies to the period of a sinewave frequency.
Then there is the time variable that applies to attack and decay of a sound, or volume changes, and the like.

One is slow enough that we can measure with our ears and a stopwatch.
The other we cannot, even though we hear it present in the pitch, or in phasing, or waveform, etc.

thanks a lot for reply on my question..
i want to address at one of your point which is

There is the time variable that applies to the period of a sinewave frequency

this is exactly what im looking for... if played a single note on a single string (eg. guitar), i know that instead of C note alone (fundamental frequency) it also consist of other overtones/harmonics within it.. the time of them to occur is what i want.. in the equation of harmonics filter bank.

this is the equation

where


as you can see,
in first equation, there are tau for window function, and i need that 'tau' for each harmonics/overtones that occur in the signal..

i already use the audacity spectrogram which is like this


the question is, how can i know the period of the harmonics/overtones(apart of fundamental frequency) to occur? as i mention earlier, i need it for my equations..

and one more thing im not understand is, is it the harmonics/overtones occur in the same initial time with fundamental frequency like has been presented in the spectrogram?

thank you sir for reply my thread.. i really appreciate it..

 

[asifadio]

oh and one more thing.. here i give you all the reader a simple explanation about my problem.. i know to get a spectrogram, one need to do STFT and from STFT the he/her get, the spectrogram is presented.. now i want to do a STFT of the signal.. not in my purpose to develop a STFT-spectrogram..

how can i do STFT while not knowing the tau or period of oscillation of harmonics/overtones within it?

 

[BradtheRad]

the question is, how can i know the period of the harmonics/overtones(apart of fundamental frequency) to occur? as i mention earlier, i need it for my equations..

A gong is a good example for this. There is the fundamental. Then there are overtones. Some are harmonics (2nd, 3rd, etc.). Some are odd multiples which add a 'noise' element.

Any overtone can come and go over a few seconds time.

None of them occur unless the gong is struck. The strike has to be t=0. But why a given overtone waits a few seconds to appear, or to fade out, requires some time in the study of vibrating metals, etc.

Although it starts to resemble chaos, the gong will still sound like itself each time it is struck.

It is hard to fit a mathematical model to this kind of sound. Supposedly a synthesizer could duplicate it by playing sine waves at the proper frequencies, at the proper amplitudes.

Most musical instruments have variations in timbre, which can be controlled by the musician. This cannot be predicted by an equation, of course. So I picked the gong sound because its sound spectrum is varied yet it has some degree of predictability.

 

[asifadio]

A gong is a good example for this. There is the fundamental. Then there are overtones. Some are harmonics (2nd, 3rd, etc.). Some are odd multiples which add a 'noise' element.

Any overtone can come and go over a few seconds time.

None of them occur unless the gong is struck. The strike has to be t=0. But why a given overtone waits a few seconds to appear, or to fade out, requires some time in the study of vibrating metals, etc.

thank you sir, this is my actual problem.. in spectrogram i've got, all the overtones are starting at the same time of the fundamental frequency (t=0).. i need to know the time of each overtones start and fade out.. is there any other method for me to achieve it instead of using spectrogram?

It is hard to fit a mathematical model to this kind of sound. Supposedly a synthesizer could duplicate it by playing sine waves at the proper frequencies, at the proper amplitudes.

Most musical instruments have variations in timbre, which can be controlled by the musician. This cannot be predicted by an equation, of course. So I picked the gong sound because its sound spectrum is varied yet it has some degree of predictability.

how the synthesizer synthesize the signal? assume that im the first one who build a synthesizer, what i will do is, get a spectrum analyzer for a timbre that i want to synthesize and combining them.. for amplitude for each frequency, i'll set it to be decrease as the frequency is getting higher.. is this the way to do it?
so, in mathematical ways, can i get the spectrum analyzer like this;

then,
extract the frequencies from there and multiple by exponentially decrease factor?

but by doing that, i neglected the time when the overtones start and fade out.. how i know the that time? did synthesizer neglected this parameter too? if i can get to know how synthesizer analyze and synthesize the signal in mathematical way, i think it would be enough

thank you sir, you just narrow down my problem to a simplest word... and sorry for bothering you with my problem..

 

[FvM]

in spectrogram i've got, all the overtones are starting at the same time of the fundamental frequency

Only at first sight, I think. They seem to have a specific structure each. Technically, a short time fourier transformation over the full audio range will probably work in bands of different window length, similarly a filter bank with constant Q has frequency dependent channel rise time.

Although a STFT spectrogram visualizes the dynamic properties of a musical sound well in some regard, it also hides detail information that's essential to recognize a specific sound.

assume that im the first one who build a synthesizer

Sure you aren't, actually you're late by half a century...
https://en.wikipedia.org/wiki/Moog_synthesizer

 

[asifadio]

Only at first sight, I think. They seem to have a specific structure each. Technically, a short time fourier transformation over the full audio range will probably work in bands of different window length, similarly a filter bank with constant Q has frequency dependent channel rise time.

actually my knowledge in signal processing is not quite enough to understand Quality Factor (Q).. but what i understand from your explanation is, STFT over a range of signal should be done by using different window length?

Although a STFT spectrogram visualizes the dynamic properties of a musical sound well in some regard, it also hides detail information that's essential to recognize a specific sound.

can you comment a bit about scalogram?

Sure you aren't, actually you're late by half a century...
https://en.wikipedia.org/wiki/Moog_synthesizer

yes, i owned one since 2005 (microKorg- when i have a band) but never look deeper than just using it..

 

[BradtheRad]

thank you sir, this is my actual problem.. in spectrogram i've got, all the overtones are starting at the same time of the fundamental frequency (t=0).. i need to know the time of each overtones start and fade out.. is there any other method for me to achieve it instead of using spectrogram?

Something you might like to study along this line (overtones coming and going) is the behavior of a plucked string.

A string has modes of vibration, starting with the fundamental, then adding portions of various harmonics.

A uniform string sounds musical. Its vibrations are multiples of the fundamental.

A guitar player needs to put on new strings every so often, because an old string loses its uniformity. Its metal wears in spots. Grime sticks to it.

The overtones get a little bit off-frequency. They are no longer 'musical'. The string no longer vibrates uniformly as it did when first manufactured. It develops irregular modes of vibration.

A detailed spectral display might show the sound consists of constant sine waves combining to create beats, etc. Or it might show sine waves coming and going.

It might be fast or it might be slow. It depends on the contour and composition of the string.

All this gets very complicated of course.

how the synthesizer synthesize the signal? assume that im the first one who build a synthesizer, what i will do is, get a spectrum analyzer for a timbre that i want to synthesize and combining them.. for amplitude for each frequency, i'll set it to be decrease as the frequency is getting higher.. is this the way to do it?

When the Moog synthesizer was invented, it was described as 'able to sound like any musical instrument'. It contained various waveform generators (sine, square, sawtooth), and filters, modulators, etc.

I think artists would have done the same thing as you describe, to get the Moog to sound as close as they could get to traditional musical instruments. It had to be difficult and time-consuming.

They also brought out sounds that had never been heard before. One might start out as a square wave, then add ring modulation, then add vibrato, all in the space of two seconds. The waveform would need a very complex equation to describe it.

You would need a series of spectral screenshots taken every tenth of a second, showing when and where frequencies come and go, to understand how sinewaves make up such a sound. One could say it is made from an arbitrary combination of effects. For the first 0.8 second the sound might obey one mathematical equation, then transition in the next 1.3 second to a different set of equations, then in the next 0.5 second to a third set of equations.