Audio Signal Generating and Processing Project

Introduction:

Listen to this piece of music.

Media:Audio_Signal_Generating_and_Processing_Project_final_verison.wav‎

Just soso, right? but this is generated by computer software by MATLAB.

- Abstract -

This project is intent to analysis different musical instrument's sound, and try to create artificial musical instrument sounds to play a piece.

- Procedure -

A record of finite number of keys on a piano keyboard was used. The original sample is here.

Media:Orginal_sound_sample.wav‎

After the first frustrating method, I decided to up/down sample the keys by right order, then place them in right key.

According to modern music theory of interval, each intervals are equally spaced, each octave is equally spaced in to 12 intervals. A octave higher means twice the frequency. So, each interval is spaced by frequency ration of $ 2^{ \frac{1}{12}} = 1.05946309 $

But here comes a problem for up/down sample, it can only up/down sample by a integer factor. One can't upsampling by 1.05946309.

However, inspect the rational number $ { \frac{18}{17}} = 1.05882353 $ , that is relatively close to 1.05946309.

Next closer fraction is $ { \frac{107}{101}} = 1.059405941 $, But this fraction doesn't change too much accuracy, but as we can see below, it increase the computation steps rapidly. So I choose $ { \frac{18}{17}} = 1.05882353 $ as the approximate factor.

Next, use this fraction, apply the following:

if a note half-step above the original is desired, then upsample by 17, then down sample by 18, call this as "move up"

In this case, the signal is preserved, but at a lower sampling frequency. If play at the original frequency, then the note half-step above is played.

if a note half-step below the original is desired, then upsample by 18, then down sample by 17, call this as "move down"

In this case, the signal is preserved, but at a higher sampling frequency. If play at the original frequency, then the note half-step below is played.

For each interval(from lower C to higher C),

take the lower C, "move up" by step recursively, then get a map of full chromatic scale, define map1, with the exact timber of the lower C;

take the higher C, "move down" by step recursively, then get a map of full chromatic scale, with the exact timber of the higher C;

if we pick higher part of the scale as map1, lower part map2, then at the junction, the timber suddenly changed, makes the sound very unnatural.

You can hear it in here Media:Audio_Signal_Generating_and_Processing_Project_Timber_before.wav

Instead, apply the following method:

a given note is contribute by both map1 and map2, and proportional to the end point.

For example, the note C# is constructed by

$ {C^\#} = { \frac{11}{12}}*map_1(C^\#) + { \frac{1}{12}}*map_2(C^\#) $

$ F = { \frac{7}{12}}*map_1(F) + { \frac{5}{12}}*map_2(F) $

$ A = { \frac{3}{12}}*map_1(A) + { \frac{9}{12}}*map_2(A) $

This take cares of the timbre difference. Minor detail is still not perfect, but maybe just change the original signal can improve it. It is a very poor recorded signal.

Error analysis:

The ratio I pick is 1.05882353 versus the accurate factor = 1.05946309;

Error factor is $ \frac{1.05946309}{1.05882353} = 1.00060403 $

Since this error accumulates, and I am generating 12 notes with one real notes, take $ 1.00060403^{12} = 1.00727253 $ as the maximum error factor.

This difference is $ log_{1.05946309}(1.00727253) = 0.12544891 $, about 1/8 of a step;

In modern music, pitch was divided in to the term "cents" to measure smaller difference in pitch. Each step contains 100 cents.

In this case, the error is within 13 cents. For pure frequency, the smallest pitch difference human ears can distinguish is about 6~7 cents.

In string musical instrument, human can distinguish about 12~20 cents.

These data need to be valid, but on my opinion, that data is the best record of all human being. I have a experiment with my music teacher, I myself can only distinguish about 1/3 of a step in string instrument(about 35 cents), and even my music teacher can only distinguish about 1/4 of a step(about 25 cents)

On the other hand, a not well toned piano can easily go off 20 cents.

So I will claim that, this approach is acceptable in pitch level.

Hence, we have a full piano keyboard by now.

The data was saved in a matrix into a .mat file.

A script was wrote, that use a special pattern of pitch and rhythm matrix to call the corresponding column of the keyboard matrix.

Then combines the different duration and pitch notes in to a song, as you heard at the beginning.

There's another method I tried, which is to generate signal directly by inspecting a musical instrument's FFT, but this method doesn't turn up good result. Documentation can be found here:

Audio_Signal_Generating_and_Processing_Project,_Previous_method

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