Audio Signal Generating and Processing Project
Introduction:
- Listen to this piece of music.
- 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.
- 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.
- if a note half-step above the original is desired, then upsample by 17, then down sample by 18, call this as "move up"
- 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