Comparing the "Tone" of musical instruments in matlab

I am trying to find a way to compare the likeness of short 500 millisecond recordings using metlab of the same note played on different instruments.

Going into detail on this specific topic: I am a music student that has been given the task to objectively determine the tone of various modern low brass instruments to determine what instrument should replace the obsolete "ophicleide" or Bass keyed bugle. I first used a visual comparison of a spectrograph of it and 6 other instruments, but that approach was too subjective.

I recorded all of the instruments with the same microphone, equipment, gain levels, and the same notes. For this reason, I believe that the signals are similar enough to use metlab tools.

I believe that comparing the fft is going to be the most accurate calculation. I tried at first a freq-domain correlation, and tested different segments of the same tone (eu, and eu2 being variables)

 corr(abs(fft(eu)),abs(fft(eu2)))    ans = 0.9963

Which is a step in the right direction, but I seem to get the opposite result when I compare different signals: (euphonium and ophicleide sound almost identical)

 corr(abs(fft(eu)),abs(fft(ophi)))  ans =   0.5242

euphonium and bass clarinet sound completely different, but this shows higher correlation

 corr(abs(fft(eu)),abs(fft(basscl)))   ans = 0.8506

I tried a normalized maximum cross-correlation magnitude formula that I found online, but I am getting the same results

 >> norm_max_xcorr_mag = @(x,y)(max(abs(xcorr(x,y)))/(norm(x,2)*norm(y,2)));
x =eu2; y = eu; 
norm_max_xcorr_mag(x,y)

 ans =   0.9638

I get a similar result when comparing the other samples

>> norm_max_xcorr_mag = @(x,y)(max(abs(xcorr(x,y)))/(norm(x,2)*norm(y,2)));
x = eu; y = basscl; 
ans =    0.6825

compared to

>> norm_max_xcorr_mag = @(x,y)(max(abs(xcorr(x,y)))/(norm(x,2)*norm(y,2)));
x = eu; y = ophi; 
norm_max_xcorr_mag(x,y)

ans = 0.3519

The euphonium and Bass Clarinet (basscl) have a completely different sound, and completely different harmonic series, but these formulas are showing closer correlation than the Euphonium and Ophicleide, whose frequency bands look almost like an identical match.

I am worried that these correlations are showing the correlation of true pitch (I am playing the same note on all of these instruments, but the ophicleide might be out of tune by up to 1 Hz) It could also be accounting for phase, or even total amplitude.

does anyone know of a better clear cut method in comparing the proportions of the harmonic overtones of these complex waveforms?

or am I barking up the wrong tree?

[Azzi Abdelmalek]

What is metlab?

[Star Strider]

It’s MATLAB ported to Babbage’s Difference Engine. Early releases were, appropriately, called metal lab, owing to its technology.

Answer 1

[Star Strider]

Disclaimer: I love early music and am attempting to teach myself the lute (that won’t stay in tune).

I suggest knnsearch to start. I would use the magnitude ( abs ) of the fft of the signals, since they are sampled at the same time and in the same conditions, and are therefore the same length and frequency resolution. The phase information may not be necessary if you are simply comparing the spectral magnitude.

If this fails to reveal the true inheritor of the ophicleide’s place in early music, consider various Feature Transformation procedures on the magnitude spectra of the candidate instruments for dimension reduction to define the important features of the various instruments. (This is the sort of procedure used for biomedical signal analysis and classification, so it is likely valid for your application.)

[Euphonium]

For the life of me I have not been able to figure out how the code wold work. I think am getting all of the nearest neighbor values in the matrix.

this is what I have:

     knnsearch (abs(fft(ophi)),abs(fft(basstuba)))

this is returning the matrix again, but I am not sure what order of the nearest neighbor is and what it has done to the matrix. Although it is comparing the FFT, I think it somehow needs to compare the ratio of the peaks. one issue is the FFT does not return one spike per overtone, but it spikes the frequencies around it, so those integers are showing up and making

It difficult for me to figure out what the data is saying. here is an image of the plot

If you could help me with a demonstration of a formula, that would be extremely helpful, as this seems beyond my pay grade. I am thinking though, would a filter be a good way to optimize the signals? maybe the freq-domain correlation is comparing the background noise floor more than the areas of high spectral magnitude?

[Star Strider]

The first argument of knnsearch is abs(fft(ophi)), The second argument is the matrix of:

FFTInst = [abs(fft(eu));
           abs(fft(eu2));
              . . .
           abs(fft(basscl))];

although it will be easier to do all the ffts at once and put them into a matrix rather than doing them in situ.

The k-th Nearest Neighbor algorithm takes the elements of (in this situation) the vector of [abs(fft(ophi))] and subtracts it from the elements of every other vector in sequence, taking the mean-squared difference (‘euclidean’ distance; other distance metrics are available) each time. The lowest mean-squared difference is the nearest neighbor. (This is essentially Classification Using Nearest Neighbors, except you’re not classifying.) You want the instrument with the magnitude spectrum closest to that of the ophicleide. In a perfect situation, this will be all you need. In reality, life (and science) are never that uncomplicated.

Ideally, background noise is random and therefore broad-band. It might affect the baseline (in which situation it is likely best to subtract the DC offset (the value of the first coefficient of the absolute value of the fft of each instrument) from the abs(fft) or the same instrument. That would normalise them all, so varying levels of background noise would not be part of the pattern-matching the knn algorithm performs.

In addition to subtracting the DC offset, normalising the magnitude spectra by dividing the DC-corrected magnitude spectrum of each instrument by the maximum magnitude of the fft for each instrument, so that the maximum for each instrument is equal to 1, would likely provide you with the most reliable data to compare the various spectra.

[Euphonium]

this is what I wrote: I also included the correlations of the signals as well

[x1, Fs] = wavread('baritone_01.wav');
[x2, Fs] = wavread('Bass clar 1_01.wav');
[x3, Fs] = wavread('basstubatone.wav');
[x4, Fs] = wavread('bbbtuba_01.wav');
[x5, Fs] = wavread('bucket_01.wav');
[x6, Fs] = wavread('cup_01.wav');
[x7, Fs] = wavread('euph bucket_01.wav');
[x8, Fs] = wavread('eupho 1_01.wav');
[x9, Fs] = wavread('funny wood_01.wav');
[x10, Fs] = wavread('funny1_01.wav');
[x11, Fs] = wavread('straight1_01.wav');
[x12, Fs] = wavread('trombonetone.wav');

 [x, Fs] = wavread('ophi 1_01.wav');

    FFTInst = [abs(fft(x1));
               abs(fft(x2));
               abs(fft(x3));
               abs(fft(x4));
               abs(fft(x5));
               abs(fft(x6));
               abs(fft(x7));
               abs(fft(x8));
               abs(fft(x9));
               abs(fft(x10));
               abs(fft(x11));
               abs(fft(x12))];

corr((abs(fft(x))),(abs(fft(x1))))
corr((abs(fft(x))),(abs(fft(x2))))
corr((abs(fft(x))),(abs(fft(x3))))
corr((abs(fft(x))),(abs(fft(x4))))
corr((abs(fft(x))),(abs(fft(x5))))
corr((abs(fft(x))),(abs(fft(x6))))
corr((abs(fft(x))),(abs(fft(x7))))
corr((abs(fft(x))),(abs(fft(x8))))
corr((abs(fft(x))),(abs(fft(x9))))
corr((abs(fft(x))),(abs(fft(x10))))
corr((abs(fft(x))),(abs(fft(x11))))
corr((abs(fft(x))),(abs(fft(x12))))

      knnsearch (abs(fft(x)), (FFTInst))

This gave me a long string, so I plotted it.

I guess it shows Nearest neighbor of every frequency. I guess I could take the mean of each system and use that to calculate a total percentage of similarity?

I normalized and did a DC offset of the audio files before running this code, but I am going to try to normalize the data itself now post fft.

by the way, the names of these files are also of mutes placed in a euphonium bell to change the sound of the instrument. That is why it is called the funny mute, or the funny wood. I am getting a little worried that if the correlation algorithem is correct, the contrabass tuba might be the second closest sound to the ophicleide, destroying my theory. However, this formula says that a bass clarinet sounds closer to a euphonium than any of the instruments do to each other. there are may high overtones, possible 3,000 more hertz that make up the bass clarinet's sound, so I wonder if the sounds are similar but 2 or 3 lower order overtones on the bass clarinet are softer, and that makes the signals look closer mathematically even if they sound extremely distinct.

[Star Strider]

I hope you did the baseline subtraction and normalisation before you did the knnsearch. Nonetheless, something is obviously wrong, unless I missed something in your latest comment. First, I may have gotten the argument order backwards in my suggested call to knnsearch. See what:

D = knnsearch(FFTInst, abs(fft(x)))

returns. It will be a vector of 12 indices in the ascending order of ‘distance’ from (similarity to) the magnitude of the fft of the ophicleide. Those indices will correspond to the rows in FFTInst. For instance, D(1) will be the index of the closest to the ophicleide, and so on for all 12.

The knnsearch function documentation states:

• IDX = knnsearch(X,Y) finds the nearest neighbor in X for each point in Y . X is an mx-by-n matrix and Y is an my-by-n matrix. Rows of X and Y correspond to observations and columns correspond to variables. IDX is a column vector with my rows. Each row in IDX contains the index of nearest neighbor in X for the corresponding row in Y .
• [IDX,D] = knnsearch(X,Y) returns an my-by-1 vector D containing the distances between each observation in Y and the corresponding closest observation in X. That is, D(i) is the distance between X(IDX(i),:) and Y(i,:).

The output of knsearch should produce a vector for the indices of the various other instruments in order of ‘nearness’ of the their fft magnitudes to that of the ophicleide. By my count you should have one column vector (or two, depending on how you called knnsearch) that has only 12 elements. What are you plotting? I have no idea what knnsearch returns without any output arguments. Usually this is simply the first output, but it seems it returned something entirely different in your call to it.

If you want the correlation coefficients, include abs(fft(x)) (the fft magnitude vector for the ophicleide), possibly as the first row in a slightly differently named FFTInst matrix (maybe FFTInstR), and then use that matrix as the argument to the corr or corrcoef functions. You need to use all the functions with output arguments to understand the information they’re giving you.

Don’t worry about your hypothesis being disproved just yet. When we get all this sorted, it may turn out that you are correct.