Spectrograms

Spectrograms provide a joint time-frequency description of a musical waveform.  A spectrogram is created by the following three step process (see Figure 1 below): 

1.  Multiply the discrete (digital) sound waveform by a succssion of compactly supported window functions.  Each of these window functions is a shifting of a single window by uniformly spaced increments in time.

2.  Each of the resulting windowed subsignals is a discrete sequence of a finite number of values, hence an FFT can be performed on each of them.  The resulting collection of FFTs is called the Gabor transform (or short-time Fourier transform) of the sound waveform.

3.  Display the magnitude-squares of the values of the Gabor transform that correspond to non-negative frequencies (since the sound waveform is real-valued, the negative frequency Gabor transform values are complex-conjugates of the positive frequency values, so their magnitudes-squared are identical to their positive counterparts, hence not displayed).  This graph of the magnitude squares is called the spectrogram for the sound waveform.

Figure 1.  Making a spectrogram.  On the left side we show a sound waveform.  The middle graph is a succession of compactly supported windows.  The graph on the right is a windowed subsignal obtained by multiplying the middle window times the sound waveform; an FFT can be applied to this windowed subsignal.  The collection of all the FFTs of the windowed subsignals is the Gabor transform of the sound waveform.  The spectrogram for this waveform is shown in Figure 2.

                                                                               Play video:     AVI     MPEG 

Figure 2.  Spectrogram of sound waveform, a sequence of four piano notes on a musical scale.  The horizontal azis has units of time (seconds) and the vertical axis has units of frequency (cycles per sec, Hz).  Darker pixels represent larger square-magnitudes.  The fundamental frequencies and their overtones for each note are clearly shown.  By clicking on either of the links below the graph you can play a video and observe how the spectrogram structures correspond to the notes you here (the choice AVI is a better quality video, but may not play with all computers; the MPEG choice is included to maximize the probability that your computer can play the video).

In the section, Musical Analysis - Examples , we use these spectrograms to analyze music.