July 28, 2006
Introducing the Spectral Evolution of the A2 Alto Note

In recent weeks, we've spoken about vibrations at different frequencies, and have counted the waves to calculate what frequencies are present. We are going to be studying the vibration of the upper A on the ALTO Kalimba. We studied the waveform for this note here.

Today, we have made a power spectrum of this note at several different places in its waveform. Any sound is made up of several different notes, or vibrational frequencies, and the power spectrum just tells us how much of each note is present.

The above plot shows the spectrum of the first 0.03 s of the note - this is basically the attack. The "x-axis" is frequency (Hz and KHz -- the farther right you go, the higher the note - 20 KHz is the highest pitch humans generally hear), and the "y-axis" is the power you have at each frequency. This note was A at 440 Hz, and we do see that the most power is in a vibration at 437 Hz (our A!). BUT, there is a LOT of purple at every frequency: this is the super-fuzzy hair we saw in the plots early on. Also notable are peaks at 858 Hz (I am guessing this is the first overtone of the air vibrating in the box, which has a fundamental vibration about one octave lower, around 440 Hz); a bunch near 4156 Hz; and something at 9184 Hz.

The above plot shows the spectrum between 0.10 and 0.20 seconds. The craziness of the attack is gone, and we see that the fundamental peak, at 444 Hz now has barely lost any energy. The harmonic at 882 Hz is down by 30 dB (dB is the unit of relative power). The stuff at 2807 is down by 35 dB, and the vibrations at 4126 and 9184 Hz are down by about 50 dB. On the kalimba, the higher the frequency of vibration in the overtones, the more quickly the overtones die out.

The high frequency components continue to fade in the 0.20 - 0.50 s spectrum, but the 884 Hz peak is still there, and the 444 Hz fundamental has hardly lost any energy.

Between 0.50 and 0.80 s, the spectrum is totally dominated by the pure tone at 445 Hz, but 75 dB down (this is WAY DOWN) from that peak is a vibration at 5100 Hz (at basically the same level as it was in the previous plot, though it is not labelled in that plot as it was a smaller feature). This plot is a demonstration of the purity of the kalimba's tone.

We've taken the spectrum for 0.10 - 0.20 s (ie, the second spectrum we showed), and played with it. We zoom in here because there is still some moderate frequency "crud" transitioning between attack and pure tone. By increasing the number of samples to 2048 per FFT, and by changing the scale on the x-axis to logrithmic (which would make a whole step interval a constant size on the plot), we can see something that is really amazing: some of the sound gets absorbed! Right around high E (660 Hz), there is lots of sound. Right around high G (725 Hz), there is lots of sound. However, in between, at high F#, there is 20 dB less signal!

What's going on here? If you've played very many old kalimbas, you know that sometimes the notes go a bit dead. Somehow (perhaps this investigation will throw some light on that), an age-related weakness on the face of the wood resonator box actually absorbs a particular note! This absorption is very narrow, just one note wide! No problem with E or G, just F#. And if you listen to the F#, it sounds sort of dead. It doesn't ring clear.

It isn't a problem with the tine: if you retune the F# to a G or an E, it sounds clear. If you take a different tine and tune it to F#, it sounds dead. So, I think the box is absorbing the vibration at F#, and sure enough, we can see an absorption feature right at F#.

Oh, we are going to have some FUN with this!