Harmonic Spectrum of a Middle C

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	Figure 4a. Table showing the Ten Octaves of the Audio Spectrum, giving a few more particulars about the musical, acoustical and subjective qualities of each octave. Also, I’ve given the approximate wavelengths of frequencies in each octave. This, coupled with a basic knowledge of how diffraction and reflection work, is primary information for understanding acoustical problems in rooms.

Power Spectrum of the same Middle C

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	Figure 4b. Power spectrum for the above sound, showing the approximate summed power levels in each octave of the audible spectrum.

Figure 4A shows the fundamental and overtones of a typical musical sound (imaginary in this case) and 4B shows the power spectrum for that sound whose fundamental, approximately 260 Hertz, is Middle C. Each successive octave has twice as many harmonics. Note that the power scale is exponential, as is the frequency scale, of course. The total power is the sum of the harmonics’ values, and the power in each octave is the sum of values of harmonics in that octave. The total power of this sound is around 550 microWatts, with the bulk being in Octave 5, and the rest evenly distributed over Octaves 4, 6, 7, 8 and 9, plus a little in 10. Even though the upper harmonics have much smaller values, there are many more of them. With so much power in the middle frequencies, this sound would be a little “tubby” or “chesty.” If you wished to reduce this quality, you would use the equalizer to cut around 500 Hz. If you wanted to emphasize edge, you might boost in octaves 8 and 9. Boosting in octave 10 would do little except bring up the high frequency noise floor (tape hiss, for instance).

From the example, it should be clear that although the pitch is Middle C, little energy is present at that frequency and much of the power of the sound is spread across several octaves further up in the spectrum.* In fact, more than 90% of the power lies above the octave in which the fundamental resides. Also, for this particular sound, there is nothing going on in the bottom three octaves.

Power phenomenon

We’ve now seen that the total power of a signal is the sum of the powers of the various signal components. By the same token, the total power is the sum of the powers of each octave. This leads to some interesting insights which are of real use when dealing with equalization. First lets look at a broadband signal with flat response, like pink noise.

Power Spectrum of Pink Noise

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	Figure 5. Power spectrum of pink noise. In this case each octave has a power level of 100 microWatts.

Figure 5 shows the spectrum of pink noise. The total power level isn’t 100 microWatts, but rather the sum of 10 octaves each with 100 microWatts, which is 1000 microWatts (or 1 milliWatt). Another, more useful way of expressing is to say that in this case the summed power is 10 dB greater than the power present in each octave.

If we cut the bottom octave of the spectrum by 10 dB (i.e. to 1/10th of its magnitude), our spectrum now has a power level of 910 microWatts (see Figure 6). If you know how to calculate decibels, you can see the overall level of the signal has only dropped by .4 dB (the power ratio between 1000 microWatts and 910 microWatts is .4 dB -- trust me!).

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	[Figure 6] Fig. 5 with Octave 1 rolled off by 10 dB.

If we return to the pink noise and boost that bottom octave by 10 dB, however, we get a much larger change in level (see Figure 7). Now the summed power is 1900 microWatts, or approximately 3 dB greater. Note that 1900 is just about double 1000, which correlates with the axiom that boosting by 3 dB is the same as doubling of the power.

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	[Figure 7] Fig. 5 with Octave 1 boosted by 10 dB.

The insight to be gained from hurting your brain thinking about this is that boosting one or more octaves (during equalization) significantly increases the overall power level, while cutting one or more octaves barely reduces the power level at all. In the next two figures, we’ll show two fairly heavily eq’d pink noise signals, one with cuts, the other with similar amounts of boosts. In these two cases we can calculate that when we cut, our level drops by 2.4 dB and when we boost our level by similar proportions, the summed level gains by 11.3 dB.

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	[Figure 8] Fig. 5 with numerous octaves cut.

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	[Figure 9] Fig. 5 with various octaves boosted in amounts equivalent to Figure 8.

The moral of this is that you can easily increase level by boosting portions of the spectrum, but you can’t cut levels very well by cutting portions of the spectrum. Interestingly, you can create any EQ curve you want by either boosting or cutting. So, if you want the EQ change to cause the instrument to stand out more, EQ by boosting. If you like the levels just fine but want to alter the timbre without changing the balance, EQ by cutting.

Additionally, this helps to explain some of the difficulties we encounter when equalizing. I’ve generally believed that “louder sounds better,” but it’s interesting to note that louder can be achieved with fader or with additive EQ. You think to yourself, “I must do something with that ___ (guitar, snare, dulcimer, whatever your problem track.)” Try boosting level to see if that helps. So often, when we EQ, we boost level and we are seduced not by the magic new timbre, but by the fact that we can now hear the instrument a little better!

By the way, I’ve deliberately glossed over the decibel business here. If you know how to do the calculations, fine. If you don’t, just trust me for the time being. I’ll explain dB some other time.