Audio Waveforms: Harmonics, Clipping

A surprise realized during church choir practice: my tenor range doesn't go above middle C, so I often drop down an octave. I used to think my second harmonic would be right up with the other tenors. Then I realized most waveforms don't have much in the way of even harmonics, so the harmonics in my downshifted voice are 3rd and 5th, up in the soprano range. This is a surprising insight about singing, contributed from engineering and math.

Decibels and why artists and front-of-house run audio so loud: I read decades ago that there is a valid reason for having the volume high. It is to get a large dynamic range. If the music volume in a hall or room is low, the dynamic range, from music peaks to background noise (crowd noise, ventilation, traffic, airplanes, construction equipment, cooling fans, bathroom water) is not very much, and quiet passages in the music (which are frequent in orchestra music) are drowned out by the background (ambient). The trouble with high volume is that everyone's hearing is being permanently damaged.

Distortion due to power-amplifier clipping, why it is so obnoxious:

The top waveform is an unclipped audio signal going to a speaker.

The middle waveform is when the power amplifier's positive supply droops down. You get clipping, shown here at +.9V. You might think the flat spot would not be heard much, since DC produces no sound.

Then I realized that the clipping can be modeled as the bottom waveform, the inverse of the peak (the part of the waveform above .9V, inverted), added to the original.

From this perspective, the third waveform, which is the distortion, is .4/2.6= 15%, a large percentage. It has sharp corners, which are very audible, and it repeats at a low, buzzy frequency. These are all characteristics that make the clipping obvious and obnoxious.

Harmonics are important when fundamental is below 1000Hz: a) 3rd harmonic of 1000Hz is 3000Hz, above which the frequencies are so high that they really aren't heard too much. Consider that the common electronic keyboard has 61 keys, highest one is only 2093Hz. b) Middle C is a very low frequency, 262Hz. Even a high soprano gets to only 1046Hz in opera. There are rich harmonics of Middle C, to the 3000/262= 11th harmonic, but not so for 1000Hz. c) On the other hand, harmonics are the only way the ear & brain can tell between different instruments and different people. (Well, concerning people, you also have mannerisms of speaking that help you identify people, like on telephone, where if you can't identify a person by the voice right away but after a few sentences you can tell who it is.) In other words, a person's voice at middle C and a musical instrument and tuning fork at middle C are producing identical fundamentals; you can tell the difference only through harmonics. (Well, there are other qualities, like attack, decay, and the different rates of decay of harmonics, like in a piano string.)

Ear and brain are remarkable frequency processors: the more you learn about how the brain works on sounds and music, the more remarkable it is. How many different people can you know, just by differences of voice and mannerisms? It must be thousands. And the brain can associate the sound with name, face, family, age. Wow. Also consider that the brain's ability to determine where a sound is coming from is partly from the time difference of sounds reaching the ears, to about 1/10 millisecond, based on 1130 ft/sec being the speed of sound and a person being able to point to a hidden sound source with a resolution of about 30 degrees.

Harmonic distortion from speakers: any speaker has a cone and a powerful permanent magnet with a narrow gap, in which a voice coil sits. The voice coil is dozens to hundreds of turns of wire, in one or more layers, glued onto a fragile insulating cylinder. The cylinder is glued onto the bottom of the cone. When you run current through the coil, the coil puts out an A.C. magnetic field that pushes or pulls the cone, creating sound waves. The speaker is a type of linear electric motor. You can read on the Eminence web site & other places how speakers have surprising sources of distortion. 1) Large, theater-size speakers that are simultaneously producing, for example, 30 Hz and 800 Hz, produce Doppler-modulated 800 Hz because sometimes the cone is moving toward the audience (on the positive half-cycle of the 30 Hz), & sometimes away from the audience (on the negative half-cycle). To counter this, and because speaker cones come in various sizes & masses for different frequency ranges, "crossovers" (passive or built into amplifiers, they are low-pass, high-pass, or band-pass filters) divide and conquer the 20 Hz to 20 kHz range. Crossovers are also needed to keep large bass voltages off the voice coil of tweeters, otherwise the little tweeter voice coils burn up. 2) Also because of the range of speaker-cone masses & sizes, speaker cones which are subjected to excessively high freq do something called "cone break-up." This is not usually destructive, it is high frequencies of vibration not making it out all the way to the edge of the cone. This is handled, again, by crossovers. 3) When the voice coil moves inward, the bottom of the coil's cylinder can bottom out at the bottom of the gap, making a thump & potentially squashing the edge of the cylinder & coil. Also, when the voice coil moves outward, the outermost turns of the coil start poking out of the gap, leaving the main part of the permanent magnet. This makes even-harmonic distortion. For 15" woofers, cone travel may be limited to +-12mm or even +-3mm, as specified by the Thiele-Small X_max parameter. 4) Every time you put in a crossover (and theater systems have up to 4 sets of speakers), there is potential overlap or underlap of the freq ranges, if the filters aren't trimmed. Plus, there is group delay in each filter that produces distortion. 5) For audio that is loud, the voice coil gets hot & the resistance increases, maybe 10%. This reduces the current & makes the speaker's sound weaker. This amounts to volume-dependent nonlinearity. 6) Cone/suspension resonance (around 40Hz for big woofers) creates an impedance peak that can take 8 ohms & make it 50 ohms at resonance. Several techniques are used to counter this resonance, which amounts to booming in woofers. A speaker makes sound even below the cone/suspension resonance, & dealing with that is partly why you have ports in bass-reflex enclosures. 7) When the frequency of sound coming from the speaker cone has a quarter wavelength less than the size of the cone, the sound starts "beaming" from the cone. It doesn't spread out, away from the axis of the speaker. Therefore, the audience at the edges doesn't hear the higher frequencies as well. This afflicts even woofers but is prominent with tweeters. If a tweeter is equipped with a horn to do impedance matching from high-Z cone to low-Z open air, the short dimension (of a rectangular horn) is best arranged horizontally; diffraction then makes the high freq spread out better to the edges of the audience. 8) Maybe not a distortion, but reflections of different freq out in the audience (from walls, people, etc.) mean a person hears different freq with varying loudness depending on postion in the room. (Rooms "color" sound.) You can sometimes hear this by moving your head as little as 4". It is more obvious with pure sine waves. In a similar way, musical instruments do not have flat spectrums, there are all kinds of resonances in instruments. The flatest sound reproduction comes from headphones! Headphones also can hurt a person's hearing--notice the freq seem to change when you put a pure sine into the amp & turn up the volume.

By the way, a voice coil has response down to D.C. If you put a 1.5V to 12V battery onto a woofer, the cone moves out or in & stays there until you remove the battery.

Audiophiles & guitarists sometimes talk about the color of the sound, the merits of brands of speakers, the goodness of types of capacitors, etc. A lot of this is imaginary, but much of it is real. Resonances & nonlinearities are prominent in music. Guitar music, in particular, is said to benefit from soft clipping of the amp, and tube amps are preferred for their higher Zout & damping of the speaker.

All in all, sound reproduction gets to be very complicated. Sound experts can make a career of it.