Monday, September 17, 2012

The Science of Music -- Part Six

With a title like “The Science of Music,” there are a lot of directions the narrative could take. We started off pretty general, describing sinusoidal waves and explaining why they are essential to understanding sound and music. From there we learned about complex waves and harmonics, still pretty general. The next discussion focused on the frequency range of human hearing. All very logical foundation topics for almost any scientific discussion of music.

Then along came the last installment, and it focused on filters and even touched on RIAA equalization. Where do you go from there? We could jump track and talk about the physics of how the violin bow excites the string vibrations when drawn or how woodwinds change resonant frequency depending on which holes are covered with which fingers. But you know that is not the direction I’m headed. I’ve mentioned a couple of times that my long term goal is to study various music recording mediums and ultimately compare phonograph records to CDs. Perhaps this series should be called “The Technology of Music.” But we’re stuck with the title now, and there’s no going back!

With that as our final destination, then it should be no surprise that we will start focusing on various electronic devices, amplifiers and what-not, that are involved in the music recording and music reproduction systems. We might call these systems by the good old name of “stereo” as in, “Can we play some music on your stereo?” even if “stereo” is a little quaint in this age of Dolby surround sound and iPods.

And that leads us to this installment where we will talk about distortion. That is, something the “stereo” does wrong that changes the music. For a formal definition of “distortion,” I offer “an alteration of the original shape (or other characteristic) of sound or music  waveform.” Distortion is usually unwanted, and many methods are employed to minimize it in practice. In some cases, however, distortion may be desirable, such as with electric guitar distortion. I will discuss both unwanted and wanted.

The addition of noise or other extraneous signals (such as  hiss, hum, and interference), is also not desired, but it is not considered to be distortion. We will save noise for another day.

There are several different types of distortion that you can get in a “stereo” system, that is when recording and amplifying and playing music. They include (1) Amplitude distortion, (2) Harmonic distortion, (3) Frequency distortion, and (4) Phase distortion. There are few more, but they are not a problem with audio frequency amplifiers and recording reproduction systems although intermodulation mixing or Modulation distortion is considered by some to be another kind of distortion. There are also specific distortions which fit in one of these four (or five) categories, but for certain reasons have special names such as “wow” and “flutter.” Finally, there are special distortions related to quantization which is what you do in order to convert analog to digital -- that is one you may encounter in CDs and DVDs.

The most important characteristic of a quality audio amplifier is something called linearity. That is, the signal is amplified in a linear manner. Small signals need to be amplified the same amount as large signals. Remember, all signals, whether they are loud or soft, are made up of sine waves that go from zero to a positive, maximum value, back to zero, and then to a maximum negative value, and then returns to zero again. If the amplifier increased parts of this wave more or less than other parts, then the wave would be distorted.

A much more serious form of amplitude distortion is a problem called “clipping.” Suppose the largest size waveform the amplifier can handle is 1 volt peak, and the sine wave input to the amplifier goes from zero to 1.2 volts peak at its maximum. The amplifier will probably amplify the wave fine up to where the voltage is 1 volt, and then can’t go higher. The result is that the amplified wave will have a “flat top.” It looks like the tops (and, as far as that goes, the bottoms) of the sine wave are clipped off like a pair of scissors cut them.

Clipped Sine Wave

 Notice that the sine wave now looks a bit like a square wave. Also, recall that a square wave contains a lot of odd numbered harmonics, the 3rd, the 5th, the 7th, etc. So, if you clip a pure sine wave, you will actually create new sine waves that are harmonics.

Although I stated that musical tones are complex waves, this distortion effect still occurs and is a problem. With the music of a symphony, clipping is going to sound very, very bad. Another name for this extreme form of amplitude distortion is called harmonic distortion because the new frequencies are harmonics of the distorted sine wave.

But, if this is a guitar amplifier playing one or two notes, then this clipping will create a very heavy sound called “fuzz tone.” And fuzz tones are devices designed to clip the waves creating this distinctive sound. They are usually found in little boxes that are at the feet of the lead guitar player. A little fuzz helps the heavy metal groove.

Typically, if you play a chord through a fuzz tone, it won’t sound too good. All the new frequencies sort of clash. But a single note or two notes strongly related, like musical fifths, can really sound “heavy” with the addition of a little fuzz -- or a lot of fuzz: “No time left for you ...” Guess Who?  Recall I said distortion is not good with music reproduction, but isn’t bad -- sometimes -- with music production.

(I know you usually play "sevenths" with a fuzz tone, but you musicians know that the seventh half-tone up is the same as the fifth half-tone down. So they're all "fifths." Musical chord theory, meet science. How do you do?)

You can get this fuzz tone effect by overdriving an amplifier. Most musicians prefer the sound of overdriven tube amplifiers rather than transistor amplifiers. I think that is because tube amps go into clipping in a little more gradual manner than solid state guitar amplifiers. When you over drive transistor circuits, it is usually pretty bad sounding, while vacuum tube amplifiers tend to handle overdriving in a more gradual, muddy manner that many prefer and some call “warm.”

Of course, some people prefer tube amplifiers just because they heard they are cool -- well, their actually hot. Any temperature applies.

Measuring Amplitude and Harmonic Distortion

Since the effect of nonlinearity in an amplifier is to create new sine waves at harmonics of the fundamental frequencies being amplified, amplitude and harmonic distortion are essentially the same thing. Amplitude distortion describes the "cause" and harmonic distortion describes the "effect." We can use that fact to measure harmonic distortion, and, thereby, indirectly, measure amplifier linearity.

What we can do is apply a pure sine wave tone to the input of an amplifier. That is where something like that HP 200CD Audio Signal Generator comes in very handy. Recall I said it produces a very pure sine wave. So if we feed in this pure sine wave into the amp, and then measure the harmonics coming out of the amplifier, that would give us a good measurement of the harmonic distortion.

To do that, we could take the output of the amplifier and use a very deep notch filter tuned to the fundamental frequency of the signal generator to attenuate or reject that input frequency. What is left is just the harmonic distortion.

Now is a good time to discuss how you measure a sine wave or complex wave signal. We’ve been looking at these waves on charts and on oscilloscopes, and we’ve been discussing things like the crest and the trough. I’ve also explained that a sine wave as an electronic signal goes from zero to a positive peak, back to zero, then to a negative peak, and back to zero again. Using an oscilloscope we can measure the value of the peak voltage and that is called “peak voltage” -- naturally.

We can also measure the voltage between the two peaks, that is, from the crest to the trough. Sine waves are symmetrical. That is, the positive peak is identical to the negative peak. With complex waves, that may not be true. So, with a sine wave, if you measure the positive peak and add the negative peak (ignoring the negative sign, other wise it would be subtraction), you get the peak-to-peak value. (More Colorado talk.) So a wave with a + 1.0 volt positive peak and a - 1.0 volt negative peak is a 2.0 v. p-p.

These are all measurements you can take with an oscilloscope. But if you are comparing different waveforms, the most useful measurement is the “effective” voltage. That is the voltage that is comparable to a DC voltage. DC is what you get from a battery. It doesn’t change (except as the battery dies out). Since a sine wave, or any wave for that matter, is always changing and has zero volts at one instant and maximum or peak volts at another, it begs the question "What is the best average voltage?" Effective means it has the same heating effect (or power) as a given DC voltage.

Using trigonometry and a little Calculus, we can calculate the effective voltage by taking the “root mean square” value. “Mean” is an average and the squaring has to do with calculating the power which is the square of the voltage and it is power that produces the heat (or -- with a guitar amp -- the volume or LOUDness). This root mean square is usually abbreviated r.m.s.

With a pure sine wave, the r.m.s. value is equal to the peak value multiplied by 0.707. It is not exactly .707. It is actually one half the square root of two (which is also 1/square root(2) , isn't math odd?) For complex waveforms, you would have to do the Fourier analysis and separate each individual sine wave frequency and amplitude and then multiply and add ... and that is what computers are for. There are also volt meters than can read the r.m.s. values of unusual waveforms. For example, the r.m.s. value of a perfect square wave is 1.0 times the peak while a pulse wave can have very low effective voltage even though the peak is high. That's what a radar wave looks like. OK, back to audio.

HP (and others) make nice volt meters that measure r.m.s. So now we can perform the experiment that measures amplifier distortion. Feed in a pure sine wave, say it is 1,000 Hz. Then run the amplifier output through a filter that attenuates the 1 kHz fundamental by over 60 db. That’s ten to the sixth or one millionth its original value. For all practical purposes, assume the 1 kHz fundamental is now gone. Measure the output of the filter and what is left is harmonic distortion. Feed in a 1 v r.m.s sine wave and, if you get .01 v out of the filter, than is called 1% total harmonic distortion since the added harmonics are 1/100 or one percent of the input sine wave. Good audio amplifiers have between .01 and .001 percent total harmonic distortion (measured as r.m.s.). If you want to test distortion that low, get a 120 db notch filter!

So that is how you measure and specify harmonic distortion, and thereby, indirectly, amplitude distortion and amplifier linearity. You may repeat the test at other frequencies including 500 Hz and 100 Hz or even higher than 1 kHz, but remember, the amplifier has its own low-pass filter, so it is the lower fundamental frequency tests that best measure the THD or Total Harmonic Distortion of the amplifier itself.

Nonlinear amplifiers can also have a problem called modulation or intermodulation. Modulation is a good thing in a radio transmitter -- you know, amplitude modulation (AM) or frequency  modulation (FM), but it typically isn’t something you want your music to do. Modulation distortion occurs when two or more sine waves "mix." This mixing in a nonlinear circuit creates "beats." These are new frequencies that are the sum and the difference of the two original frequencies. For example, if 1,000 Hz and 1,100 Hz intermodulate,you get 100 Hz and 2,100 Hz. These are not harmonics and note that one is a low frequency, not an over tone or higher tone. I won’t get into any more details, but simply state that amplifier nonlinearity can cause the creation of new frequencies and you are not going to like the sound. Harmonic overtones from harmonic distortion can be pleasing to the ear, but these modulation produced tones are real bad. That is why two notes next to each other on the piano, like C and C# sound so bad when played together. More deep music theory happening there. However, like all art, breaking the rules can be good. Piano discord on "I Want to Tell You" -- good. Discord of the Beatles themselves -- not good.

Frequency Distortion

Frequency distortion is something we’ve already discussed in the last installment. It simply means that not all the original frequencies are amplified equally. Is this really distortion? If I like my Beethoven symphonies with extra bass and less treble, and I adjust the tone controls accordingly, then I’ve distorted the signal, but I LIKE IT!!

It is a true distortion, however. Certain parts of a “stereo” system are particularly limited in frequency response. That includes the microphone that recorded the music or speech, the loudspeaker in the system, and even the room where either the microphone or loudspeaker is in. In addition, long cables to the speakers can cause a loss of high frequencies. This is frequency distortion.

Again you can compensate with tone controls. Graphic equalizers are particularly useful in adjusting the frequency response to match the room. Most rooms have resonances that will make certain bands of frequencies much louder, and that is not an effect you typically want in your music.

Phase Distortion

The final class of distortion to be discussed is phase distortion. Remember that amplitude and frequency are the key attributes we try to preserve. But there are also phase relationships between the different component sine wave that need to be preserved. This is especially an issue to maintain stereo “separation.” That is, the ability to reproduce a good sound image that is interpreted by the human ear and mind to give a three dimensional perspective to the sound.

There are several theories that explain the ability of humans and animals to locate the source of a sound. In general it is due to having two ears. A sound from the left will reach the left ear before the right ear. It will likely be louder in the left ear too. There are a lot of other issues involved such as the addition of sight and an awareness of surroundings. The ears will hear different sounds in terms of both amplitude and also delay effects. These delays mean the left ear and the right ear may hear a complex wave with a different phase. So timing and phase are involved. Good amplifiers will preserve the relative amplitude of a sound between the two stereo speakers, but what is the impact of phase?

Recall that phase was how much ahead or behind a second wave is compared to the first wave. Also recall that audible frequencies are from 20 Hz to 20 kHz and that wavelength is a reciprocal function of frequency. That is, the 20 Hz wave has the longest wavelength and 20 kHz the shortest. Compare the wavelength of these frequencies in the open air: 20 Hz = 56 ft.; 100 Hz = 10.1 ft.; 1 kHz = 1.1 ft.; and 10 kHz = 0.11 ft. With the human ears separate by less than one foot, even taking Big Head Todd into account, then the phase difference with low or bass tones will be very small. BUT, the phase difference with high frequency tones is quite different at ear separation distances.

For that reason, the human hearing system, ears and brain, only have the ability to locate high frequency tones. That’s why bats and submarines use extremely high pitched sounds to navigate by “sonar.” And stereo systems, those with two or even more speakers, really only produce a stereo image with the high frequency tones.


Further, most of the “power” in the musical spectrum is at the low frequencies. Remember the ear doesn’t hear low tones as well as 1 kHz waves, so bass notes must be amplified much more than treble. So the power requirement for “bass” is much higher than for “treble.” Further, bass reproduction requires bigger, and therefore more expensive speakers. Put all these scientific facts together and it leads to the “sub-woofer.” Since your ear can’t separate the bass on the left from the bass on the right, just combine the two channels into a single channel, equip that channel with a powerful amplifier and big speakers, and you’ve got it. Use a low-pass filter set around 100 Hz to separate the bass signal from the rest of the music, and feed that to this sub-woofer system. Then the “stereo” speakers, which are now only producing the mid-range and high frequency signals, can be smaller. That is the basis of modern “surround sound” systems.

The Dolby corporation coined some terms: “2.1” implies two speakers (stereo) with one sub-woofer; “5.1” is two stereo speakers, one middle speaker located between the stereo speakers, two speakers behind the listener, and one sub-woofer; and “7.1” adds two more speakers opposite the ears. The goal is to reproduce the sound in a movie theater where we simulate movement on the screen with sound moving around the auditorium using multiple speakers. (Some theater surround systems have 10 or 20 pairs of speakers along the walls to produce this effect.) George Lucas founded THX sound systems to allow home theaters to match this movie sound with a lot less speakers. The THX name is from one of his early movies, "THX 1138" with Robert Duvall -- a big name in Hollywood. THX and Dolby are also big names, and big standards, in modern surround sound systems.

So, to get back to distortion, phase distortion would shift the relative phase of the component sine waves, and both change wave shape as well as modify the stereo imaging. We may do this on purpose if we’re trying to create surround sound effects out of normal stereo signals. Shifting phase is done with reactive components such as capacitors and inductors. Recall that those passive components were also used to make filters. It is important in filter design to control any phase shift and very effective filters, such as the one I described for measuring total harmonic distortion, must be designed very carefully to minimize phase shift. You can translate “designed very carefully” to “expensive.”

Other Distortion

I mentioned earlier that vacuum tubes went into overload and clipped in a little “rounder” or “softer” manner than solid state devices. In general, vacuum tubes are more linear than transistors. Modern transistor audio amplifiers are very linear, but they obtain that linearity using something called feedback -- negative feedback. One problem with feedback is it can create certain types of amplitude distortion, especially when the signal is changing rapidly, such as high frequencies and transients, which is the name of quickly changing wave shapes, for example, when you strike a drum or cymbal. One of these transient induced distortions is called SID or slew-induced distortion and the other is called TIM or transient intermodulation distortion.

Slew-induced Distortion (SID, or sometimes: Slew-rate Induced Distortion) is caused when an amplifier or transducer is required to change output (or displacement), that is: "slew," faster than it is able to do so without error. Transducer is the name of a device that converts the mechanical energy of sound or music to or from electrical signals. Some example transducers are guitar pickups and microphones as well as loudspeakers. Being physical, and having mass, they can have problems with transients which are require very rapid movements. Special design of both microphones and speakers is required to capture and reproduce these very high frequency phenomenon.

During these displacement signals or transients, other signals may suffer considerable gain distortion, leading to Intermodulation distortion. A loud snare drum wave may actually distort the guitar sound in a recording when amplified in the "stereo." Transient Intermodulation Distortion (TIM) is a form of modulation of the signal during these rapid changes and also may cause compression or reduction of the peaks which would add harmonics. Basically the sudden changing input causes nonlinearity in the amplifier response that can momentarily create modulation effects. It has to do with the impact of the sudden change on the feedback network and delays in that feedback and phase changes in that feedback and ...  Modern amplifier design is very complex. Limiting high frequency response to 20 kHz helps since the higher frequencies contribute a lot to this complexity, but even 20 kHz response requires some good designs by some smart engineers. That is why the best stereo equipment can be very expensive.

Modern solid state stereo amplifiers minimize these effects, but they can still exist. A lot of purists point to the fact that vacuum tube audio amplifiers have much less negative feedback in their design, and so they are less effected by these transient distortion effects. However, solid state devices, due to their very small size, can handle the high frequencies in a transient or rapidly changing wave better. So, as they say, the jury is still out, and you should sample both with your own ears to decide which is better -- vacuum tube or solid state. Finally, remember, the weakest parts of the chain are things like microphones, phono cartridges, and especially loud speakers, not to mention the furniture and window coverings and carpet in the listening room. Are you ready to rebuild your home entertainment room?

Wow and Flutter, Temolo and Vibrato

Finally there are some effects where frequencies of fixed tones vary. This can be on purpose with vibrato added to an organ tone or tremolo effects. If they are intended, that is fine, but the amplifier should not add them on their own. Tape players such as reel-to-reel, cartridge (remember the eight track) and cassettes can have problems where the tape speed varies and that is called “wow” and “flutter.”

"Flutter" is a rapid variation of signal amplitude, frequency, or phase. In recording and reproducing equipment, the deviation of frequency caused by irregular mechanical motion, such as that of capstan rotation speed in a tape transport mechanism, during operation can be the cause. "Wow" is a relatively slow form of pitch variation which can affect both phonograph records and tape recorders.

We also purposely add these "flutters" to the music. Tremolo and Vibrato are terms you hear (no pun intended). Technically there is a difference between “vibrato” and “tremolo.” Vibrato is a wavering of the amplitude of the signal and tremolo is an actually change of frequency. Violinists introduce tremolo by moving the finger holding down the string rapidly back and forth. This changes the pitch slightly.There are several reasons that violinists do that. First is, since the violin has no frets, it is difficult to finger at exactly the right point on the string to produce a given note. A little tremolo increases the chance you will hear the correct pitch. In addition, violin sounds are very nearly pure sine waves, and we've mentioned that pure sine waves are not very pleasing to the ear. A little tremolo adds variations and that makes the violin note much more musical to the ear. Some singers add a lot of tremolo to their voice; another trick to hit the right pitch.

Similarly, with electronic organs, the notes can be rather pure. So organs often have vibrato and tremolo added to enrich the sound. Vibrato is pretty easy to produce in an electronic organ. Tremolo is a bit harder to do, especially in the old days with only tube circuits, since each circuit cost a lot to build. Current solid state organs have a lot more circuits since transistors and integrated circuits are dirt cheap.

With the old tube organs, this tremolo issue was more of a big deal, so Leslie invented a speaker cabinet that had rotating horns over the speakers. As the horns spin they add a Doppler effect to produce a real tremolo to the musical tones.

Doppler effect is what you hear when a train blows its whistle as it passes by. If you listen to a train whistle as the train approaches, it seems higher pitched than when the train is moving away from you. That is because, when the train was approaching, the wavelength of the sound was shortened by the train's motion toward you. Shorter wavelength is higher frequency. As the train moves away, the opposite effect occurs and the wavelengths are lengthened by the moving of the train, which lowers the pitch.

The rotating horns in the Leslie speaker cabinet have a similar effect and the result is a true tremolo. The best part of the Leslie design is that it has two speeds and the belt that drives the speaker cones has some slippage. So, when the change speeds, the effect is a gradual and noticeable change from slow to fast or from fast to slow. Modern electronic organs can duplicate the Leslie sound (although not perfectly), and can even mimic the gradual change from slow speed to fast. But, give me a real Leslie and a Hammond B-3 any day!!

Finally, quantization noise is part of the analog to digital conversion process. In analog-to-digital conversion, the difference between the actual analog value and quantized digital value is called quantization error or quantization distortion. This error is either due to rounding or truncation of the binary number produced. With digital systems, there can also be "jitter" caused by variations in the clocking circuits in the analog-digital converter. I'll get into that more when we get to digital music. Some consider this quantization error noise and others refer to it as a distortion. And that is all the noise and distortion we need for now. So, until next time, when I’ll “Bring in ‘da Noise,” again, good-bye.

Originally written on Feb. 21, 2012 during a visit to my Dad's home in Hillsboro, Oregon and posted on Facebook. During my two week visit with my dad, I wrote an article a day. I started with a long series on the Science of Photography which had thirteen individual articles. I then started this series on the Science of Music. It isn't finished and I have a lot more to say. I hope to add to this series in the future.

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