Monday, September 17, 2012

The Science of Music -- Part Four

In the last three articles I’ve discussed waves to the nth degree. I spoke often of the two essential characteristics (or parameters) of waves. That is, frequency and amplitude. There is also a third attribute called phase, and -- in addition -- frequency can also be represented by the reciprocal characteristic called period or wavelength. Amplitude is easily changed either through amplification or just some interesting acoustic designs such as the shape of an auditorium or the shape of a guitar body.

Now it is time to focus (no, not going back to photography) on frequency. That is what I frequently do in my spare time. Actually, the two main parameters of amplitude AND frequency will be the point of this discussion. We’re about to dive deep into frequency response charts and equalization and filter curves. This will require we add another item to our warehouse of knowledge -- a mathematical item -- straight out of high school math -- called “logarithms.”

“Logs,” as they are called by people who find it easy as falling off of the same will not be explored in depth. There’s a lot about logs that was very interesting before the invention of the modern calculator. Before calculators there were log tables and slide rules (which were created with “built-in” logs). Logs are still very important to advanced math, but I’ll bet they don’t get the focus in high school that they got back in the last century when they were invaluable aids to calculation.

We just want to understand logarithmic scales, and we won’t even have to get into the math to do that. For reasons I’ll soon explain, logarithmic scales are often used to compare both sound (and music) amplitude and frequency, and logs are at the core of decibels. That’s a tenth of a bel (deci = tenth), and the "bel" was named after Alexander Graham Bell -- famous for both the telephone and the cracker. (Is there corn in a graham cracker? Because there sure is a lot of corn is this writing!) Interesting that the "bel" is spelled with only one "l." I might have to write an article about that ... but not today.

Today we’re going to talk about the human ear and its sensitivity to both amplitude and frequency. I assume most have heard (no pun intended) about the frequency range of human hearing from 20 Hz to 20 kHz (“kHz” is 1,000 Hz.) That is a wide range of frequencies. In addition, we’ve learned that the harmonics of fundamental tones are multiples of the fundamental frequency of the note. This is also true of the “keys” on a piano, that is, the frequency of the musical notes.

Consider the note "A" that is three octaves below middle C. The frequency of that low note is 55 Hz. The next "A" up on the musical scale (that is, up one octave) is twice that frequency or 110 Hz. The next "A" higher is 220 Hz. After that, A just above middle C, is 440 Hz. It keeps going like that with each note of A up the scale twice the note below.

As I’ve said, that is called an “octave.” “Octave” actually means “eight,” and that is because there are 8 “full” tones between two similar named notes on the scale. For example, in the scale of C major, there is C - D - E - F - G - A - B - C. And these "same named" notes, obviously, are very closely related. That was one of the earliest discoveries in music, and the Greeks, who founded a lot of the math we've been chopping through, noticed this double / half relationship. The precision and order implied by this simple doubling fact influenced their thinking and this is apparent in both their math and their philosophy of science ... and that topic is definitely due for a long series of discussions ... but, again, not today.

I won’t get anymore into the “music” since this is a series on “science,” but I’m sure most musicians out there are very aware of this octave relationship. It is no surprise that all the tones called “A” on the keyboard (or guitar or trumpet) are multiples or harmonics of each other. When you think about what we’ve learned about complex waves containing multiple frequencies, it makes good sense. We will learn more about that later too. Often, when playing music or "jamming," the correct note to play will be an A, and the decision to play a low A or a high A depends on what you want to do with the melody, make it go up or make it go down. Oh, wait, I said I wasn't going to get into music today, just science. Sorry!

So it is natural if you are looking at the frequency of notes on the scale or notes that the human ear can hear, you would use a measurement that has more “resolution” for low frequencies than higher frequencies. Such a measurement is called a “logarithmic” scale.

(Warning: My first use of the word "scale" means the notes on the keyboard or staff in music. My second use of the word "scale" means the numbers on the bottom or side of a graph or chart. Sorry about the confusion.)

One example would be a chart with 20 Hz on the bottom and a given distance to the right of the scale (or “x-axis") is 40 Hz. Assume it is one inch along the bottom of the chart from 20 Hz to 40 Hz. Next is 80 Hz at the same distance, one inch from the 40. Next is 160 Hz at the same distance, etc. Equal distance for each “doubling.”

That would be a logarithmic scale with powers of two. That would work fine for a frequency chart, but most log scales are power of ten. That is they would go from 10 Hz to 100 Hz to 1,000 Hz to 10,000 Hz in equal distance on the graph or scale. This logarithmic method of charting matches the characteristics of frequency in that it is the doubling of frequency that matters most.

In addition to our hearing and discernment of frequency being “logarithmic,” it turns out the ear’s sensitivity to amplitude or loudness is also logarithmic. We note changes in volume that are multiples. Basically, in terms of measured amplitude, the ear responds to a percentage of change, not an absolute amount of change. With most people, the smallest change they can detect in sound volume or amplitude is about a 26% increase in power. (And that 26% change is the value of 1 decibel -- how convenient.)

So, if we are going to make a chart with loudness or amplitude on the “y” axis (up and down) and frequency on the “x” axis (left and right). That is called a log-log scale and most all frequency charts will have this double logarithmic scale. Here is an example of a frequency chart for a stereo amplifier.

Frequency Response Graph

Notice that the frequencies, from 10 Hz to 100,000 Hz is logarithmic and the scale changes as you move to the right. It goes from 10 to 100 to 1,000 to 10,000 to 1000,000. It may not be obvious that the amplitude scale on the left is also logarithmic. That is because, instead of showing signal strength as volts (or milli-volts: thousandths of a volt) it shows amplitude in decibels or “db.”

As I said, decibels are themselves a logarithmic measurement. Ten db increase is time ten. Twenty db increase is times 100. And thirty db increase is times 1,000.  By the way, with a power scale, 3 db is half / double. We will see (or is it "hear") sound (and music) amplitude often given in decibels or db because db’s are a logarithmic measure and the ear’s response to volume or loudness or amplitude is also logarithmic. A small change in the amplitude of a soft sound is detected, but it takes a large change in a loud sound to be noticed, but I repeat myself.

By the way, I’m using “volume,” “loudness,” and “amplitude” as the same thing. They are subtly different, and I’ll explain that too -- when it is time.

The frequency response chart I included shows variations in the amount of amplification through the range of human hearing. This is due to the design of the amplifier. The goal is for an amplifier to have very “flat” response curves. That means the output at different frequencies is about the same, and the curve would be a straight line or “flat.” Such perfection may not be possible, but high quality amplifiers come very close to attaining this flat response. (It is usually the speakers that cause the most variation with frequency.) But all audio amplifiers drop off at the lowest frequencies and the highest frequencies.

There is no point amplifying frequencies above the range of human hearing since most dogs, or dolphins, or bats don't have credit cards and don't buy a lot of stereo equipment. Plus, amplifying high frequencies that can’t be heard would waste power and even cause interference. We don’t want your audio amplifier picking up the local radio or TV station.

On the low end, most amplifiers only go down to about 20 Hz. For one thing, the design of vacuum tube amplifiers required something called “capacitive coupling” which made it difficult to amplify very low audio frequencies that you couldn’t really hear anyway. However, modern movie music has some really low tones fed to “sub-woofers” that you more “feel” than hear. With some solid state amplifiers, called “direct coupled,” can go down to 5 Hz or even lower, possibly all the way down to 0 Hz or DC. And that can really shake the seats.

(Think for a moment what is “all the way down." An octave lower than 20 Hz is 10 Hz. Next down is 5 Hz. Then 2.5 Hz. Then 1.25 Hz. It’s a LONG WAY DOWN.)

There can be problems with these direct coupled amplifiers because they can put out a DC voltage, and that is not what you want in your loudspeaker. It can cause both excessive heating in the speaker and distortion. The DC can be adjusted to zero, but that is a hassle, so there are very few true DC output amplifiers, except in laboratory equipment.

Also, any of these very low frequency response amplifiers can have problems with record turntables and something called “rumble,” which is the very low frequency component added by the rotating turntable. So, if you have a direct coupled amplifier in a modern sound system used for movies and all that jazz, then buy a good quality turntable or you’ll think there’s a Harley in the background when you play records.

Now we’ve seen what a amplifier frequency response curve looks like. What does the frequency response of the human ear look like? You’ll be surprised.

Equal Loudness Contour

Several things to note. First, ears are not manufactured by RCA or Pioneer. So they aren’t all the same. As we know from experience, human ears vary. Some are big. Some are small. Some have a lot of piercings. This chart is an average chart and it is used for certain studies and calibrations and tests. Second, note that the frequency response varies with loudness. Electronic amplifiers usually respond the same to all levels of signals. That is called a “linear” design. The human ear is decidedly non-linear. If you compare the two charts above carefully, you'll see the amplifier system has about the same graph at different sound levels, but the shape of the curves for the human ear are quite different at different amplitudes. Hmmm.

Note that when the line on the chart goes up, it takes a louder signal to hear -- the ear is less sensitive to that frequency. Our hearing is best at the point where the graph is the lowest or nearest the bottom of the graph. With very soft sounds, our best hearing is at about 3 - 4 kHz. At loud sounds it is fairly flat from 500 - 3,000 Hz. No coincidence that that is the primary range of frequencies of speech. (Also note they didn't test high frequencies at 100 db for fear of damaging the ears of the test subjects. So it is just estimated.)

The hearing response of the ear is also affected by age, and old people, like yours truly, can't hear so well in the upper ranges. What's that? ... Never mind!

Also note the relatively low response to bass sounds below 100 Hz and how the ear hears less and less as the volume drops off. I said that there were some differences between some of the terms being used. “Amplitude” is an engineering measurement either in volts or power units -- typically measured in db. “Volume” is what we call the “gain” control on your amplifier and volume is roughly the same as amplitude.

“Loudness,” on the other hand, is about how the ear hears it, and you can see that bass tones must be much louder to be heard as the same level as 1 kHz -- and even more so at low volume levels. That's right, our response to bass does go up with volume. That’s why some old stereos had a “loudness” button. You press that button, and bass is boosted. In theory, that was for listening to music at a soft level. You pressed the “loudness” button to increase the bass to make up for the ear’s non-linear response to bass. Of course, I always turned up the volume AND pressed the “loudness” button because I LOVE BASS. What did you say? I said ... oh never mind.

Some amplifiers will label the gain control "volume" and some will label it "loudness," although the second label tends to imply the amplifier changes its frequency response automatically; as you turn up the volume, it turns down the bass. Or your amplifier may have buttons or knobs to adjust the frequency response or even a row of little controls to adjust the gain for each octave. That will be the next topic, how all these tone controls and equalizers work, as well as the equalization of phonograph records (RIAA) and other topics relating to tape recorders and CD players. At least, we'll get started putting these "wave" and frequency ideas to work.

Well, gentle readers, that’s enough for today. I’ll come back to this topic next time as we talk about equalization, tone controls, and filters. We will learn what they are, how they work, and why we use them. You’ll see some more of these log-log frequency charts. Until then, turn up the volume, press the loudness button, turn on the sub-woofer, and FEEL THE NOISE ...

Come on feel the noise
Girls rock your boys
We'll get wild, wild, wild
Wild, wild, wild.

So you think I got an evil mind
I tell you honey
I don't know why
I don't know why

So you think my singin's out of time
It makes me money
I don't know why
I don't know why
Oh no

Come on feel the noise
Girls rock your boys
We'll get wild, wild, wild
Wild, wild, wild.

Now you’ve got it. Get your guitar. Go to the closet. Put on your sister’s dress, maybe a little eye shadow. I’ll make a rock star out of you yet.

Originally written on Feb. 18, 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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