Monday, September 17, 2012

The Science of Music -- Part One

There were still topics I could have written about in the Science of Photography series. I could have described lenses and issues such as chromatic aberration or lighting and color temperature or lens bokeh and aesthetic blur. Lot’s more photography to talk about, but, instead, I’m starting a new series I call “The Science of Music.” Again, there are many topics to discuss including scales and chords and tunings, but my goal is quite focused this time. I want to describe enough about music (and sound) to make a transition from analog to digital. I promised a friend long ago to write about music CDs and how they work and why many aficionados, including my friend, prefer records.

So let’s start building a body of knowledge of music and what it is from a physical perspective and take the journey from records to tapes to CDs to whatever may be next. We can talk abut compression and expansion and tone controls and dynamic range and frequency response and timbre and color and tonal richness. There won’t be any Bach or Beethoven (or Beatles), but plenty from Edison to Berliner to Goldmark to Ray Dolby and Bill Lear to Amar G. Bose and Henry Kloss. Of course, we’ll also talk about Jonathan Livingston Digital, the person who invented the modern CD or “digital” recording. What, there was really a guy named “Digital?” No, I just made that up, but it made you think ... didn’t it?

Let’s start with the fundamentals. That would be something from nature called “simple harmonic motion.” It’s all around us. Take a spring held at one end with a weight on the other end. Pull the weight down and it will start to oscillate or “bounce” up and down. If you carefully plot the oscillations, they will plot out a simple harmonic motion.

Or, consider a pendulum or a swing. Pull the swing all the way back and let go, it will swing to and fro in simple harmonic motion. Let’s analyze this swing motion carefully. When you let it go, it was still for just an instant, but then gravity started it falling. As it passed though the bottom of the swing motion it was going the fastest it will go. It then starts to swing up on the other side of the arc and now gravity makes it slow down until it reaches a point almost exactly as high as the point you let it go. At that instant it stops, and then starts falling back the other way. Again, as it passes through the bottom of the swing it is going maximum velocity and it climbs back up to the point where you let it go.

This isn’t quite simple harmonic motion because there is some friction. The wind resistance robs the swing of a little bit of energy and it won’t come back to quite the point you let it go. It will swing back and forth several times, each time a little less height until the motion dies out. It is call “damped” harmonic motion because the friction slowly dampens out or reduces the amount of swing.

Now, think of a guitar string and give it a pluck. Simple harmonic motion? No, in this case it is not. That’s because the guitar string adds harmonic overtones. We’ll get back to that in an article or two. For now, let’s stay with the simple harmonic motion, although now you know why I started with that physical phenomenon. It is related to sound and music.

Before we try to make the connection, let’s talk about the pendulum or bouncing spring. Pendulums are used in clocks to keep time. They have a mechanism that is typically a wound spring or an electric motor that gives the pendulum a tiny push each time to keep it from dampening out. That is true harmonic motion and it produces a very simple motion as I described earlier. Physicists model that motion with a mathematical equation that contains the sine function. You remember the sine (and the cosine and the tangent, etc.) from Trigonometry ... Remember? ... High school? ... Math? ... Ah yes, now it all comes back.

You probably first learned about the trigonometric functions as ways to work with right triangles. This is the same sine function, but now we work with harmonic and circular motion.

The complete formula that describes simple harmonic motion of the spring is "y = sin(x)." The formula can get a little trickier if if we add amplitude and phase and throw in a few greek letters for flavor. A version I like is "y = A sin(t)." That formula ignores an important characteristic called phase which I’ll address at some future time. For now, let’s focus on two important characteristics of this sine wave or “sinusoidal” wave.

First is something called “amplitude.” You could also call it something simple and non-scientific like “volume” or “loudness” and you would adjust this amplitude with the “volume control” or “gain” on your radio or guitar amp or sound system. In my formula, the amplitude is represented by "A."

The second important characteristic is the frequency. It is the “t” in my little equation, but to really form a good mathematical equation you would have to write the value of “t” in some form such as “radians per second.” Most of us think of frequency as cycles per second which are called “Hertz” (abbreviated "Hz") after Heinrich Hertz (1857-1894), the German physicist. Musicians call frequency "pitch."

There are different ways to indicate this as part of the sine function, but we don't need to get into that in great detail. Don't get me wrong, I'd love to dig into the math, but we're here to learn the science of music so we can understand records and tapes and CDs, not to get involved in the math any more than needed. I'll try to keep the mathematics at the 20,000 foot level, and hopefully it will make sense and add to your understanding, not snow you in like a Colorado blizzard.

It is important, however, for you to understand that this fundamental building block of all music (and sound) is a simple sine wave. AND NOW YOU DO!!!

For a basic, “pure” sine wave, that’s all you need to know: the amplitude and the frequency. If you have more than one sine wave or you’re measuring in respect to some instant in time, there is a third parameter called phase, but I said I’d save that for later.

Now here is some simple math. If sin(x) is a particular frequency, say 440 Hz, then sin(2x) is twice the frequency. We’ll use that idea later too. Not too painful, eh? Well, that was math. It didn't hurt at all. At least not yet!

Getting back to the guitar string, I’ll bet you’re wondering why the sine wave formula doesn’t work for it. Well, it does, but we have to get fancy. You see, if you pluck a guitar string, you don’t get simple harmonic motion, you get a very complex motion (and sound) that contains more than one sine wave at different frequencies. We’ll get into that soon, although it may have to wait for the next installment. It is a good thing there are these extra frequencies, because a very pure sine wave is not very musical or enjoyable.

There are no musical instruments that create pure tones (at least before the invention of synthesizers and some modern electronic instruments.)

By definition, a pure tone is a tone with a sinusoidal waveform -- a pure sine wave.

A pure sine wave is an artificial sound. Hermann von Helmholtz (1821-1894) is thought to have created the first pure sine wave tone with the "Helmholtz siren," a mechanical device that forces compressed air through holes in a rotating plate. This is presumably the closest thing to a sine wave that was heard before the invention of electronic oscillators. If you remember the start to the old black and white TV series called "Outer Limits," the opening shot was an oscilloscope showing a sine wave with a noise in the background that we all associate with electronics and old sci-fi movies, but not exactly musical.

(As an aside you may have heard of the Helmholtz Resonator in loudspeakers. Loudspeaker enclosures often use the Helmholtz resonance of the enclosure to boost the low frequency response. So that old guy actually helped with modern high fidelity sound systems.)

Another aside: By the way, two guys in their garage in Palo Alto, California developed a simple circuit to produce a very pure, audio frequency sine waves and went on to found a giant electronics corporation.

No, their names weren’t Steve Wozniak and Steve Jobs. This was 40 years earlier and their names were David Packard and William Hewlett and they founded Hewlett-Packard where the two Steve’s worked many years later. Also, that first oscillator called the 200A was first produced in 1939. In 1969, yours truly was using the upgraded version called the 200CD and I once owned a 200CD of my very own. The HP signal generators were famous for producing such a pure sinusoidal waves and the amplitude remained quite constant as you adjusted the frequency. Many a stereo had its technical data produced in the laboratory using one of the HP audio signal generators. It was the foundation building block of HP today. And most feel HP was an essential core of what is now called "silicon valley," although Stanford University had a part to play in building the valley too.

By the way, a pure sinusoidal musical note is not a nice thing at all. “[Pure] Sine waves are generally uncomfortable to the ear, and may cause noise-induced hearing loss at lower volumes than other noises. Sound localization is often more difficult with sine waves than with other sounds; they seem to ‘fill the room’.” -- to quote Wikipedia.

But sine waves are where it all starts. So, hang onto your hats, turn up the volume, and wait until the next installment to learn how sine waves can be combined into beautiful musical tones. In the case of electronic synthesizers, they actually are combined to produce musical notes and many electronic keyboards and organs have been doing this combining for years. For example, the harmonic draw-bars on the original Hammond Organ adjust the mixture of sine waves, and Robert Moog really gave you power to build up these waves with his instrument's knobs, and dials.

But all traditional musical instruments create the waveforms complete with pleasing overtones and harmonics whether from the vibrating strings of a violin, piano, or guitar or from vibrating air in a horn, woodwind, or flute. There are a few other examples such as a xylophone or bell or drums and other percussion instruments that get the tones from vibrating chunks of metal, wood, or glass. But, in all those cases, the natural tones are a combination of sine waves. The lowest frequency of all these sine waves is called the "fundamental tone."

And that is why we started with the sine wave. It is in combination that we get the beautiful music. And that combination will be the subject of our next installment of “The Science of Music.”

Originally written on Feb. 15, 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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