Just Intonation, Part 1 Music Technology Pages Return to MusicWords Home

Just Intonation, Part 4: The Acoustics of Intonation

In order to explain the theory that underlies just intonation, we need to start with a little background check. You probably know most of this stuff already, but we need to be clear about it before we go on.

Sound consists of vibrations in the air or some other medium. (I'll have a little more to say about what happens when air vibrates when I get around to writing the sampling tutorial.) Another term for vibrations is waves. Sound travels through the air more or less the way ripples travel across the surface of a pond, though it travels a lot faster.

In fact, we can draw a picture of a sound wave that looks a lot like a cross-section of a water wave. This type of diagram is our old friend the waveform, which you've probably seen in the pages of Keyboard, if not on your computer screen. When we draw a waveform, we're diagramming changes in air pressure (or the equivalent in some other medium, such as electrical fluctuations in a wire) over the course of time.

These vibrations, or changes in air pressure, come in two flavors: periodic and random. Random vibrations, in which there is no repeating pattern, are technically known as "noise." In periodic vibrations, on the other hand, the waveform repeats. In the physical world no two repetitions of a sound wave are exactly alike, but they're similar enough that our ears can easily detect the similarity and repetition.

If you pluck a guitar string, you'll be able to see sound waves being generated. The string will vibrate back and forth rapidly. These vibrations will be transmitted to the air, either through the body of the instrument (in an acoustic guitar) or through pickups and an amp (in an electric guitar). The string will be vibrating too rapidly for you to see individual cycles of the vibration, but it should be obvious that it's making approximately the same physical motion over and over, which will result in a repeating, periodic sound wave in the air.

Think about how the string moves. At a certain moment, it's stretched as far out to the side as it can go without snapping. Tension pulls it back through the center position -- the place where it will eventually come to rest. As it keeps moving, it will reach a point where it's stretched out to the other side. Then it will rebound back through the rest position again. Before long it will be back at the point where it started.

The movement I've just described, a complete pattern of travel from one side to the other, and then back, is called a cycle. And because the movement of the string repeats rapidly, it makes sense to talk about it in terms of how many cycles the string goes through in a single second. The term cycles per second gives us a way of talking about how the string is behaving.

We could just as easily talk about cycles per minute or cycles per hour, but the numbers would be so large they'd be unwieldy. Typically, sound waves occur in a range between 20 cycles per second and 20,000 cycles per second. The standard term for "cycles per second" is Hertz. This is abbreviated "Hz." So we usually say that the range of human hearing is between 20Hz and 20,000Hz. The term "kiloHertz" means "thousands of Hertz," so 20,000Hz can also be written 20kHz.

Sound waves that vibrate slower than 20Hz are called subsonic. Those above 20kHz are called supersonic. Such sounds exist in the air, but we can't hear them.

Another term that's often used is frequency. In acoustical theory, the frequency of a periodically vibrating object is measured in cycles per second, or Hz. Frequency is not quite the same thing as pitch, because pitch is a subjectively perceived phenomenon -- that is, dependent on our ears -- rather than absolute. Our perception of pitch depends to some slight extent on other factors besides frequency, but for practical purposes, the two terms mean pretty much the same thing.

Pitch

As a musician, you probably know the difference between a high pitch and a low one. Beginners who play stringed instruments are sometimes confused by this terminology, because they think of the outer end of the neck, where the tuning pegs are located, as being "high." But the way the words are usually used, stopping the string in this area produces lower pitches. In scientific terms, a high note is one that has a higher frequency -- that is, more cycles per second.

Though we can't count the cycles in a pitched sound consciously, our ears are very good at counting them and sending our brains some very precise information about the count. Not only can we tell when one sound is higher or lower in pitch than another -- that is, when we're hearing more or fewer cycles in every second -- we can tell when we're hearing exactly twice as many or half as many, or when we're hearing exactly 50% more or 50% fewer.

If you have any musical training at all, you know what an octave is. The octave is so basic to our perception of frequency that two notes an octave apart both have the same name. Two notes named G, for instance, will always be one or more octaves apart (unless they're zero octaves apart, in which case the interval is called a unison). And when we say two notes are an octave apart, what we mean in terms of counting vibrations is that the higher note has exactly twice as many vibrations per second as the lower one.

For example, if we start with a note that has a pitch of 50Hz, a note with a pitch of 100Hz will be an octave higher. The next octave up will be a note at 200Hz, and so on. We can easily construct tables that show such octave relationships:

What's interesting about such a table is that it isn't linear; it's exponential. That is, in the lower end of the frequency range, two notes that are an octave apart are separated by a very small number. In the upper end of the range, two notes an octave apart are separated by a much larger number. If we're planning to put 12 notes into every octave -- which is not necessary, but a useful place to start -- it's easy to see that two adjacent notes will be a lot closer to one another in the lower range than in the upper range. At least, they'll appear to be closer together if we look at the raw frequency numbers.

But our ears don't hear in raw numbers. They hear in ratios. The interval of the octave is a ratio of 2:1 (a doubling in frequency), and our ears hear any octave interval as being essentially like any other octave interval, whether the actual pitches are 25Hz and 50Hz, 400Hz and 800Hz, or 2,713Hz and 5,426Hz. Likewise, the interval of a perfect fifth is a ratio of 3:2, and any two tones whose frequencies have a 3:2 relationship will be perceived as a perfect fifth, whether the frequencies are 100Hz and 150Hz (50Hz apart), 524Hz and 786Hz (262Hz apart), or whatever.

It's easy to confirm that your ear hears intervals as ratios by playing a few notes on a piano. If you play a C and D in the lower register, followed by a C and D in the upper register, your ears will perceive the two C-D intervals as being essentially identical, even though the upper-register C and D are a lot further apart in terms of raw frequency numbers.

The ability of our ears to identify intervals is what makes just intonation -- or, for that matter, any intonation system -- work. In the next section of this tutorial, we'll take a closer look at interval ratios.

Harmonics and Overtones

A semi-legendary ancient Greek named Pythagoras is supposed to have discovered the harmonic relationships in plucked strings. Whether or not he existed, what he discovered was this: If you pluck a string, and then stop it at the mid-point and pluck it again, the second tone will be an octave higher than the first one.

In modern terms, we'd say that the length of the vibrating string has an inverse relationship to the frequency. If the length of the vibrating portion of the string is 1/2 of the length of the open string, the frequency is 2/1 of the frequency of the open string.

If you stop the string high up on the neck, so that only 1/3 of its length can vibrate, the frequency will be 3/1 times the frequency of the open string. And if you stop the string 1/3 of the way along its length, leaving the upper 2/3 of the string to vibrate, the frequency of the resulting tone will have a ratio of 3/2 compared to the frequency of the unstopped string -- a perfect fifth.

Pythagoras was quite taken with these mathematical relationships. He built a whole scale using the interval of the perfect fifth as its basis. Or so legend has it. If your synth has a setting for a Pythagorean scale, it's built using nothing but octaves and fifths, though that scale was developed during the Renaissance, I believe, not by Pythagoras.

Overtones

It wasn't until the 19th Century that a mathematician named Helmholtz made a crucial discovery about the nature of sound waves. (Maybe he only formulated a fact that had been known empirically for a long time. Whatever. He still gets the credit for it.) Helmholtz discovered that any tone could be described mathematically as the sum of one or more sine waves, each of which had its own frequency and amplitude.

If you have a synthesizer handy, you can set it up to play pure sine waves. They're really, really muted, and not very interesting musically. We won't have much more to say about them. I'm certainly not going to try to explain the term sine, which comes from trigonometry. Suffice it to say that a complex sound wave (trumpet, snare drum, etc.) can always be deconstructed mathematically into a bunch of sine waves.

With most instruments that produce pitched tones, the sine waves within the tone turn out to have the simple mathematical relationships to one another that Pythagoras discovered. They're whole-number multiples of the base frequency. To take a simple example, if a plucked string is vibrating at a frequency of 100Hz, its tone (which is much more interesting than a simple sine wave) can be analyzed as being made up of sine waves at 100Hz, 200Hz, 300Hz, 400Hz, 500Hz, and so on.

These sine waves are all present in the tone we hear. Each of them has its own amplitude (loudness), and each may change loudness in a different way during the course of the tone. In synthesizer terms, we would say each sine wave has its own amplitude envelope. Differences in the relative amplitudes of various sine waves during the course of a tone are what allow us to tell the difference between a violin and a clarinet, for example.

The sine waves that make up a tone are called harmonics. They're also called overtones and partials. These terms have slightly different meanings; technically, the lowest sine wave in the tone (100Hz in our example above) is called the fundamental. The next sine wave up the stack is called the first overtone, because it's over the fundamental. But the fundamental is the first harmonic, so the first overtone is the second harmonic. Just to make matters a little more confusing, if the sine waves in a tone happen not to be whole-number multiples of one another (for example, if the tone has sine waves at 137Hz, 169.2Hz, 301Hz, and so on), the sine waves are not harmonics, but they're still called partials.

A few instruments, such as bells, have partials that aren't even close to being whole-number multiples of one another. The sounds of these instruments are referred to as clangorous. Many other real-world instruments produce partials that are close to being whole-number multiples of the fundamental, but not quite on. The piano's overtones, for instance, tend to be a bit sharp. This fact accounts for much of the expressive power of the tones: They tend to sound lively or interesting when there are little imperfections in the overtones. This, incidentally, is why synthesizer designers go to great lengths to add programming features with which we can mess up the mathematically perfect overtones produced by an oscillator. Detuning, chorusing, and similar techniques are all ways of creating movement among the overtones.

For the purposes of this tutorial, though, we're going to assume that the partials of the tones we talk about are harmonics, and we're going to avoid using the term "overtone" where possible.

One reason why this whole discussion is relevant to the topic of just intonation is because the individual tones within the harmonic series define a series of intervals. For example, if the fundamental is, for convenience, at 100Hz, the overtones at 300Hz and 400Hz will have a frequency ratio of 4:3. We can easily construct a table that shows these relationships:

Note that I've assigned note names starting with C purely to make it easier to see the relationships in the table. The frequency of a C is not actually 100Hz. Nevertheless, the other note names in the right column are not assigned arbitrarily; those are the notes you'll actually hear in relation to a C fundamental if you isolate the harmonics in the series. More or less. What you'll hear won't be an equal-tempered E or B-flat or F-sharp: You'll hear a note of a similar pitch, whose relationship with the fundamental is one of the intervals used in just intonation. The E is quite a bit lower than the E in the equal-tempered scale, and the B-flat is even further from the equal-tempered B-flat. These pitches contain the seeds of a new tuning system.

Knowing about overtones is also useful because it helps explain the complex sound we sometimes get when we play two tones together. For instance, let's suppose we have a tone whose fundamental is 83Hz. Its overtones will be at 166Hz, 249Hz, 332Hz, and so on. Another tone might have a fundamental of 50Hz, giving it overtones at 100Hz, 150Hz, 200Hz, 250Hz, 300Hz, and so on. Looking at these numbers, it might strike you that 249 is almost the same as 250. When the two tones are played together, the 249Hz overtone will create an interference pattern (a beat) with the 250Hz tone. The beating will have a frequency of 1Hz. By listening closely to the overtones, you can easily hear this beating.