The first thing musicians must do before they can play together is "tune". For musicians in the standard Western music tradition, this means agreeing on exactly what pitch is an "A", what is a "B flat" and so on. Other cultures not only have different names for notes and different scales, they may even have different notes based on a different tuning system. In fact, our own tuning system, which is called equal temperament, was not always used even in Europe. All tuning systems are based on the physics of sound. But they all are also affected by the history of their music traditions, as well as by the tuning peculiarities of the instruments used in those traditions.

Almost all music traditions recognize the octave. When one note has a frequency that is exactly two times the frequency of a second note, then the first note is one octave higher than the second note. A simple mathematical way to say this is that the ratio of the frequencies is 2:1. Two notes that are exactly one octave apart sound good together because their frequencies are related in such a simple way. If a note had a frequency, for example, that was 2.11 times the frequency of another note (instead of exactly 2 times), the two notes would not sound so good together. In fact, most people would find the effect very unpleasant and would say that the notes are not "in tune" with each other.

To find other notes that sound "in tune" with each other, we look for other sets of pitches that have a "simple" frequency relationship. These sets of pitches with closely related frequencies are often written as a harmonic series. Because the notes of a harmonic series are so closely related, the harmonic series is not just a useful idea constructed by music theory. It is often found in "real life". For example, all the notes that a bugle can play are part of a harmonic series. And every musical note you hear is not a single pure frequency, but is actually a blend of the pitches of a harmonic series.

Figure 1: Here are the first sixteen pitches in a harmonic series that starts on a C natural. The series goes on indefinitely, with the pitches getting closer and closer together. A harmonic series can start on any note, so there are many harmonic series, but all of them have the same set of intervals and the same frequency ratios.

A Harmonic Series

What does it mean to say that two pitches are "closely related"? It doesn't mean that their frequencies are almost the same. Two notes whose frequencies are almost the same - say, the frequency of one is 1.005 times the other - sound bad together. Again, when we hear them, we say they are "out of tune". Notes that are closely related have frequencies that can be written as a ratio of two small whole numbers; the smaller the numbers, the more closely related the notes are. (See Sound, Physics, and Music and The Harmonic Series for more on why this is so.)Two notes that are exactly the same pitch, for example, have a frequency ratio of 1:1.

For more examples, look at the harmonic series in the figure above. The number below a note tells you the relationship of that note's to the frequency of the first note in the series - the fundamental. For example, the frequency of the third note is three times the frequency of the first, and the frequency of the fifteenth note is fifteen times the frequency of the first. In the examplef, the fundamental is a C. That note's frequency times 2 gives you another C; times 2 again (4) gives another C; times 2 again gives another C (8), and so on. These are the red notes in the figure. Now look at the blue notes. The first one is the G that is number 3 in the series. 3 times 2 is 6; and number 6 in the series is also a G. So is number 12 (6 times 2). Check for yourself the other octaves in the series. You will find that the ratio for one octave is always 2:1, just as the ratio for a unison is always 1:1. So notes with this small-number ratio of 2:1 are so closely related that we give them the same name and base the structure of our tuning system on this octave relationship.

The next closest relationship is the one based on the 3:2 ratio, the interval of the perfect fifth (for example, the C and G in the above harmonic series). The next lowest ratio, 4:3, gives the interval of a perfect fourth. Again, these pitches are so closely related and sound so good together that their intervals have been named "perfect", and all major and minor chords are based on the perfect fifth. (See Triads and Naming Triads.)

Pythagorean Intonation

The Pythagorean system is so named because it was actually discussed by the famous Greek mathematician and philosopher, who lived in the sixth century B.C. Pythagoras understood the simple arithmetical relationship involved in intervals of octaves, fifths, and fourths. He and his followers believed that numbers were the ruling principle of the universe, and that musical harmonies were a basic expression of the mathematical laws of the universe. Their model of the solar system involved the earth and other celestial spheres revolving around the sun, each sphere making music as it revolved.

In the Pythagorean system, all tuning is based on the interval of the pure fifth. Pure intervals are the ones found in the harmonic series, with very simple frequency ratios. So a pure fifth will have a frequency ratio of exactly 3:2. Using a series of perfect fifths (and assuming perfect octaves, too, so that you are filling in every octave as you go), you can eventually fill in an entire chromatic scale.

Figure 2: You can continue this series of perfect fifths to get the rest of the notes of a chromatic scale; the series would continue F sharp, C sharp, and so on.

Pythagorean Intonation

The main weakness of the Pythagorean system is that as you continue on your series of perfect fifths, you eventually come back to the note that you started on. (See The Circle of Fifths.) For example, starting on C, a series of perfect fifths would be: C, G, D, A, E, B, F sharp, C sharp, G sharp, D sharp (or E flat), A sharp (or B flat), F, C. And the C that you arrive at after a series of pure fifths is a little higher than the C that is several perfect octaves from your beginning C. So instruments that use Pythagorean tuning have to use eleven pure fifths and a smaller "wolf" fifth in order to keep all octaves pure. Keys that avoid the wolf fifth sound just fine on instruments that are tuned this way, but keys in which the wolf fifth is heard often must be avoided. To avoid some of the worst problems with wolf notes, some harpsichords and other keyboard instruments were built with split keys for D sharp/E flat and for G sharp/A flat. The front half of the key would play one note, and the back half the other (differently tuned) note.

Pythagorean tuning was widely used in medieval and Renaissance times. Major seconds and thirds are larger in Pythagorean intonation than in equal temperament, and minor seconds and thirds are smaller. Some people feel that using such intervals in medieval music is not only more authentic, but sounds better too, since it was in fact conceived in this tuning system.

More modern Western music, on the other hand, does not sound pleasant using Pythagorean intonation. The thirds are simply too far away from the pure third. In medieval music, the third was considered a dissonance and was used sparingly, but modern harmonies are built on thirds (see Triads).

Just Intonation

Just intonation is the system of tuning that is used (often unconsciously) by musicians who can make small tuning adjustments quickly. This includes vocalists, most wind instruments, and many string instruments. Look again at the harmonic series.

Figure 3: Both the 9:8 ratio and the 10:9 ratio in the harmonic series are written as whole notes. 9:8 is considered a major whole tone and 10:9 a minor whole tone. The difference between them is less than a quarter of a semitone.

As the series goes on, the ratios get smaller and the notes closer together. Standard notation writes all of these "close together" intervals as whole steps (whole tones) or half steps (semitones), but they are of course all slightly different from each other. For example, the notes with frequency ratios of 9:8 and 10:9 and 11:10 are all written as whole steps. To compare how close (or far) they actually are, turn the ratios into decimals.

Whole Step Ratios Written as Decimals

• 9/8 = 1.125
• 10/9 = 1.111
• 11/10 = 1.1

Note:

In case you are curious, the size of the whole tone of the "mean tone" system is also the mean, or average, of the major and minor whole tones.

These are fairly small differences, but they can still be heard easily by the human ear. Just intonation uses both the 9:8 whole tone, which is called a major whole tone and the 10:9 whole tone, which is called a minor whole tone, in order to construct both pure thirds and pure fifths. Because chords are constructed of thirds and fifths (see Triads), this makes typical Western harmonies particularly pleasing to the ear.

The problem with just intonation is that it matters which steps of the scale are major whole tones and which are minor whole tones, so an instrument tuned exactly to play with just intonation in the key of C major will have to retune to play in C sharp major or D major. For instruments, like voices, that can tune quickly, that is not a problem, but it is unworkable for piano and other slow-to-tune instruments.

There are times when tuning is not much of an issue. When a good choir sings in harmony without instruments, they will tune without even thinking about it. All chords will tend to use pure fifths and thirds, as well as seconds, fourths, sixths, and sevenths that reflect the harmonic series. Instruments that can bend most pitches enough to fine-tune them during a performance - and this includes most orchestral instruments - also tend to play the "pure" intervals. This can happen unconsciously, or it can be deliberate, as when a conductor asks for an interval to be "expanded" or "contracted".

But for many instruments, such as the piano, harp, bells, harpsichord, xylophone - any instrument that cannot be fine-tuned quickly - tuning is a big issue. A harpsichord that has been tuned using the Pythagorean system or just intonation may sound perfectly in tune in one key - C major, for example - and fairly well in tune in a related key - G major - but badly out of tune in a "distant" key like D flat major. Adding split keys or extra keys can help, but also makes the instrument more difficult to play.

Equal temperament was invented to solve this problem. In equal temperament, only octaves are pure. The octave is divided into twelve equally spaced half steps, and all other intervals are measured in half steps. This gives, for example, a fifth that is a bit smaller than a pure fifth, and a major third that is larger than the pure major third. The differences are smaller than the "wolf tones" found in other tuning systems, but they are still there.

Because of this, it took equal temperament centuries to become the accepted tuning method. Bach's "Well-Tempered Klavier" is a collection of beautiful brilliantly-written counterpoint, but it was also a plea and an advertisement for equal temperament; the entire collection cannot be played well except on an instrument that uses equal temperament.

In a way, equal temperament is a compromise between the Pythagorean approach and the mean-tone approach. Neither the third nor the fifth is pure, but neither of them is terribly far off, either. Because equal temperament divides the octave into twelve equal half-steps, the frequency ratio of each half step is the twelfth root of 2. (There is a review of powers and roots in Music and Math if you need it.)

Figure 4: In equal temperament, the ratio of frequencies in a semitone is the twelfth root of two. Every interval is then simply a certain number of semitones. Only the octave (the twelfth power of the twelfth root) is a pure interval.

In equal temperament, the only pure interval is the octave. (The twelfth power of the twelfth root of two is simply two.) All other intervals are given by irrational numbers based on the twelfth root of two, not nice numbers that can be written as a ratio of two small whole numbers. In spite of this, equal temperament works fairly well, because most of the intervals it gives actually fall quite close to the pure intervals. To see that this is so, let's write both kinds of interval (equal temperament and pure tones) as decimals. You can find these decimals for yourself using a calculator.

Figure 5: Look again at the harmonic series figure (above) to see where the ratios come from for the pure intervals. The ratios for equal temperament are all multiples of the twelfth root of two.

Comparison of Frequency Ratios for Equal Temperament and Pure Harmonic Series

Except for the unison and the octave, none of the raitios for equal temperament are exactly the same as for the pure interval. Many of them are reasonably close, though. In particular, perfect fourths and fifths and major thirds are not too far from the pure intervals. The intervals that are the furthest from the pure intervals are the major seventh, minor seventh, and minor second.

Because equal temperament is now so widely accepted as standard tuning, musicians do not usually even speak of intervals in terms of ratios. Instead, tuning itself is now defined in terms of equal-temperament, with tunings and intervals measured in cents. A cent is 1/100 (.01) of an equal-temperament semitone. In this system, for example, the major whole tone discussed above measures 204 cents, the minor whole tone 182 cents, and a pure fifth is 702 cents.