Can someone please explain what overtones are and why they are important? I read a lot on the web about them but didn't understand. Also how can they be used when writing for orchestration?
What is an overtone?
When you pluck a guitar string, there are a variety of ways it could vibrate1:
Each vibration will produce a different pitch, because they have different frequencies. For example, the top-left corner has the lowest frequency because it has the smallest number of crests/valleys--its crests are highly infrequent.
In most cases, when you play an instrument, the pitch we hear is the one with the lowest frequency. In other words, we hear the pitch produced by the top-left vibration. But when we listen a little more closely (as you've probably done), we can hear higher pitches being produced from the string at the same time. That's because the string isn't just vibrating with the lowest frequency. It is also simultaneously vibrating with some of the higher frequencies shown in the image above. Those higher frequencies are what folks often refer to as the "harmonics."
So in essence, the "harmonics" are higher pitches/tones that occur when you strike a string, blow air through a wind instrument, etc. They are produced because strings, air, etc. don't vibrate in simple, singular ways. Rather, they vibrate with many different frequencies at once, producing a combination shape like this2:
While it's true that the string vibrates with all of these different frequencies at the same time, it should be noted that the harmonics occur in varying strengths/intensities. In many cases only the first few upper harmonics are audible at all, and the others simply aren't produced or are too faint to hear. To test this, go to an acoustic piano and very gently depress the C5 key. Press down so gently that it doesn't create a sound at all when the hammer strikes. While holding down the C5 key, quickly strike and release the C4 note. What you'll notice is that the C5 string continues to ring out even after the C4 string is muted. The reason this happens is: the C4 string vibrates with its fundamental frequency, but it also vibrates with its 2nd harmonic, and the 2nd harmonic is C5. So when C4 vibrates, its 2nd harmonic excites the C5 string through a phenomenon called resonance.
Harmonics are important for many reasons. To name a few: they create an extra harmonious sound when certain chords are played. Piano hammers are placed in specific locations to avoid exciting dissonant harmonics. They are used by some brass instruments to reach notes that wouldn't be available without harmonics.
To clarify the terminology: the lowest frequency produced on the string/in the air is called the "first harmonic" or the "fundamental frequency." That's the frequency we predominantly hear. The higher harmonics (those above the first) are the extra notes that we hear ringing faintly with higher pitches.
This description presents a simplified model, which is fairly accurate for most instruments. In reality, harmonics and overtones are two different things. "Overtone" has a broader meaning: "any higher frequency above the fundamental which is simultaneously produced." To qualify as a harmonic frequency, there's an extra condition: "any higher frequency above the fundamental which is an integer multiple of the fundamental frequency."
So "overtone" is an umbrella term, and there are two types of overtones: (1) "harmonic" overtones (which are integer multiples of the fundamental frequency) and (2) overtones that aren't integer multiples of the fundamental frequency (I'll call these disharmonious overtones). Everything I've discussed above has been about harmonic overtones. This is a fine way to answer the question, I think, because in the case of most instruments, the overtones deviate only slightly from the true harmonic frequencies. So in many cases, the overtones are close enough to be considered harmonic frequencies. The higher overtone frequencies may not be exactly twice the fundamental frequency, or 3x the fundamental frequency, but they're pretty close.
For some other instruments, though, the overtone frequencies will differ more dramatically from the true harmonics. For example, on timpani, the first overtone is 1.6 times the fundamental frequency. This dissonance is usually small, though, and many instruments are designed to minimize it.
Overtones are harmonics, or upper partials; extra notes which sound when a fundamental note is played. The words are almost synonymous, but not exactly.
When a string is played, or a note blown, the loudest sound we usually hear is the fundamental - the one we would sing back, or use to identify what the note is. It has, to greater or lesser degrees, other notes that can also be perceived. The first there is an octave above the fundamental, the next a perfect fifth above that, and the next is two octaves above the original. From then on, the harmonics come along at smaller and smaller intervals, usually quieter as they go on.
Each instrument has a different mix of these upper partials, which actually gives it its unique timbre - tone. Violins are rich in them.
A simple way to find them is to use a guitar. By touching the string while plucking it, harmonics can be played. Touch exactly 1/2 way along the string gives the first (octave). 1/3 of the length gives octave and a fifth, 1/4 gives the next octave. Carefully working out exactly where along the string will give 1/5, 1/6, etc.
Bugles use the harmonics to produce all of their notes, as when the embouchure is tightened, higher notes from the harmonic range are made.
Any sound or other vibration produced by a process that repeats perfectly at some frequency f by be decomposed mathematically into a combination of sine waves at frequencies that are integer multiples of f. Sounds produced by processes that don't repeat perfectly can be decomposed into sine waves, but they won't all be multiples of a common frequency. In practice, many instruments produce vibrations that don't repeat quite perfectly, but are close enough that they can be decomposed into waves that are mostly multiples of a common frequency or very close to it.
Some natural processes that produce sound have obvious mechanisms that will produce multiple frequencies. A plucked string, for example, will have many resonant modes as discussed in another answer, but other instruments may not have any obvious means of generating overtones. A kazoo, for example, will produce many overtones even though it has no natural resonance at any of the frequencies involved. In many cases, the effect may be modeled by simple harmonic distortion [such a model could be used with a kazoo].
If a device that produces simple harmonic distortion is fed a sound consisting of one or more frequencies, every frequency it produces will be the sum of integer multiples of the input frequencies. For example, if a distorting amplifier were fed a combination of 200hz and 250Hz signals, each frequency in its output, in Hz, would be some multiple of 200 plus some multiple of 250. All such frequencies would be multiples of 50, so the output would be some combination of multiples of 50Hz. If something like a kazoo is fed a combination of frequencies that are all multiples of 440Hz, it will output will likewise consist of such frequencies.
Most musical instruments contain some resonating components which produce natural harmonics in aforementioned fashion (like a vibrating string) but also contain other components which introduce distortion. For example, the top of a violin changes shape somewhat as it vibrates, and this will introduce harmonic distortion which adds frequency components beyond those generated by the string itself. For instruments which are only used to play a single note at a time, such frequency components will be multiples of the note's fundamental pitch, so such distortion will often be pleasant. Nonetheless, harmonic distortion often forms a key part of what makes various instruments sound as they do.
The existing answers are great for technical explanations but might be missing something from the compositional (orchestration) standpoint. What you need to learn about is how different combinations of instruments and tones can sound good together because they reinforce each other (consonance) or interfere with each other (dissonance).
This happens at a pitch level according to the harmonic series, such as
C1 - C2 - G2 - C3 - (E3*) - G3 - (Bb3*) - C4 (etc.)
*These pitches are not exactly "in tune" to a Western ear because we use a scale in which the frequencies are adjusted (tempered) to allow us to play in multiple keys and create major, minor and other modes - which would not happen if we used exact frequencies.
On a variable pitch instrument like a violin, fretless bass, trombone, synthesizer, or the human voice, you can produce notes that exactly correspond to the harmonic series for the the fundamental pitch (C1 in the above example). Most other instruments can only be played at certain pre-determined pitches according to the physical capabilities of the instrument, although with wind instruments the performer can slightly vary the pitch (i.e. intonation) with embouchure and breath control. The instrument as a whole can also be tuned to a given pitch and keyboard instruments (piano, organ) have to be tuned in advance.
Generally, the more skilled the performer, the better they are at playing "in tune" with themselves and others. As a composer, then, you can write music that utilizes these frequency ratios to create consonance or dissonance according to what you are trying to achieve. There is a tendency to associate dissonance with negative emotions and consonance with more positive ones, although this is not a hard and fast rule.
That's why usually you don't have two low notes close together, for example, because they produce overtones that interfere with each other and create an ugly sound. If you do create a dissonance, be sure to do it intentionally and not by accident.
Lastly, as mentioned already, instruments have a timbre which contains the fundamental note and various overtones. This is not as easy to control and actually forms part of the characteristic sound of the instrument. However, you also have to be conscious of how the timbre / overtones of one instrument will interact with another one. See this link for some examples of good /bad instrument combinations (in the opinion of that author).
Hopefully this helps with the "why it's important" part from a composer's standpoint.
Any note, other than a pure sine wave, heard in an anechoic chamber, will contain additional frequencies to the fundamental one. The laboratory demonstration of a string vibrating as a whole, in halves, thirds etc. is an ideal case. In real life, a string will not possess absolute uniformity and negligible mass. It will also be attached to other structures with their own resonances. Therefore (and fortunately, for making interesting sounds) many of the overtones will be shifted from their theoretical place in the 'harmonic series'. An extreme case is the church bell, which has a strong overtone a major 7th above the fundamental! But, broadly speaking, instrumental sounds consist of a mix of frequencies corresponding pretty closely to the harmonic series. How can this be used in orchestration?
Well, you could borrow the trick organ builders use to produce the sound of a 32' pipe without actually having to build such an expensive and bulky item. Two ranks, an octave and a twelfth above the desired pitch, can cause an interesting effect. Read about it here:
The same technique could be used in instrumental scoring. Write a pair of flutes, a perfect 5th apart. Can you hear the 'virtual' lower octave in this example?
Acoustic phenomenon or perceptual one? How far out-of-tune can the notes be before the effect stops working? (How far out-of-tune do ANY notes have to be before they lose their harmonic function?) We're getting into a a fascinating area here...
Simply put, overtones are higher frequencies that are played at the same time as the tonic. (Likewise, undertones are played at the same time, but at lower frequencies than the tonic.)
Overtones and undertones make an A 440 sung by a human sound different than an A 440 played by an oboe, or by any other instrument.