18

What are overtones and what are harmonics?

How do they relate?

I know that the first harmonic is the tone A at 440 HZ and the second harmonic is at 880HZ an octave higher, it is also the first overtone, is this just a normal tone? Is the base frequency all the difference. How are overtones different from simple tones, for instance if I play a 440 HZ A as a normal tone is that then different from an overtone?

14

In the context of acoustic analysis:

Overtone: any resonant frequency above the fundamental frequency.

Harmonic: resonant frequency that is an integer multiple of the fundamental frequency.

For almost any^* musical instrument, any time you play a sound, you get a whole series of overtones. The feature of pitched instruments that makes them pitched is that the important overtones are the harmonic ones. When you play an A4 note, it is composed of harmonics with frequencies 440, 880, 1320,1760, 2200, 2640... When you play A5 (880Hz fundamental) it is composed of 880, 1760, 2640... Note that the set of harmonics in the 880Hz sound is only a subset of those in the 440Hz sound. Our ability to identify the separation between the harmonics is an integral component of how we perceive pitch, as shown by sounds where the fundamental is missing. Thus, the different sets of harmonics are an integral part of how we here these two sounds a being different.

The idea that a musical tone is comprised of exactly the harmonics is a (useful) idealization -- real instruments have some degree of inharmonicity so the peaks may differ slightly from exact integer ratios. Another aspect of this idealization is to think of overtones as specific frequencies; really each of the overtones has some spread to it; but for pitched, and especially for long sustained notes, this spread is very small compared to the separation between the peaks.

When the term overtone is used in less technical contexts it tends to connote to the perceivable presence of other pitches within a given pitch, i.e. you play a A, and you can pull out and hear that there is also an E in it. (Note that a E4 note has harmonics 660, 1320, 1980, 2640... and that half of these are present in the A note)

Outside of acoustical analysis the phrase "harmonic" can also be used in different ways:

  1. as in "harmonious" with its indication of consoance, and
  2. as in "playing a (natural) harmonic" on a stringed instrument -- in this the player gently presses a finger at a point of the string corresponding to one of the nodes of one of the harmonics of the full string -- this damps out the overtones that are not "harmonics of that harmonic".

This answer has been significantly edited to incorporate components of JCPederosa's and Caleb Hines's answers.

^* The exceptions that I have in mind are some simple additive synthesizers that might only have one or two sine wave components in them, and some forms of percussion instruments (like shakers) that only have weak spectral peaks.

  • So the tone we hear from an instrument, is actually the sum of all of its harmonics? – Phanest Nov 9 '14 at 20:52
  • Yes -- you can compare a synthesized sine wave at a given frequency to an instrument's that has that as it's fundamental frequency. For sustained notes, the difference in timbre is in terms of how much of the different overtones are present (the sine signal does not have any overtones). – Dave Nov 9 '14 at 22:13
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The difference is quite simple, and we might be over-complicating it in other answers.

  • Overtone: any resonant frequency above the fundamental frequency.

  • Harmonic: resonant frequency that is an integer multiple of the fundamental frequency.

A harmonic is a type of overtone. All resonant frequencies above the fundamental are overtones, but only the ones that are integer multiples of the fundamental are harmonics.

Note that an overtone is defined as being above the fundamental, so the overtone series starts counting after the fundamental. The harmonic series starts counting from the fundamental.

Here we have 4 different overtone series as example.

Harmonics:   1st   2nd   3rd   4th   5th    
            |     |     |     |     |
            20    40    60    80    100    
                  |     |     |     |
Overtones:        1st   2nd   3rd   4th

Harmonics:   1st         2nd         3rd
            |           |           |
            20    33    40    42.5  60 
                  |     |     |     |
Overtones:        1st   2nd   3rd   4th

Harmonics:   1st                     2nd
            |                       |
            20    33    56    42.5  60 
                  |     |     |     |
Overtones:        1st   2nd   3rd   4th 

Harmonics:   1st                     
            |                       
            20    33    56    42.5  69.8 
                  |     |     |     |
Overtones:        1st   2nd   3rd   4th 

Further reading:

5

In general, whenever there is a sound, there is more than one frequency occurring simultaneously (exception: A pure sine wave tone has a single frequency). But any periodic function, such as a sound wave, can be written (via a Fourier transform) as a sum of individual sine waves, each with their own frequency and amplitude. If you plot the amplitude of each frequency, you get a spectrum. Each of these (potentially infinite) frequencies that make up a sound is called a partial.

For a burst of white noise, every possible frequency has roughly the same (non-zero) amplitude. For a pure sine wave tone, every frequency but one has zero amplitude.

For pitched sounds, such as musical notes, you typically have series of discrete peaks (partials) at varying frequencies. The lowest peak is called the fundamental, and usually has the greatest amplitude. This corresponds to the pitch that you hear. All other partials -- not counting the fundamental -- are called overtones.

Any partial that is a multiple of the fundamental (including the fundamental, which is one times itself) is a harmonic partial -- or just a harmonic. Thus for a fundamental frequency of F, the harmonic partials are F, 2F, 3F, 4F, ... Usually, pitched instruments are designed in such a way that they amplify only the harmonic partials, and dampen any non-harmonic partials. If, for physical reasons, an instrument peaks at a frequency that is slightly off of the harmonic series, say at 5.01F instead of 5F, the difference is called inharmonicity (this is why, for example, smaller upright pianos have an inferior sound). Percussion instruments, on the other hand, have more complex modes of vibration, and will typically include additional inharmonic overtones. For unpitched instruments like drums, there may not even be a distinct fundamental.

For a pitched instrument, in which all the overtones are harmonic partials, the numbering of overtones and harmonics will be off by one, because the fundamental is not an overtone. Thus you have:

  • F = 1st harmonic = fundamental
  • 2F = 2nd harmonic = 1st overtone
  • 3F = 3rd harmonic = 2nd overtone, and so on...

So in summary:

  • Overtones = All partials that are above the fundamental.
  • Harmonics = All partials that are multiples of the fundamental.
  • Best answer for this question! – neevek Apr 11 at 8:51
1

Harmonics are the pure sine components of periodic signals. The fundamental (which would count as the 1st harmonic using the straightforward numbering system but is actually almost never explicitly called that) has the same frequency as the fundamental period of what you are looking at. The 2nd harmonic has twice the frequency, the 3rd harmonic three times the frequency and so on.

Now there is a difference when disharmonicity comes into play. A number of tone generators don't actually produce actually periodic signals since the overtones are generated by modes and those modes may not be perfect harmonics.

Vibrating strings look like skipping ropes from the side. Now one can swing a skipping rope in "2nd harmonic" mode where one of the rope turners' hands is up when the others is done. If you have a really long rope and really good skippers and turners, then they can skip alternatingly.

It turns out that when the string is thick, its ends are comparatively stiff which shortens the effective string length. And the bendier the whole shape is, the more it is effectively shortened. So the overtones of thick strings tend to have a higher frequency than the proper harmonics would have. So the various modes (and thus the overtones) of the string are approximately harmonics but not perfectly so.

An interesting other case are free reeds like in a harmonica or accordion. The reed travels through a reed plate with slots, a metal reed (basically a rectangular strip of metal) blocking the slot is mounted on one side and sort of punches hard holes into the air stream when bending back and forth through the slot. Since the punching-holes-in-the-air-stream act is periodic, the resulting overtone-rich sound does not have any disharmonicity.

However, the reed action itself has higher modes: the fundamental mode is just bending the reed everywhere in the same direction, but there are some higher modes where the reed bends back and forth. Now free reeds are not just flat but are profiled in a manner where bending the reeds stronger will still result in the same frequency (accordion) or will result in bent pitches (harmonica). Because of that profile, the metal overtones are usually so far away from the harmonics that they don't get excited.

However, when tuning such reeds, one scratches or files the reed in different places depending on whether it needs to go up or down in pitch. In the process of tuning it may happen that one of the vibrating modes comes close to an actual harmonic. In that case, it will get excited and interfere with the air harmonics from the "punching a hole in the airstream" act.

The reed will sound unclean, and it will suffer quite more mechanical stress due to the additional mode it vibrates with in almost-resonance. Tuning the reed off and then back again might get rid of the problem.

So with free reeds (actually, the non-free ones as well), you don't want any overtones in the reed action itself. Instead the overtones are a result of the non-sine way in which the regular periodic reed action occurs.

For that kind of process, there is no disharmonicity. A coupled resonant air column (like in organ pipes or oboe etc but not harmonica or accordion) might have its own modes and consequently disharmonic overtones, however.

  • you say that the 2nd harmonic has twice the frequency, the next has 3x, etc.(1st para.) The 3rd harmonic is an octave and a fifth away from the fundamental. If the frequency is double, treble etc, the notes will be octaves of the original. – Tim Nov 11 '14 at 11:08
-2

A non-scientific answer : each sound has a basic frequency. This is called the first harmonic. Why it's a harmonic and not a basic fundamental, I don't know ! Other sounds emanate from the sound source. Those that are twice, thrice, etc. the frequency are called harmonics or overtones, the same thing.

Partials, or inharmonic overtones also exist within a sound. Rather than being 'octave copies' of the basic frequency, they are 5ths, etc. As a string is split into fractions (on gtr, for example, touching a string at fractions along its length) it produces the different overtones. As in - 1/2 way - 2nd harmonic. 1/3 way - fifth. 1/4 way - 3rd harmonic. 1/4 way - maj. 3rd.It is entirely possible, further along a guitar string, to play an octave scale, albeit not spot on, frequency wise, starting around the 2 1/4 fret going to the 1st fret.All partials, or inharmonic overtones which are found within the basic open string note, although rather in the background, aurally

A lot of these sounds can be heard when a note is played on an instrument. The proportion of each varies with each instrument. This gives each its own timbre (tone). Bells have maybe the most partials, which makes them sound like they are 'more out of tune' than, say, a flute, which has fewer.

Distortion produced by overdrive, on guitars, enhance harmonics, or overtones. This becomes a problem when whole chords are played, as each component note then has its partials magnified. All the sounds blend into noise. By using only a I and its V, we're left with overtones which sound good together, without clashing. Thus the ubiquitous 5th chord, beloved of a lot of guitarists.

Scientific answers, with graphic explanation, will, I hope, follow.

A simple straight answer is that including the fundamental, all are harmonics :440 Hz, 880, 1760, etc.(on A, say). Overtones do not include the basic. Overtones = 880, 1760, etc. So the 1st harmonic is 440, whereas the 1st overtone is 880: 1st overtone =2nd harmonic.I am quoting here - there are other quotes which say that 1st harmonic =1st overtone (as 880 Hz in the example).

EDIT: The overtones are called 'upper partials'- the fundamental note being a partial, not an upper partial.'Upper partial' is a synonym for 'harmonic', which is not quite correct, since though all harmonics (except the fundamental) are 'upper partials', not all upper partials are harmonics.Going back to the bell mentioned earlier, it has upper partials which do not correspond to the harmonic series.

  • @MatthewRead - thanks. What I allude to is that a fundamental harmonic is rarely referred to as a harmonic. Most musicians would believe that an open string on a guitar, for example, is not a harmonic, whereas touching above 12th fret IS. Not a misnomer per se, but it comes close ! Or is it just a well kept secret? – Tim Nov 9 '14 at 10:21
  • 2
    I don't think your distinction between partials and harmonics is correct. – Pat Muchmore Nov 9 '14 at 13:29
  • @Pat Muchmore - I've edited, but am not much the wiser. Upper partials in Google is mostly teeth ! I think a scientific explanation may give better reasoning. Can you help? – Tim Nov 9 '14 at 14:10
  • I tend to think of harmonic (acoustic analysis) and harmonic (stringed instrument technique) as being homonyms. Note that "playing a harmonic" does not produce a sound that is just a single sinusoidal component. – Dave Nov 9 '14 at 16:48

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