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In acoustics the basic vibration is the 'first harmonic'. The term overtone is used to refer to any resonant frequency above the fundamental frequency. 
The 'second harmonic' (twice the fundamental frequency) is the first overtone. Persons who count differently are wrong. In counting − harmonics are not overtones. 
There are integer multiples of a certain frequency (fundamental), that are called
harmonics, partial tones (partials) or overtones. It is important to note that the
term 'overtones' does not include the fundamental frequency. The first overtone is
therefore already the second harmonic or the second partial. The term overtone
should never be mixed with the other terms, as the counting is unequal. The term harmonic has a precise meaning  that of an integer (whole number) multiple of the fundamental frequency of a vibrating object. A harmonic frequency is a multiple of a fundamental frequency, also called "harmonic". Especially when it comes to counting, do not say: "overtones are harmonics". Musicians prefer the term overtones and physicists prefer the term harmonics. Sound engineers are somewhat uncertain between these two terms. Harmonics and overtones are also called resonant frequencies. 

Comparison to fundamental 
Interval to previous tone 
Frequency ratio 
Tone example 
Frequency in Hz 
fundamental frequency  fundamental  1:1  C  65 
double frequency  octave  2:1  c  130 
trifold frequency  fifth  3:2  g  195 
fourfold frequency  fourth  4:3  c'  260 
fivefold frequency  major third  5:4  e'  325 
sixfold frequency  minor third  6:5  g'  390 
sevenfold frequency  7:6  natural seventh  455  
eightfold frequency  8:7  c''  520  
ninefold frequency  large whole tone  9:8  d''  585 
tenfold frequency  small whole tone  10:9  e''  650 
elevenfold frequency  11:10  alphorn F5  715  
twelvefold frequency  12:11  g''  780  
thirteenfold frequency  13:12  845  
fourteenfold frequency  14:13  910  
fifteenfold frequency  15:14  b''  975  
sixteenfold frequency  minor second  16:15  c'''  1040 
The frequency ratio results in a natural tone series, not related to the equal stretched temperament. 
Harmonics  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16 
Partials  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16 
Overtones  Fundamental  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15 
Frequency  f  2·f  3·f  4·f  5·f  6·f  7·f  8·f  9·f  10·f  11·f  12·f  13·f  14·f  15·f  16·f 
Example Hz  65  130  195  260  325  390  455  520  585  650  715  780  845  910  975  1040 
Tone name  C2  C3  G3  C4  E4  G4  Bb4  C5  D5  E5  F#5  G5  Ab5  Bb5  B5  C6 
gray = evennumbered harmonics, (1f), 2f, 4f, 6f..., e.g. triode valve (tube) and
white = oddnumbered harmonics, 1f, 3f, 5f..., e.g. organ pipe closed at top
and clarinet all integer harmonics, e.g. saw tooth wave. The frequency spectrum of a symmetric square wave signal (pulsepause ratio of 1:1) has exclusively oddnumbered harmonics of 3, 5, 7… or evennumbered overtones 2, 4, 6… The sound spectra of clarinets tend to have strong odd harmonics (fundamental, 3rd, 5th, 7th etc) and weak even harmonics (2nd, 4th, 6th etc), at least in their lowest registers. Tympanic membranes or bells have a large number of individual vibrations, which are not simply the exact multiples of single fundamental frequency. These are not harmonic overtones and are often called partial tones or partials. Overtones whose frequencies are not an integer multiple of the fundamental are called inharmonic and are often perceived as unpleasant. Inharmonics are not the same as Enharmonics. Bells have more clearly perceptible partials than most instruments. An "exciter" is a special equalizer, which creates new overtones. The processed signal is added to the original input signal. 
 
Evennumbered harmonics  Oddnumbered harmonics  
are oddnumbered overtones  are evennumbered overtones 
Typical "warm" tube sound, particularly triodes contain predominantly in the spectrum evennumbered multiples of the fundamental frequency, and thus outstanding evennumbered harmonics, or evennumbered partial tones 2, 4, 6… One can also say, tube amplifiers at high levels (distortion) contain strong oddnumbered overtones  that are evennumbered partials or harmonics. Organ pipes closed at the top (gedackt), which are half as long as open organ pipes of the same pitch, have a slightly dull and hollow sound. The spectrum shows predominantly oddnumbered multiples of the fundamental frequency and thus outstanding oddnumbered harmonics, or oddnumbered partial tones 3, 5, 7… One can also say, closed, covered, stopped organ pipes, and also the clarinet contains mostly evennumbered overtones  that are oddnumbered partials or harmonics. Because a clarinet acts like a closed tube resonator, it theoretically produces only oddnumbered harmonics, that are evennumbered overtones. But there are also some evennumbered harmonics content; look at: http://www.phys.unsw.edu.au/jw/clarinetacoustics.html#harmonics 
Notice: All harmonics are overtones is correct, but . . . Evennumbered overtones are oddnumbered harmonics, or partial tones. Oddnumbered overtones are evennumbered harmonics, or partial tones. Harmonics do not have the same counting like the overtones. Do not mix overtones with harmonics. It is recommended to mention odd and even multiples of the fundamental frequency, because that coincides with the counting of the harmonics and the partial tones, and if possible avoid the mention of even and odd overtones, which have a different counting, what shows often typical errors. Avoid the word combination "harmonic overtones". 
The predominant oddnumbered harmonics or alternatively evennumbered overtones of a clarinet The 2nd and the 4 the harmonic are very weak, but important 
Courtesy of Joe Wolfe: http://www.phys.unsw.edu.au/jw/flutes.v.clarinets.html
The words "oddnumbered overtones" are not correct for a clarinet.
Only even overtones or odd harmonics are right here.
What is the right answer? Which typical overtones are found in a clarinet in addition to the fundamental? ● Even overtones 2, 4 and 6 or odd harmonics 3, 5, and 7? Which typical harmonics are found in a clarinet in addition to the fundamental? Odd harmonics 3, 5 and 7 or even overtones 2, 4, and 6? Which "melodious" overtones are produced by a triode electode tube in addition to the fundamental with a slight overdrive (distortion)? ● Odd overtones 1, 3 and 5 or even harmonics 2, 4, and 6? Which "melodious" harmonics are produced by a triode electrode tube in addition to the fundamental with a slight overdrive (distortion)? Even harmonics 2, 4 and 6 or odd overtones 1, 3, and 5? No wonder that in books and on the Internet many of these statements are rather unwisely. One should have his head quite clear in thinking, and looking at the figure above. 
Clarinets odd harmonics 1 3 5 7 9 11 Flutes even harmonics 2 4 6 8 10 12 14 16 18 
Length calculation for an open pipe:
Length calculation for a closed pipe: 
Frequency of concert pitch A4 and the counting of harmonics and overtones
Frequency  Order  Harmonics  Overtones 
1 × f = 440 Hz  n = 1  1st (odd) harmonic  fundamental 
2 × f = 880 Hz  n = 2  2nd (even) harmonic  1st (odd) overtone 
3 × f = 1320 Hz  n = 3  3rd (odd) harmonic  2nd (even) overtone 
4 × f = 1760 Hz  n = 4  4th (even) harmonic  3rd (odd) overtone 
4 × f = 2200 Hz  n = 5  5th (odd) harmonic  4th (even) overtone 
Symmetrical amplifier push/pull circuits must cancel even order harmonics, like 2, 4, 6… and therefore promote odd order harmonics, like 1, 3, 5… when driven into the nonlinear range. 
Brass instruments: trumpet, baritone horn, trombone, flugelhorn, cornet, tenor horn (alto horn), horn, euphonium, tuba, bass tuba, cimbasso, alphorn, conch, didgeridoo, natural horn, keyed bugle, keyed trumpet, serpent, ophicleide, shofar, vladimirskiy rozhok, vuvuzela 
"Open ends" displacement "Fixed ends" 
f_{n} = c / λ_{n} = n · f_{1} = n · c / (2 L) 
f_{n} = (2n − 1) · f_{1} = (2n − 1) · c / (4 L) 
Open pipe with all harmonics or also with all overtones 
Closed pipe with unevenharmonics and therefore only with even overtones 
In figures of waves is almost never indicated whether the pressure variations
in the air (sound pressure) or the displacement of the air particles
is meant. Here you see the displacement. Notice: To a sound displacement node a sound pressure antinode corresponds. The effect of waves reports our hearing, but only the sound pressure deviations (fluctuations, variations) move effectively our eardrums. 
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