Harmonics, Overtones, and the Fundamental

Calculations of Harmonics from Fundamental Frequency

 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.

"Overtones" = Harmonics minus 1, or "Harmonics" = Overtones + 1

 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.

 Overtones ▼ Harmonics Overwaves Partial tones Partials Enter: Frequency in Hz Fundamental frequency in Hz = 1st harmonic ↓ Solution: Frequencies in Hz 1st overtone 2nd harmonic 2nd overtone 3rd harmonic 3rd overtone 4th harmonic 4th overtone 5th harmonic 5th overtone 6th harmonic 6th overtone 7th harmonic 7th overtone 8th harmonic 8th overtone 9th harmonic 9th overtone 10th harmonic 10th overtone 11th harmonic 11th overtone 12th harmonic 12th overtone 13th harmonic 13th overtone 14th harmonic 14th overtone 15th harmonic 15th overtone 16th harmonic
 Odd-numbered harmonics: Even-numbered harmonics 1st harmonic = fundamental tone 2nd harmonic = octave, 3rd harmonic = fifth above octave 4th harmonic = 2nd octave 5th harmonic = third above 2nd octave 6th harmonic = fifth above 2nd octave 7th harmonic = minor seventh above 2nd octave 8th harmonic = 3rd octave 9th harmonic = whole tone above 3rd octave 10th harmonic = third above 3rd octave

 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 = even-numbered harmonics, (1f), 2f, 4f, 6f..., e.g. triode valve (tube) and white = odd-numbered 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 (pulse-pause ratio of 1:1) has exclusively odd-numbered harmonics of 3, 5, 7… or even-numbered 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.

Even-numbered harmonics                |            Odd-numbered harmonics
are odd-numbered overtones            |            are even-numbered overtones

 Typical "warm" tube sound, particularly triodes contain predominantly in the spectrum even-numbered multiples of the fundamental frequency, and thus outstanding even-numbered harmonics, or even-numbered partial tones 2, 4, 6… One can also say, tube amplifiers at high levels (distortion) contain strong odd-numbered overtones - that are even-numbered 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 odd-numbered multiples of the fundamental frequency and thus outstanding odd-numbered harmonics, or odd-numbered partial tones 3, 5, 7… One can also say, closed, covered, stopped organ pipes, and also the clarinet contains mostly even-numbered overtones - that are odd-numbered partials or harmonics.   Because a clarinet acts like a closed tube resonator, it theoretically produces only odd-numbered harmonics, that are even-numbered overtones. But there are also some even-numbered harmonics content; look at: http://www.phys.unsw.edu.au/jw/clarinetacoustics.html#harmonics

 Notice:  All harmonics are overtones is correct, but . . .                 Even-numbered overtones are odd-numbered harmonics, or partial tones.                 Odd-numbered overtones are even-numbered 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 odd-numbered harmonics or alternatively even-numbered 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 "odd-numbered 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: Calculation without flute mouth correction.

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"

 fn = c / λn = n · f1 = n · c / (2 L) fn = (2n − 1) · f1 = (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.