Harmonics overtones fundamental partials frequency distortions THD relationship between frequencies of overtones and fundamental frequency frequencies stopped flute clarinet even-numbered harmonics odd even - sengpielaudio
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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         
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
 Interval to
 previous tone
 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 f f f f f f f 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.
Harmonics and overtones comparison
Even numbered harmonics        Odd numbered harmonics
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:
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
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".
Harmonische Oberwelle Vergleich
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.
Vergleich Klarinette Floete
odd harmonics
1 3 5 7 9 11

even harmonics
2 4 6 8 10 12 14 16 18

Length calculation for an open pipe:
Formula open pipe
Length calculation for a closed pipe:
Formula 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
Harmonics Intervals

              "Open ends"                  displacement         "Fixed ends"
Offene Pfeife
fn = c / λn = n · f1 = n · c / (2 L)
                Gedackte Pfeife
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.
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