# A clarinet has no even harmonics. What would produce no odd harmonics?

According to this link, the waveforms of clarinets do not have even-numbered components in their harmonic series:

A closed cylindrical air column will produce resonant standing waves at a fundamental frequency and at odd harmonics... As can be seen from a sample waveform, the even harmonics missing from the tone

Is there an example of a system that would produce no odd harmonics?

• In this question do you mean the clarinet originally had all the harmonics in its output, in the first place, but somehow, due to some filtering, lost those odd ones, named as the missing harmonics? I don't think so. Those missing harmonics are the result of acoustical charactheristic of the device and are responsible for, together with the present harmonics, creating the timbre of the clarinet. Feb 5, 2015 at 12:13

The clarinet has no even numbered harmonics up from its lowest pitch because of its shape. It is open at one end (the bell) and effectively closed at the other (the mouthpiece). This causes the pressure wave to be maximal at the mouthpiece and minimal (atmospheric pressure) at the bell and the sound wave to have a node at the top and a peak at the opening.

If we picture a wave in its full sinusoidal up-and-down cycle we can envision that the part of the wave resonating in the clarinet at the lowest pitch / longest wave is 1/4 of the full wave.

All higher harmonics must vibrate with this node + peak pattern, which prevents any even multiples of the fundamental pitch from sounding, as they would have either a node or a peak at both ends of this air column.

This web page covers this and includes an image of the wave harmonics.

https://www.phys.unsw.edu.au/jw/clarinetacoustics.html

Producing "only" even harmonics is equivalent to producing all harmonics.

Say your original signal contains all harmonics:

1f, 2f, 3f, 4f, 5f, ...

Now remove all the odd harmonics:

2f, 4f, 6f, 8f, 10f, ...

The original 1f has been removed, so now you will hear 2f as the fundamental, and 4f will sound like the 2nd harmonic, 6f will sound like the 3rd harmonic, etc. You're back to 1f, 2f, 3f, 4f, 5f, ... except an octave higher. I think this might be something that happens in instruments, like "overblowing"? But I'm not sure. Anyway it will just produce a harmonic sound an octave higher, but with a different timbre.

Somewhat related: https://dsp.stackexchange.com/a/6178/29

• however according to your reasoning, finally we end up with a new harmonics series with 1f,2f,3f,4f.. which now contains even and odd harmonics ? contradicting the original purpose ? Feb 5, 2015 at 16:33
• Yes, that's my point. "Producing only even harmonics" is the same thing as "producing both even and odd harmonics an octave higher". So is it really meaningful to say that an instrument produces only even harmonics? Feb 5, 2015 at 16:36
• I think it's quite natural to assume that the OP means even harmonics on top of the fundamental frequency, i.e. $f_0$, $2f_0$, $4f_0$, etc. Feb 5, 2015 at 17:16
• first of all, both the question and the quota in it states that "the instrument is producing only the Odd harmonics", not the even. Hence your initial claim on the explanation of "producing only even harmonics is equivalent to..." isn't much relevant to the problem. Nevertheless, CTFT for a periodic signal $x(t) = \sum{a_k e^{-j\omega_0kt} }$ has a0 as DC (even), $a_1,a_{-1}$ as 1st harmonic (odd), $a_2,a_{-2}$ as 2nd harmonic (even) and so on. Hence an even only spectral can indeed be considered as a new signal with half the period, twice the fundamental of the original signal. Feb 5, 2015 at 18:50
• The question is asking for instruments that "produce no odd harmonics". The fundamental is an odd harmonic, so it would not be produced by such an instrument. You could intentionally design an instrument that only produced harmonics 1, 2, 4, 6, 8, but that's not a natural sequence like 1, 2, 3, 4, 5 or 1, 3, 5, 7, 9. Feb 5, 2015 at 19:43

I do not know of a physical system which produces only even harmonics, but you could think of devising a system which doubles the frequency of the incoming periodic signal (containing even and odd harmonics), and adds it to a low-pass filtered version containing only the fundamental frequency. From the low-pass filtered signal you get $f_0$ (the fundamental frequency), and from the frequency doubled signal you get $2f_0$, $4f_0$, etc.