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Essentially all instruments produce overtones, which are frequencies other than the dominant frequency of the note.

-- How do harmonics work?

The use of "essentially" there got me thinking. Are there any instruments which do not produce overtones?

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    No overtones == exact sinewave. A synthesizer producing a sine wave (as some do) will thus fit the bill. (also silence has no overtones, so technically a completely broken and thus silent instrument will also fit the bill!) – abligh Jun 2 at 12:12
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    Are you allowing electronic instruments? A theremin produces a sine wave I believe. – marcellothearcane Jun 2 at 19:50
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    @marcellothearcane I think digital theremins are often sampled, and analogue ones use some circuits to get a more interesting waveform. I'm not a hundred percent sure though. – Nobody Jun 2 at 20:39
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    I have heard that in theory, a perfectly hemispherical bell (possibly made from an ideal, vanishingly thin material) will vibrate with a perfect sine wave. I haven't checked the calculations on this myself, though. – Arthur Jun 3 at 8:00
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    There is another way to make an instrument appear to have no overtones, and that is to only play notes in the top octave of your hearing. Since the first harmonic is one octave above the fundamental, it will then be out of your hearing. This is why many instruments with big ranges sound rather plain at the top - they simply aren't producing harmonics you can hear. This is why it can be quite difficult to tell the difference between one instrument and another in the your top octave. So if its bottom note was in your top octave, it would appear to have no harmonics at all. – David Robinson Jun 3 at 14:28

10 Answers 10

31

A tuning fork comes close, though amplifying it by placing it on some resonating object - a wooden table, piano case, or try your head :-) - will add some harmonics.

The sound-producing element of a Fender Rhodes electric piano is essentially a tuning fork, though other parts of the instrument are designed to 'dirty up' the pure tone.

The tone of a flute, especially in the higher register, is close to a sine wave.

Note that we're talking about the sustain portion of a note. Both tuning fork and flute produce much more complex sounds as a note is attacked. You could mistake a tuning fork for a flute if the attack portion of a note was chopped off. I don't think you'd confuse the two if the attack was also heard though!

This principle was put to good use in 'Hybrid Synthesisers' like the Roland D50 or Yamaha SY range. A short sampled attack was followed by a synthesised sustain and release. It combined a remarkable degree of realism and controllability with economical use of sample memory.

So your answer is: although some instruments have a sustain close to a sine wave, I can't think of one outside the test bench that lacks a more complex attack.

  • Hope you don't mind me dropping a link there. I searched around to verify that tuning forks do indeed produce a sine-wave, and found this. – user45266 Jun 3 at 0:56
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    "The tone of a flute, especially in the higher register, is close to a sine wave." Is that really true, or is it just because our ear fails to pick up many of the harmonics? – Arthur Jun 3 at 8:01
  • @Arthur I wouldn't be surprised if spectral analysis of flute tone showed a decrease in higher harmonics regardless of the human hearing range, but I bet that does contribute to the perceived effect... <img src="ux1.eiu.edu/~cfadd/3050/Adventures/chapter_12/x_12.10.jpg" alt="Image result for flute timbre"/> – user45266 Jun 3 at 17:13
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I've heard it claimed that human whistling comes very close to being a perfect sine wave:

Spectrograph

The video here seems to show only one peak on the spectrograph, supporting a nearly perfectly sinusoidal waveform.

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    For an instrument, flute and piccolo are very close to whistling and to fairly pure sign waves with breath noise on top. – Todd Wilcox Jun 2 at 3:43
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    @ToddWilcox flutes, at least in low register, produce something closer to triangle than sine spectrum-wise. The linear (cylindrical or conical) pipes support overtones quite well, they just aren't excited as strongly as in reed woodwinds. Whistling (and, I'd suspect, ocarina) is different, because the resonance chamber isn't tube-shaped at all. – leftaroundabout Jun 2 at 10:57
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As far as I know every instrument produces overtones. Some might think that unpitched percussion don't have overtone, but they produce them as well.

However, there are some electronic instruments, such as synthesizers (sine waves) which can be played without producing any overtones, but every acoustic instrument does.

If I'm correct the ocarina might be the instrument which come as close as possible to creating 'no overtones'. In fact, they do create overtones as well, but because of their shape, the overtones are actually many octaves above the keynote scale.

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    The whole reason we call those instruments "unpitched" is because of their numerous inharmonic overtones. +1 – user45266 Jun 1 at 20:25
  • Sine waves from synths and unpitched percussion would have been my guess as well. Nice answer – Shevliaskovic Jun 2 at 8:05
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    @Shevliaskovic Synths yes (or at least maybe) - unpitched no. As user45266 says: unpitched instruments have many overtones but they are not integer multiples of the fundamental as usual. It is these non-integer harmonics that make it "untuned". The ear cannot make sense of these harmonics. – badjohn Jun 3 at 15:35
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It is worth looking at the reason WHY there are so few instruments that produce sine waves. It is clearly fairly difficult, from the point of view of physics, to make a sine wave without electronics, but people could have tried to get close if they wanted to.

The psycho-acoustic answer is that few attempted this because it does not sound interesting. It is notable that, of the examples suggested, most are:

Not designed for human entertainment (e.g. tuning fork)
Designed as part of something (e.g. one stop on a synthesizer, to be played with others)
Designed with other features to make the sound more interesting (e.g. theramin)

One instrument that gets fairly close is the Stylophone. This produces a sine wave - in theory - simply because this was the cheapest sound to aim for in an electronic instrument. Any deviation from the sine wave is not caused by aesthetic considerations, but by an over-riding desire for cheapness in the design brief. That is to say, the overtones are caused entirely by the cheap amplifier, cheap speaker and cheap plastic case.

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    Circuits to produce clean sine waves whose amplitude is consistent over a range of frequencies are nowhere near as cheap as circuits to produce pulses at an adjustable rate. Pulse waves and square waves are much cheaper and easier to produce, and are what I'd expect from a simple stylophone circuit. – supercat Jun 3 at 19:39
  • @supercat But the problem with pulses is that you would get a huge amount of higher harmonics and you would need to attenuate these. Doing this in a way that produced a similar tone at different frequencies would have been very difficult. In my experience a harmonic oscillator was the standard frequency generator before ICs and I am reasonably certain this is what the link means by a voltage control oscillator, even though I cannot find this stated specifically. – David Robinson Jun 3 at 21:25
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    An LC harmonic oscillator using a tuned coil can be a very stable way of generating a continuous reasonably-clean sine wave at one particular amplitude and frequency, but there are trade-offs between speed of start-up, ability to adjust the output frequency, and purity of the output waveform. Toys like otamatones use relaxation oscillators (such oscillators have a distinctive, decidedly non-sinusoidal sound), and I would expect that stylophones would too. – supercat Jun 3 at 22:25
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    You win. @supercat. I have managed to find a website where it has been reverse engineered and it was a relaxation circuit powered by a Programmable Uni-junction Transistor. Not surprisingly, it was replaced with a 555 at some stage after the invention of the most successful chip ever. – David Robinson Jun 4 at 0:05
  • I'd not seen the actual schematic for the original Stylophone, but it's interesting to note that the output of the vibrato circuit is much closer to sinusoidal than the audio output of the Stylophone (possible because it only has to operate at one rate). To help it start quickly, though, the loop gain is set high enough to cause distortion which is visible on the scope trace. – supercat Jun 4 at 15:39
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You can create pure a sine wave with some electronic generators. Another way is to use software. I created a series of pure sine waves in wav files at various frequencies for a hearing test. They don't sound like any real instrument that I have ever heard. So, that says that no instrument that I have heard produces a pure sine wave. Of course, I have not heard all instruments but I have heard many. The closest might be a flute but it still was recognisably different. I do not find a pure sine wave appealing.

Note that there is more to the difference between the sounds of various instruments than the harmonics: e.g. attack, decay, stability of pitch, etc. Back in the days of cassette tapes I had a tape of piano music stretch badly. It no longer sounded at all like a piano, it sounded like a musical saw. The harmonics would not have been changed (much) by the stretching. It indicated that an essential part of the piano sound is the stability of the pitch. For that reason, since then, I always used solo piano music to assess turntables. It is a long time since I did that though as I was an immediate convert to CDs. Partly due to this experience.

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    I can confirm your observations about piano: it's an instrument that always makes it obvious when a turntable isn't spinning at a perfectly steady rate. – jberryman Jun 2 at 20:10
  • One reason your sine waves may not sound like any real instruments is because musical instruments can be distinguished by any other factors. (Check this out: youtu.be/thD6TNUoyIk) – user45266 Jun 3 at 1:05
  • @user45266 Did you read my second paragraph? – badjohn Jun 3 at 6:41
  • @badjohn Yes. Just wanted an excuse to drop that link there :) – user45266 Jun 3 at 14:56
6

frequencies other than the dominant frequency of the note

Any finite wave has frequencies other than the dominant frequency. Single frequency is only possible for a sinusoid that has lasted since forever with constant amplitude and will continue to do so.

For any finite wave you will be able to perceive (with your ear or any physical measuring device) a bundle of neighbouring frequencies like seen in the image included in another answer. The width of the bundle is limited by duration of the signal.

NOTE: This answer does not discuss overtones in the common meaning of the term (multiples of fundamental frequency) but the definition from the question (quoted above).

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    Yeah, though this is a bit pedantic. By making the tone last long you can easily push all the “side channel” content below the audibility threshold. – leftaroundabout Jun 2 at 11:05
  • @leftaroundabout That's why I added "much" in my answer. – badjohn Jun 2 at 12:40
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    Is this really true from the point of view of a listener? I'd find a little more explanation useful here. – topo Reinstate Monica Jun 2 at 20:27
  • @topomorto It is true from a mathematical point of view, not from a listener's point of view. It's trivial to develop sounds whose harmonics due to this effect are so far outside the human hearing range that we can effectively ignore them. But when doing other things (such as working with pulsed lasers), these extra harmonics are an incredibly important nuance. – Cort Ammon Jun 3 at 4:35
  • @CortAmmon what I'm wondering is whether it really is philosophically true from a mathematical point of view, or if it's only true from the point of view of a certain analysis technique. It's possible to generate a finite number of cycles of a sine wave of any frequency; that will be heard as a sine wave persisting for a certain duration. If there's any energy at a different frequency, where has it come from? – topo Reinstate Monica Jun 3 at 5:45
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There are already quite a number of "instruments" listed in other answers but I think a subset of organs may reproduce an sinusoid approximate.

On the electrical side of things the Hammond organ used a spinning tonewheel and electrical pickup to generate near sines. Each key had several wheels spinning at multiples of the fundamental frequency. You could adjust valves controlling the strength (amplitude) of each harmonics -- an early prototype of additive synthesizers. Hence I will argue that the Hammond organ, unlike other instruments, was designed with sinusoidal production in mind. You could also argue that the Hammond was simply an attempt to replicate the fuller feel of true pipe organ.1 A live demonstration of can be found on youtube (with accompanying spectrogram).

There's also the original Telharmonium, a gargantuan factory-sized machine that produced near sines in a similar way.

On the Aerophone side of the things, there are certain pipes which are highly sinusoidal including the Tibia pipes of which you can hear a bit in the first 30 seconds of this video.

1You could also argue that the Hammond was simply an attempt to replicate the fuller feel of true pipe organ.

2

See also the lasso d'amore at which reference is stated "the timbre of the notes [...] are 'almost all fundamental,' according to Fourier analysis (similar to sine waves)." It is possible to play the instrument at a speed so near the transition from one resonance to another that two simultaneous pitches are produced. (This tends to be possible at higher speeds, at which it is difficult to prevent having different speeds in different arcs of the arm movement producing different tones from each arc.)

  • Ouch! Almost as horrifying as those vuvuzelas :-) – Carl Witthoft Jun 3 at 13:40
0

A pure sine wave is the only instrument that plays a tone without any overtones. This isn't a strange coincidence. An instrument's timbre is the consequence of its unique overtones - which ones it has, which ones are loudest, whether some overtones are slightly flat or sharp, and how the overtones mutate over time. Since there's only one timbre profile that can come from having no overtones, it shouldn't come as a surprise that there's only one sound that fits the bill. And when you strip all overtones from a sound wave, a sine wave is exactly what you get.

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By far, I'm no expert in this, but here's my best shot.

Timbre is the result of a specific series of overtones sounding off louder than others. We are looking for a timbre that only has the fundamental sounding off and nothing sounding above it. I suppose anything that could produce a single sine wave would be your answer. Perhaps an organ with only one tone sounding?

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